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Symmetry in Network Coding Formalization, Graph-theoretic Characterization, and Computation Jayant Apte John Walsh Department of Electrical and Computer Engineering Drexel University, Philadelphia ISIT, 05 Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Outline Motivation, Context, & Bigger Picture Formalization via Groups Group Action Network Symmetry Group Symmetries of Network Codes and Rate Regions Graph Theoretic Characterization Two Special Network Coding Problem Classes Automorphism Groups Characterization for k-unicast Networks Characterization for MSNC Networks Computation Applications Computation of Polyhedral Bounds on Rate Regions A generalization of partition-symmetrical entropy functions Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

General Network Coding Problem Sources S Network Sinks T Y s Paths... In(i) i Out(i).. s.. t e E U..... U e R e K T β(t) A[R,R,] T B[H(Y ),H(Y ),] T Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Rate Region Rate region: all possible rate and source entropy vectors satisfying all network constraints. Collect the N network random variables and their joint entropies. Define Γ N :N -dim. cone, region of valid entropy vectors. (revisit later) Constraints from network A: L = {h Γ N : h Y S =Σ s S h Ys } () L = {h Γ N : h X Out(k) Y s =0} () L = {h Γ N : h X Out(i) X In(i) =0} () L = {(h T, R T ) T R N + E + : R e h Ue, e E} () L 5 = {h Γ N : h Y β(t) U In(t) =0}. (5) Rate region (cone) in terms of edge rates and source entropies (derived from [Yan, Yeung, Zhang TranIT 0]): R (A) = proj RE,H(Y S )(con(γ N L ) L 5 ) (6) Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Rate Region Example A(, ) network and its rate region R (A) Rate region: a cone with dimensions of all variables in the network Y s Y s g g U U t U t Y t Y Y g R H(Y ) s Y Y R H(Y ) R + R H(Y )+H(Y )+H(Y ) R + R H(Y )+H(Y ) R + R +R H(Y )+H(Y )+H(Y ) Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 5 / 55

Rate Regions through Algorithms & Software NETWORK CODING PROBLEMS X X X R e MAJOR RECENT FOCUS OF ASPITRG: CAPACITY REGION & Codes that achieve it A H(X ). H(X K ) R. R E 0 X,X X,X X Y E E E EXAMPLE: X,X D D D D X X X X, Y R + R H(X) R + R H(X) R + R H(X) R + R +R H(X)+H(Y ) R +R +R H(X)+H(Y ) R + R + R H(X)+H(Y ) R +R +R H(X)+H(Y ) Jayant Apte, Congduan Li, and John MacLaren Walsh, Algorithms for Computing Network Coding Rate Regions via Single Element Extensions of Matroids, in IEEE International Symposium on Information Theory (ISIT 0), Jul. 0, pp. 06 0. [Online]. Available: http://www.ece.drexel.edu/walsh/ Apte_ISIT_0.pdf Congduan Li, Jayant Apte, John MacLaren Walsh, Steven Weber, A new computational approach for determining rate regions and optimal codes for coded networks, in The 0 IEEE International Symposium on Network Coding (NetCod 0), Jun. 0. [Online]. Available: http://www.ece.drexel.edu/walsh/linetcod.pdf Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 6 / 55

Listing Networks & Calculating Their Rate Regions List of All Networks up to a Certain Size All Their Rate Regions Our Software Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 7 / 55

Listing Networks & Calculating Their Rate Regions List of All Networks up to a Certain Size All Their Rate Regions Our Software group networks into equivalence classes (only differing in labels) Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 8 / 55

Listing Networks & Calculating Their Rate Regions List of All Networks up to a Certain Size All Their Rate Regions Our Software group networks into equivalence classes (only differing in labels) list only canonical representative of each equivalence class, (Li NetCod '5 & Trans IT sub) Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 9 / 55

Listing Networks & Calculating Their Rate Regions List of All Networks up to a Certain Size group networks into equivalence classes (only differing in labels) All Their Rate Regions list only canonical representative of each equivalence class (Li NetCod '5 & Trans IT sub) Our Software network symmetry group (today's talk) determines size of the equivalence class, & is constructed in enumeration Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 0 / 55

Listing Networks & Calculating Their Rate Regions List of All Networks up to a Certain Size All Their Rate Regions Our Software network symmetry group (today's talk) determines size of the equivalence class, & is constructed in enumeration network symmetry group can be exploited to reduce complexity of rate region calculation (Apte,Walsh NetCod'5) Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Rate Region Database Network Coding Rate Region Database rate region extremal codes converse proof rate region extremal codes converse proof rate region extremal codes converse proof Database of ~7000 Rate Regions of >00k Networks (Li Trans IT sub 0) Database of ~65k Rate Regions of > 7M or 55B and bounds for ~.8T more (Li Trans IT sub 05) Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Rate Region Database Network Coding Rate Region Database rate region extremal codes converse proof rate region extremal codes converse proof rate region extremal codes converse proof Database of ~7000 Rate Regions of >00k Networks (Li Trans IT sub 0) Database of ~65k Rate Regions of > 7M or 55B and bounds for ~.8T more (Li Trans IT sub 05) What now? Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Rate Region Database Network Coding Rate Region Database rate region extremal codes converse proof rate region extremal codes converse proof rate region extremal codes converse proof Database of ~7000 Rate Regions of >00k Networks (Li Trans IT sub 0) Database of ~65k Rate Regions of > 7M or 55B and bounds for ~.8T more (Li Trans IT sub 05) What now? Submit 65,000 transactions papers? Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Rate Region Database Network Coding Rate Region Database rate region extremal codes converse proof rate region extremal codes converse proof rate region extremal codes converse proof Database of ~7000 Rate Regions of >00k Networks (Li Trans IT sub 0) Database of ~65k Rate Regions of > 7M or 55B and bounds for ~.8T more (Li Trans IT sub 05) What now? Analyze, learn, and explain something! Can't read and remember 65,000 network proofs. Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 5 / 55

Rate Region Database Network Coding Rate Region Database rate region extremal codes converse proof rate region extremal codes converse proof rate region extremal codes converse proof Database of ~7000 Rate Regions of >00k Networks (Li Trans IT sub 0) Database of ~65k Rate Regions of > 7M or 55B and bounds for ~.8T more (Li Trans IT sub 05) Investigate Structure through Network Hierarchy Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 6 / 55

Hierarchy: Embedding Operators & Forbidden Embedded Networks Minor-closed graphs: finite number of forbidden minors [RobertsonSeymour98-00] Rota s conjecture in matroid theory: finite number of forbidden minors for F q representability [Oxley0] Embedding Operators w/big Small rate region expression [Li Allerton 0, Li Trans IT sub 0] Forbidden embedded networks for F q codes sufficiency. Sources Network Destinations Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 7 / 55

Example: Forbidden embedded networks Goal: minimal forbidden networks for sufficiency Scalar binary codes considered k =,, ; E =,,, 760 non-isomorphic MDCS 9 sufficient, 58 insufficient minimal forbidden minors [Li, et. al TransIT sub 0] 58 / 760 insufficient 9 / 760 sufficient forbidden embedded networks Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 8 / 55

Hierarchy: Combination Operators Combination operators [Li ITW sub 05, Li Trans IT sub 05] Rules for combining smaller networks into large ones. Rate region of large expressed in terms of those of small. Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 9 / 55

Hierarchy: Combination Operators: Ex. [Li ITW sub 05] 5 6 X X X X X 5 X 6 0 R R 7 R 9 R 0 R 0 R R 6 R 7 R R 5 0 9 R R 6 5 R 7 8 {,} X {,,} {} R R A : X R 5 {6} {5} R 5 6 {6} X X 5 R A : ' ' 8 7 9 {,} {,} {5} {6} A : 0 {,5} 5 6 X 5 X 6 8 6 7 R 8 R 6 R7 {,} {} {} {} {} {6} {6} {5} R i H(X ),i=, R 6 H(X 6 ) R H(X ) R + R H(X )+H(X ) R 7 + R 8 H(X 6 ) R H(X ) R + R +R 5 H(X )+H(X ) R 6 + R 7 H(X 5 )+H(X 6 ) R + R H(X )+H(X ) R 6 + R 7 +R 8 H(X 5 )+H(X 6 ) R 6 A : 5 R i H(X ),i=, R + R H(X )+H(X ) R i H(X ),i=,, R + R i H(X )+H(X ),i=, R + R i + R 5 H(X )+H(X ),i=, R 6 H(X 6 ) R i + R 8 H(X 6 ),i=7, 7 R 6 + R i H(X 5 )+H(X 6 ),i=7, 7 R 6 + R i +R 8 H(X 5 )+H(X 6 ),i=7, 7 R 9 H(X ) R 0 i= H(X i) R H(X 5 ) R H(X )+H(X 5 ) R 0 + R + R H(X )+H(X 5 )+H(X 6 ) X A : ' ' ' X X X 6 9 0 R 9 R 0' 0 R 0 R R 8 7 9 0 {','} {','} R 9 H(X ) {5'} {6'} {'5'} R 0 H(X )+H(X )+H(X ) R H(X 5 ) 6' A 5 : 5' X 5 ' X R H(X )+H(X 5 ) R 0 + R + R H(X )+H(X 5 )+H(X 6 ) Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 0 / 55

Hierarchy: Both Embedding & Combination Ops [Li Trans IT sub 05] Network Cap: no beyond Large size Combinations reach a portion of large networks Moderate size Small size Embeddings enlarge the reachable portion Tiny size Use operators together to get RR for big networks. Partial Network Closure. Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Hierarchy: Both Embedding & Combination Ops [Li Trans IT sub 05] Start with the single (, ), single (, ), and the four (, ) networks; These 6 tiny networks can generate new 65 networks w/ small cap! 05.65789:.6:;75/< 99==5>7.5=705.65; ;?9@0.8!"#"$!"#%$!%#%$!"#"$!"#%$!%#%$!&#"$ % % % % % %!&#%$ ' &' &' ' &' &'!(#($ " " " ) &* &+!(#"$ &" &+ &+ "' &"& &**!(#%$ ',- &'& ' *&+ +%)!"#($ ( " ( % &' &&!"#"$ (% (% (% %( "*" )""!"#%$ ' &"* &"* ' ("+& *%)&!%#($ ' ' " ' ' "!%#"$ ' ' &- ' ' %%!%#%$ ' ' (*" ' ' %%"'.// %+ (,( *+) )) "%'' &&+"* Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Hierarchy: Both Embedding & Combination Ops [Li Trans IT sub 05] With the increase of cap size, number of new networks increases! 05.65789:.6:;75/< 99==5>7.5=705.65; ;?9@0.8!"#"$!"#%$!%#%$!"#"$!"#%$!%#%$!&#"$ % % % % % %!&#%$ ' &' &' ' &' &'!(#($ " " " ) &* &+!(#"$ &" &+ &+ "' &"& &**!(#%$ ',- &'& ' *&+ +%)!"#($ ( " ( % &' &&!"#"$ (% (% (% %( "*" )""!"#%$ ' &"* &"* ' ("+& *%)&!%#($ ' ' " ' ' "!%#"$ ' ' &- ' ' %%!%#%$ ' ' (*" ' ' %%"'.// %+ (,( *+) )) "%'' &&+"* Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Hierarchy: Both Embedding & Combination Ops [Li Trans IT sub 05] Embedding operations are important in the process! 05.65789:.6:;75/< 99==5>7.5=705.65; ;?9@0.8!"#"$!"#%$!%#%$!"#"$!"#%$!%#%$!&#"$ % % % % % %!&#%$ ' &' &' ' &' &'!(#($ " " " ) &* &+!(#"$ &" &+ &+ "' &"& &**!(#%$ ',- &'& ' *&+ +%)!"#($ ( " ( % &' &&!"#"$ (% (% (% %( "*" )""!"#%$ ' &"* &"* ' ("+& *%)&!%#($ ' ' " ' ' "!%#"$ ' ' &- ' ' %%!%#%$ ' ' (*" ' ' %%"'.// %+ (,( *+) )) "%'' &&+"* Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Group Action Action of a finite group on a set Definition a The (left) group action of a group G S n on set S is the map φ : G S S :(g, x) φ(g, x) thatsatisfies: G φ((g h), x) =φ(g,φ(h, x)), g, h G, x S G φ(e, x) =x, x S a denotes subgroups, S n denotes symmetric group Denote φ(g, x) asx g and, φ(g, X )asx g for x S, andx S For X S, anelementg G is said to stabilize X setwise if X g = X The collection of all group elements g G that setwise stabilize a subset X S forms a group called stabilizer subgroup, denoted as G X. Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 7 / 55

Group Action on Network Constraints Example Recall network constraints L 5 = L L L L L 5 Let [n] be the set of subscripts of RVs in X n and ˆL be the set of sets of constraints arising from different network coding problems The natural action of G S n on [n] induces following sequence of actions: Act ([n]) induced Act(X n ) induced Act( X n ) induced Act( ˆL) Let g =(, )(, )(7, 8) G S 8 (natural action on [8] and X 8 ) {, } {, }, {,,, 8} {,,, 7} (induced action on X 8 ) {h {,} = h {,,,8} } {h {,} = h {,,,7} } {h 8 R 8 } {h 7 R 7 } (induced action on ˆL) Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 8 / 55

Rate Region Rate region: all possible rate and source entropy vectors satisfying all network constraints. Collect the N network random variables and their joint entropies. Define Γ N :N -dim. cone, region of valid entropy vectors. (revisit later) Constraints from network A: L = {h Γ N : h Y S =Σ s S h Ys } (7) L = {h Γ N : h X Out(k) Y s =0} (8) L = {h Γ N : h X Out(i) X In(i) =0} (9) L = {(h T, R T ) T R N + E + : R e h Ue, e E} (0) L 5 = {h Γ N : h Y β(t) U In(t) =0}. () Rate region (cone) in terms of edge rates and source entropies (derived from [Yan, Yeung, Zhang TranIT 0]): R (A) = proj RE,H(Y S )(con(γ N L ) L 5 ) () Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 9 / 55

Network Symmetry Group Definition (Network Symmetry Group) The network symmetry group G I of a MSNC instance I =(G =(V, E), S, T,β) is the subgroup of S n, n = S + E, that stabilizes L 5 setwise under its induced action on ˆL. Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Network Symmetry Group Example 5678 9:;7<7:;7:67! " /! ) #! $"%)( 0"%)0,0&%*0'%+0-%.! " /! ) #! $"%)(! " #! $"%&%'(! ) #! $)%*%+( ), " :;7 6:=5>9:=! ) #! $)%*%+(! $&%*( #! $,%&%*(!, #! $,%-%.(! $'%-( #! $)%'%-(! " #! $"%&%'(! $&%*( #! $,%&%*(!, #! $,%-%.(! $+%.( #! $"%+%.(! $+%.( #! $"%+%.(! $'%-( #! $)%'%-( 5>=7 6:=5>9:= & ' * Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Network Symmetry Group Example 5678 9:;7<7:;7:67! " /! ) #! $"%)( 0"%)0,0&%*0'%+0-%.! " /! ) #! $"%)(! " #! $"%&%'(! ) #! $)%*%+( ), " :;7 6:=5>9:=! ) #! $)%*%+(! $&%*( #! $,%&%*(!, #! $,%-%.(! $'%-( #! $)%'%-(! " #! $"%&%'(! $&%*( #! $,%&%*(!, #! $,%-%.(! $+%.( #! $"%+%.(! $+%.( #! $"%+%.(! $'%-( #! $)%'%-( 5>=7 6:=5>9:= & ' * Network Symmetry Group of Butterfly Network is of order with (,)()(,5)(6,7)(8,9) being the only non-trivial symmetry Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

Symmetries of Network Codes Definition A network code ({f e }, {g t }) for a MSNC instance I is an assignment of a function f e to each edge e E and a function g t to each sink t T. Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 5 / 55

Symmetries of Network Codes Definition A network code ({f e }, {g t }) for a MSNC instance I is an assignment of a function f e to each edge e E and a function g t to each sink t T. Theorem Let G I be the NSG associated with MSNC instance I. Then,forany g G I,ifanetworkcode({f i }, {g t }) satisfies L 5,sodoes ({f i g }, {g t g }) for every g G I. Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 5 / 55

Symmetries of Rate Regions Theorem If ω, r is an achievable source information rates and edge rates vector pair for a MSNC instance I, soare[ω s g s S], [R e g e E] for every g G I. Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 6 / 55

Symmetries of Rate Regions Theorem If ω, r is an achievable source information rates and edge rates vector pair for a MSNC instance I, soare[ω s g s S], [R e g e E] for every g G I. Example +-*5+.6$7.*/,.*,'%7.*+ ********* +-*5+.6$7.*/,.*,'%7. #$%&'(!!"! #$%&'(!!"! 8"8! #$%&'(! #$%&'( #$%&')*$+,-*%./*&.,0/*'-.( Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 6 / 55

Two Special Problems MSNC Multisource Network Coding Described by tuple (G, S, T,β) β : T S \ allows arbitrary subset demands Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 8 / 55

Two Special Problems MSNC Multisource Network Coding Described by tuple (G, S, T,β) k-unc k-unicast Network Coding Described by tuple (G, S, T,β) β : T S \ allows arbitrary subset demands β : T S is a bijection Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 8 / 55

Automorphism Groups Several combinatorial objects can be described as collections of subsets of some set eg. graphs, digraphs, linear codes and so on Automorphism group of a combinatorial object is the group containing all permutations that keep the collection of subsets invariant Example )%$'*-%&&!'"('&"'#('&!'#((! " $%&!'"'#( )*+,*$'* -*+,*.*/0/+0*05*"*,67,,*05*) # :68;9*/0.+,*.*#<* =>?6.+0,*05*&!'"'#( )%$'*-%&8!'"9'8!'#9(! " # $%&!'"'#( )*+,*$'* -*+,*.*,67,*05 :68;9*/0.+,*.*+@+A**.@*8"'#9 Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 0 / 55

NSG via Graphs for k-unicast Construction. Start with the underlying DAG G =(V, E). Add a feedback edges F = {(t,β(t)) t T}to create circulation graph G c =(V, E F c ). Take line graph: vertices are edges of G c, edges are length directed paths in G c to create dual circulation graph G c. Associate random variables with vertices of G c Theorem The network symmetry group of a k-unc instance is G I = G Fc Aut(G c ), the group of permutations of X n induced by the subgroup of automorphism group of its dual circulation graph that setwise stabilizes the feedback edge set. Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

NSG via Graphs for k-unicast Example!-5-',6#*,&7 &'(&")*#'+,-.(*/0 @ > @ >?? A A C B C B --------)',7-.(*/0-+8 &'(&")*#'+,-.(*/0 $>D?% 9(+:)7;-<=;;7#('76 $ADB% $?DA% $ADC% $@D?% $>DB% $CD>% $@DC% symmetries are the subgroup of!"#$ % that setwise stabilizes the feedback edges {(6,),(5,)} $BD@% Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

NSG via Graphs for k-unicast Example.*/0*")'&)!5!"#!$%&'"()*+#&,- 9 7 9 7 8 8 : : < ; < ; ********%")5*+#&,-*(6!"#!$%&'"()*+#&,- =:>;? =8>:? =7>8? =:><? =9>8? =7>;? )5'@(#/*ABB5'#A*+#($,***** =<>7? **!()'&")*"C5)'"'A **&)C =9><? =;>9? Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 / 55

NSG via Graphs for MSNC Networks A slightly more complicated characterization. Start with underlying DAG G =(V, E). Add new vertices S, edges E S and F c to create circulation graph G c. Take line graph to create dual circulation graph Gc. Compute AUT(Gc ) 5. Obtain network symmetry group as an induced group Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 6 / 55

NSG via Graphs for MSNC Networks A slightly more complicated characterization. Start with underlying DAG G =(V, E). Add new vertices S, edges E S and F c to create circulation graph G c. Take line graph to create dual circulation graph G c. Compute AUT(G c ) 5. Obtain network symmetry group as an induced group Theorem The network symmetry group of a MSNC instance is the group of permutations induced onto E E S by the subgroup of Aut(G c ) that stabilizes E S setwise. Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 6 / 55

NSG via Graphs for MSNC Networks Example,-./0.-56-78 97:;65<-.=96>? & ( ) *! " & ( ) * &+ (+ )+ *+! " # $ % # $ % &' && &( &) &' && &( &) Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 7 / 55

NSG via Graphs for MSNC Networks Example +,-./0-,5,67 & ( ) *! " # $ % &' && &( &) The Network Symmetry group is generated by: g =(, )(, )(5, 7)(6, 8) (9, )(0, )(, ) (, )(5, )(6, )(8, 9) g =(, )(7, 8)(, )(5, 6) (7, 8)(9, 0)(, )(, ) g =(, )(5, 6)(, 5)(, 6) (7, 9)(8, 0)(, )(, ) A group of order 8 Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 8 / 55

Computing NSGs One can use off-the-shelf (di)graph automorphism group computation software for computing NSGs nauty: No Automorphisms, Yes? implements Brendan McKay s canonical labeling algorithm We used SageMath s implementation of the same algorithm Can also be enumerated along with canonical networks [Li, Walsh, Weber, NetCod 05, Trans IT sub 05] Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 50 / 55

Computation of Polyhedral Bounds on Rate Regions Let H n R n +n Act ([n]) induced Act(X n ) induced Act( X n ) induced Act(H n) Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 5 / 55

Computation of Polyhedral Bounds on Rate Regions Let H n R n +n Act ([n]) induced Act(X n ) induced Act( X n ) induced Act(H n) Let L 5 be the intersection of hyperplanes and halfspaces in H n associated with network constraints Γ n L 5 is stabilized setwise under action of NSG on H n NSG gives polyhedral symmetry group of Γ n L 5 NSG gives polyhedral symmetry group of proj ω,r (Γ n L 5 ) Extend to any outer/inner polyhedral bound Γ k n, k {in,out} provided that Γ k n is stabilised setwise under induced action of S n on H n e.g. ZY, DFZ non-shannon outer bounds, matroid inner bounds, subspace inner bounds Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 5 / 55

Ageneralizationofpartition-symmetricalentropyfunctions Let Fix G (S) fors H n be the subset of S fixed under the induced action of G on H n Let p be the partition of X n arising from orbits of G I,Σ p be the partition respecting subgroup of S n Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 5 / 55

Ageneralizationofpartition-symmetricalentropyfunctions Let Fix G (S) fors H n be the subset of S fixed under the induced action of G on H n Let p be the partition of X n arising from orbits of G I,Σ p be the partition respecting subgroup of S n Qi Chen and R. W. Yeung [arxiv] defined following regions: Ψ p =Fix Σp (Γ n ), Ψ p =Fix Σp (Γ n) () For a MSNC instance I, one can also define: Ψ I =Fix G I(Γ n ), Ψ I =Fix G I(Γ n) () Theorem For a MSNC instance I with NSG G I and associated orbits in X n denoted as p, Ψ p Ψ I and Ψ p Ψ I Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 5 / 55

Conclusions Defined a network symmetry group Showed that it gives symmetries of rate regions & codes Showed how to calculate it with graph automorphism based tools Mentioned applications in enumeration [Li Walsh Weber Trans IT Sub 05], rate region calculation [Apte Walsh NetCod 5], and partition-symmetrical entropy functions Provided context in computational network coding research Jayant Apte, John Walsh (Drexel University) Symmetry in Network Coding ISIT, 05 55 / 55