On Multi-source Networks: Enumeration, Rate Region Computation, and Hierarchy

Transcription

1 On Multi-ource Network: Enumeration, Rate Region Computation, and Hierarchy Congduan Li, Student Member, IEEE, Steven Weber, Senior Member, IEEE, and John MacLaren Walh, Member, IEEE arxiv: v [cit] Jul 05 Abtract Thi paper invetigate the enumeration, rate region computation, and hierarchy of general multi-ource multi-ink hyperedge network under network coding, which include multiple network model, uch a independent ditributed torage ytem and index coding problem, a pecial cae A notion of minimal network and a notion of network equivalence under group action are defined An efficient algorithm capable of directly liting ingle minimal canonical repreentative from each network equivalence cla i preented and utilized to lit all minimal canonical network with up to 5 ource and hyperedge Computational tool are then applied to obtain the rate region of all of thee canonical network, providing exact expreion for 744,9 newly olved network coding rate region correponding to more than trillion iomorphic network coding problem In order to better undertand and analyze the huge repoitory of rate region through hierarchy, everal embedding and combination operation are defined o that the rate region of the network after operation can be derived from the rate region of network involved in the operation The embedding operation enable the definition and determination of a lit of forbidden network minor for the ufficiency of clae of linear code The combination operation enable the rate region of ome larger network to be obtained a the combination of the rate region of maller network The integration of both the combination and embedding operator i then hown to enable the calculation of rate region for many network not reachable via combination operation alone I INTRODCTION Many important practical problem, including efficient information tranfer over network [4], [5], the deign of efficient ditributed information torage ytem [6], [7], and the deign of treaming media ytem [8] [0], have been hown to involve determining the rate region of an abtracted network under network coding Yan et al celebrated paper [] ha provided an exact repreentation of thee rate region of network under network coding Their eential reult i that the rate region of a network can be expreed a the interection of the region of entropic vector [], [3] with a erie of linear (in)equality contraint created by the network topology and Support under National Science Foundation award CCF and 488 i gratefully acknowledged C Li, S Weber and J W Walh are with the Department of Electrical and Computer Engineering, Drexel niverity, Philadelphia, PA SA ( congduanli@drexeledu, weber@coedrexeledu, and jwalh@coedrexeledu) Preliminary reult were preented at Allerton 04 [], NetCod 05 [], and ITW 05 [3]

2 the ink-ource requirement, followed by a projection of the reult onto the entropie of the ource and edge variable However, thi i only an implicit decription of the rate region, becaue the region of entropic vector Γ N i till unknown for N 4 Neverthele, a we have previouly demontrated in [4] [6], through the ue of appropriate inner and outer bound to Γ N that we will review in II-C, thi implicit formulation can be ued to develop algorithm by which a computer can very rapidly calculate the rate region, it proof, and the cla of capacity achieving code, for mall network, each of which would previouly have taken a trained information theorit hour or longer to derive While the development of thi rate region calculation, code election, and convere proof generation algorithm [6] i not the focu of the preent paper, it involve developing technique to derive and project polyhedral inner and outer bound decription for contrained region of entropic vector When it come to rate region, V and II-C of the paper will focu more on exactly what wa calculated, what can be calculated, and, later in the paper, what can be learned from the reulting rate region, rather than the exact computation by which the rate region were reached The rate region algorithm deign and pecialization, which involve a eparate and parallel line of invetigation, are left to dicuion by another erie of paper [7] [9] The ability to calculate network coding rate region for mall network rapidly with a computer motivate an alternative, more computationally thinking oriented, agenda to the tudy of network coding rate region At the beginning, one goal i to demontrate the method power by applying the algorithm to derive the rate region of a many network and application of network coding a poible To do thi, in II we firt lightly generalize Yeung labeled directed acyclic graph (DAG) model for a network coding problem ( [3], Ch ) to a directed acyclic hypergraph context, then demontrate how the enlarged model handle a pecial cae the wide variety of other model in application in which network coding i being employed, including, but not limited to, index coding, multilevel diverity coding ytem (MDCS), and ditributed torage With the lightly more general model in hand, the firt iue in the computationally thinking oriented agenda i network generation and enumeration, ie, how to lit all of the network falling in thi model cla In order to avoid repetitive work, thereby reaching the larget number of network poible with a contant amount of computation, it i deirable to undertand preciely when two intance of thi model (ie, two network coding problem) are equivalent to one another, in the ene that the olution to one directly provide a olution to another Thi notion of network coding problem equivalence, which provide a very different approach but i in the ame high level pirit a the tranformation of network channel coding problem to network coding problem in [0] and the tranformation of network coding problem to index coding problem in [], will be reviited at multiple point of the paper, beginning in thi preent context of enumeration, but alo playing an important role in the dicuion of hierarchy later The firt notion of equivalence we develop i that of minimality, by removing any redundant or unneceary part in the network intance Partially owing to the generality of the network coding problem model, many valid intance of it include within a network part which can be immediately detected a extraneou to the determination of the intance rate region In thi ene, an intance i directly reducible to another maller intance by removing

3 3 completely unneceary and unhelpful ource, node, or edge In order to provide the mallet poible intance by not including thee extraneou component, we formalize in III the notion of network minimality, liting a erie of condition which a network coding problem decription mut obey to not contain any obviouly extraneou ource, node, or edge The next notion of equivalence look to ymmetry or iomorphim between problem decription Beginning by oberving that a network mut be labeled to pecify it graph, ource availability, and demand to the computer, yet the underlying network coding problem i inenitive to the election of thee label, we define in IV-B a notion of network coding problem equivalence through the iomorphim aociated with the election of thee label We review that the proper way to formalize thi notion i through identifying equivalence clae with orbit under group action A naïve algorithm to provide the lit of network coding problem to the rate region computation oftware would imply lit all poible labeled network coding problem, tet for iomorphim, then narrow the lit down to only thoe which are inequivalent to one another, keeping only one element, the canonical repreentative, of the equivalence cla However, the key reaon for formalizing thi notion of equivalence i that the number of labeled network coding problem intance explode far fater than the number of network coding problem equivalence clae Hence, we develop a better technique for generating lit of canonical network coding problem intance by harneing technique and algorithm from computational group theory that enable u to directly lit the minimal canonical repreentative of the network coding problem equivalence clae a decribed in IV-C With the lit of all minimal canonical network coding problem up to a certain ize in hand, we can utilize our algorithm and oftware to calculate the rate region bound, the Pareto optimal network code, and the convere proof, for each, building a very large databae of rate region of network coding problem up to thi ize Owing to the variety of the model, even for tiny problem, thi databae quickly grow very large relative to what a human would want to read through For intance, our previou paper applying thi computational agenda to the narrower cla of MDCS problem [6], yielded the rate region of 6,868 equivalence clae of MDCS problem and bound for 49 more MDCS problem equivalence clae, while the databae developed in thi paper contain the rate region of 744,9 equivalence clae of network coding problem Thee equivalence clae of network correpond to olution for 9,050,490 network coding problem with graph pecified via edge dependence and,38,64,63,9 network coding problem pecified in the typical node repreentation of a graph While it i poible to ue the databae to report tatitic regarding the ufficiency of certain clae of code a will be done in V, in order to more meaningfully enable human to learn from the databae, a well a from the computational reearch, one mut utilize ome notion of network tructure to organize it for analyi Our method of endowing tructure on the et of network coding problem i through hierarchy, in which we explain the propertie and/or rate region of larger network a being inherited from maller network (or vicevera) Of coure, part of a network coding problem i the network graph, and further, network coding and entropy i related to matroid, and thee nearby field of graph theory and matroid theory have both undergone a thorough tudy of hierarchy which directly inpire our approach to it In graph theory, thi notion of hierarchy i achieved by recognizing maller graph within large graph which can be created by deleting or contracting the larger graph

4 4 edge, called minor, and i directly aociated with a crowning achievement Namely, the celebrated well-quaiordering reult of graph theory [], [3], howed that any minor cloed family of graph (ie, one for which any minor of a graph in that family i alo in the family) ha at mot a finite lit of forbidden minor, which no graph in that family can contain While the familie of minor cloed graph are typically infinite, they are then, in a ene, capable of being tudied through a finite object which i their forbidden minor: if a graph doe not have one of thee forbidden minor, it i then in the family In matroid theory, which in a certain ene extend graph theory, one ha a imilar notion of hierarchy endowed through matroid minor, generated through matroid contraction and deletion While one can generate minor cloed familie of matroid with an infinite erie of forbidden minor, the celebrated, and poibly recently proved, Rota conjecture [4], [5], tated that thoe matroid capable of being repreented over a particular finite field have at mot a finite lit of forbidden minor In thi paper, inpired by thee hierarchy theorie in graph and matroid, we aim to derive a notion of network minor created from a erie of contraction and deletion-like operator that hrink network coding problem, called embedding operator, a well a operation for building larger network coding problem from maller one, called combination operator Thee operator work together to build our notion of minor and a ene of hierarchy among network coding problem Developing a notion of network coding problem hierarchy i important for everal reaon Firt of all, a explained above, even after one ha calculated the rate region of all network up to a certain ize, it i of interet to make ene of thi very large quantity of information by tudying it tructure, and hierarchy i one way of creating a notion of tructure Second of all, the computational technique for proving network rate region can only handle network with ten of variable, the um of the number of ource and number of hyperedge in the graph, and hence are limited to direct computation of fairly mall problem intance If one want to be able to utilize the information gathered about thee mall network to undertand the rate region of network at cale, one need method for putting the maller network together into larger network in a way uch that the rate region of the larger network can be directly calculated from thoe of the maller network Our embedding operator, defined and dicued in VI, extend the erie of embedding operation we had for MDCS problem in [6], and augment them, to provide method for obtaining mall network from big network in uch a way that the rate region of the maller network, and it propertie, are directly inherited from the larger network Our combination operator, dicued in VII, work in the oppoite direction: they provide method for putting together maller network to make larger network in uch a way that the rate region of the larger network can be directly calculated from the rate region of the maller network Both of our lit of operator are mall and omewhat imple, however, when they work together, they provide a very powerful way of endowing hierarchical tructure in network coding problem In particular, the joint ue of the combination and embedding operator provide a very powerful way of obtaining rate region of large network from mall one, a well a decribing the propertie of familie of network coding problem, a we demontrate in VIII They open the door to many new avenue of network coding reearch, and we hall decribe briefly ome of the related future problem for invetigation in IX

5 5 Source S Network Sink T Path In(i) i Out(i) Y t Y (t) e e R S e T Fig : A general network model A II BACKGROND: NETWORK CODING PROBLEM MODEL, CAPACITY REGION, AND BONDS The cla of problem under tudy in thi paper are the rate region of multi-ource multi-ink network coding problem with hyperedge, which we hereafter refer to a the hyperedge MSNC problem For eae of reading the paper, a notation table i preented in Table I A network coding problem in thi cla, denoted by the ymbol A, include a directed acyclic hypergraph (V, E ) [6] a in Fig, coniting of a et of node V and a et E of directed hyperedge in the form of ordered pair e = (v, A ) with v V and A V \v The node V in the graph are partitioned into the et of ource node S, intermediate node G, and ink node T, ie, V = S G T Each of the ource node S will have a ingle outgoing edge (, A ) E The ource node in S have no incoming edge, the ink node T have no outgoing edge, and the intermediate node G have both incoming and outgoing edge The number of ource will be denoted by S = K, and each ource node S will be aociated with an independent random variable Y, S, with entropy H(Y ), and an aociated independent and identically ditributed (IID) temporal equence of random value For every ource S, define Out() to be it ingle outgoing edge, which i connected to a ubet of intermediate node and ink node A hyperedge e E connect a ource, or an intermediate node to a ubet of non-ource node, ie, e = (i, F ), where i S G and F (G T \ i) For brevity, we will refer to hyperedge a edge if there i no confuion For an intermediate node g G, we denote it incoming edge a In(g) and outgoing edge a Out(g) For each edge e = (i, F ), the aociated random variable e = f e (In(i)) i a function of all the input of node i, obeying the edge capacity contraint R e H( e ) The tail (head) node of edge e i denoted a Tl(e) (Hd(e)) For notational implicity, the unique outgoing edge of each ource node will be the ource random variable, e = Y if Tl(e) =, denoting E S = {e E Tl(e) =, S } to be the variable aociated with outgoing edge of ource, and E = E \ E S to be the non-ource edge random variable For each ink t T, the collection of ource thi ink will demand will be labeled by the non-empty et β(t) S Thu, a network can be repreented a a tuple A = (S, G, T, E, β), where β = (β(t), t T ) Note that, though thi commonly ued node-repreentation of a network i convenient for undertanding the network topology, we will ue a more concrete repreentation in IV for enumeration of network intance For convenience, network with K ource and L = E edge are referred a (K, L) intance

6 6 TABLE I: Notation table A A, B, C, D a network intance ( II) general et [x a a A ] vector with element x a indexed by/ for each a A β demand of ink node ( II) d, d, f, π mapping or function e, E an (hyper)edge or encoder, et of all (hyper)edge or encoder ( II) F F q head node of a hyperedge ( II) finite field of order q ( II-B) g, G an intermediate node, et of intermediate node ( II) G, S, g,, g k acting group, ymmetric group, group generated by g,, g k ( IV-A) H( ), h, h A Hd(e), Tl(e) In(g), Out(g) entropy function, an entropy vector, coordinate in h aociated with A ( II) head node, tail node of edge e ( II) i, j, k, l general index term I incoming, outgoing edge of node g ( II) independent et ( II-C) K, L, N number of ource, intermediate (hyper)edge/encoder, and total variable in a network L i, L A (N = K + L) ( II) et aociated with network contraint ( II-B) M, M a matroid, ground et of a matroid ( II-C) M minimal(a ), minimal A A(R (A )) N O increaed dimenion of pace in calculating rate region ( II-B) function to reduce a network A to it minimal repreentation and aociated rate region operator ( III) collection of variable in a network ( II-B) collection of network in an equivalence cla under our definition ( IV-B) p, P error probability, probability function ( II-B) P i (X ) all ize i ubet of a given et X ( IV-C) Q, W : edge encoding for a network topology and ink demand ( IV-A) r M Γ N, Γ N, Γ N, Γ q N, Γq N,N, Γ q N,, Γlinear N R e, R rank function of a matroid M ( II-C) region of entropic vector on N variable, it cloure, Shannon outer bound, inner bound from F q-repreentable matroid on N element, inner bound from F q- repreentable matroid on N element,inner bound from F q-repreentable matroid on infinity number of element, inner bound from linear ubpace arrangement ( II-C) edge capacity on edge e, rate vector ( II-B) r, ω, Proj r,ω ( ) dimenion aociated with all edge capacitie and all ource entropie, projection operator with the projecting dimenion are aociated with r, ω ( II-B) R c(a), R (A), R o(a), R,q, R N q, Rq, R linear rate region or bound aociated with Γ N, Γ N, Γ q N, Γq N,N, Γ q N,, Γlinear N ( II-B, II-C), S a ource node, et of all ource ( II) t, T a ink node, et of all ink ( II) T, edge variable, upport et of ( II-B) V canonical repreentative, ie, tranveral, output from Leiterpiel algorithm ( IV-C) et of all node in a network ( II) V, V a vector pace, multiple vector pace X, Y random variable ( II) X, Y vector of variable ( II-B) X, Y Z general et, upport et on Y ( II-B, IV) collection of network intance ( IV-C)

7 7 Y g 5 6 t Y g Y 4 g t Y Y 9 Y 0 g 5 6 g Y 0 4 g t Y Y 9 t Y Fig : A normal MSNC in [3] can be viewed a a pecial intance of the hyperedge MSNC A our focu in the manucript will be on rate region of a very imilar form to thoe in [], [3], thi network coding problem model i a cloe to the original one in [], [3] a poible while covering the multiple intance of application in network coding in which the ame meage can be overheard by multiple partie Thee application include index coding, wirele network coding, independent ditributed ource coding, and ditributed torage The implet and mot direct model change to incorporate thi capability i to witch from directed acyclic graph to directed acyclic hypergraph A we hall ee in II-B, thi mall change to the model i eaily reconciled with the network coding rate region expreion and it proof from [], [3] A Special Network Clae The network coding problem model jut decribed ha been elected to be general enough to include a variety of model from the application of network coding a pecial cae A few of thee pecial cae that will be of interet in example later in the paper are reviewed here, including a decription of the extra retriction on the model to fall into thi ubcla of problem Example (Yan Yeung Zhang MSNC): The network model in [3], where the edge are not hyperedge and the output of a ource node can be multiple function of the ource, can be viewed a a pecial cla of network in our model Thi i becaue the ource can be viewed a intermediate node and a virtual ource node connecting to each of them can be added For intance, a mall network intance of the model in [3], a hown in Fig, can be viewed a a hyperedge network intance introduced in thi paper, by adding virtual ource node, to ource,, repectively Example (Independent Ditributed Source Coding): The independent ditributed ource coding (IDSC) problem, which were motivated from atellite communication ytem [7], can be viewed a a pecial cla of network in our model They are three-layer network, where ource are connected with ome intermediate node and thoe intermediate node will tranmit coded meage to ink For intance, the IDSC problem in Fig 3 can be converted to a hyperedge multi-ource network coding problem A a pecial cla of IDSC problem with decoding prioritie among ource, the multi-level diverity coding ytem [6], [8] are naturally a cla of network in our current general model In the experimental reult ection V, we will not only how reult on general hyperedge network, but alo ome reult on IDSC problem

8 8 Y Y E E 3 E 3 D D D 3 Y Y Y Y Y t t g Y 3 t 3 Y Y Y Y Fig 3: An IDSC problem can be viewed a a hyperedge MSNC Fig 4: An index coding problem i a three-layer hyperedge MSNC with only one intermediate edge Example 3 (Index Coding): Since direct acce to ource a ide information i allowed in our network model, index coding problem are alo a pecial cla of our model with only one intermediate edge That i, a K-ource index coding problem can be viewed a a (K, ) hyperedge MSNC and vice vera For intance, an index coding problem with 3 ource, a hown in Fig 4, i a (3, ) intance in our model B Rate Region Having defined a network coding problem, we now define a network code and the network coding rate region Definition : An (n, R) block code, with R = [τ,, τ K, R,, R L ] R E +, conit of a erie of mutually independent ource Y (n), uniformly ditributed in Y = {,, nτ }, and block encoder and decoder i) The block encoder, one for each e E, are function that map a block of n ource obervation from all ource in E S In(Tl(e)), and the incoming meage aociated with the edge E In(Tl(e)), to one of nre different decription in e = {0,,, η e }, where η e = nre, f (n) e : E S In(Tl(e)) Y i E In(Tl(e)) ii) The block decoder, one for each ink t T, are function d (n) t : e In(t) E e In(t) E S Y i e, e E () β(t) Y, t T ()

9 9 Denote by (n) e e the random meage on edge e E, which i the reult of the encoding function f (n) e Further, we can define the probability of error for each ink t T a [ p (n),err t (R) = P d (n) t ( (n) In(t) (n) ) [Y β(t)] ], (3) and the maximum over thee a p (n),err (R) = max t T p(n),err t (4) Definition : The rate region of a network A, denoted a R c (A), i the cloure of the et of all achievable rate vector R, where a rate vector R R c (A) i achievable if there exit a equence of encoder {f (n) = [f e (n) e E ]} and decoder {d (n) = [d (n) t t T ]} uch that p (n),err (R) 0 a n The rate region R c (A) can be expreed in term of the region of entropic vector, Γ N, a in [], [3] The dicuion on Γ N and it bound i deferred to II-C For the hyperedge MSNC problem, define a et N = {Y, e S, e E } with ingle letter random variable aociated with ource and edge, repectively, and define N = N = K + L Then, if we collect joint entropie of all non-empty ubet of N h = [h A A N ], we have h Γ N into a vector A will be hown in II-C, Γ N i in the pace of RN Note that the edge capacitie, r = [R e e E ], are extra variable aociated with each edge Therefore, we will conider the pace in R M, where M = N +L, L = E We define L i, i =, 3, 4, 5 a network contraint repreenting ource independence, coding by intermediate node, edge capacity contraint, and ink node decoding contraint repectively: L = {h R M : h YS = Σ S h Y } (5) L 3 = {h R M : h Out(g) (Y S In(g) E In(g)) = 0, g G } (6) L 4 = {[h T, r T ] T R M + : R e h e, e E } (7) L 5 = {h R M : h Yβ(t) In(t) = 0, t T } (8) and we will denote L 3 = L L 3, L 4 5 = L 4 L 5 and L A = L L 3 L 4 L 5 Note that we do not have L contraint (which repreent the coding function at each ource) a in [], due to our different notation with e = Y if Tl(e) = Further, Γ N and L i, i =, 3, 5 are viewed a ubet of R M with indexed by r T uncontrained, ince they actually are in the pace of R N L 4 i alo viewed a ubet of R M, with the unreferenced dimenion (ie all non-ingleton entropie) left uncontrained The following extenion of the rate region from [] characterize our lightly different rate region formulation R c (A) for our lightly different problem Theorem : The expreion of the rate region of a network A i R c (A) = Proj r,ω (con(γ N L 3) L 4 5), (9) where con(b) i the conic hull of B, and Proj r,ω (B) i the projection of the et B on the coordinate [ r T, ω T ] T where r = [Re e E ] and ω = [H(Y ) S ]

10 0 Proof: We preent a ketch of the proof here and a detailed proof in Appendix A Firt oberve that the proof of Theorem in [] can be extended to network preented above, with hyperedge and intermediate node having direct acce to ource Some difference include: I) the hyperedge model potentially make one edge variable connected with more than one node and thu be involved in more than one intermediate node contraint (L 3 ) Therefore, it may contrain more on the edge variable in the region of entropic vector; II) the coding function for each intermediate (hyper)edge may encode ome ource edge with ome other non-ource edge together; III) the decoding at ink node may be a function of ome ource edge and non-ource edge a well; IV) there i only one outgoing edge for each ource and it carrie the ource variable itelf The difference will not detroy the eence of the proof in [] For the convere and achievability proof, we view the edge capacitie a contant (recall that our rate vector include both ource entropie and edge capacitie), and then conider the convere and achievability of the aociated ource entropie, which become eentially the proof in [] While the analytical expreion determine, in principle, the rate region of any network under network coding, it i only an implicit characterization Thi i becaue Γ N i unknown and even non-polyhedral for N 4 Further, while Γ N i a convex cone for all N, Γ N i already non-convex by N = 3, though it i alo known that the cloure only add point at the boundary of Γ N Thu, the direct calculation of rate region from (9) for a network with 4 or more variable i infeaible On a related note, at the time of writing, it appear to be unknown by the community whether or not the cloure after the conic hull i actually neceary in (9), and the uncertainty that neceitate it incluion muddle a number of otherwie imple proof and idea For thi reaon, ome of the dicuion in the remainder of the manucript will tudy a cloely related inner bound to R c (A) decribed in the following corollary In all of the cae where the rate region ha been computed to date thee two region are equivalent to one another Corollary : The rate region R c (A) of a network A i inner bounded by the region R (A) = Proj r,ω con(γ N ) L A (0) Proof: Clearly R c = Proj r,ω (con(γ N L 3) L 4 5) Proj r,ω (con(γ N L 3) L 4 5) Next, oberve that interecting with L 3 i equivalent to requiring certain information inequalitie (which are non-negative for all entropic vector) to be identically zero, and a conic combination of uch entropic vector thu can only yield uch an information inequality identically zero if the ame information inequality wa identically zero for each entropic vector Hence con(γ N L 3) = con(γ N ) L 3, and thu, Proj r,ω (con(γ N L 3) L 4 5) = Proj r,ω (con(γ N ) L A ) Thi complete the proof Again, both R c (A) and it cloely related inner bound R (A) are not directly computable becaue they depend on the unknown region of entropic vector and it cloure However, replacing Γ N with finitely generated inner and outer bound, a decribed in the following corollarie, tranform (9) into a polyhedral computation problem, which involve applying ome linear contraint onto a polyhedron and then projecting down onto ome coordinate The cloure would be unneceary if Γ N = con(γ N ), ie if every extreme ray in Γ N had at leat one point along it that wa entropic (ie in Γ N ) At preent, all that i known i that Γ N ha a olid core, ie that the cloure only add point on the boundary of Γ N

11 Corollary : Let A Γ N region i given by be ome finite et of entropic vector, then a polyhedral inner bound to the rate R c (A) R (A) Proj r,ω (con(a ) L A ) () Proof: It i clear that Proj r,ω (con(a L 3 ) L 4 5) will be an inner bound to Proj r,ω (con(γ N L 3) L 4 5) and hence R (A) Furthermore, for uch a finite et A L 3 mut alo be a finite et, and hence con(a L 3 ) = con(a L 3 ) i a cloed polyhedral cone Additionally, oberve that every equality in L 3 can be viewed a etting a non-negative definite information inequality quantity to zero, and ince every point in A mut thu lie in only the non-negative half pace thee equalitie generate, the extreme ray of con(a L 3 ) mut be thoe extreme ray of con(a ) in L 3, implying con(a L 3 ) = con(a ) L 3 Putting thee fact together we oberve that the inner bound Proj r,ω (con(a L 3 ) L 4 5) = Proj r,ω (con(a ) L 3 L 4 5) = Proj r,ω (con(a ) L A ) Similarly, polyhedral cone Γ out N region Corollary 3: Let Γ out N rate rate region i given by Γ out N Proof: Since Γ N Γ N Γout N outer bounding the convex cone Γ N yield polyhedral outer bound to the rate be a cloed polyhedral cone that contain Γ N, then a polyhedral outer bound to the R c (A) Proj r,ω (Γ out N L A ) (), Γ N L 3 Γ N L 3 Γ out N L 3 Thu, con(γ N L 3) con(γ out N L 3) = L 3 Hence R (A) = Proj r,ω (con(γ N L 3) L 4 5) Proj r,ω (Γ out N Thee corollarie inpire u to ubtitute Γ N L A) with uch cloed polyhedral outer and inner bound Γout,and Γ in N = con(a ), A Γ N to Γ N, repectively, to obtain an outer and inner bound on the rate region: R out (A) = proj r,ω (Γ out N L A ), (3) R in (A) = proj r,ω (Γ in N L A ) (4) If R out (A) = R in (A), we know R c (A) = R (A) = R out (A) = R in (A) Otherwie, tighter bound are neceary In thi work, we will ue (3) and (4) to calculate the rate region Typically the Shannon outer bound Γ N and ome inner bound obtained from matroid, epecially repreentable matroid, are ued We will briefly review the definition of thee bound in the next ubection, while detail on the polyhedral computation method with thee bound are available in [4], [5], [7], [8] N C Contruction of bound on rate region An introduction to the region of entropic vector and the polyhedral inner and outer bound we will utilize from it can be found in greater detail in [6] Here we briefly review their definition for accuracy, completene, and convenience ) Region of entropic vector Γ N : Conider an arbitrary collection X = [X,, X N ] of N dicrete random variable with joint probability ma function p X To each of the N non-empty ubet of the collection of

12 random variable, X A := [X i i A ] with A {,, N}, there i aociated a joint Shannon entropy H(X A ) Stacking thee ubet entropie for different ubet into a N dimenional vector we form an entropy vector h = [H(X A ) A {,, N}, A ] (5) By virtue of having been created in thi manner, the vector h mut live in ome ubet of R N +, and i aid to be entropic due to the exitence of p X However, not every point in R N + i entropic ince, for many point, there doe not exit an aociated valid ditribution p X All entropic vector form a region denoted a Γ N It i known that the cloure of the region of entropic vector Γ N i a convex cone [] Elementary inequalitie on Shannon entropie hould form a fundamental outer bound on Γ N, named the Shannon outer bound Γ N ) Shannon outer bound Γ N : We oberve that elementary propertie of Shannon entropie indicate that H(X A ) i a non-decreaing ubmodular function, o that A B {,, N}, C, D {,, N} H(X A ) H(X B ) (6) H(X C D ) + H(X C D ) H(X C ) + H(X D ) (7) Since they are true for any collection of ubet entropie, thee linear inequalitie (6), (7) can be viewed a upporting halfpace for Γ N Thu, the interection of all uch inequalitie form a polyhedral outer bound Γ N for Γ N and Γ N, where Γ N := h h A h B A B RN h C D + h C D h C + h D C, D Thi outer bound Γ N i known a the Shannon outer bound, a it can be thought of a the et of all inequalitie reulting from the poitivity of Shannon information meaure among the random variable While Γ = Γ and Γ 3 = Γ 3, Γ N Γ N for all N 4 [], and indeed it i known [9] that Γ N i non-polyhedral for N 4 The inner bound on Γ N we conider are baed on repreentable matroid We briefly review the baic definition of matroid and repreentable matroid 3) Matroid baic: Matroid theory [4] i an abtract generalization of independence in the context of linear algebra and graph to the more general etting of et ytem There are numerou equivalent definition of matroid, however, we will preent the definition of matroid utilizing rank function a thi i bet matched to our purpoe Definition 3: A et function on a ground et M, r M : M {0,, M }, i a rank function of a matroid M if it obey the following axiom: ) Cardinality: r M (A ) A ; ) Monotonicity: if A B M then r M (A ) r M (B); 3) Submodularity: if A, B M then r M (A B) + r M (A B) r M (A ) + r M (B) A ubet with rank function r M (A ) = A i called an independent et of the matroid Though there are many clae of matroid, we are epecially intereted in one of them, repreentable matroid, becaue they can be related to linear code to olve network coding problem a dicued in [4], [5]

13 3 4) Repreentable matroid: Repreentable matroid are an important cla of matroid which connect the independent et to the notion of independence in a vector pace Definition 4: A matroid M with ground et M of ize M = N and rank r M (M ) = r i repreentable over a field F if there exit a matrix A F r N uch that for each et A M the rank r M (A ) equal the linear rank of the correponding column in A, viewed a vector in F r Note that, for an independent et I, the correponding column in the matrix A are linearly independent There ha been ignificant effort toward characterizing the et of matroid that are repreentable over variou field ize, with a complete anwer only available for field of ize two, three, and four For example, a matroid M i binary repreentable (repreentable over a binary field) iff it doe not have the matroid,4 a a minor Here, a minor i obtained by erie of operation of contraction and deletion [4] k,n i the uniform matroid on the ground et M = {,, N} with independent et I equal to all ubet of {,, N} of ize at mot k For example,,4 ha a it independent et I = {,,, 3, 4, {, }, {, 3}, {, 4}, {, 3}, {, 4}, {3, 4}} (8) Another important obervation i that the firt non-repreentable matroid i the o-called Vámo matroid, a well known matroid on ground et of ize 8 That i to ay, all matroid are repreentable, at leat in ome field, for N 7 5) Inner bound from repreentable matroid: Suppoe a matroid M with ground et M of ize M = N and rank r M (M ) = r i repreentable over the finite field F q of ize q and the repreenting matrix i A F r N q that r M (B) = rank(a :,B ), B M, the matrix rank of the column of A indexed by B Let Γ q N uch be the conic hull of all rank function of matroid with N element and repreentable in F q Thi provide an inner bound Γ q N Γ N, becaue any extremal rank function of Γq N i by definition repreentable and hence i aociated with a matrix repreentation A F r N q, from which and r random variable u uniformly ditributed in F q, we can create the random variable [X,, X N ] = ua, u niform(f r q), (9) whoe element have joint entropie h A = r M (A ) log q, A M Hence, all extreme ray of Γ q N are entropic, and Γ q N Γ N Further, if a vector in the rate region of a network i (a projection of) an F q-repreentable matroid rank, the repreentation A can be ued a a linear code to achieve that rate vector, and thi code i denoted a baic calar F q code For an interior point in the rate region, which i the conic hull of projection of F q -repreentable matroid rank, the code to achieve it can be contructed by time-haring between the baic calar code aociated with the rank involved in the conic combination Thi code i denoted a calar F q code Detail on contruction of uch a code can be found in [5] and [6] One can further generalize the relationhip between repreentable matroid and entropic vector etablihed by (9) Suppoe the ground et M = {,, N } and a partition M = {,, N} We define a partition mapping π : M M uch that i M π(i ) = M, and π(i ) π(j ) =, i, j M, i j That i, the et M i

14 4 partitioned into N dijoint et Suppoe the variable aociated with M are X,, X N Now we define for n M the new vector-valued random variable Y n = [X i i π (n)] The aociated entropic vector will have entropie h A = r M ( n A π (n)) log q, A M, and i thu proportional to a projection of the original rank vector r keeping only thoe element correponding to all element in a et in the partition appearing together Thu, uch a projection of Γ q N form an inner bound to Γ N, which we will refer to a a vector repreentable matroid inner bound Γ q N,N A N, Γ q N, i the conic hull of all rank of ubpace on F q The union over all field ize for Γ q N, i the conic hull of the et of rank of ubpace Similarly, if a vector in the rate region of a network i (a projection of) a vector F q -repreentable matroid rank, the repreentation A can be ued a a linear code to achieve that rate vector, and thi code i denoted a a baic vector F q code The time-haring between uch baic vector code can achieve any point inide the rate region [5], [6] 6) Dimenion function of linear ubpace arrangement: A tated above, Γ q N,N become a tighter and tighter inner bound on Γ N a N Thi conider the increae in dimenion but doe not conider the field other than F q Actually, if we conider all poible F q field and let N, we will get the inner bound aociated with all linear code, denoted by Γ linear N, which i tighter than Γ q N, for a fixed F q Specifically, conider a collection of N linear vector ubpace V = (V,, V N ) of a finite dimenional vector pace, and define the et function d : V N +, where d(a ) = dim ( i A V i) for each A {,, N} i the dimenion of the vector pace generated by the union of ubpace indexed by A For any collection of ubpace V, the function d i integer valued, and obey monotonicity and ubmodularity Additionally, for every ubpace dimenion function d, there i an aociated entropic vector Indeed, one can place the vector forming a bai for each V i, over all i, ide by ide into a matrix A, which when utilized in (9), will yield random ubvector having the deired entropie Thu, the conic hull of dimenion of linear ubpace arrangement form an inner bound on Γ N, we denote it by Γlinear N Integrality, monotonicity, and ubmodularity are neceary but inufficient for for a given et function d : V N + to be dimenion function of ubpace arrangement That i, there exit additional inequalitie that are neceary to decribe the conic hull of all poible ubpace dimenion et function A dicued in [30], Ingleton inequality [3] together with the Shannon outer bound Γ 4, completely characterize Γ linear 4 For N = 5 ubpace [3] found 4 new inequalitie in addition to the Ingleton inequalitie that hold, and prove thi et i irreducible and complete in that all inequalitie are neceary and no additional non-redundant inequalitie exit For N 6, [3], [33] there are new inequalitie from N to N, and Γ linear N remain unknown All the bound dicued in thi ection could be ued in (3) and (4) to calculate bound on rate region for a network A If we ubtitute the Shannon outer bound Γ N into (3), we get R o (A) = proj r,ω (Γ N L A ) (0) Similarly, we ubtitute the repreentable matroid inner bound Γ q N, the vector repreentable matroid inner bound

15 5 Γ q N,N, Γq N, and the linear inner bound Γlinear N into (4), to obtain R,q (A) = proj r,ω (Γ q N L A), () R N q (A) = proj r,ω (Γ q N,N L A), () R q (A) = proj r,ω (Γ q N, L A), (3) R linear (A) = proj r,ω (Γ linear N L A ) (4) We will preent the experimental reult utilizing thee bound to calculate the rate region of variou network in V However, before we do thi, we will firt aim to generate a lit of network coding problem to which we may apply our computation and thereby calculate their rate region In order to tackle the larget collection of network poible in thi tudy, in the next ection we will eek to obtain a minimal problem decription for each network coding problem intance, removing any redundant ource, edge, or node III MINIMALITY REDCTIONS ON NETWORKS Though in principle, any network coding problem a decribed in II form a valid network coding problem, uch a problem can include network with node, edge, and ource which are completely extraneou and unneceary from the tandpoint of determining the rate region To deal with thi, in thi ection, we how how to form a network intance with equal or fewer number of ource, edge, or node, from an intance with extraneou component We will how the rate region of the intance with the extraneou component i trivial to calculate from the rate region of the reduced network Network coding problem without uch extraneou and unneceary component will be called minimal We firt define a minimal network coding problem, then how, via Theorem, how to map a non-minimal network to a minimal network, and then form the rate region of the non-minimal network directly from the minimal one Definition 5: An acyclic network intance A = (S, G, T, E, β) i minimal if it obey the following contraint: Source minimality: (C) all ource cannot be only directly connected with ink: S, Hd(Out()) G ; (C) ink do not demand ource to which they are directly connected: S, t T, if t Hd(Out()) then / β(t); (C3) every ource i demanded by at leat one ink: S, t T uch that β(t) ; (C4) ource connected to the ame intermediate node and demanded by the ame et of ink hould be merged:, S uch that Hd(Out()) = Hd(Out( )) and γ() = γ( ), where γ() = {t T β(t)}; Node minimality: (C5) intermediate node with identical input hould be merged: k, l G uch that In(k) = In(l); (C6) intermediate node hould have nonempty input and output, and ink node hould have nonempty input: g G, t T, In(g), Out(g), In(t) ;

16 6 Edge minimality: (C7) all hyperedge mut have at leat one head: e E uch that Hd(e) = ; (C8) identical edge hould be merged: e, e E with Tl(e) = Tl(e ), Hd(e) = Hd(e ); (C9) intermediate node with unit in and out degree, and whoe in edge i not a hyperedge, hould be removed: e, e E, g G uch that In(g) = e, Hd(e) = g, Out(g) = e ; Sink minimality: (C0) there mut exit a path to a ink from every ource wanted by that ink: t T, β(t) σ(t), where σ(t) = {k S a path from k to t}; (C) every pair of ink mut have a ditinct et of incoming edge: t, t T, i j, In(t) In(t ); (C) if one ink receive a uperet of input of a econd ink, then the two ink hould have no common ource in demand: If In(t) In(t ), then β(t) β(t ) = ; (C3) if one ink receive a uperet of input of a econd ink, then the ink with uperet input hould not have direct acce to the ource that demanded by the ink with ubet input: If In(t) In(t ) then t / Hd(Out()) for all β(t) Connectivity: (C4) the direct graph aociated with the network A i weakly connected To better highlight thi definition of network minimality, we explain the condition involved in greater detail The firt condition (C) require that a ource cannot be only directly connected with ome ink, for otherwie no ink need to demand it, according to (C) and (C0) Therefore, thi ource i extraneou The condition (C) hold becaue otherwie the demand of thi ink will alway be trivially atified, hence removing thi recontruction contraint will not alter the rate region Note that other ource not demanded by a given ink can be directly available to that ink a ide information (eg, a in index coding problem), a long a condition (C3) i atified The condition (C3) indicate that each ource mut be demanded by ome ink node, for otherwie it i extraneou and can be removed The condition (C4) ay that no two ource have exactly the ame path and et of demander (ink requeting the ource), becaue in that cae the network can be implified by combining the two ource a a uper-ource The condition (C5) require that no two intermediate node have exactly the ame input, for otherwie the two node can be combined The condition (C6) require that no node have empty input except the ource node, for otherwie thee node are uele and extraneou from the tandpoint of atifying the network coding problem The condition (C7) require that every edge variable mut be in ue in the network, for otherwie it i alo extraneou and can be removed The condition (C8) guarantee that there i no duplication of hyperedge, for otherwie they can be merged with one another The condition (C9) ay that there i no trivial relay node with only one non-hyperedge input and output, for otherwie the head of the input edge can be replaced with the head of the output edge The condition (C0) reflect the fact that a ink can only decode the ource to which it ha at leat one path of acce, and any demanded ource not meeting thi contraint will be forced to have entropy rate of zero The condition (C) indicate the trivial requirement that no two decoder hould have the ame input,

17 7 for otherwie thee two decoder can be combined The condition (C) imply tipulate that implied capabilitie of ink node are not to be tated, but rather inferred from the implication In particular, if In(t) In(t ), and β(t) β(t ), the decoding ability of β(t) i implied at t : puruing minimality, we only let t demand extra ource, if any The condition (C3) i alo neceary becaue the availability of i already implied by having acce to In(t), hence, there i no need to have direct acce to We next how that the rate region of the network with extraneou component can be eaily derived from the network without extraneou component, and vice vera Following the ame order of the contraint (C) (C4), we give the action on each reduction and how the rate region of the network with thoe extraneou component can be derived Theorem : Suppoe a network intance A = (S, G, T, E, β), with rate region and bound R l (A), l {c,, q, (, q), o}, i a reduction from another network A = (S, G, T, E, β ), with rate region bound R l (A ), l {c,, q, (, q), o}, by removing one of the redundancie pecified in (C) (C4) in Def 5 Then, by defining R \A = Proj \A R to be the projection of R excluding coordinate aociated with A, and ω be the ource rate of, we have the following Source minimality: (D) If S, Hd(Out( )) G =, A will be A with removed and R l (A ) := { R R \ R l (A), ω = 0 } l {c,, q, (, q), o} (5) if t T uch that β(t ) and / In(t ), while R l (A ) := { R R \ R l (A), ω 0 } l {c,, q, (, q), o} (6) otherwie Furthermore, R l (A) = Proj \ R l (A ) l {c,, q, (, q), o} (7) (D) if S, t T, uch that t Hd(Out( )) and β(t ), A will be A with In(t) = In(t ) \ and β(t) = β(t ) \ Further, R l (A ) = R l (A) for all l {c,, q, (, q), o} (D3) if S, uch that t T, Y / β(t ), A will be A with removal of the redundant ource and R l (A ) := { R R \ R l (A), H(Y ) 0 } l {c,, q, (, q), o} (8) and R l (A) = Proj \ R l (A ) l {c,, q, (, q), o} (9) (D4) if, S uch that Hd(Out()) = Hd(Out( )) and γ() = γ( ), A will be A with ource, merged and R l (A ) = { } R [R T \{, }, H(Y ) + H(Y )] T R l (A) l {c,, q, o} (30)

18 8 3 Y3 Y g t Y Y g t Y t3 Y 3 (C): ource 3 doe not connected (C): ink t3 ha direct acce to (C3): ource Y3 i not demanded with any intermediate node, and Y, the demand of Y i trivially by any ink, and thu i redundant thu i extraneou atified and thu t3 i redundant Y Y3 3 Y g t Y Y g4 g g3 t Y Y 3 g Y t3 Y t Y g t Y Y t3 Y 3 3 (C4): ource Y, Y3 have exactly the (C5): node g, g have ame input, (C6): node g3, g4 and ink t have ame output and demander, and thu and thu can be combined empty input/ output, and thu are re- can be combined Y dundant g t Y Y Y t Y Y Y g g g 3 3 Y t Y t Y Y g 0 t Y t Y g 0 g t3 Y 3 (C7): edge i not connected (C8): edge, 3 have exactly (C9): node g0 ha exactly one to any other node, and thu i the ame input and output node, input and one output, and they can redundant and thu can be combined be combined Y g t Y Y g Y g t Y t3 Y Y 3 Y Y t Y + t Y g g t Y t3 Y Y g 3 t Y Y t3 Y 3 C0: ink t3 ha no acce to but (C): ink t, t have exactly (C): t decode Y from, demand Y, o the only way to at- the ame input and thu can be hence t alo can decode Y, thu ify it i i ending no information combined into one ink node there i no need to lit Y in β(t ) Y g t Y Y g Y g t Y t Y 3 t3 Y 4 3 Y Y3 Y4 g 3 g 3 t Y t3 Y 4 t4 Y 3 5 t5 Y 4 (C3): t decode Y from, (C4): each connected compo- thu t alo can decode Y, thu nent can be viewed a a eparate there i no need to keep direct network intance acce of t to Y Fig 5: Example to demontrate the minimality condition (C) (C4)

19 9 ie, replace H(Y ) in R l (A) with H(Y ) + H(Y ) to get R l (A ), l {, q, o} Furthermore, Node minimality: R l (A) = { R \{} R R l (A ), ω = 0 } l {c,, q, (, q), o} (3) (D5) If k, j G uch that In(k ) = In(j ), A will be A with k, j merged o that In(k) = In(k ) = In(j ), Out(k) = Out(k ) Out(j ), and G = G \j Further, R l (A) = R l (A ) for all l {c,, q, (, q), o} (D6) If g G uch that In(g) =, or Out(g) =, A will be A with removal of the redundant node() g and R l (A ) = R l (A) for all l {c,, q, (, q), o} Similarly, if t T uch that In(t ) =, A will be A with removal of the redundant node() t and the deletion of any ource it demand, R l (A ) = { R R \β(t ) R l (A), ω = 0 β(t ) }, and R l (A) = Proj \β(t ) R l(a )for all l {c,, q, (, q), o} Edge minimality: (D7) If e E uch that Hd(e ) =, A will be A with removal of edge e, and R l (A) = Proj \e R l (A ) for all l {c,, q, (, q), o} R l (A ) = {R R \e R l (A), R e 0}, (3) (D8) If e, e E with Tl(e) = Tl(e ), Hd(e) = Hd(e ), A will be A with edge e, e merged a e and { } R l (A ) = R [R T \{e,e }, R e + R e ] T R l (A) ; l {c,, q, o} (33) ie, replace R e in R (A) with R e + R e to get R (A ) Furthermore, { } R l (A) = [R T \{e,e }, R e] T R R l (A ), R e = 0 l {c,, q, (, q), o} (34) (D9) If e, e E, g G uch that In(g ) = e, Hd(e) = g, Out(g ) = e, then A will be A with the node g removed and a new edge e i replacing e, e by directly connecting Tl(e) and Hd(e ) Further, { } R l (A ) = R [R T \{e,e }, min{r e, R e }] T R l (A) ; (35) ie, replace R e in R l (A) with min{r e, R e } to get R l (A ) Accordingly, { } R l (A) = [R T \{e,e }, min{r e, R e }] T R R l (A ) (36) for all l {c,, q, (, q), o} Sink minimality: (D0) If t T, S, uch that β(t ) but / σ(t ), then A will be A with deleted, R l (A ) = {R R \ R l (A), H(Y ) = 0}, (37) and R l (A) = Proj \ R l (A ) (38) for all l {c,, q, (, q), o}

20 0 (D) If t, t T, t t, uch that In(t) = In(t ), then A will be A with ink t, t merged and R l (A ) = R l (A) for all l {c,, q, (, q), o} (D) If t, t uch that In(t) In(t ) and β(t) β(t ), then A will be A with removal of β(t) β(t ) from β(t ) and R l (A ) = R l (A) for all l {c,, q, (, q), o} (D3) If t, t, β(t) uch that In(t) In(t ) and t Hd(Out( )), then A will be A with removal of from In(t ) and R l (A ) = R l (A) for all l {c,, q, (, q), o} Connectivity: (D4) if A i not weakly connected and A, A are two weakly diconnected component, then R l (A ) = R l (A ) R l (A ) for all l {c,, q, (, q), o} Proof: In the interet of conciene, for all but (D4) and (D8) we will only briefly ketch the proof for the expreion determining R (A ) from R (A), a the map in the oppoite direction and the other rate region bound follow directly from parallel argument (D) hold becaue i not communicating with any node other than poibly ink If there i a ink that demand it that doe not have direct acce to it, then thi ink can not uccefully receive any information from it, ince doe not communicate with any intermediate node Hence, in thi cae ω = 0 and every other rate i contrained according to R (A) becaue the remainder of the network ha no interaction with Alternatively, if every ink that demand ha direct acce to it, any non-negative ource rate can be upported for, and the remainder of the network i contrained a by R (A) becaue no other part of the network interact with (D) hold becaue the demand of at ink t i trivially atified if it ha direct acce to The contraint ha no impact on the rate region of the network In (D3) if a ource i not demanded by anyone, it can trivially upport any rate When two ource have exactly the ame connection and are demanded by ame ink a under (D4), they can be imply viewed a a combined ource for R l with l {c,, q, o}, ince the exact region and thee bound enable imple concatenation of ource Since the ource entropie are variable in the rate region expreion, it i equivalent to make a the combined ource, which ince the previou ource were independent, will have an entropy which i the um of their entropie Moving from R l (A ) to R l (A) i then accomplihed for any l {c,, q, (, q), o} by oberving that A can be viewed a A with ω = 0 An intermediate node can only utilize it input hyperedge to produce it output hyperedge, hence when two intermediate node have the ame input edge, their encoding capabilitie are identical, and thu for puruing minimality of repreentation of a network, thee two node having the ame input hould be repreented a one node Thu, (D5) i neceary and the merge of node with ame input doe not impact the coding on edge or the rate region, a the aociated contraint L A = L A If the input or output of an intermediate node i empty, a in (D6) it i incapable of affecting the capacity region If, a in the econd cae covered by (D6) the input to an ink node i empty, any ource which it demand can only be reliably decoded if they have zero entropy

21 (D7) i clear becaue an edge to nowhere can not effect the ret of the capacity region and i effectively uncontrained itelf (D8) can be hown a follow If R (A) i known and when edge e in A i repreented a two parallel edge e, e o that the network become A, then the contraint on e, e in A i imply to make ure the total capacity R e + R e can allow the information to be tranmitted from the tail node to head node Simple concatenation of the meage among the two edge will achieve thi for thoe bound l {c,, q, o} allowing uch concatenation Therefore, replace the R e in R l (A) with R e +R e will obtain the rate region R l (A ) for any l {c,, q, o} Moving from R l (A ) to R l (A) i accomplihed by recognizing that A i effectively A with R e = 0 nder the condition in (D9), an intermediate node g ha exactly one input edge e and exactly one output hyperedge e, and the input e i an edge (ie g i it only detination) The rate coming out of thi node can be no larger than the rate coming in ince the ingle output hyperedge mut be a determinitic function of the input edge It uffice to treat thee two edge a one hyperedge connecting the tail of e to the head of e with the rate the minimum of the rate on the two link If a ink demand a ource that it doe not have acce to, the only way to atify thi network contraint i the ource entropy i 0 Hence, (D0) hold The removal of thi redundant ource doe not impact the rate region of the network with remaining variable (D), imilar to (D3), oberve that two ink node with ame input yield the exact ame contraint L A a L A with the two ink node merged (D) i eay to undertand becaue the decoding ability of β(t) at ink node t i implied by ink t The non-neceary repeated decoding contraint will not affect the rate region for thi network (D3), imilar to (D), oberve that the ability of t to decode implie that t can decode it a well, and hence, adding or removing the direct acce to at t will not affect the rate region (D4) i obviouly true ince the weakly diconnected component can not influence each other rate region Fig 5 contain example illutrating thee reduction In general, we can define a minimality operator A = minimal(a ) on network, which check the minimality condition (C) (C4) on A one by one, in the order (C), (C), (C6), (C5), (C3), (C4), (C7) (C4) If any of the condition encountered i not atified, the network i immediately reduced it according to the aociated reduction in Theorem, and the reulting reduced network i checked again for minimality by tarting again at condition (C), if needed, until all minimality condition are atified Furthermore, define the aociated rate region operator R (A ) = minimal A A(R (A)) which move through each of the reduction tep applied by minimal(a ) to the network A in revere order, utilizing the expreion for the rate region change under each reduction, thereby obtaining the rate region of A from A Accordingly, let R (A) = minimal A A(R (A )) be the rate region operator which move through each of the reduction tep applied by minimal(a ) to the network A in order, utilizing the expreion for the rate region change under each reduction, thereby obtaining the rate region of A from A Thi network minimality operator and it aociated rate region operator will come in ue later in the paper However, we next dicu the enumeration of minimal network of a particular ize

22 Y g t Y I) Y g t Y Y $ Y Y g t Y II) Y g Y 6$ Y t Y III) Y g Y t Y g t Y Fig 6: A demontration of the equivalence between network coding problem via iomorphim: network I and II are equivalent becaue II can be obtained by permuting Y, Y However, network III i not equivalent to I or II, becaue the demand at the ink do not reflect the ame permutation of Y, Y a i neceary on the ource ide IV ENMERATION OF NON-ISOMORPHIC MINIMAL NETWORKS Even though the notion of network minimality ( III) reduce the et of network coding problem intance by removing part of a network coding problem which are ineential, much more need to be done to group network coding problem intance into appropriate equivalence clae Although we have to ue label et to decribe the edge and ource in order to pecify a network coding problem intance (identifying a certain ource a ource number one, another a ource number two, and o on), it i clear that the eence of the underlying network coding problem i inenitive to thee label For intance, it i intuitively clear that the firt two problem depicted in Fig 6 hould be equivalent even though their labeled decription differ, while the third problem hould not be conidered equivalent to the firt two In a certain ene, having to label network coding problem in order to completely pecify them obtruct our ability to work efficiently with a cla of problem Thi i becaue one unlabeled network coding problem equivalence cla typically conit of many labeled network coding problem In principle, we could go about invetigating the unlabeled problem by exhautively liting labeled network coding problem obeying the minimality contraint, teting for equivalence under relabeling of the ource and node or edge indice, and grouping them together into equivalence clae However, liting network by generating all variant of the labeled encoding become infeaible rapidly a the problem grow becaue of the large number of labeled network in each equivalence cla A a more feaible

23 3 alternative, it i deirable to find a method for directly cataloguing all (unlabeled) network coding problem equivalence clae by generating exactly one repreentative from each equivalence cla directly, without iomorphim (equivalence) teting In order to develop uch a method, and to explain the connection between it olution and other iomorphim-free exhautive generation problem of a imilar ilk, in thi ection we firt formalize a concie method of encoding a network coding problem intance in IV-A With thi encoding in hand, in IV-B, the notion of equivalence clae for network coding problem intance can be made precie a orbit in thi labeled problem pace under an appropriate group action The generic algorithm Leiterpiel [34], [35], for computing orbit within the power et of ubet of ome et X on which a group G act, can then be applied, together with ome other tandard orbit computation technique in computational group theory [36], [37], in order to provide the deired non-iomorphic network coding problem lit generation method in IV-C A Encoding a Network Coding Problem Though, a i conitent with the network coding literature, we have thu far utilized a tuple A = (S, G, T, E, β) to repreent a network intance, thi encoding prove to be inufficiently parimoniou to enable eay identification of equivalence clae A will be dicued later, the commonly ued node repreentation of a network, a key component of the A = (S, G, T, E, β) encoding, unnecearily increae the complexity of enumeration Hence, we repreent a network intance in an alternate way for enumeration Specifically, a network intance with K ource and L edge that obey the minimality condition (C-C4) i encoded a an ordered pair (Q, W ) coniting of a et Q of edge definition Q {(i, A ) i {K +,, K + L}, A {,, K + L} \ {i}, A > 0}, and a et W of ink definition W {(i, A ) i {,, K}, A {,, K + L} \ {i}} Here, the ource are aociated with label {,, K} and the edge are aociated with label {K +,, K + L} Each (i, A ) Q indicate that the edge i E i encoded excluively from the ource and edge in A, and hence repreent the information that A = In(Tl(i)) Furthermore, each ink definition (i, A ) W repreent the information that there i a ink node whoe input are A and which decode ource i a it output Note that there are L non-ource edge in the network, each of which mut have ome input according to condition (C6) We additionally have the requirement that Q = L, and, to enure that no edge i multiply defined, we mut have that if (i, A ) and (i, A ) are two different element in Q, then i i A the ame ource may be decoded at multiple ink, there i no uch requirement for W A i illutrated in Figure 7a and 7b, thi edge-baed definition of the directed hypergraph included in a network coding problem intance can provide a more parimoniou repreentation than a node-baed repreentation, and a every edge in the network for a network coding problem i aociated with a random variable, thi repreentation map more eaily to the entropic contraint than the node repreentation of the directed acyclic hypergraph doe Additionally it i beneficial becaue it i guaranteed to obey everal of the key minimality contraint In particular, the repreentation enure that there are no redundant node (C5), (C), ince the intermediate node are aociated directly the element of the et {A i, (i, A ) Q} and the ink node are aociated directly with

24 4 {A i, (i, A ) W } Repreenting Q a a et (rather than a multi-et) alo enure that (C8) i alway obeyed, ince uch a parallel edge would be a repeated element in Q B Expreing Network Equivalence with a Group Action Another benefit of the repreentation of the network coding problem a the ordered pair (Q, W ) i that it enable the notion of network iomorphim to be appropriately defined In particular, let G := S {,,,K} S {K+,,K+L} be the direct product of the ymmetric group of all permutation of the et {,,, K} of ource indice and the ymmetric group of all permutation of the et {K +,, K + L} of edge indice The group G act in a natural manner on the element of the et Q, W of edge and ink definition In particular, let π G be a permutation in G, then the group action map π((i, A )) (π(i), π(a )) (39) with the uual interpretation that π(a ) = {π(j) j A } Thi action extend to an action on the et Q and W in the natural manner π(q) {π((i, A )) (i, A ) Q} (40) Thi action then extend further till to an action on the network (Q, W ) via π((q, W )) = (π(q), π(w )) (4) Two network (Q, W ) and (Q, W ) are aid to be iomorphic, or in the ame equivalence cla, if there i ome permutation of π G uch that π((q, W )) = (Q, W ) In the language of group action, two uch pair are iomorphic if they are in the ame orbit under the group action, ie if (Q, W ) {π((q, W )) π G} =: O (Q,W ) In other word, the equivalence clae of network are identified with the orbit in the et of all valid minimal problem decription pair (Q, W ) under the action of G We elect to repreent each equivalence cla with it canonical network, which i the element in each orbit that i leat in a lexicographic ene Note that thi lexicographic (ie, dictionary) order i well-defined, a we can compare two ubet A and A by viewing their member in increaing order (under the uual ordering of the integer {,, L + K}) and lexicographically comparing them Thi then implie that we can lexicographically order the ordered pair (i, A ) according to (i, A ) > (j, A ) if j < i or i = j and A < A under thi lexicographic ordering Since the element of Q and W are of the form (i, A ), thi in turn mean that they can be ordered in increaing order, and then alo lexicographically compared, enabling comparion of two edge definition et Q and Q or two ink definition et W and W Finally, one can then ue thee ordering to define the lexicographic order on the network ordered pair (Q, W ) The element in an orbit O (Q,W ) which i minimal under thi lexicographic ordering will be the canonical repreentative for the orbit A key baic reult in the theory of group action, the Orbit Stabilizer Theorem, tate that the number of element in an orbit, which in our problem i the number of network that are iomorphic to a given network, i equal to the

25 5 ratio of the ize of the acting group G and it tabilizer ubgroup G (Q,W ) of any element elected from the orbit: {π((q, W )) π G} = O(Q,W ) G = G(Q,W ), G (Q,W ) := {π G π((q, W )) = (Q, W )} (4) Note that, becaue it leave the et of edge, decoder demand, and topology contraint et-wie invariant, the element of the tabilizer ubgroup G (Q,W ) alo leave the et of rate region contraint (5), (6), (7), (8) invariant Such a group of permutation on ource and edge i called the network ymmetry group, and i the ubject of a eparate invetigation [8], [9] Thi network ymmetry group play a role in the preent tudy becaue, a depicted in Figure 7a and 7b, by the orbit tabilizer theorem mentioned above, it determine the number of network equivalent to a given canonical network (the repreentative we will elect from the orbit) In particular, Fig 7a how the orbit of a (, ) network (Q, W ) whoe tabilizer ubgroup (ie, network ymmetry group) i imply the identity, and hence ha only one element In thi intance, the number of iomorphic labeled network coding problem in thi equivalence cla i then G = S {,} S {3,4} = 4 in the edge repreentation, a hown at the left Even thi tiny example demontrate well the benefit of encoding a network coding problem via the more parimoniou repreentation (Q, W ) v the encoding via the node repreentation hypergraph (V, E ) and the ink demand β( ) Namely, becaue the ize of the group acting on the node repreentation i S {a,b} S {c,d,e,f,g} = 40, and, a the tabilizer ubgroup in the node repreentation ha the ame order (), the number of iomorphic network repreented in the node baed repreentation i 40 By contrat, Fig 7b how the orbit of a (, ) network (Q, W ) whoe tabilizer ubgroup (ie, network ymmetry group) i the larget poible among (, ) network, and ha order 4 In thi intance, the number of iomorphic labeled network coding problem in thi equivalence cla i G G (Q,W ) = in the edge repreentation The tabilizer ubgroup in the node repreentation ha generator {(a, b)(d, f)(e, g)}, {(d, e)(f, g)}, which ha the ame order of 4, and hence there are 40 4 = 60 iomorphic network coding problem to thi one in the node repreentation C Network Enumeration/Liting Algorithm Formalizing the notion of a canonical network via group action on the et of minimal (Q, W ) pair enable one to partly develop a method for directly liting canonical network baed on technique from computational group theory To olve thi problem we can harne the algorithm Leiterpiel, looely tranlated nake and ladder [34], [35], which, given an algorithm for computing canonical repreentative of orbit, ie, tranveral, on ome finite et X under a group G and it ubgroup, provide a method for computing the orbit on the power et P i (X ) = {B X B = i} of ubet from X of cardinality i, incrementally in i In fact, the algorithm can alo lit directly only thoe canonical repreentative of orbit for which ome tet function f return, provided that the tet function ha the property that any ubet of a et with f = alo ha f = Thi tet function i ueful for only liting thoe ubet in P i (X ) with a deired et of propertie, provided thee propertie are inherited by ubet of a uperet with that property

26 6 # 3 4 ` # 3 4 ` # 3 4 ` # 3 4 ` a c b d # a b c d e f g # 3 4 ` # 3 4 ` # 3 4 ` # Q W {(3, {, }), (4, {, 3})} {(, {3, 4}), (, {3}), (, {4})} {(3, {, }), (4, {, 3})} {(, {3}), (, {4}), (, {3, 4})} 3 {(3, {, 4}), (4, {, })} {(, {3, 4}), (, {3}), (, {4})} 4 {(3, {, 4}), (4, {, })} {(, {3}), (, {4}), (, {3, 4})} Subgroup of S {,} S {3,4} tabilizing (G, T )=h()i a e f g a b Subgroup of S {a,b} S {c,d,e,f,g} tabilizing (V, E) = h()i Iomorph in Node Repreentation (40) Iomorph in Edge and Sink Definition Repreentation (4) (a) All iomorphim of a (, ) network with empty ymmetry group 3 4 a c d e f g b # a b c d e f g b b a a Subgroup of S {a,b} S {c,d,e,f,g} tabilizing (V, E) # Q W = h{(a, b)(d, f)(e, g)}, {(d, e)(f,g)}i {(3,{,}),(4,{,})} {(,{,3}),(,{, 4}),(,{,3}),(,{,4})} The tabilizer ubgroup i of order 4 Subgroup of S {,} S {3,4} tabilizing (G, T )=S {,} S {3,4} (b) All iomorphim of a (, ) network with full ymmetry group Fig 7: Example of (, ) network with all edge iomorphim (left) and all node iomorphim (right) The intance indice are marked by # and the label are marked by l

27 7 To ee how to apply and modify Leiterpiel for network coding problem enumeration, let X be the et of poible edge definition X := {(i, A ) i {K +,, K + L}, A {,, K + L} \ {i}} (43) For mall to moderately ized network, the orbit in X from G and it ubgroup can be readily computed with modern computational group theory package uch a GAP [37] or PERMLIB [36] Leiterpiel can be applied to firt calculate the non-iomorphic candidate for the edge definition et Q, a it i a ubet of X with cardinality L obeying certain condition aociated with the definition of a network coding problem and it minimality (cf C C4) Next, for each non-iomorphic edge-definition Q, a lit of non-iomorphic ink-definition A, alo contrained to obey problem definition and minimality condition (C C4), can be created with a econd application of Leiterpiel The peudo-code for the reulting generation/enumeration i provided in Alg A outlined above, in the firt tage of the enumeration/generation algorithm, Leiterpiel i applied to grow ubet from X of ize i incrementally in i until i = L Some of the network condition have the appropriate inheritance propertie, and hence can be incorporated a contraint into the contraint function f in the Leiterpiel proce Thee include no repeated edge definition: If B C X and C ha the property that no two of it edge definition (i, A ) and (i, A ) have i = i, then o doe B Hence, the contraint function f in the firt application of Leiterpiel incorporate check to enure that no two edge definition in the candidate ubet define the ame edge acyclicity: If B C X and C i aociated with an acyclic hyper graph, then o i B Hence, the contraint function f in the firt application of Leiterpiel check to determine if the ubet in quetion i acyclic At the end of thi firt Leiterpiel proce, ome more canonical edge definition et Q can be ruled a non-minimal owing to (C), requiring that each ource appear in the definition of at leat one edge variable For each member of the reulting narrowed lit of canonical edge definition et Q, we mut then build a lit of canonical repreentative ink definition W Thi i done by firt creating the (Q-dependent) et of valid ink definition X := {(i, A ) a directed path in Q from i to at leat one edge in A } (44) which are crafted to obey the minimality condition (C0) that the created ink (defined by it input which i the et of ource and edge in A in the ink definition (i, A )) mut have at leat one path in the hyper graph defined by Q to the ource i it i demanding, and (C) that i can not have a direct connection to the ource it i demanding Leiterpiel i then applied to determine canonical (lexicographically minimal) repreentative of ink definition et W, utilizing the aociated tabilizer of the canonical edge definition et Q being extended a the group, with the tet function f handling the minimality condition (C) and (C3), during the iteration Thi econd application of Leiterpiel to determine the lit of canonical ink definition et W for each canonical edge definition et Q doe not have a definite cap on the cardinality of each of the canonical ink definition et W

28 8 Input: number of ource K, number of non-ource edge L Output: All non-iomorphic network intance Z Initialization: Z = ; Let X := {(i, A ) i {K +,, K + L}, A {,, K + L} \ {i}}; Let f be the condition that (i, A ), (i, A ) uch that i = i ; Let f be the condition of acyclicity; Let acting group G := S {,,K} S {K+,,K+L} ; Call Leiterpiel algorithm to incrementally get all candidate tranveral up to L: T L = Leiterpiel(G, P f,f L (X )); for each Q T L do end if Q obey (C) then Let X := {(i, A ) i {,, K}, A {,, K + L} \ {i}, a directed path in Q from i to at end leat one edge in A }; Let f be the condition (C); Let f be the condition (C3); Let acting group G := S {,,K} S {K+,,K+L} ; Call Leiterpiel algorithm to incrementally get all candidate canonical repreentative, ie, tranveral, up to no new element can be added obeying (C,C3): T K = Leiterpiel(G, P f,f K (X )); for each W T K do end if (Q, W ) obey (C3 C7) and (C4) then end Z = Z (Q, W ); Algorithm : Enumerate all non-iomorphic (K, L) network uing Leiterpiel algorithm Rather, ubet of all ize are determined incrementally until there i no longer any canonical ubet that can obey the contraint function aociated with (C) and (C3) Each of the candidate canonical ink definition et W (of all different cardinalitie) are then teted together with Q with the remaining condition, which do not have the inheritance property neceary for incorporation a contraint earlier in the two tage of Leiterpiel proceing Any pair of canonical (Q, W ) urviving each of thee check i then added to the lit of canonical minimal non-iomorphic network coding problem intance An additional pleaant ide effect of the enumeration i that the tabilizer ubgroup, ie, the network ymmetry group [9], are directly provided by the econd Leiterpiel Harneing thee network ymmetry group provide

29 9 a powerful technique to reduce the complex proce of calculating the rate region for a network coding problem intance [8] Although thi method directly generate the canonical repreentative from the network coding problem equivalence clae without ever liting other iomorph within thee clae, one can alo ue the tabilizer ubgroup provided by Leiterpiel to directly enumerate the ize of thee equivalence clae of (Q, W ) pair, a decribed above via the orbit tabilizer theorem Experiment ummarized in Table II how that the number of iomorphic cae i ubtantially larger than the number of canonical repreentive/equivalence clae, and hence the extra effort to directly lit only canonical network i worthwhile It i alo worth noting that a node repreentation, utilizing a node baed encoding of the hyper edge, would yield a ubtantially higher number of iomorph D Modification to Other Problem Type A final point worth noting i that thi algorithm i readily modified to handle liting canonical repreentative of pecial network coding problem familie contained within our general model, a decribed in II-A For intance, IDSC problem can be enumerated by imply defining Q to have each edge acce all of the ource and no other edge, then continuing with the ubequent ink enumeration proce It i alo eaily adapted to enumerate only directed edge and match the more retrictive contraint decribed in the original Yan, Yeung, and Zhang [] rate region paper E Enumeration Reult for Network with Different Size By uing our enumeration tool with an implementation of the algorithm above, we obtained the lit of canonical minimal network intance for different network coding problem ize with N = K + L 5 While the whole lit i available [38], we give the number of network problem intance in Table II, where Z, ˆ Z, ˆ Z n repreent the number of canonical network coding problem (ie, the number of equivalence clae), the number of edge decription of network coding problem including ymmetrie/equivalence, and the number of node decription of network coding problem including the ymmetrie/equivalence, repectively A we can ee from the table, the number of poibilitie in the node repreentation of the network coding problem explode very quickly, with the more than trillion labeled node network coding problem covered by the tudy only neceitating a lit of coniting of roughly 750,000 equivalence clae of network coding problem That aid, it i alo important to note that the number of non-iomorphic network intance increae exponentially fat a network ize grow For intance, the number of non-iomorphic general network intance grow from 333 to 485, 890 (roughly, an increae of about 500 time), when the network ize grow from (, ) to (, 3) To provide an illutration of the variety of network that are encountered, Fig 8 depict all 46 of the 333 canonical minimal network coding problem of ize (, ) obeying the extra contraint that no ink ha direct acce to a ource A a pecial cla of hyperedge multi-ource network coding problem, it i eaier to enumerate IDSC network, defined in Example in II Since we aume that all encoder in IDSC have acce to all ource, we only need to conider the configuration at the decoder, which additionally are only afforded acce to edge from intermediate

30 30 Y g Y g t Y Y g t Y Y g t Y Y g t Y Y t Y Y Y Y g g Y t Y Y g Y t Y Y g Y Y t g t Y Intance # Intance # Intance #3 Intance #4 Intance #5 Y g t Y t Y Y g Y g t Y Y t Y Y Y g Y g t Y g t Y Y Y g t Y Y g Y t 3 Y Y g g t Y Y Intance #6 Intance #7 Intance #8 Intance #9 Intance #0 Y g t Y Y g t Y Y g t Y Y g t Y t Y Y g t Y Y Y g t Y g Y t Y Y g t Y g t Y Y g t 3 Y Intance # Intance # Intance #3 Intance #4 Intance #5 Y g t Y Y g t Y Y Y g t Y Y g t Y Y g t Y Y g t Y Y Y g t Y Y g t Y Y g t Y Y g t Y Intance #6 Intance #7 Intance #8 Intance #9 Intance #0 t Y Y g Y g Y g Y g t Y Y Y g t Y t Y t Y Y t Y Y Y g t 3 Y Y g Y g Y g t Y Y Y g t Y Y Intance # Intance # Intance #3 Intance #4 Intance #5 Y g t Y Y Y g t Y Y g t Y Y g t Y Y g t Y Y Y g t Y Y g t Y Y Y g t Y Y g t Y Y g t Y Intance #6 Intance #7 Intance #8 Intance #9 Intance #30 Y g t Y Y g t Y Y g t Y Y g t Y t Y Y g t Y Y g t Y Y g t Y Y g t Y Y g t Y Y g t 3 Y Intance #3 Intance #3 Intance #33 Intance #34 Intance #35 t Y Y g Y t Y t Y t Y Y t t Y Y Y Y g g g g t Y Y g t 3 Y Y t Y Y Y t Y Y t Y Y t 3 Y Intance #36 Intance #37 Intance #38 Intance #39 Intance #40 Y g Y t Y g t Y Y Y g t Y Y g t Y Y g t Y Y Y g t Y Y Y g t Y Y g t Y Y g t Y g t Y Intance #4 Intance #4 Intance #43 Intance #44 Intance #45 t Y Y g t Y Y g t 3 Y Intance #46 Fig 8: All 46 non-iomorphic network intance of (, ) network with the contraint that ink do not have direct acce to ource

31 3 TABLE II: Number of network coding problem of different ize: Z repreent the number of non-iomorphic network, Z ˆ repreent the number of iomorphic network with edge iomorphim, and Zˆ n repreent the number of iomorphic network with node iomorphim (K, L) Z ˆ Z Z ˆ n (,) (,3) (,4) (,) 6 (,) (,3) (3,) (3,) (4,) Total TABLE III: Lit of number of IDSC configuration Z n i the number of configuration in the node repreentation including iomorphim, Z i the number of configuration in the edge repreentation including iomorphim, and Z i the number of all non-iomorphic configuration (K, L) 3 Z n Z Z Z n Z Z node Thee extra contraint are eaily incorporated into Algorithm by removing the edge definition, retricting to the unique one aociated with the IDSC problem, and enumerating excluively the ink definition We give the enumeration reult for K =, 3 and L =, 3 IDSC network in Table III, while the full lit i available in [39] From the table we ee that, even for thi pecial type of network, the number of non-iomorphic intance grow very quickly For intance, the number of non-iomorphic IDSC intance grow from 33 to 79 (roughly, a factor of 6 increae), when the network ize grow from (, 3) to (3, 3) V RATE REGION RESLTS FOR SMALL NETWORKS With the lit of minimal canonical network coding problem provided by the algorithm in the previou ection in hand, the next tep in our computational agenda wa to determine each of their rate region with computational tool In thi ection, we decribe a databae we have created which contain the exact region of all general network with ize N = K + L 5 and all IDSC network with ize K =, 3 and L =, 3 A Databae of Rate Region for all network of ize N = K + L 5 We begin by decribing the experimental reult we obtained by running our rate region computation oftware on all general hyperedge network intance of ize N = K + L 5 Thee problem conit of 744, 9 canonical

32 3 TABLE IV: Sufficiency of code for network intance: Column 3 8 how the number of intance that the rate region inner bound match with the Shannon outer bound (K, L) Z R, (A) R N+ (A) R N+ (A) R N+3 (A) R N+4 (A) Rlinear N (, ) (, 3) (, 4) (, ) (, ) (, 3) (3, ) (3, ) (4, ) Total: minimal network, repreenting 9, 050, 490 network in the edge (Q, W ) encoding and network in the tandard node repreentation, a indicated in Table II For each non-iomorphic network intance, we calculated everal bound on it rate region: the Shannon outer bound R o, the calar binary repreentable matroid inner bound R,, the vector binary repreentable matroid inner bound R N+,, R N+4, and linear inner bound R N linear A indicated in II, if the outer bound on the rate region matche with an inner bound, we not only obtain the exact rate region, but alo know the code that uffice to achieve any point in it The general code contruction from repreentable matroid follow a imilar proce in [5], [6], where rate region and achieving code are invetigated for MDCS Though it i infeaible to lit each of the 744, 9 rate region in thi paper, a ummary of reult on the matche of variou bound i hown in Table IV The full lit of rate region bound can be obtained at [38] and can be re-derived uing [40] Several key obervation kay be made from Table IV Firt of all the Shannon outer bound i proved to be tight for all network of ize N = K + L 5 Additionally, the reult how that linear code are ufficient to exhaut the entire capacity region for all of them, a indicated in column and 8 in Table IV Furthermore, we invetigate the number of network whoe rate region are achievable by imple linear code, eg, binary code (column 3 7 in Table IV), and find that imple binary code are capable of exhauting mot of the capacity region For all (, ) and (, ) network, calar binary code uffice However, thi i not true in general even when there are only one or two edge variable For example, there are ome intance in (3, ), (4, ), (, ) and (3, ) network for which calar binary code do not uffice A we can ee from Table IV, a the vector binary inner bound get tighter and tighter (ie, a we move to the right in Table IV), the exact rate region i etablihed for more and more intance That i, with tighter and tighter binary inner bound, more and more intance are found for which binary code uffice

33 33 Y Y g t g t Y Y 3 Y Y 3 Y 3 3 g 3 t 3 Y Y 3 3 R H(Y ) R + R 3 H(Y )+H(Y 3 ) R + R 3 H(Y )+H(Y 3 ) R + R H(Y )+H(Y )+H(Y 3 ) R + R +R 3 H(Y )+H(Y )+H(Y 3 ) Fig 9: Block diagram and rate region R (A) for the (3, 3) network intance A in Example 4 Fig 0: Comparion of rate region R (A) (which equal to R o(a)) and R(A) 7 for the (3, 3) network intance A in Example 4, when ource entropie are (H(Y ), H(Y ), H(Y 3)) = (,, ) and the cone i capped by R + R + R 3 0: the white part i the portion that calar binary code cannot achieve The ratio of R(A) 7 over R (A) i about 9957% for thi choice of (H(Y ), H(Y ), H(Y 3)) In order to provide a ample of the ort of reult available in the databae [38], the following example how the variou inner bound on the rate region of a repreentative (3, 3) problem Example 4: A 3-ource 3-encoder hyperedge network intance A with block diagram and rate region R (A) hown in Fig 9

34 34 Fig : Comparion of rate region R 7 (A) and R,(A) for the (3, 3) network intance A in Example 4 when ource entropie are [H(Y ), H(Y ), H(Y 3)] = [,, ] and the cone i capped by R + R + R 3 0: the white part i the portion that calar binary code cannot achieve The ratio of R,(A) over R 7 (A) i about 994% for thi choice of [H(Y ), H(Y ), H(Y 3)] Firt, calar binary code do not uffice for thi network The calar binary coding rate region i R + R + R 3 H(Y ) + H(Y ) + H(Y 3 ) R, = R (A) R + R + R 3 H(Y ) + H(Y ) + H(Y 3 ) (45) One of the extreme ray in the Shannon outer bound on rate region i [R, R, R 3, H(Y ), H(Y ), H(Y 3 )] = [,,, 0, 0, ] Thi extreme ray cannot be achieved by calar binary code becaue no calar code can encode a ource with entropy of two into a variable with entropy of at mot one Fig 0 illutrate the gap between R (A) and R, (A) with a particular ource entropy aignment When ource entropie are [H(Y ), H(Y ), H(Y 3 )] = [,, ] and the cone i capped by R + R + R 3 0, there i a clear gap between the two polytope, though the inner bound occupie more than 99% of the exact rate region for thi choice of [H(Y ), H(Y ), H(Y 3 )] Second, vector binary code from 7 bit do not uffice for thi network either The vector binary coding rate region i R 7 = R (A) {R + R + R 3 H(Y ) + H(Y ) + H(Y 3 )} (46)

35 35 TABLE V: Sufficiency of code for IDSC intance: Column 3 and 4 how the number of intance that the rate region inner bound match with the Shannon outer bound (K, L) Z R, (A) R N+ (A) (, ) (, 3) (3, ) (3, 3) One of the extreme ray in the Shannon outer bound i [R, R, R 3, H(Y ), H(Y ), H(Y 3 )] = [,,,,, 0] Thi extreme ray cannot be achieved by binary code from 7 bit becaue the empty ource Y 3 take one bit a well when we aign bit to variable in general Hence, at leat 8 bit are neceary (+++++=8), a will be hown later In our inner bound calculation, every variable in the network need to have at leat one aociated element from the repreentable matroid, even though it entropy can be zero, like Y 3 in thi cae Though thi inner bound i till looe in the ene of matching with the exact rate region, it i tighter than the calar binary inner bound R, (A) Thi i illutrated in Fig by chooing a particular ource entropy tuple When ource entropie are [H(Y ), H(Y ), H(Y 3 )] = [,, ] and the cone i capped by R + R + R 3 0, there i a clear gap between the two polytope, though the calar inner bound take more than 99% pace of the tighter vector binary inner bound for thi choice of [H(Y ), H(Y ), H(Y 3 )] However, vector binary code from 8 bit uffice for thi network and thu R 8 (A) = R (A) One can contruct vector binary code to achieve all extreme ray in the Shannon outer bound on the rate region For intance, the extreme ray [R, R, R 3, H(X), H(Y ), H(Z)] = [,,,,, 0] can be achieved by the vector binary code a follow: = [Y + Y (), Y () ], = Y () + Y (), 3 = Y (), where Y (), Y () are the two bit in ource Y B Databae of Rate Region for mall IDSC intance Here, experimental reult on thouand of IDSC (defined in Example in II-A) intance are preented eparately We invetigated rate region for 9 non-iomorphic minimal IDSC intance repreenting 530 iomorphic one Thee include the cae when (K, L) = (, ), (, 3), (3, ), (3, 3) Similarly, for the rate region of each noniomorphic IDSC intance, we calculated it Shannon outer bound R o, calar binary inner bound R,, and the vector binary inner bound R N+, where N = K + L = K + E A ummary of reult on the number of intance for which the variou bound agree i hown in Table V The exact rate region, their convere, and the code that achieve them for all 9 non-iomorphic cae can be obtained at [39] and can be re-derived uing [40] For the non-iomorphic IDSC intance we conidered, the Shannon outer bound i alway tight on the rate region, and the exact rate region are obtained Scalar binary code alo only uffice for the intance with L = but not for all intance with L = 3 However, vector binary code from binary

36 36 matroid on N + variable uffice for all the 9 intance Thu, for the IDSC problem up to K 3, L 3, vector binary code uffice After obtaining thee maive databae of all rate region for mall network, our next quetion i how to learn from them, and further, how to ue them to olve more (larger) network For thi purpoe, we will develop in the following two ection notion of network hierarchy that enable u to relate network of different ize, their rate region, and their propertie with one another VI NETWORK EMBEDDING OPERATIONS In thi ection, we propoe a erie of embedding operation relating maller network to larger network in a manner uch that one can directly obtain the rate region of the maller network from the rate region of the larger network Thee operation will be elected in a manner that, due to thi mapping, propertie of the larger network can be conidered to be inherited from mall network embedded within it In particular, we will how that if a certain cla of code i inufficient to exhaut the rate region of a mall network embedded in a larger one, then thi cla of code will be inufficient to exhaut the rate region of the larger one a well A Definition of embedding operation The firt operation i ource deletion When a ource i deleted or removed, the ource doe not exit in the new network and the decoder that previouly demanded it will no longer demand it after deletion Fig a illutrate the deletion of a ource When ource k i deleted, t will no longer require k A particular example i hown in Fig 3a After deleting the ource, the minimality condition are checked to make ure the obtained network i minimal, and if not, an aociated minimal network i found via a erie of reduction according to Thm Definition 6 (Source Deletion (A\k)): Fix network A = (S, G, T, E, β) If ource k S i deleted, then, the new network i minimal(a ), where A = (S, G, T, E, β ) with S = S \ k and β = (β(t) \ k, t T ) The next operation we conider i edge contraction When an edge i contracted, the edge will be removed, and it head node are given direct acce to all the input of the tail node Fig b demontrate the contraction of an edge A it how, when edge e i contracted, the head node it connect to will directly have acce to all the input of it tail node Minimality condition need to be checked after thi operation A particular example i hown in Fig 3b Definition 7 (Edge Contraction (A/e)): Fix network A = (S, G, T, E, β) If edge e i contracted, then, the new network i minimal(a ), with A = (S, G, T, E, β) where E = E \(e In(Tl(e))) e In(Tl(e)) {Tl(e ), Hd(e) Hd(e )} Finally, we define edge deletion When an edge i deleted, it i imply removed from the graph, and the reulting graph i then checked and, if neceary, further reduced, for minimality Fig c demontrate the deletion of an

37 37 Y k k K kt Y (t) K kt Y (t)\k (a) Source deletion: when ource k i deleted, it end nothing to the network Decoder that previouly required Y k will no longer require it e e e e (b) Edge contraction: when e i contracted, the head node directly have acce to input of Tl(e) (c) Edge deletion: when delete e, it head node no longer receive information from e Fig : Definition of embedding operation on a network edge, and Fig 3c give a particular example of the operation With conideration of minimality condition, we formally define the edge deletion Definition 8 (Edge Deletion (A\e)): Fix network A = (S, G, T, E, β) If edge e i deleted, then, the maller network intance minimal(a ), with A = (S, G, T, E, β) where E = E \ e Baed on thee operation, we make precie the notion of an embedded network, or a network minor Definition 9 (Embedded Network): A network A i aid to be embedded in another network A, or i a minor of A, denoted a A A, if A can be obtained by a erie of operation of ource deletion, edge deletion/ contraction on A Similarly, we ay that A i an extenion of A, denoted A A With thi definition in hand, we et out in the next ubection on determining the relationhip between the rate region and propertie of a large network and the rate region and propertie of a mall network embedded within it B Inheritance of Rate Region & their Propertie nder Embedding Operation In thi ection we will prove a erie of theorem that explain both how to obtain the rate region of an embedded network, under the operator defined in the previou ubection, from that of a larger extenion network, a well a how certain propertie of the rate region can be viewed a inherited under embedding operation A particularly intereting rate region property we will conider i the ufficiency of a cla of linear code to exhaut the entire capacity region Note that in each of the theorem below, the network A will refer to the network in the definition of the aociated operator (ource deletion, edge contraction, and edge deletion) before the minimal( ) operator i

38 38 Y g t Y t Y Y Y g Y 3 t 3 Y g g 3 t t t 3 Y Y Y Y g g t t 3 Y Y (a) Source deletion example: when i deleted, it hyperedge i removed, and the ink t which previouly Y, Y will now demand only Y When minimality i conidered, it will be oberved that the new ink t ability to decode Y ha been implied by t Thu, t i removed a well At thi point and 3 have become parallel edge, which are then merged Y g Y g t Y t Y g t Y t Y t Y t Y Y g 3 Y t 3 Y Y g Y t 3 Y Y t 3 Y Y (b) Demontration of edge contraction on a network: when e 3 i contracted, the input of g will be directly available to t 3 When minimality i conidered, Y i now trivially decoded at t 3 due to direct acce to it, and thu Y i removed from β(t 3 ) In addition, g i removed Y g t Y Y g t Y Y g t Y Y t Y g t 3 3 Y Y Y g t Y Y t 3 Y g t Y (c) Demontration of edge deletion on a network: when e i deleted, t, t 3 have no acce to Then t, t are combined ince they have the ame input after deleting Fig 3: Example to how the embedding operation on a network applied Theorem 3: Suppoe a network A = minimal(a ) i a minimal form of a network A created by deleting ource k from another network A = (S, G, T, E, β), ie, A = A \ k Then for every l {, q, (, q), o} ) R l (A ) = minimal A A (Proj ω\h(yk ),r ({R R l (A) H(Y k ) = 0}) (47) Proof: We will prove R l (A ) = Proj ω\h(yk ),r ({R R l (A) H(Y k ) = 0}), ince the remainder of the theorem hold from the minimality reduction in Thm Select any point R R (A ) Then there exit a conic combination of ome point in R (A ) that are aociated with entropic vector in Γ N uch that R = α j r j, where α j 0, j For each r j, there exit random r j R (A )

39 39 [ variable Y (j) \k, (j) i, i E, where Y (j) \k = [ h (j) = H(A ) i in Γ N, where N = Y (j) i A ] i S \ k { Y (j), uch that the entropy vector }], e (j) S \ k, e E N i the number of variable in A Furthermore, their entropie atify all the contraint determined by A Define Y (j) k to be the empty ource, H(Y (j) k ) = 0 Then the entropie of random variable {Y (j) \k, (j) i, i E } Y (j) k will atify the contraint in A with H(Y (j) k ) = 0 and the entropy vector [ { }] h (j) = H(A ) A Y (j), e (j) S, e E will be in Γ N ince adding an empty variable doe not make an entropic vector to be non-entropic Denote r j = [r (j) j, H(Y k ) = 0], then r j R (A) Hence, by uing the ame conic combination, we have an aociated rate point R = α j r j {R R (A) H(Y k ) = 0} Thu, we r j R (A) have R (A ) Proj ω\h(yk ),r({r R (A) H(Y k ) = 0}) If R i achievable by F q code (calar or vector), there exit a contruction of ome baic F q code (calar or vector) to achieve it Since letting Y (j) be empty doe not affect the other ource and code, the ame contruction of baic F q code will alo achieve the point R with H(Y ) = 0 Thu, R l (A ) Proj ω\h(yk ),r({r R l (A) H(Y k ) = 0}), l {q, (, q)} On the other hand, if we elect any point R {R R (A) H(Y k ) = 0}, then, there exit a conic combination of ome point in R (A) {H(Y k ) = 0} aociated with entropic vector in Γ N, ie, R = α j r j, α j { } r j R (A) {H(Y k )=0} 0, j For each r j, there exit random variable Y (j) S, (j) E uch that their entropie atify all the contraint determined by A Furthermore, ince α j 0, the only conic combination make H(Y k ) = 0 i the cae that { } H(Y (j) (j) k ) = 0 We can drop H(Y k ) becaue the entropie of Y (j) \k, (j) E atify all contraint determined by A and the entropic vector projecting out Y (j) k i till entropic ing the ame conic combination, R = α j r j = Proj ω\h(yk ),r R R (A ) Thu, we have Proj ω\h(yk ),r({r R (A) H(Y k ) = Proj \H(Yk ) r j R (A) 0}) R (A ) If R i achievable by F q code C, then the code to achieve R could be the code C with deletion of row aociated with ource Y k, ie, C = C \Yk,: Thu, Proj ω\h(yk ),r({r R l (A) H(Y k ) = 0}) R l (A ), l {q, (, q)} Furthermore, for any point R R o (A ), there exit an aociated point h Γ N and a rate vector r = [R e e E ] uch that R = Proj ω\h(yk ),r [h, r ] L A Clearly, if we increae the dimenion of h by adding a variable Y k with zero entropy, ie, H(Y k ) = 0, we have the new entropy vector in Γ N That i, if we define [ ] h = h A {Y, e S,e E } A {Y, e S, e E }, then h Γ N Since H(Y k ) = 0, the network contraint in A will be atified given that the zero entropy doe not break the conditional entropie aociated with network contraint Hence, there exit an aociated point R R o (A) with H(Y k ) = 0 Therefore, we have R o (A ) Proj ω\h(yk ),r({r R o (A) H(Y k = 0)}) Reverely, uppoe a point R R o (A) i picked with H(Y k ) = 0 There exit a vector h Γ N and a rate vector r = [R e e E ] uch that R = Proj ω,r [h, r] L A Since the network contraint L A with H(Y k ) = 0 will be L A, and Proj ω\h(yk ),r [h, r] Γ N L A, we have Proj ω\h(yk ),rr R o (A ) Therefore, we have Proj ω\h(yk ),r({r R o (A) H(Y k = 0)}) R o (A ) Theorem 4: Suppoe a network A = minimal(a ) i a minimal form of a network A obtained by contracting

40 40 e from another network A = (S, G, T, E, β), ie, A = A/e Then ) R l (A ) = minimal A A (Proj ω,r\re R l (A), l {, q, o} (48) ) R,q (A ) minimal A A (Proj ω,r\re R,q (A), (49) Proof: We will prove R l (A ) = Proj ω,r\re ({R R l (A)}) for l {, q, o}, and for the calar cae, R,q (A ) ) minimal A A (Proj ω,r\re R,q (A), ince the remainder of the theorem hold from the minimality reduction in Thm Select any point R R (A ) Then there exit a conic combination of ome point in R (A ) that are aociated with entropic vector in Γ N uch that R = α j r j, where α j 0, j For each r j, there exit random r j R (A ) variable Y (j) S, (j) i, i E \ e, uch that the entropy vector [ { h (j) = H(A ) A i in Γ N, where N = Y (j), (j) i S, i E \ e N i the number of variable in A Furthermore, their entropie atify all the contraint determined by A In the network A, define e (j) to be the concatenation of all input to the tail node { } of e, e (j) = (j) In(Tl(e)) Then the entropie of random variable Y (j) S, (j) E will atify the contraint in A, and additionally obey H( e (j) ) = H( (j) In(Tl(e)) ) Hence, h(j) = [ { H(A ) A Y (j) }],, (j) i S, i E }] Γ N That i, r j = [r j, R e H( (j) In(Tl(e)) )] R (A) By uing the ame conic combination, we have an aociated rate point R = { α j r j R R (A) R e H( (j) In(Tl(e)) } ) Thu, we have r j R (A) R(A ) Proj ω,r\re ({R R(A) R e H( In(Tl(e)) )}) Proj ω,r\re R(A) (50) If R i achievable by general F q code, ince concatenation of all input i a valid F q vector code, we have R q (A ) Proj ω,r\re ({R R q (A) R e H( In(Tl(e)) )}) Proj ω,r\re R q (A) (5) However, we cannot etablih ame relationhip when calar F q code are conidered, becaue for the point R, the aociated R with H( e ) may not be calar F q achievable On the other hand, if we elect any point R {R R (A)}, then, there exit a conic combination of ome point in R (A) aociated with entropic vector in Γ N, ie, R = α j r j, α j 0, j For each { } r j R (A) r j, there exit random variable Y (j) S, (j) E uch that their entropie atify all the contraint determined by { } A Since the entropie of Y (j) S, (j) i i E \ e atify all contraint determined by A (becaue they are a ubet of the contraint from A) and the entropic vector projecting out e i till entropic Thu, by letting R e (j) to be uncontrained, we have Proj ω,r\re r j R (A ) Further, by uing the ame conic combination, R = Proj \Re α j r j = Proj ω,r\re R R (A ) Thu, we have Proj ω,r\re ({R R (A)}) R (A ) r j R (A) If R R (A) i achievable by F q code C, either calar or vector, then the code to achieve R = Proj ω,r\re R R (A ) could be the code C with deletion of column aociated with edge e, ie, C = C :,\e, becaue the code on edge e i not of interet Thu, we have Proj ω,r\re R l (A) R l (A ), l {q, (, q)}

41 4 Furthermore, for any point R R o (A ), there exit an aociated point h Γ N and a rate vector r = [R i i E \ e] uch that R = Proj ω,r\re [h, r ] L A Clearly, if we increae the dimenion of h by adding a variable e which i the vector of all input variable to the tail node of e, ie, e = [ i i In(Tl(e))] and H( e ) = H( i, i In(Tl(e))), we have the new vector in Γ N That i, if we define h A {Y h =, i S,i E }, e / A e A h A {Y, i S,i E } { i i In(Tl(e))} for A {Y, i S, i E }, then h Γ N Further, we let R e to be uncontrained, ie, R e = Since H( e ) R e, the network contraint in A will be atified given that the other contraint will not be affected Hence, there exit an aociated point R R o (A) with H( e ) R e, where R e i uncontrainted Therefore, we have R o (A ) Proj ω,r\re ({R R o (A)}) Reverely, uppoe a point R R o (A) i picked with R e uncontrained There exit an aociated vector h Γ N and a rate vector r = [R i i E ] uch that R = Proj ω,r [h, r] L A Since R e i uncontrained, we will have H( e ) uncontrained a well Since the network contraint L A with R e uncontrained will be L A, and Proj ω,r\re have Proj ω,r\re ({R R o (A)}) R o (A ) (5) [h, r] Γ N L A, we have Proj ω,r\re R R o (A ) Therefore, we Theorem 5: Suppoe a network A = minimal(a ) i a minimal form of a network A obtained by deleting e from another network A = (S, G, T, E, β), ie, A = A \ e Then ) R l (A ) = minimal A A (Proj ω,r\re ({R R l (A) R e = 0}), l {, q, (, q), o} (53) Proof: We will prove R l (A ) = Proj ω,r\re ({R R l (A) R e = 0}), ince the remainder of the theorem hold from the minimality reduction in Thm Select any point R R (A ) Then there exit a conic combination of ome point in R (A ) that are aociated with entropic vector in Γ N uch that R = α j r j, where α j 0, j For each r j, there exit random r j R (A ) variable Y (j) S, (j) i, i E \ e, uch that the entropy vector [ { h (j) = H(A ) A Y (j), (j) i S, i E \ e i in Γ N, where N = N i the number of variable in A Furthermore, their entropie atify all the contraint determined by A Let e (j) be the empty et or encoding all input with the all-zero vector, e (j) = and further { } let R e = 0 Then the entropie of random variable Y (j) S, (j) E will atify the contraint in A, and additionally [ { }] obey H( e (j) ) R e = 0 Furthermore, the vector h (j) = H(A ) A Y (j), (j) i S, i E Γ N That i, r j = [r j, R e = 0] R (A) By uing the ame conic combination, we have an aociated rate point R = α j r j {R R (A) R e = 0} Thu, we have R (A ) Proj ω,r\re ({R R (A) R e = 0}) r j R (A) If R i achievable by general F q linear vector or calar code, there exit a contruction of baic linear code to achieve it Since all-zero code i a valid F q vector and calar linear code, we have R l (A ) Proj ω,r\re ({R R l (A) R e = 0}), l {q, (, q)} }]

42 4 On the other hand, if we elect any point R {R R (A) R e = 0}, then, there exit a conic combination of ome point in R (A) {R e = 0} aociated with entropic vector in Γ N, ie, R = α j r j, α j { } r j R (A) {R e=0} 0, j For each r j, there exit random variable Y (j) S, (j) E uch that their entropie atify all the contraint determined by A Furthermore, ince α j 0, the only conic combination make R e = 0 i the cae that R e (j) = 0 and { } further H( e (j) ) = 0 We can drop H( e (j) ), ie, R e, becaue the entropie of Y (j) S, (j) \e atify all contraint determined by A and the entropic vector projecting out e (j) i till entropic ing the ame conic combination, R = Proj \Re α j r j = Proj ω,r\re R R (A ) Thu, we have Proj ω,r\re ({R R (A) R e = 0}) r j R (A) R (A ) If R i achievable by F q code C, then the code to achieve R could be the code C with deletion of column aociated with edge e, ie, C = C :,\e Thu, Proj ω,r\re ({R R l (A) R e = 0}) R l (A ), l {q, (, q)} Furthermore, for any point R R o (A ), there exit an aociated point h Γ N and a rate vector r = [R i i E \ e] uch that R = Proj ω,r\re [h, r ] L A Clearly, if we increae the dimenion of h by adding a variable e with zero entropy, ie, H( e ) = 0, we have the new entropy vector in Γ N That i, if we define [ ] h = h A {Y, i S,i E } A {Y, i S, i E }, then h Γ N Further, we let R e = 0 Since H( e ) = R e = 0, the network contraint in A will be atified given that the zero entropy (capacity) doe not break the conditional entropie aociated with network contraint Hence, there exit an aociated point R R o (A) with H( e ) = R e = 0 Therefore, we have R o (A ) Proj ω,r\re ({R R o (A) R e = 0}) Reverely, uppoe a point R R o (A) i picked with R e = 0 There exit a vector h Γ N and a rate vector r = [R i i E ] uch that R = Proj ω,r [h, r] L A Since R e = 0, we will have H( e ) = 0 Since the network contraint L A with H( e ) = R e = 0 will be L A, and Proj ω,r\re we have Proj ω,r\re ({R R o (A) R e = 0}) R o (A ) [h, r] Γ N L A, we have Proj ω,r\re R R o (A ) Therefore, Corollary 4: Conider two network A, A, with rate region R (A), R (A ), uch that A A If F q vector (calar) linear code uffice, or Shannon outer bound i tight for A, then ame tatement hold for A Equivalently, if F q vector (calar) linear code do not uffice, or Shannon outer bound i not tight for A, then ame tatement hold for A Equivalently, if R l (A) = R (A), then R l (A ) = R (A ), for ome l {o, q, (, q)} Proof: From Definition 9 we know that A i obtained by a erie of operation of ource deletion, edge deletion, edge contraction Theorem 3 5 indicate that ufficiency of linear code, vector or calar, and the tightne of Shannon outer bound are preerved for each ingle embedding operation For vector cae, if R q (A) = R (A), (47), (48), (53) directly give R q (A ) = R (A ) for ource deletion, edge contraction, and edge deletion, repectively Similar argument work for the tightne of the Shannon outer bound For calar code ufficiency, (47) and (53) indicate the ame preervation of ufficiency of calar code for ource and edge deletion, repectively For edge contraction and aumption of if R,q (A) = R (A), (48) and (49) indicate R (A ) R,q (A ) Together with the traightforward fact that R,q (A ) R (A ), ince calar F q code achievable rate region mut be ubet of the entire rate region, we can ee R,q (A ) = R (A ) hold for edge contraction a well Having introduced the embedding operation, which give maller network from larger network, we next

43 43 (a) Source merge: the merged ource will erve for the new larger network (b) Sink merge: input and output of the ink are unioned, repectively (c) Intermediate node merge: input and output of the node are unioned, repectively (d) Edge merge: one extra node and four aociated edge are added to replace the two edge Fig 4: Combination operation on two maller network to form a larger network Thickly lined node (edge) are merged introduce ome combination operation to get larger network from maller one VII NETWORK COMBINATION OPERATIONS In thi ection we propoe a erie of combination operation relating maller network with larger network in a manner uch that the rate region of the larger network can be eaily derived from thoe of the maller one In addition, the ufficiency of a cla of linear network code i inherited in the larger network from the maller one Throughout the following, the network A = (S, G, T, E, β) i a combination of two dijoint network A i = (S i, G i, T i, E i, β i ), i {, }, meaning S S =, G G =, T T =, E E =, and β (t ) β (t ) =, t T, t T A Definition of Combination Operation The operation we will define will merge network element, ie, ource, intermediate node, ink node, edge, etc, and are depicted in Fig 4 Since each merge will combine one or everal pair of element, with each pair containing one element from A and the other from A, each merge definition will involve a bijection π indicating which element from the appropriate et of A i paired with it argument in A We firt conider the ource merge operation, in which the merged ource will function a identical ource for both ub-network, a hown in Fig 4a A ink requiring ource involved in the merge will require the merged ource intead

44 44 Y g Y t g t Y Y Y Y g t 3 Y g 3 3 Y 4 g 3 Y g 3 t 4 Y t 5 Y Y g 4 Y 4 4 g 6 4 t 6 Y 3 Y 4 6 t Y t t 3 Y t 4 t 5 t 6 Y Y Y Y 4 Y Y 4 (a) Demontration of ource merge on two network: ource, 3 are merged to, o will end information to both ub-network Y g t Y Y g t Y t Y Y Y g 3 t 3 Y Y 3 4 Y 3 Y 4 4 g 3 5 t 4 t 5 g 4 6 t 6 3 Y 3 Y 3 Y 4 4 Y 4 Y 3 Y 4 t Y Y Y 3 g 3 4 g 3 t 3 Y 5 t 5 Y 4 g 4 t 6 6 Y 3 Y 4 (b) Demontration of ink merge on two network: ink t, t 4 are merged to t, o their input and demand are combined 3 4 (c) Y g g Y Y Y Y 3 Y 4 4 g 3 g t t t 3 t 4 t 5 t 6 Y Y Y 3 Y 4 Y 3 Y 4 Y Y g Y Y Y 3 t 3 Y Y 3 g 3 g 4 t t 5 6 t 4 t 5 t 6 Y Y 3 Y 4 Y 3 Y 4 Demontration of node merge on two network: node g and g 4 are merged to node g, o their input and output are alo combined 3 Y g t Y g t Y Y Y t 3 Y 3 Y 3 4 g 3 t 4 Y 3 g g Y 0 g Y 3 3 g 3 3 t 3 Y Y Y Y t t Y t 4 Y 3 5 t 5 Y 4 5 t 5 Y 4 4 Y 4 g 4 6 t 6 Y 3 Y 4 4 Y 4 g 4 6 t 6 Y 3 Y 4 (d) Demontration of edge merge on two network: when, 4 are merged, one extra node and four edge are added to replace, 4 in the two network, repectively Fig 5: Example to demontrate combination of two network

45 45 Definition 0 (Source Merge (A S ˆ = A π( S ˆ )) Fig 4a): Merging the ource S ˆ S from network A with the ource π( ˆ S ) S from a dijoint network A, will produce a network A with i) merged ource S = S S \ π( ˆ S ), ii) G = G G, iii) T = T T, iv) E = (E E \ A ) B, where A = {e E E Tl(e) B = {(, F F ) S ˆ π( S ˆ )} include the edge connected with the ource involved in the merge, ˆ S, (, F ) E, (π(), F ) E } include the new edge connected with the merged ource, and v) updated ink demand β (t) t T β(t) = ) ( ) (β (t) \ π(ŝ) π π(ŝ) β (t) t T Fig 5a demontrate the ource merge in a network example Similar to ource merge, we can merge ink node of two network, a demontrated in Fig 4b When two ink are merged into one ink, we imply union their input and demand a the input and demand of the merged ink Definition (Sink Merge (A T ˆ + A π( T ˆ )) Fig 4b): Merging the ink T ˆ T from network A with the ink π( T ˆ ) T from the dijoint network A will produce a network A with i) S = S S ; G = G G, ii) T = T T \ π( ˆ T ), iii) E = E E A \ B, where A = {(g, F F ) g G, F ˆ T, F T, (g, π(f ) F ) E } update the head node of edge in A with new merged ink, B = {(g, F ) E F π( ˆ T ) } include the edge connected to ink in π(ˆ(t )), and v) updated ink demand β(t) = β i (t) t T i \ ˆ T, i {, } β (t) β (π(t)) t T ˆ Fig 5b demontrate the ink merge in a network example (54) Next, we define intermediate node merge When two intermediate node are merged, we union their incoming and outgoing edge a the incoming and outgoing edge of the merged node, repectively, a illutrated in Fig 4c Definition (Intermediate Node Merge (A g + A π(g)) Fig 4c): Merging the intermediate node g G from network A with the intermediate node π(g) G from the dijoint network A will produce a network A with i) S = S S, ii) G = G G \ π(g), iii) T = T T, iv) E = E E A B \ C \ D, where A = {(g, F \ π(g) g) g G, (g, F π(g)) E } update the head node of edge in A that have π(g) a head node, B = {(g, F ) (π(g), F ) E } update the tail node of edge in A that have π(g) a tail node, C = {e E Tl(e) = π(g)} include the edge in A that have π(g) a tail node, D = {e E π(g) Hd(e)} include the edge in A that have π(g) a head node; and v) updated ink demand Fig 5c demontrate the node merge in a network example β (t) t T β(t) = (55) β (t) t T

46 46 Finally, we define edge merge A demontrated in Fig 4d, when two edge are merged, one new node and four new edge will be added to create a "cro" component o that the tranmiion will be in the new component intead of the two edge being merged Definition 3 (Edge Merge (A e+a π(e)) Fig 4d): Merging edge e E from network A with edge π(e) E from dijoint network A will produce a network A with i) S = S S, ii) G = G G g 0, where g 0 / G, g 0 / G, iii) T = T T, iv) E = (E \e) (E \π(e)) {(Tl(e), g 0 ), (Tl(π(e)), g 0 ), (g 0, Hd(e)), (g 0, Hd(π(e)))}; and v) updated ink demand given by (55) It i not difficult to ee that thi edge merge operation can be thought of a a pecial node merge operation Suppoe the edge being merged are A e, A π(e) If two virtual node g, g are added on e, π(e), repectively, plitting them each into two edge, o that e, π(e) go into and flow out g, g, repectively, then, the merge of g, g give the ame network a merging e, π(e) Fig 5d demontrate the edge merge in a network example B Preervation Propertie of Combination Operation Here we prove that the combination operation enable the rate region of the mall network to be combined to produce the rate region of the reulting large network, and alo preerve ufficiency of clae of code and tightne of other bound Theorem 6: Suppoe a network A i obtained by merging S ˆ with π( S ˆ ), ie, A S ˆ = A π( S ˆ ) Then R l (A) = Proj((R l (A ) R l (A )) L 0 ), l {, q, (, q), o} (56) { with L 0 = H(Y ) = H(Y π() ), S ˆ }, and the dimenion kept in the projection are (H(Y ), S ) and (R e, e E ), where S, E repreent the ource and edge et of the merged network A, repectively Remark : The inequality decription of the polyhedral cone Proj((P P ) L 0 ) for two polyhedral cone P j, j {, } can be created by concatenating the inequality decription for P and P, then replacing the variable H(X π() ) with the variable H(X ) for each Proof: Select any point R R (A) Then there exit a conic combination of ome point in R (A) that are aociated with entropic vector in Γ N uch that R = r j R (A) ˆ S random variable Y (j) S, (j) i, i E \ e, uch that the entropy vector [ { h (j) = H(A ) A Y (j) α j r j, where α j 0, j For each r j, there exit, (j) i S, i E i in Γ N, where N i the number of variable in A Furthermore, their entropie atify all the contraint determined by A When decompoing A into A, A, let iid copie of variable Y (j), S ˆ work a ource π( S ˆ ) S The aociated edge connecting variable {Y (j), (j) i S, i E }, {Y (j) S ˆ and node in G will then connect π( S ˆ ) and node in G Then the random, (j) i S, i E } will atify the network contraint determined }]

47 47 by A, A, and alo L 0 Thu, R Proj((R (A ) R (A )) L 0 ) Similarly, if R i achievable by F q code, vector or calar, the ame code applied to the part of A that i A, A will achieve R, R, repectively Putting thee together, we have R l (A) Proj((R l (A ) R l (A )) L 0 ), l {, q, (, q)} Next, if we elect two point R R (A ), R R (A ) uch that H(Y ) = H(Y π() ), S ˆ, then there exit conic combination R = α i,j r i,j for i =,, and for each r i,j there exit a et of variable aociated r i,j R (A i) with ource and edge Since H(Y ) = H(Y π() ) and ource are independent and uniformly ditributed, we can let the aociated variable Y (j) of all variable {Y (j) and Y (j) π() be the ame variable Then, after combination, the entropy vector, e (j) S, e E } will be in Γ N Furthermore, their entropie, together with the rate vector from R, R, will atify all network contraint of A, and there will be an aociated point r = r r with L 0 ing the ame conic combination, we will find the aociated point R = R R L 0 Hence, Proj((R (A ) R (A )) L 0 ) R (A) Now uppoe there exit a equence of network code for A and A achieving R, R By uing the ame ource bit a the ource input for in A and π() in A for each Ŝ, we have the ame effect a uing thee ource bit a the input for in the ource merged A and achieving the aociated rate vector R, implying R R l (A), l {q, (, q)}, and hence R,q (A) Proj((R,q (A ) R,q (A )) L 0 ) and R q (A) Proj((R q (A ) R q (A )) L 0 ) Together with the tatement above, thi prove (56) for l {, q, (, q)} Furthermore, by (3), any point R R o (A), i the projection of ome point [h, r] Γ N L A, where h Γ N and r = [R e e E ] Becaue the Shannon inequalitie and network contraint in Γ N L A form a uperet (ie, include all of) of the network contraint in Γ N L (A i ), the ubvector [h i, r i ] of [h, r] aociated only with the variable in A i (with Y π() being recognized a Y for all Ŝ) are in Γ N i L (A i ) and obey L 0, implying R Proj((R o (A ) R o (A )) L 0 ), and hence R o (A) Proj((R o (A ) R o (A )) L 0 ) Next, if we elect two point R R o (A ), R R o (A ) uch that H(Y ) = H(Y π() ), ˆ S, then there exit [h i, r i ] Γ Ni L (A i ), where h i Γ Ni and r i = [R e e E i ], uch that R i = Proj ri,ω i [h i, r i ], i {, } with h X = h X π() for all Ŝ Define h whoe element aociated with the ubet A of N = S E i h A = h A N + h A N h A π( ˆ S ) where N i = S i E i, i {, } By virtue of it creation thi way, thi function i ubmodular and h Γ N Since the two network are dijoint, the lit of equalitie in L 3 (A) i imply the concatenation of the lit in L 3 (A ) and L 3 (A ), each of which involved inequalitie in dijoint variable N and N, and the ame thing hold for L 4 with conideration of r i Furthermore, ince h i L (A i ) and h Y = h Y π(), ˆ S, h obey L (A) The definition of h, together with h i L (A i ), i {, } and h Y = h Y π(), ˆ S, implie that h L (A) Finally h L (A ) and h L (A ) imply h L 5 (A) Putting thee fact together we oberve that [h, r] Γ N L A, o R R o (A), implying R o (A) Proj((R o (A ) R o (A )) L 0 ) Theorem 7: Suppoe a network A i obtained by merging ink node ˆ T with π( ˆ T ), ie, (A ˆ T + A π( ˆ T )) Then R l (A) = R l (A ) R l (A ), l {, q, (, q), o} (57) with the index on the dimenion mapping from {e E Hd(e) π( T ˆ )} to {e E Hd(e) T ˆ, Tl(e) G }

48 Proof: Conider a point R R l (A) with conic combination of R = r j R l (A) 48 α l,j r l,j, where α l,j 0 for any j and l {, q, (, q), o} Each r l,j ha aociated random variable or the aociated code Due to the independence of ource in network A, A, and the fact that their ource and intermediate node are dijoint, the variable arriving at a merged ink node from A will be independent of the ource in A and the variable arriving at a merged ink node from A will be independent of the ource in A In particular, Shannon type inequalitie imply the Markov chain H(Y S In(t) E, In(t) E ) = H(Y S In(t) E ) and H(Y S In(t) E, In(t) E ) = H(Y S In(t) E ) for all t T (even if the aociated entropie are only in Γ N and not necearily Γ N ) Thi then implie, together with the independence of the ource, that H(Y β(t) In(t) ) = H(Y β(t) S In(t) E ) + H(Y β(t) S In(t) E ), howing that the contraint in L 5 (A) imply the contraint in L 5 (A ) and L 5 (A ) Furthermore, given the dijoint nature of A and A, the contraint in L i (A), are imply the concatenation of the contraint in L i (A ) and L i (A ), for i {, 3, 4 } Furthermore, the joint independence of all of Y S, Y S imply the marginal independence of the collection of variable Y S and Y S, o that L (A) implie L (A i ), i {, } Thi how that r l,j r l,j r l,j and further R R l (A ) R l (A ), and hence R l (A) R l (A ) R l (A ), l {, q, (, q), o} Next, conider two point R i R l (A i ), i {, } for any l {q, (, q), o} By definition thee are projection of [h i, r i ] Γ q N i, L (A i), [h i, r i ] Γ q N i L (A i ), [h i, r i ] Γ Ni L (A i ), repectively, for i {, }, where h i Γ Ni and r i = [R e e E i ] Define h with value aociated with ubet A N of h A = h A N + h A N, then it i eaily verified that the reulting [h, r, r ] Γ q N, L A, [h, r, r ] Γ q N L A, [h, r, r ] Γ N L A, repectively, (imply ue the ame code from A and A on the correponding part of A) Since R = Proj ω,r [h, r, r ], we have proven R R l (A), and hence that R l (A) R l (A ) R l (A ) Further, for two point R i R (A i ), i {, }, there exit a conic combination of r i j, R i = αj i ri j, r { } { } i j R (Ai) with aociated random variable Y (j), (j) i S, i E, Y (j), (j) i S, i E atifying the network contraint determined by A, A Due to the independence of ource and dijoint of edge variable, the union of variable in A, A will atify the network contraint in the merged A With the ame conic combination, we have R = [αj r j, α j r j ] R (A) Thu, R (A) R (A ) R (A ) r i j R (Ai) Theorem 8: Suppoe a network A i obtained by merging g and π(g), ie, A g + A π(g) Then R l (A) = R l (A ) R l (A ), l {, q, (, q), o} (58) with dimenion/ indice mapping from {e E Hd(e) = π(g)} to {e E Hd(e) = g, Tl(e) G } and from {e E Tl(e) = π(g)} to {e E Tl(e) = g, Hd(e) G } Proof: Conider a point R R l (A) for any l {, q, (, q)} and all random variable aociated with each component r l,j in the conic combination R = α l,j r l,j, where α l,j 0 for any j and l {, q, (, q), o} r j R l (A) The aociated variable atify L i (A), i =, 3, 4, 5 Partition the incoming edge of the merged node g in A, In(g), up into In (g) = In(g) E the edge from A, and In (g) = In(g) \ In (g), the new incoming edge reulting from the merge Similarly, partition the outgoing edge Out(g) up into Out (g) = Out(g) E and

49 49 Out (g) = Out(g) \ Out (g) The L 3 contraint dictate that there exit function f e uch that for each e Out(g), e = f e ( In(g), In(g)) Define the new function f e via f e( f e ( In(g), 0) In(g), In(g)) = f e (0, In(g)) e Out (g) e Out (g) ie, et the poible value for the incoming edge from the other part of the network (poibly erroneouly) to a particular contant value among their poible value let label it 0 The network code uing thee new function f e will utilize the ame rate a before The contraint and the topology of the merged network further dictated that Ini(g) were expreible a a function of S i, i {, } In the remainder of the network (moving toward the ink node) after the merged node, at no other point i any information from the ource in the other part of the network encountered, and the decoder at the ink node in T need to work equally well decoding ubet of S, regardle of the value of S Since the erroneou value for the In(g) ued for f e, e Out (g) wa till a valid poibility for ome (poible other) value() of the ource in S, the ink mut till produce the correct value for their ubet of S A parallel argument for T how that they till correctly decode their ource, which were ubet of S, even though the f e were changed to f e Note further that (59) will till be calar/vector linear if the original f e were a well However, ince the f e no longer depend on the other half of the network, the reulting code can be ued a eparate code for A and A, given the aociated rate point R i by keeping the element in R aociated with A i, i {, } (or the aociated rate point r i l,j by keeping element in r l,j) in the natural way, implying that R R l (A ) R l (A ) Thi then implie that R l (A) R l (A ) R l (A ) for all l {, q, (, q)} The oppoite containment i obviou, ince any rate point or code for the two network can be utilized in the trivial manner for the merged network Thi prove (58) for l {, q, (, q)} Next, conider any pair R i Γ Ni L (A i ), where h i Γ Ni R o (A i ) i {, }, which are, by definition, projection of ome [h i, r i ] and r i = [R e e E i ], i {, } Defining h whoe element aociated with the ubet A N i h A = h A N + h A N, where the interection repect the remapping of edge under the intermediate node merge, we oberve that [h, r, r ] Γ N L A, and hence it projection R R o (A), proving R o (A) R o (A ) R o (A ) Finally, conider a point R R o (A), which i a projection of ome [h, r] Γ N L A, where h Γ N r = [R e e E ] For every A N i, define h i A = h A S 3 i h S3 i, and define h with h A = h A N + h A N and R = proj ω,r h We ee that h i L (A i ), i {, }, becaue conditioning reduce entropy and entropy i nonnegative, but all of the conditional entropie at node other than g were already zero, while at g, the conditioning on the ource from the other network will enable the ame conditional entropy of zero ince the incoming edge from the other network were function of them Thi how that R R o (A ) R o (A ) Owing to the independence of the ource proj ω h = proj ω h, while proj r h proj r h due to the fact that conditioning reduce entropy The coordinate convex nature then implie that R R o (A ) R o (A ) howing that R o (A) R o (A ) R o (A ) and completing the proof (59) and

50 50 Theorem 9: Suppoe a network A i obtained by merging e and π(e), ie, A e + A π(e) Then R l (A) = Proj \{e,π(e)} ((R l (A ) R l (A )) L 0), l {, q, (, q), o} (60) where L 0 = { R (Tl(j),g0) R j, R (g0,hd(j)) R j, j {e, π(e)} }, and projection dimenion \{e, π(e)} mean projecting out dimenion aociated with e, π(e) Furthermore, R(A q ) R(A q ) and L 0 are viewed in the dimenion of N + N + 4 with aumption that all dimenion not hown are uncontrained Proof: A oberved after the definition of edge merge, one can think of edge merge a the concatenation of two operation: i) plit e in A and π(e) in A each up into two edge with a new intermediate node (g and π(g), repectively) in between them, forming A and A, repectively, followed by ii) intermediate node merge of A g + A π(g) It i clear that L 0 decribe the operation that mut happen to the rate region of A i, i {, } to get the rate region of A i, becaue the content of the old edge e or π(e) mut now be carried by both new edge after the introduction of the new intermediate node Applying Thm 8 to A and A yield (60) With Theorem 6 9, one can eaily derive the following corollary regarding the preervation of ufficiency of linear network code and tightne of Shannon outer bound Corollary 5: Let network A be a combination of network A, A via one of the operation defined in VI If F q vector (calar) linear code uffice or the Shannon outer bound i tight for both A, A, then the ame will be true for A Equivalently, if R l (A i ) = R (A i ), i {, } for ome l {o, q, (, q)} then alo R l (A) = R (A) Now we have defined both embedding and combination operator, and have demontrated method to obtain the rate region of network after applying the operator Next, we would like to dicu how to ue them to do network analyi and olve large network VIII RESLTS WITH OPERATORS In thi ection, we demontrate the ue of the operator defined in VI and VII We will firt dicu the ue of network embedding operation ( VI) to obtain the forbidden network minor and to predict code ufficiency for a larger network given the ufficiency i known for the maller embedded network Then, we dicu the ue of combination operation ( VII) to olve large network Finally, we dicu how to ue both combination and embedding operation together to obtain even more olvable network A e Embedding Operation to Obtain Network Forbidden Minor One natural ue of embedding operation i to obtain rate region for maller embedded network given the rate region for a larger network, a hown in VI, ince the rate region of embedded network are projection of the rate region of the larger network with ome contraint We can ue the embedding operator in a revere manner From Corollary 4, we oberve that if a cla of linear code uffice for a larger network, then it will uffice for the maller network embedded in it a well Equivalently, the inufficiency of a cla of code for a network i inherited by the larger network that have thi network

51 5 Y g t Y Y Y g t Y Y I) g Delete 4 5 g 5 4 II) g 4 g 4 g 3 g 3 Y 3 t Y Y 3 t Y Contract Y g t Y Y g t Y Y IV) g 3 g t Y Delete Y III) 5 g g 4 3 Y 3 t Y Fig 6: ing embedding operation to predict the inufficiency of calar binary code for a large network: ince the large network I and intermediate tage network II, III contain the mall network IV a a minor, the inufficiency of calar binary code for network IV, which i eaier than network I to ee, predict the ame property for network III, II and I embedded inide Thi i imilar to the forbidden minor property in matroid theory, which tate that if a matroid i not F q repreentable, then neither are it extenion Therefore, if we know a mall network ha the property that a cla of linear code doe not uffice, then we can predict that all network containing it a a network minor through ome embedding operation, will have the ame property, without any calculation For intance, in Fig 6, the mall network IV i a (, 3) network for which calar binary code do not uffice, becaue when H(Y ) =, H( ) = H( 3 ) = H( 5 ) =, there i no calar olution (but there i a vector olution) Since network I, II, III contain it a a minor (through the operation in the figure), we know that calar binary code will not uffice for them, either Thi i verified by our computation in V A in matroid theory, we may have a collection of mall network that we know hould be "forbidden" a a minor in larger network, in the ene of enuring the ufficiency of a cla of linear code For intance, in [6], it wa hown that for the thouand of MDCS network for which calar binary code do not uffice, there are actually only forbidden network minor For general hyperedge MSNC problem, we alo built imilar relationhip a hown in Fig 7 A Table IV how, the number of intance that calar binary code do not uffice for (, 3), (3, ), (, ), (3, ), (, 3) network are 5, 0, 3, 064, 44484, repectively Thoe inufficient intance hould not be a minor of any larger network that calar binary code uffice, and hence are forbidden minor To obtain a minimal lit of network forbidden minor, we build a hierarchy among them A Fig 7 how, there are out of 064 inufficient (3, ) network actually containing maller forbidden network minor, of which 4867 can be related by ource or edge deletion and 563 can be related by edge contraction Similarly, there are out of inufficient (, 3) network actually containing maller forbidden network minor, of which 444 can be related by ource or edge deletion and can be related by edge contraction Thoe network

52 5 (3, ) 3987 in total (, 3) in total 064 inufficient 833 ufficient inufficient ufficient have embedded minor have no embedded minor 6083 have embedded minor 8430 have no embedded minor 4867 related by ource or edge deletion 563 related by edge contraction 444 related by ource or edge deletion related by edge contraction 5 inufficient 3 inufficient 3 inufficient 0 inufficient (3, ) (, ) (, ) (, 3) Fig 7: Relation between network of different ize that calar binary code do not uffice The deletion operation conider both ource and edge deletion, while the contraction operation only conider edge contraction that do not have maller forbidden network minor are new one B e Combination Operation to Solve Large Network A hown in VII, the rate region of a combined network can be directly obtained from the rate region of the network involved in the combination We how that the ufficiency of a cla of linear code are preerved after combination Actually, if we know one of the network involved in the combination ha the property that a cla of linear code doe not uffice, the combined network will have the ame property, ince that mall network i embedded in the combined network and it can be obtained by deleting the other network Fig 8 demontrate the idea of olving large network obtained by combination operation It how that the rate region of the large combined network can be obtained directly from the rate region of maller network In addition, ufficiency of linear code i preerved Example 5: A (6, 5) network intance A can be obtained by combining five maller network A,, A 5, of which the repreentation are hown in Fig 8 The combination proce i I) A = A {t, t } + A {t, t }; II) A 3 = A e 4 + A 3 e 7 with extra node g 0 and edge e 4, e 7 ; III) A 45 = A 4 g 0 + A 5 g 0 ; IV) A = A 3 {X,, X 6 } = A 45 {X,, X 6 } From the oftware calculation and analyi [40], [4], one obtain the rate region below the 5 mall network According to the theorem in VII, the rate region R (A) for A obtained from R (A ),, R (A 5 ), i depicted next to it Additionally, ince calculation howed binary code and the Shannon outer bound uffice for A i, i {,, 5}, Corollary 5 dictate the ame for network A

53 53 Fig 8: A large network and it rate region created with the operation in thi paper from the 5 network below it C e combination and embedding operation to generate olvable network The combination operator provide a method for building large network directly from maller network in uch a way that the rate region of the large network can be directly obtained from thoe of the mall network The embedding operator provide a method for obtaining a mall network from a large network in uch a way that the rate region of the mall network can be expreed in term of the rate region of the large network Thee fact indicate that we can obtain network by integrating combination and embedding operation If we tart with a lit of olved network, all network obtained in the following combination and/or embedding proce will be olvable Note that one can apply thee operation, epecially the combination operation, an infinite number of time to obtain an infinite number of network For demontration purpoe, we would like to limit the ize of network involved in the proce A VII how, it i not difficult to predict the wort cae network ize after combination Since the combined network may have redundancie, the wort cae here mean there i no redundancy after combination o that the network ize i eay to predict For intance, after merging k ource of a (K, L) with k ource of a K, L network, the network ize after merging will be K + K k + L + L in the wort cae We define the wort cae partial cloure of network a the network obtained in the combination and embedding proce uch that no network involved exceed the ize limit An algorithm to generate the wort cae partial cloure of network i hown in Algorithm A it how, a long a the predicted network doe not exceed the preet ize limit, it will be counted a a new network and will be ued a a eed, a long a it i not iomorphic to the exiting one