Tight Bounds for Maximal Identifiability of Failure Nodes in Boolean Network Tomography

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1 Tight Bounds for axial Identifiability of Failure Nodes in Boolean Network Toography Nicola Galesi Sapienza Università di Roa Fariba Ranjbar Sapienza Università di Roa arxiv: v1 [cs.ds] 28 Dec 2017 Abstract We study axial identifiability, a easure recently introduced in Boolean Network Toography to characterize networks capability to localize failure nodes in end-to-end path easureents. Under standard assuptions on topologies and on onitors placeent, we prove tight upper and lower bounds on the axial identifiability of failure nodes for specific classes of network topologies, such as trees, bounded-degree graphs, d- diensional grids, in both directed and undirected cases. Aong other results we prove that directed d-diensional grids with support n have axial identifiability d using nd onitors; and in the undirected case we show that 2d onitors suffice to get identifiability of d 1. We then study identifiability under ebeddings: we establish relations between axial identifiability, ebeddability and diension when network topologies are odelled as DAGs. Through our analysis we also refine and generalize results on liits of axial identifiability recently obtained in [11] and [1]. Our results suggest the design of networks over N nodes with axial identifiability Ω( log N) using 2 log N onitors and heuristics to place onitors and edges in a network to boost axial identifiability. Index Ters Boolean Network Toography, Node Failure Localization, axial Identifiability, Hypergrids, Ebeddings, Diension I. INTRODUCTION onitoring a network to localize corrupted coponents is essential to guarantee a correct behaviour and the reliability of a network. In any real networks direct access and direct onitoring of the individual coponents is not possible (for instance because of liited access to the network) or unfeasible in ters of available resources (protocols, counications, response-tie etc.). A well-studied approach to localization of failing coponents is network toography. Network toography focuses on detecting the state of single coponents in the network by running a easureent process along the network. The process starts by sending packets (containing suitable data to capture interesting failures) fro specific source-onitor nodes and terinates receiving another data packet on other specific target-onitor nodes. easureent is done along a set of end-to-end paths, each one starting and ending with a onitor node. In this work we focus on the proble of detecting node states (failing/working), using a boolean network toography approach [4], [5] where the received data at each onitor is one bit (failure (1) /working (0)), capturing the presence or the absence of a failure along a path. We are interested in identifying (uniquely) failure nodes. Receiving a 0 (working state) at an end onitor of a path eans that each node in the path is working properly. Then the localization of failing nodes in a set of paths P (or a network view as a set of paths) is captured by the solutions to the following boolean syste: ( ) x v b p p P v p where b is a vector of boolean values (corresponding to final easureent in the paths) and x v are boolean variables, one for each node v 1. Any solution to this syste is a possible placeent of node-failures satisfying the easureents. Since working paths (b p = 0) entail x v = 0 for all v p, we can assue that the syste is (in fact equivalent to) a boolean forula in Conjunctive Noral For on positive variables. A. The Proble A set of (non onitor) nodes failing siultaneously is a failure set. Each solution to Eq. 1 captures a failure set that can occur in the network according to the easureents. But as readily seen solutions to Eq. 1 are often ultiple. a et al. in [13] proposed a paraeter easuring the ability of a network of capturing the axiu nuber of siultaneous failure nodes which are uniquely identifiable. This is called axial identifiability (Definition II.3). Recently the study of axial identifiability received a lot of attention due to its iportance in applications in boolean network toography [1], [9] [13], [16]. In this work we focus on this easure and we study upper and lower bounds for axial identifiability of specific classes of network topologies. B. Previous Works and Contributions Identifiability as defined in [13] (Definition 1) captures the cobinatorial property that to separate two sets U and W (of failure nodes) one wants to exhibit a easureent path in P touching nodes of exactly one of the two sets. The axial size of sets in P of failure nodes one can guarantee identifiability for, is then a easure of the ability of P to identify failure sets uniquely. For link failure identifiability [11] there are structural results on graphs proving necessary and sufficient conditions for identifiability etrics based on links. In the work [1] the authors studied upper bounds on the axiu identifiability of failure nodes with a given nuber 1 Notice we are not including onitor nodes in the boolean syste (1)

2 of onitoring paths in several concrete and applied network scenarios (f.i. consistency routing see [1] and Definition V.1) with the ai of design optial networks for node failure identifiability. We explore what axial identifiability of node failure requires and iplies in ters of network topologies. Under standard assuptions on topologies and on onitors placeent, we prove upper bounds on the axial identifiability of specific classes of network topologies, such as trees, boundeddegree graphs, d-diensional grids, in both directed and undirected cases. Instead of checking experientally the optiality of the upper bounds, we give an algorithic/cobinatorial analysis of lower bounds for axial identifiability obtaining tight bounds. This approach naturally leads to algoriths to devise network topologies with a guarantee of reaching a precise axial identifiability of failure nodes. Since d- diensional grids plays an iportant role in our results, we start the study of axial identifiability of node failure under ebeddings of DAGs. We establish relations between axial identifiability, ebedabbility and diension of networks odelled by DAGs. Results which also entail algorithic approaches to approxiate the axial identifiability of failure nodes in any (directed acyclic) network topology. C. Results and Organization We study axial identifiability µ for several classes of network topologies both directed and undirected. First we reforulate an equivalent definition of axial identifiability (Definition II.3) in ters of set operations, which entails a clean cobinatorial approach to prove lower and upper bounds for µ. After soe preliinaries in Section II on basic notions, in Subsection II-C we explain our network odels and soe assuptions we work with. The two assuptions siple-pathfree and balanced-onitor trees allow to avoid trivial and uninteresting cases in studying axial identifiability on the network topology associated to a set of easureent paths (see Leas II.6,II.7,II.10,II.11). In Subsection II-D we start proving a structural result: naely that independently fro the topology, µ is always (strictly) upper bounded by the ax between the nuber of input and output onitors. In particular this iplies that with 2 onitors there is no identifiability of failures at all. This result generalizes (but also siplifies the proof of) the analogous result of [11] stating that link-failure identifiability needs strictly ore than two onitors independently of the topology. In Section III we consider directed topologies. We prove that for trees axial identifiability is exactly 1 (Theore III.3). After looking to the case of bounded degree graphs and proving that µ is upper bounded by the inial indegree (Lea III.4), we consider the question whether there are (directed) topologies with axial identifiability strictly greater than 1. Theore III.9 gives a positive answer proving that for directed n n grids, axial identifiability is exactly 2. In Theore III.10 we lift to d-diensional directed grids for d > 2 and generalizing previous theore we prove that axial identifiability is exactly d. According to our odels, there are nd onitors in such grids. Section IV is focused on undirected topologies. Since we loose directions, onitor placeent and assuptions on the odels here are slightly different fro the directed case (see Subsection II-C3). After obtaining (in Theore IV.3 and Lea IV.4) the sae bounds for the directed case for trees and bounded-degree graphs, we look at the case of grids. We can exploit the greater nuber of paths in the undirected hypergrid and prove in Theore IV.6 that reducing the nuber of onitors fro nd to 2d, axial identifiability in d- diensional grids reains at least d 1 and at ost d. Section V starts the study of the relations between identifiability, ebeddings and diension of DAG and posets. We observe that if a DAG G is ebeddable into a d-hypergrid then its diension (see Preliinaries) is bounded by the axial identifiability of the hypergrid. We explore other possible connections: we present a siple exaple proving that is not possible to relate precisely axial identifiability with ebeddability. Nevertheless we explore two directions: (1) restricting the class of topologies we want to ebed and (2) restricting the apping that defines the ebedding. In the first case we prove in Theore V.2 that the axial identifiability of a (directed) network G which satisfies routing consistency (see [1] and Definition V.1) is upper bounded by the axial identifiability of any other DAG G where G is ebeddable to. In the second case we consider ebeddings which preserve soe distance properties (Definition V.1). We prove (Theore V.4) that if a DAG G is ebeddable into a DAG G by a distance preserving ebedding then they have sae axial identifiability. Finally we obtain a general result (Theore V.7) proving that if a DAG G is closed under transitivity and ebeddable into another DAG G, axial identifiability can only increase. This is sufficient to prove that for any DAG G its axial identifiability is lower bounded by the diension of its transitive closure. In Section VI we discuss applications of our results. In Subsection VI-A we propose an algorith to build a network on N nodes where axial identifiability is (log N) using only 2 log N onitors. In Subsection VI-B we propose an heuristic to boost axial failure identifiability of a network P already built. The idea is to siulate in P an hypergrid of diension d log N. In the final version we will present soe experients validating the heuristics. Finally in Subsection VI-C we refine and siplify (in Theore VI.1 and Corollary VI.2) results on liits on axial identifiability recently showed in [1]. In the last Section VII we point out soe directions raising fro this approach we think worthy to be further explored. Due to page liitations soe proofs are oitted in this draft. Oitted proofs can be found in the appendix. II. PRELIINARIES A. Sets, Graphs, Paths, Posets, Ebeddings For a set U let χ U be its indicator variable, i.e. χ U (x) = 1 iff x U. U V = (U \ V ) (V \ U) is the syetric

3 difference between U and V. Notice that χ U V = χ U χ V. In a graph G = (V, E), V is a set of nodes and E V V. G is undirected if pairs in E are unordered, i.e. (uv) E iff (vu) E. Otherwise G is directed. G is DAG if a directed graph with no cycle. A path p in G fro a node u to a node v is a sequence of edges p = (u 1 u 2 ), (u 2 u 3 )... (u k 1 u k ) such that u 1 = u and u k = v and (u i u i+1 ) E for all i [k 1]. If G is a DAG, then we identify the path p also with sequence of nodes u 1... u k For a node u in G, N(u) is the set of neighbourhood of G, i.e. {v V (uv) E}. The degree of u, deg(u), is the cardinality of N(u). The degree of G is (G) = ax u V deg(u). We also consider the inial degree δ(g) of G. If G is directed then we distinguish N i (u), the set of neighbourhood v of u s.t. (vu) E, fro N o (u), the neighbourhood v of u s.t. (uv) E. For all degree easures on G we distinguish the in-degree i (G) and δ i (G) and the out-degree o (G), and δ o (G). In a DAG G, we denote by S, the source nodes, i.e. nodes u with in-degree 0, and by T the target nodes v with out-degree 0. Definition II.1 (Directed hypergrids). Let d N + and n N, n 4. The (directed) hypergrid of diension d over support [n], H n,d, is the graph with vertex set [n] d and where there is a directed edge fro a node x = (x 1, x 2,..., x d ) to a node y = (y 1, y 2,..., y d ) if for soe i [d] we have y i x i = 1 and x j = y j for all j i. In the case of undirected hypergrid in H n,d there is an edge between a node x and a node y if for soe i [d] we have x i y i = 1 and x j = y j for all j i. In the case of siple grids over n nodes, i.e. d = 2, we use the notation H n. i is the set of nodes x = (x 1, x 2,..., x d ) such that x i = 1. A border node is a node of H n,d which is also in soe i. We consider directed rooted trees T n over n N + nodes (fro now on we oit that n N + ). An (undirected) tree is an acyclic graph with no cycle where any two nodes u and v are connected by a path. We consider: (1) downward directed trees T n where the root of T n is the only source node and the leaves of T n are the only target nodes (i.e. i (T n ) 1) (2) upward directed trees T n, where the root is the only target node and the leaves are source nodes (i.e. 0 (T n ) 1). Let P be a set of paths over nodes V. For a node v V, let P P (v) be the set of paths in P passing through v (we will oit index P fro now on since it will be always clear). For a set of nodes U, P(U) = u U P(u). Hence if U V, P(U) P(V ). Each DAG G = (V, E) is equivalent to a poset with eleents V and partial order G, where u G v if v is reachable fro u in G. Eleents u and v are coparable if u G v or v G u, and incoparable otherwise. We write u G v if u G v and u v. A apping f fro a poset G = (V, E) to a poset G = (V, E ) is called an ebedding if it respects the partial order, that is, all u, v V are apped to u, v V such that u G v iff u G v. IF G is ebeddable into G we write G G. It is iediate to see that ebeddings are one-toone (i.e. injective) appings. Definition II.2 (Poset diension [7]). Let G be a poset with n eleents. The diension of G, di(g) is the sallest integer d such that G H n,d. Dushnik and iller [7] proved that for any n > 1, the hypergrid H n,d has diension exactly d. B. Identifiability A preliinary definition of identifiability was introduced in [13] and refined and studied in [1], [11], [12] aong others. We forulate identifiability in ters of set operations. Let P be a set of paths over n nodes V, Definition II.3 (k-identifiability [11]). P is k-identifiable iff the following property holds: for all U, W V, with U W and U, W k, it holds that P(U) P(W ). onotonicity of identifiability (a property noticed in several works [1], [11]), i.e. that k-identifiability iplies k - identifiability for k < k, is trivial fro our definition. Notice oreover that the definition (given in [1], [11]) of k- identifiability with respect to a node set S, is equivalent to our definition refining U W with (U S) (V S). Definition II.4 (axial identifiability). Let P be a set of paths. The axial identifiability µ(p) of P is the ax k 0 such that P is k-identifiable. We want to copute, possibly exactly, the value for the axiu identifiability for sets of paths foring specific topologies. Following Definition II.3, to prove that a set P is not k-identifiable it is sufficient to find two distinct node sets U and W of cardinality at ost k, and prove that P(U) P(W ) = Notice that by the onotonicity property of identifiability, this iplies that µ(p) k 1, hence upper bounds on µ(p). Lower bounds on µ can be proved by arguing that for all distinct node sets U and W of cardinality at ost µ(p), P(U) P(W ), that is one can always build a path touching exactly one between U and W. Lower bounds on µ(p) are hence interesting since to prove the we have to provide algoriths to build paths that distinguish between any two (failure) node sets of cardinality at ost µ(p). C. Physical Networks and Topology odels Boolean Network Toography concerns with localizing failure nodes in a set of end-to-end easureent paths. We discuss the assuptions we do odelling a physical network or a set of easureent paths and then with a graph topology. We assue the paths P be defined on a set of nodes V and and a set of onitor nodes. We assue the paths to be either all directed or all undirected. onitor nodes are considered different fro nodes in V. Associated to P we consider a graph G P = (V, E), where the edges E encode the wires between nodes V in P. We do not adit ulti-edges in E. i.e. if two different paths pass through the sae edge (uv), we count it only once in E. G will be directed or undirected according to P.

4 1) ONITORS: We assue to know if each onitor is a input-onitor (denoted by ) where the easureent signal is routed out and output-onitors (denoted by ) where the boolean failure/correct essage is received. onitor nodes will be not part of the topology and hence not counted in V. But in order to easure identifiability in soe cases we want to count in the degree of a node u V the nuber of onitors it was linked to in the physical network. 2) DIRECTED TOPOLOGIES: In the case of a directed set P the inforation about the onitor nodes will be explicit in G P. We consider two subsets S (sources) and T (targets) of V, including respectively the nodes in P linked to input-onitors and output-onitors. For directed topologies the (in)-degree of a source node u S is exactly the nuber of onitors u was linked to in P. Analogously for target nodes. Notice that this odel works also if we want to design a real network P fro a logical odel G. 3) UNDIRECTED TOPOLOGIES: The difference with the directed case is that we no longer have source and target nodes in G. So forally in this case our graph G P = (V, E) is copleted by a set of nodes Z V. Z are the nodes in G P linked to a onitor in P. In the undirected case we choose not to save in the degree of a node u the info about the nuber of onitors it was linked to in P (though we ight do that). The reason is that in an undirected network essages can flow in each direction, while the link onitor-nodes are always only in one direction. We assue that in graph G, fro each node u it is possible to reach two different nodes in Z. As said, the degree is relevant only for real nodes, so we erge different onitors linked to the sae node in the physical network into one onitor in the logical topology. For instance if P was ade by two paths p 1 = ( 1 u)(uw)(w 1 ) and p 2 = ( 2 u)(uw)(w 2 ), its logical odel is a graph ade by the only edge (uv). P 1 2 u v 1 2 Notice that, as for the directed case, also here each tie we consider a path in G P, this is a path starting and ending in a node in Z, hence aong the paths in G P there are all the paths in P. In the rest of the paper we often use to say for undirected topologies a node u linked to a onitor eaning a node u in G P linked to a onitor in P. The following easure allow to keep track of differences between P and G P : Definition II.5. The hitting nuber of G P, h(g P ) is defined as follows: for an edge e in G P, h(e) = {p P : e p} and h(g P ) = ax e GP h(e). 4) PATHS: Our graphs encode a odel for a set of endto-end easureent paths, in such a way that all and only the paths in G P correspond to paths in P. But since we do G P u v not keep onitor nodes explicitly in G P and we can have siple easureent paths like (u)(u) we generalize the definition of path in graphs. A degenerate path in G P is a path ade by only one node u, iff u in P is linked to both an input- and an output-onitor. 5) SIPLE PATH FREE (SPF) ODEL: We call a path p in a graph G = (V, E) a siple path if p := (u 0 u 1 )... (u k u k+1 ) and N(u i ) = {u i 1, u i+1 } for any i [k]. A weakness of axial identifiability is that if the graph G underlying the network includes a siple path, then µ(g), the axial identifiability of G is upper bounded by 1. Lea II.6. If G = (V, E) is undirected and connected and δ(g) 2, then µ(g) 1. Proof. Let u V be the node with degree 2 in G. Let w one of its neighbours. Set U = {u} and W = {u, w}. Clearly P(U) P(W ). But since fro w and u pass exactly the sae paths in G, we have that P(W ) P(U). Hence P (U) = P (W ) and then P(U) P(W ) = 0. We have found two sets U and W of cardinality at ost 2 such that P(U) P(W ) =. Hence G is not 2-identifiable and therefore µ(g) 1. We are interested in finding a relation between the topology of a network and its axial identifiability. For this reason we assue that our graph (both directed and undirected) do not contain nodes with degree less than or equal to 2. Notice that (by our assuption on onitors) a node linked to a onitor ight have degree 2. The siple path free assuption for undirected graphs G P = (V, E) and Z V associated to a set of easureent paths P is hence as follows: all nodes u V \ Z have deg(u) 3. In the case of directed graphs, the siple path free assuption requires that there is no node u in G such that both deg i (u) 1 and deg o (u) 1. Previous Lea has an analogous for the case of directed graphs, that justify this assuption, which can be proved the sae way. Lea II.7. If G = (V, S, T, E) is directed and there exists a node u such that both deg i (u) 1 and deg o (u) 1, then µ(g) 1. The assuption of looking at SPF topologies in analyzing logical topologies in Network Toography is not new ( [3], [6]). The reason is that a siple path in a physical network p := (u 0 u 1 )... (u k u k+1 ) can be copressed into a single edge (u 0 u k+1 ) in the corresponding logical network. Usually the nodes copressed are network devices where no branching of traffic occurs and therefore do not appear in the logical topology. 6) ONITOR-BALANCED (B) TREES: In order to avoid siple case where identifiability ight be 0 we ake another assuption on our odel for tree topologies. Consider the following definition. Definition II.8. Let T n be a SPF undirected tree. We say that T n is an input tree if there is a node in T n linked to a sourceonitor. We say that T n is an output tree if there a node in T n linked to a target-onitor.

5 A tree can be both an input and an output tree. Given a tree T n and one of its edges e = (uv), let T e (u) (resp.t e (v)) be the subtree of T n obtained fro cutting the edge (u, v) and taking the tree rooted at u (resp. v). Definition II.9. A SPF undirected tree T n is balanced if for each node u in T n 1) there are w, z N(u) such that T (uw) (w) and T (uz) (z) are both input trees; 2) there are w, z N(u) such that T (uw) (w) and T (uz) (z) are both output trees Previous definition is otivated by the fact non-balanced trees are very weak fro identifiability perspective: Lea II.10. If T n is SPF tree but not B, then µ(t n ) < 1. Proof. If T n is not balanced there will always be a node u in T n and a neighbour w of u where exactly the sae paths are passing through. Since T n is SPF, then for all v V \ Z, deg(u) 3. Assue you have a node u with three neighbours w 1, w 2, w 3 such that T 1 = T (uw1) (w 1 ), T 2 = T (uw2) (w 2 ) are input trees and T 3 = T (uw3) (w 3 ) is an output tree. Then fro u and w 3 pass exactly the sae paths. Analogously if T 1 and T 2 are output and T 3 input trees. Fix U = {u} and W = {w 3 }. We have found two disjoint sets of nodes of cardinality 1 where exactly the sae paths are passing through. Hence P(U) P(W ) =. Hence µ(t n ) < 1. oreover in SPF-B trees, there are at least two different paths passing through each edge. Lea II.11. If T n is SPF-B undirected tree, then each edge e belongs to at least two different paths. Proof. Since T n is SPF, then deg(u) 3 for all u V \Z. Fix 3 neighbours w i, i = 1, 2, 3 of u and let e i = (uw i ) and T i = T ei (w i ). It is easy to argue to argue that by B assuption at least two of the have the sae label (input/output) and the other has the opposite label. Hence there is an at least an e i such that two different (end-to-end) paths are touching e i. D. Node Failure Identifiability needs ore than 2 onitors As entioned above we distinguish between input and output onitors. Only for the following result (where we do no talk specifically about directed or undirected graphs), we do not consider any of the assuptions we ade before. Theore II.12. Let G be a (connected) graph odelling a set of easureent paths P. Let be the nuber of input onitors and the nuber of output onitors in P. Then independently of the topology G, µ(g) < ax(, ). Proof. If every onitor is linked to one and only one node in G, then we fix U be the set of nodes in G linked to the input onitors and W be the set of nodes in G linked to the output onitors. Now, it is obvious that U, W ax(, ). Since our topology is connected, there is no way of separating U fro W with a path going fro an input onitor to an output onitor. We will always touch both. Thus P(U) P(W ) = and µ(g) < ax(, ). Assue now onitors are linked to ore than one node in G. For each onitor i, we select one node u aong the one in G i is linked to and add it to the set U, and we select another node w linked to an output onitor and add it to the set W. We know there is a path fro u to w since G is connected. Now as above there exist U and W of cardinality at ost ax(, ) such that P(U) P(W ) =. Therefore we have µ(g) < ax(, ). Since to run easureents in a set of paths we need at least one input and one output onitor, previous theore iediately iplies that two onitors in a set of easureent give no identifiability at all. Corollary II.13. If a set of easureent paths P has only two onitors, then axial identifiability is 0. A. Trees III. IDENTIFIABILITY FOR DIRECTED TOPOLOGIES We start proving that in ters of identifiability directed trees are very weak, even assuing the SPF hypothesis that they do not contain siple paths. Lea III.1. Let T n a SPF directed tree. Then µ(t n ) 1. Proof. Consider a node u in T n. Since T n is SPF u has either in-degree 2 or out-degree 2. According to whether the tree is downward (paths fro root to leaves) or upward (path fro leaves to root), one of the two cases can happen: 1 z t t u w t t Fix W = {u, w} and U = {u}. P(U) P(W ). oreover in both cases each path passing through w is also touching u, hence P({w}) P({u}). Therefore P(W ) P(U). Hence P(U) = P(W ) and therefore P(U) P(W ) = Lower bounds for axial identifiability for directed trees follows by: Lea III.2. Let T n be a SPF tree. Then µ(t n ) 1. Proof. Let u and w two distinct nodes in T n. Let U = {u} and W = {w}. Each node in T n is on soe (end-to-end) path fro the root to a leaf. If u and w lie on different paths, then clearly there are paths in in P(U) but not in P(W ). Hence P(U) P(W ). Assue there is a root-to-leaf path p touching both u and w and wlog p eets before w. Let p w be the subpath of p truncated at node w. Let w 1 N o (w) 1 z t t u w t t

6 be the neighbour of w lying on p. Since T n is SPF there is necessarily another node w 2 N o (w) and in T n there is a path q fro w 2 to a leaf. Hence the concatenation of p w with q is a path fro the root to a leaf touching w but not u. Hence in P(W ) but not in P(U). Hence P(U) P(W ). Theore III.3. Let T n be a directed SPF tree. Then µ(t n ) = 1. B. Bounded in-degree graphs The proofs given for trees can be extended to bounded degree graphs where axial identifiability is upper bounded by the inial in-degree of G. Reeber that δ i (G) and i (G) are respectively the inial and the axial in-degree of G. Lea III.4. Let G = (V, E) be a SLF directed graph. Then µ(p) δ i (G). Proof. Let w be a node in G with deg i (w) = δ i (G). Define W = N i (w) and U = N i (w) {w}. It is clear that each path passing through w is passing through a node in N i (w), hence P({w}) P(N i (w)). Therefore P(U) = P(W), which proves the clai since U = δ i (G) + 1. C. Grids H 4 We have seen in the previous Subsection that directed trees have axial identifiability exactly 1. Can we find topologies whose axial identifiability is strictly greater than 1? We start with 2-diensional grid H n. Since directed grids H n have inial in-degree bounded by 2 by Lea III.4 we have. Lea III.5. Let n 3. Then µ(h n ) 2. Let S be the set of sources nodes in H n and T is the set of target nodes in H n. Given a node u in H n, we also define T (v) the set of nodes in H n reachable fro v and S(u), the set of nodes in H n u is reachable fro. { S(u) = {u}, u S S(u) = (v,u) H n S(v), u V \ S T (v) = {v}, v T T (v) = (v,u) H n T (u), v V \ T The following Leas give a way to build paths avoiding specific nodes. Lea III.6. Let n 3. Let u be a node in H n and w S(u) with w u. There is path p w u fro a node in S to u not touching w. Proof. By induction on = S(u). If = 1, then S(u) = {u} and u is in S and since u w, then p w u is the (degenerate) path ade by the only node u. Assue S(u) = > 1. Then N i (u) = 2, Hence there is w 1 N i (u) such that w 1 w. Since N i (u) = 2, then S(w 1 ) S(u), and hence S(w 1 ) <. By induction we have a path p w w 1 fro S to w 1 avoiding w. Then define as p u w, the path concatenating p w w 1 with u. A siilar proof holds also for the nodes reachable fro u. We oit details in this version. Lea III.7. Let n 3. Let u be a node in H n and w T (u) with w u. There is path qu w fro u to a node in T not touching w. Lea III.8. Let n 3. µ(h n ) 2. Proof. Let V be the set of nodes of H n. We have to prove that for any U, W V with U W and such that U, W 2, P(U) P(W ). It is sufficient to find a path p H n fro S to T touching exactly one between U and W. We split in the following cases: 1) at least one between U and W has cardinality 1; 2) both U and W have cardinality 2. Case 1 Assue wlog that W = {w}. Since U W, then there is a node u U \ W, such that w u. w can be either in: (1) S(u); or (2) in T (u); or (3) in V \ (S(u) T (u)). In case (3) any path p fro S to T passing through u is not touching w and proves the clai. In case (1) we use Lea III.6 to have a path p w u fro S to u avoiding W. oreover, any path p fro u to T is avoiding w, then the coposition of p w u with p proves the clai. In Case (2) any path p fro S to u avoids w, and Lea III.7 guarantees a path qu w fro u to T avoiding w. Hence the coposition of the paths p and qu w proves the clai. Case 2 Observe that though U W, they ight share a node. So there ight be two cases: (A) U W = 1 and (B) U W = 0. In case (A) we fix u to be the node of U not in W. In case (B) say U = {u 0, u 1 } we fix u to be the node in U not reachable in H n by the other node in U, i.e. the u i such that u i S(u 1 i ). Notice that this node always exists since the nodes in U cannot reach each other in H n. As in case (1) we divide in three cases according to the position of W wrt u. i. W S(u); ii. W T (u); iii. S(u) W 1 and T (u) W 1; In case (iii) a siilar arguent as above works. Since S(u) W 1, then either (if S(u) W = 0) any path fro S to u avoids W, or (if S(u) W = 1) we can apply Lea III.6 to find a path p u fro S to u avoiding W. Using T (u) W 1, a siilar arguent works for finding a path q u fro u to T avoiding W. Hence the coposition of p u and q u is a path fro S to T passing fro u but avoiding W.

7 In case (i) we further distinguish two cases and fix the w as follows: (A) U W = 1. w is the only node in U W. (B) w is any node in W. Denote by v be the other node in W. Since u w, then by Lea III.6 there is a path p w u fro S to u avoiding w. oreover, since W S(u) any path q u fro u to T avoids W. Hence the path p u concatenation of p w u with q u touches U and, avoids W, unless v p u. If v p u then we odify p u into a new path p v touching W but avoiding U. We first identify a node z on p u. Assue u to be the node u = (x 1, x 2 ) with x 1, x 2 [n]. Since p w u is ending at u and P is directed, there is a first node z in p w u such that starting fro z all the nodes in p w u lie either on the sae row (x 1 ) or on the sae colun (x 2 ) of u. Let p w z be the subpath obtained fro truncating p w u to the node z. Notice that by the definition of z, z u 2. Clearly u T (z), then we can use Lea III.7 on z and u to find a path qz u fro z to T avoiding u. Since w S(u), then w T (z) and by the construction of z, S(u) T (z) is the set of nodes (a path) in p w u fro z to u, oreover recall that p w u avoids w. Since in Case (A) w U, the path p v concatenating p w z with qz u goes fro S to T, touches W but avoids U. S(u) pu z u pu Case (i).a q u z T (z) T (u) In case (B), i.e. when w U the other node of U, say u 1, ight belong to p v and hence in this case p v does not avoid U. To handle this last case we build another path p v fro S to T touching v and avoiding the whole U. Let t z be the subpath of p v fro S to z. Let q be the subpath of p v starting at z and ending in u 1. Assue that u 1 has coordinates (x 2, y 2 ). Siilarly as done for z, there is a first node z 1 in q such that starting fro z 1 all the nodes in q lie either on the sae row (x 2 ) or on the sae colun (y 2 ) of u 1. z 1 u since q (being a subpath p v ) avoids u. z 1 u 1 by the definition of z 1. Let q z,z1 be the subpath obtained fro truncating q at z 1. Using Lea III.7 on z 1 and u 1 there is a path qz u1 1 fro z 1 to T avoiding u 1. Hence the path p v obtained fro concatenating t z with q z,z1 and then with qz u1 1 goes fro S to T, touches v but avoids both u and u 1, hence U. Clai is proved. In case (ii) a syetric arguent of case (i) works. We oit the details in this extended abstract. Together previous theore and Lea III.5, iplies the following Theore III.9. Let n N, n 3. Then µ(h n ) = 2. 2 If v is a source node and v N i(u), then z is v itself. D. Hypergrids In this section we extend previous results to hypergrids. Reeber that our odel of H n,d assues that in the physical network each border node is linked to a onitor(actually one onitor for each diension i [d] such that x i ). Theore III.10. Let d, n N, d > 2 and n 3. Then µ(h n,d ) = d. Since H n,d has in-degree exactly d, then Lea III.4 iplies iediately that µ(h n,d ) d. First observe that a node u = (u 1,..., u d ) is in S iff u i = 1 for soe i [d] and is in T iff u i = n for soe i [d]. { S(u) = {u}, u S S(u) = v N S(v), u V \ S i(u) T (v) = {v}, v T T (v) = u N o T (u), v V \ T For the lower bound, we lift Lea III.6 and Lea III.7 to diension d > 2. Their proof is siilar to the previous case. We prove the first one. Lea III.11. Let d, n N, d > 2 and n 3. Let u be a node in H n,d and X = {w 1,..., w j } S(u) be a set of j < d pairwise distinct nodes all distinct fro u. Then there is path p X u fro a node in S to u avoiding (any node of) X. Proof. Let u be a node in H n,d. Let X = {w 1,..., w j } S(u) fulfilling the hypothesis. By induction on = S(u). If = 1, then S(u) = {u} and u is in S and since u is different fro all the nodes in X, then the (degenerate) path p X u, ade by the only node u proves the clai. Assue S(u) = > 1. Then u is neither in S nor in T. Hence N i (u) = d. Then, since j < d, there is w N i (u) such that w X. Since w N i (u), then S(w) S(u), and hence S(w) <. By induction we have a path p X w fro S to w avoiding X. Then define as p X u, as the path concatenating p X w with u through w. A siilar proof holds also for the nodes reachable fro u in H n,d we oit the proof in this version. Lea III.12. Let d, n N, d > 2 and n 3. Let P be a H n,d. Let u be a node in P and X = {w 1,..., w j } T (u) be a set of j < d pairwise distinct nodes all distinct fro u. Then there is path p X u fro u to a node in T avoiding (any node of) X. Proof. (of Theore III.10) Let V = [n] d. We have to prove that for any U, W V with U W and such that U, W d, P(U) P(W ) 1. It is sufficient to find a path p H n,d fro S to T touching exactly one between U and W. As for the case of siple grids we split in two cases: 1) at least one between U and W has cardinality at ost d 1; 2) both U and W have cardinality d. Using previous Leas the proof is siilar to the case d = 2 and we defer its sketch to the appendix.

8 E. iniizing the nuber onitors on directed hypergrids Our odel of H n,d assues that each border node x is linked to one onitor in the physical network for each diension i [d] such that x i. Is it possible to reduce the nuber of onitors but still get sae lower bounds? An ideal placeent of onitors in H n,d would have only 2d onitors on the nodes (1,..., 1) and (n,..., n). Notice however that in the directed case Lea III.4 applies. Hence if we reove onitors fro the border nodes, the inial in-degree dropoff to 1 and then µ(h n,d ) 1. In the next subsection, when we analyze undirected grids, we show that axial identifiability can be d 1 with only 2d onitors. IV. IDENTIFIABILITY FOR UNDIRECTED TOPOLOGIES A. Trees For the case of undirected trees even assuing onitorbalancing, we can prove that SPF trees have axial identifiability exactly 1. Lea IV.1. Let T n µ(t n ) 1. an undirected SPF-B tree. Then Proof. We prove that T n is not 2-identifiable. We have to show two sets W and U of cardinality at ost 2 such that P(U) P(W ) =. Let w a node in T n and v N(w) such that T (v) is an input-tree. There ust exists another neighbour of w, z N(w), with z v such that T (z) is an output-tree 3. Fix U = {v}, W = {v, w}. P(U) P(W ). oreover each path passing through w and z is also touching v, hence P({w}) P({v}). Therefore P(W ) P(U). Hence P(U) = P(W ) and therefore P(U) P(W ) =. Lea IV.2. Let T n be a balanced a SPF-B tree. Then µ(t n ) 1. Proof. (sketch) Let u and w two distinct nodes in T n. Fix U = {u} and W = {w}. If w and u lie on different (end-to-end) paths, the clai is trivially proved. The BT assuption iplies (see Lea II.11) that fro each node there are at least two different (end-to-end) paths passing through. Hence fro u there is another path which is not touching w (otherwise there would be a cycle in a tree). Theore IV.3. Let T n be a balanced a SPF-B tree. Then µ(t n ) = 1. B. Bounded degree Recall Subsection II-C3. Let G = (V, E) be an undirected graph and Z V. We assue each node in G reachable fro two distinct nodes in Z. 1 δ(g) (G) V. As for the directed case axial identifiability only depends on the inial degree of G and not where onitor are placed. Lea IV.4. Let G = (V, E) undirected. Then µ(g) δ(g) 3 Indeed if T (v) is also an output-tree and there is only one output-tree, there ust be another input tree rooted at another neighbour t of w. Then we can choose t instead of w. Proof. Let u V be such that deg(u) = δ(g). Fix U = N(u) and W = {u} N(u). Each path touching u is passing through at least a node in N(u). Hence P({u}) P(N(u)). Hence P(W ) = P({u}) P(N(u)) = P(N(u)) = P(U) and then P(U) P(W ) =. We have found two sets U, W of cardinality at ost δ(g) + 1 such that P(U) P(W ) =. Hence µ(g) δ(g). C. Hypergrids Recall Subsection II-C3 and notice that in the undirected case onitors are no longer counting in the degree of nodes of the graph odelling the network. Hence if H n,d is undirected, δ(h n,d ) = d (exactly on the corner nodes). Using Lea IV.4, we then have the upper bound µ(h n,d ) d. If we assue that all border nodes are linked to onitors that is Z = i [d] i, then each (end-to-end) path in the directed H n,d is also a path for the undirected H n,d. Hence Theore III.10 gives a lower bound also in this case. Therefore: Theore IV.5. Let H n,d be undirected and Z = i [d] i. Then µ(h n,d ) = d. Clearly in the undirected case we have any ore paths in H n,d than in the directed case. All these paths are becoing relevant to obtain a axial identifiability of d greatly iniizing the nuber of onitor nodes needed. While previous theore assues dn onitors, we show in the next Subsection that only 2d onitors suffice to get axial identifiability d 1 in the case of undirected d-diensional grids. D. iniizing the nuber of onitors on undirected hypergrids We prove that 2d onitors are sufficient to reach µ d 1 in undirected H n,d Theore IV.6. Let H n,d be undirected and n 2 d+1. Let S = {(1,..., 1),... (n,..., n, 1)} and T = {(n,..., n),..., (1,... 1, n)} be the set of nodes linked to input and output onitors. Then µ(h n,d ) d 1. The section is devoted to the proof of the theore for the case d = 2. The proof of Theore IV.6 is along the sae lines and its proof can be found in the appendix Definition IV.7. Let x = (x 1, x 2 ) be a node in in H n. S(x) = {v = (y 1, y 2) 1 y i x i, i [2]} T (x) = {v = (y 1, y 2) x i y i n, i [2]} Theore IV.8. Let H n,2 be undirected and n N, s.t. n 2 3. Let S = {(1, 1), (n, 1)} the set of input onitors and let T = {(n, n), (1, n)} be the set of output onitors. Then µ(h n,2 ) 1. Proof. We have to prove that for any U, W V with U W such that U, W 1, then P(U) P(W ).

9 It is sufficient to find a path p fro S to T touching exactly one between U and W. Assue W = {w}, U = {u}. Since U W, then u w. According to the position of W, we divide in the three following cases: 1. w V \ (S(u) T (u)) 2. w S(u) 3. w T (u); In case 1, any path fro S to T passing through u is not touching w and proves the clai. In case 2, if w N((1, 1)), any path fro the other input onitor ((n, 1)) passing through u to an output onitor is not touching w. If w / N(u), we prove it by induction on S(u) =. If S(u) = 1, then S(u) = {u} and the path p ade by u fro S to u avoids w. Notice that this case iplies u = (1, 1). Now consider S(u) =. Then we fix any u N(u) S(u). Therefore S(u ) S(u) and S(u ) <. By induction, there is a path p u fro S to u avoiding w. So the path p, the coposition of p u and u u, fro S to u avoids w as well. oreover, any path p fro u to T is avoiding W. Therefore the coposition of p and p is touching u but avoiding W and the clai is proved. If w N(u), then we fix u N(u) such that u w. If w / S(u ), then any path p u fro S to u avoid w. the path p, the coposition of p u and u u, fro S to u avoid w as well. if w S(u ), since w / N(u ) by the sae arguent as above, there is a path p u fro S to u avoiding w. Hence the path p, coposition of p u and u u, avoids w. oreover, any path p fro u to T is avoiding W. So the coposition of p and p touches u but avoids W. In case 3, a syetric arguent of case 2 works. V. IDENTIFIABILITY, EBEDDINGS, DIENSION Recall that the diension di(g) of a DAG G is the sallest d N such that G H n,d i.e. G is ebeddable in H n,d. Hence if G H n,d, then di G d. Since µ(h n,d ) d, then di G µ(h n,d ). We explore possible relationships between the diension of directed graphs and axial identifiability. The next exaple shows that in the ost general for ebeddability can destroy identifiability. Consider the following two graphs. G 1 w 1 u 1 u 2 û 1 w 2 ŵ 1 ŵ 2 G 2 G 1 is ebeddable in G 2 (f.i. f(x) = ˆx). But while in G 1 there is a path (an edge) fro u 1 to u 2 avoiding W = {w 1, w 2 }, in G 2 no path connecting u 1 to u 2 is avoiding f(w ) in G 2. Still in soe cases we can use ebedabbility to say soething on identifiability. Consider the following Definition of routing consistency given in [1]. Definition V.1. ( [1]) A set of paths P is routing consistent if any two distinct paths p and p in P and any distinct nodes û 2 u and w traversed by both paths (if any) p and p follow the sae subpath between u and w. Though a weak class of topologies (any edge-cut disconnect the graph into two coponents) these graphs are eployed in any practical routing protocols (see [1]). For these topologies (in the directed case) we can prove the following result. Theore V.2. Assue that G is routing consistent and G G, then µ(g) µ(g ). Proof. Assue that µ(g ) k. We prove that µ(g) k. Since µ(g ) k, there are two sets U, W V such that U W, at least one of the, wlog say U, has cardinality k + 1, and P G (U ) P G (W ) =. Fix U = f 1 (U ) and W = f 1 (W ). By injectivity of f, U has cardinality k + 1 and U W (since otherwise U W = ). Assue by contradiction that P G (U) P G (W ). That is, there exists a path p in G fro S to T touching nodes in only one between U and W, say U. Let p = (u 1 u 2 )... (u r u r+1 ), r 1. Hence u i u i+1 for all i [r]. Let u i = f(u i). Clearly if u i U, then u i U. Since f is an ebedding (i.e. x y iff f(x) f(y)), then u i u i+1, u 1 S and u k+1 T. Hence there are paths p i in G fro u i to u i+1 and the path p = p 1,... p k is a path fro S to T in G. If all nodes in p are in V \ W this is a contradiction with the fact P G (U ) P G (W ) =. Then there is a i [r] such that in p i is touching a node w W. Hence we have that in G, u i w u i+1. Since f is an ebedding and since u i = f 1 (u i ), this eans that in G, u i f 1 (w) u i+1. Then in G there is a path fro u i to u i+1 passing through f 1 (w). This contradicts the routing consistency of G since between u 1 and u 2 there is another path, the edge that is in p. The previous exaples shows that restricting the class of graphs one can still hope to bound identifiability using ebedabbility. In the next two results we restrict ebeddings, obtaining siilar relationships but for broader classes of topologies. Assue that f is an ebedding between two DAGs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ). Let us say that f is distance-reducing (d.r) if for all x, y V 1, d G1 (x, y) d G2 (f(x), f(y)). Here d G (x, y) is the length of the shortest path between x and y in G. We call f distance preserving (d.p.) if d G1 (x, y) = d G2 (f(x), f(y)). Theore V.3. Let G and G be two DAGs such that G f G, where f is a (d.r.)-ebedding. Then µ(g) µ(g ). Proof. Assue µ(g) k, we prove that µ(g ) k. Since µ(g) k, there are two sets U, W V such that U W, at least one of the, wlog say U, has cardinality k + 1, and P G (U) P G (W ) =. Fix U = f(u) and W = f(w ). By injectivity of f, U has cardinality k+1 and clearly U W (since otherwise U W = ). Assue by contradiction that P G (U ) P G (W ). That eans that there exists a path p fro S to T touching node of only one between U and W, say U. Let p = (u 1, u 2)... (u ru r+1), r 1. Hence u i u i+1 for all i [r]. Let u i = f 1 (u i ). Since f is an ebedding u i U, u i u i+1 and u 1 S and u r+1

10 T. oreover since f distance reducing and (u i, u i+1 ) is an edge in G, then (u i, u i+1 ) is an edge in G. Bu then then path in G p = (u 1, u 2 )... (u r u r+1 ) is a path fro S to T touching only nodes in U. This is a contradiction with the fact P G (U) P G (W ) =. It is straightforward to see that if f is distance-preserving, then equality holds. Theore V.4. Let G and G be two DAGs such that G f G, where f is a (d.p.)-ebedding. Then µ(g) = µ(g ). For a general DAG G, we cannot obtain a direct relation between identifiability and diension and, as previous exaple shows, we cannot relate identifiability of two graphs which are ebeddable one into the other. Nevertheless we can lower bound identifiability of a DAG G using ebeddability. Let G be the transitive closure of a DAG G. Since G and G are defined on the sae set of nodes and each path in G is a path in G, it follows: Lea V.5. µ(g) µ(g ). Now following the idea of Theore V.3 it is not difficult to prove the following theore (see appendix) Theore V.6. Let G be a DAG closed under transitivity. Assue that G G. Then µ(g) µ(g ). All together these result iply: Theore V.7. Let G be a DAG and assue that G G. Then µ(g) µ(g ). Hence by definition of diension of a DAG G we get Corollary V.8. Let G be a DAG. Then µ(g) di G. VI. APPLICATIONS AND EXPERIENTS A. Designing an optial network wrt failure axial identifiability Assue we have to design a network over N 4 nodes and we ai to have axial identifiability of failure nodes. Theore IV.6 suggests how to set edges between the nodes in the network and how to place onitors in such a way to to reach an identifiability of at least log N. Fix d = log N and n = N 1/d. Then N = n d and it is easy to see that for N 4 it holds that n 2 d+1 (as required by the theore). Assue wlog that all values are integers (otherwise approxiate the to sallest integer values such that N n d ). Assign an address to each node in N as a d-diensional vector in [n] and place edges between nodes in N following the construction of H n,d. Finally place input and output onitors (2 log N) linked to those nodes of N, as required by Theore IV.6. B. Adding edges to boost node failure identifiability Assue we have a network with very low axial identifiability of failure nodes (for instance due to a sall inial in-degree). We explore the idea to add edges to get better ax identifiability. We propose the following algorith whose ain idea is that of trying to approxiate a hypergrid of diension d (a paraeter to be tuned), choosing the appropriate input and output onitors and adding edges (accooding to certain heuristic) in order to increase the δ. Experients on concrete networks will appear on the full version. Algorith 1 AGrid Input: G = (V, E), d Output: G = (V, E ), I V, I V /* Select input and output onitors*/ 1: for i = 1... d do 2: (x, y) nodes at ax shorter distance in G 3: I = I {x}; I = I {y}; 4: V = V {x, y} /* Boost inial degree to d */ 5: for all v V do 6: choose u.a.r d N(v) nodes w i in V \ N(v) 7: for all w i do 8: E = E (v, w i ) C. Upper bounding µ in ters on nuber of nodes and paths Our bounds can be used to prove ore refined upper bounds on axial identifiability than those obtained in [1] (Theores IV.1 and IV.2). We consider the undirected case but siilar results can be proved for the directed case. Theore VI.1. Let G be a network defined over n nodes and paths p i each of length (i.e. nuber of edges) l i. Let l = i l i. Then µ(g P ) in{n, 2 nl }. Proof. Assue a graph G has n nodes and inial degree δ(g) d. Then there are at least nd/2 paths in G. This is because we count at least the paths of length 1 in G. Now assue that G is defined over n nodes and paths and l edges in total. Let d be the inial degree in G. Since we have l edges in G, it follows by previous observation that ndl/2. Hence it ust be that d 2/nl. Since oreover d n, the result follows by Lea IV.4. This result can be lifted to a set P of easureent paths using the hitting nuber of G P (see Definition II.5). Corollary VI.2. Let P be a set of easureent paths over n nodes and paths of total length l. Let h be the hitting nuber of G P. Then µ(g P ) in{n, 2h nl }. VII. FURTHER RESEARCH We shortly address (see the appendix for a coplete discussion) three directions related to our approach. What can we say on boolean network toography and failure identifiability using algorith for coputing diension of DAGs [17] [19]? It is a well known result [15] that planar graphs over n nodes can be ebedded into a (n 2) (n 2) grid. Our results iplies a lower bound. What else can be said on upper bounds for failure axial identifiability in planar graphs [8]? A k-tc-spanner [2] of a graph G is a graph H with a sall diaeter that preserves the connectivity of the

11 original graph. Are k-tc-spanners useful to identify axial failure identifiability of a network? REFERENCES [1] Novella Bartolini, Ting He, and Hana Khafroush. Fundaental liits of failure identifiability by boolean network toography. In INFOCO IEEE, [2] Piotr Beran, Arnab Bhattacharyya, Elena Grigorescu, Sofya Raskhodnikova, David P. Woodruff, and Grigory Yaroslavtsev. Steiner transitiveclosure spanners of low-diensional posets. Cobinatorica, 34(3): , [3] Tian Bu, Nick G. Duffield, Francesco Lo Presti, and Donald F. Towsley. Network toography on general topologies. In AC, editor, Proceedings of SIGETRICS, pages 21 30, [4] Nick G. Duffield. Siple network perforance toography. In Proceedings of the 3rd AC SIGCO Internet easureent Conference, IC 2003, iai Beach, FL, USA, October 27-29, 2003, pages AC, [5] Nick G. Duffield. Network toography of binary network perforance characteristics. IEEE Trans. Inforation Theory, 52(12): , [6] Nick G. Duffield and Francesco Lo Presti. Network toography fro easured end-to-end delay covariance. IEEE/AC Trans. Netw., 12(6): , [7] Ben Dushnik and E. W. iller Source. Partially ordered sets. Aerican Journal of atheatics,, 63(3): , [8] Stefan Felsner and Willia T. Trotter. Posets and planar graphs. Journal of Graph Theory, 49(4): , [9] Ting He, Novella Bartolini, Hana Khafroush, InJung Ki, Liang a, and To La Porta. Service placeent for detecting and localizing failures using end-to-end observations. In 36th IEEE International Conference on Distributed Coputing Systes, ICDCS 2016, Nara, Japan, June 27-30, 2016, pages IEEE Coputer Society, [10] Ting He, Athanasios Gkelias, Liang a, Kin K. Leung, Ananthra Swai, and Don Towsley. Robust and efficient onitor placeent for network toography in dynaic networks. IEEE/AC Trans. Netw., 25(3): , [11] Liang a, Ting He, Kin K. Leung, Ananthra Swai, and Don Towsley. Inferring link etrics fro end-to-end path easureents: Identifiability and onitor placeent. IEEE/AC Trans. Netw., 22(4): , [12] Liang a, Ting He, Ananthra Swai, Don Towsley, and Kin K. Leung. Network capability in localizing node failures via end-to-end path easureents. IEEE/AC Trans. Netw., 25(1): , [13] Liang a, Ting He, Ananthra Swai, Don Towsley, Kin K. Leung, and Jessica Lowe. Node failure localization via network toography. In Carey Williason, Aditya Akella, and Nina Taft, editors, Proceedings of the 2014 Internet easureent Conference, IC 2014, Vancouver, BC, Canada, Noveber 5-7, 2014, pages AC, [14] Cristan Neri. Identificabilità di congestioni di rete su specifiche classi di grafi, Sapienza University, [15] Walter Schnyder. Ebedding planar graphs on the grid. In Proceedings of the First Annual AC-SIA Syposiu on Discrete Algoriths, SODA 90, pages , Philadelphia, PA, USA, Society for Industrial and Applied atheatics. [16] Dian Zad Tootaghaj, Ting He, and To La Porta. Parsionious toography: Optiizing cost-identifiability trade-off for probing-based network onitoring,. IFIP Perforance, [17] Willia T. Trotter and John I. oore Jr. The diension of planar posets. J. Cob. Theory, Ser. B, 22(1):54 67, [18] Javier Yáñez and Javier ontero. A poset diension algorith. J. Algoriths, 30(1): , [19] ihalis Yannakakis. The coplexity of the partial order diension proble. SIA Journal on Algebraic Discrete ethods, 3(3): , VIII. APPENDIX Proof. (of Theore III.10) Case 1 Let W = {w 1,..., w d 1 }. Since U W, then there is a node u U \ W, such that w i u, i [d 1]. Now according to the position of W, we have four cases: i. W is neither in S(u) nor in T (u) ii. W S(u) iii. W T (u) iv. W S(u) d 1 and W T (u) d 1; In case i, any path fro S to T passing through u is not touching W and proves the clai. In case ii, we use lea III.11. Then there is a path p w u fro S to u avoiding W. Furtherore, any path p fro u to T is avoiding W. Therefore, the coposition of p w u with p proves the clai. In case iii, by lea III.12 there is a path p w u fro u to T avoiding W and of course any path p fro S to u avoids W. Then the coposition of p and p w u proves the clai. In case iv, we use lea III.11. Then there is a path p w u fro S to u avoiding W. oreover by lea III.12, there is a path qu w fro u to T avoiding W. Hence the coposition of p w u and qu w proves the clai. Case 2 Although U W, they ight share soe nodes. So there are two cases: (A) U W = 0 and (B) U W d 1. In case (A), let U = {u 1,..., u d }. We fix u to be the node in U not reachable in H n,d by other nodes in U. In case (B), we fix u be the node of U not in W. Like in case 1, we divide in three cases as following: i. W S(u); ii. W T (u); iii. S(u) W d 1 and T (u) W d 1; In case (iii), since S(u) W d 1, then we can apply lea III.11 to find a path p u fro S to u avoiding W. Using T (u) W d 1 a siilar arguent works for finding a path p u fro u to T avoiding W. Then the coposition of p u and p u is a path fro S to T passing through u but avoiding W. In case (i), first we fix w 1,..., w d 1 W this way: For case (A), let the be any d 1 nodes in W. For case (B), if U W = d 1; w 1,..., w d 1 are the nodes in U W and let X = {w 1,..., w d 1 }. If U W < d 1; Let w 1,..., w i (i < d 1) be the nodes in U W. The rest of d 1 i nodes that we need are any nodes in W \ (U W ). Now we have w 1,..., w d 1 and let X = {w 1,..., w d 1 }. Now let v be the other node in W. Since u w i, i [d 1], then by lea III.11 there is a path p X u fro S to u avoiding X. Furtherore, since X S(u) any path p u fro u to T avoids X. Hence the path p u concatenation p X u of with p u touches U and avoids W unless v p u. If v p u, then we odify p u into a new path p v touching W but avoiding U. We first identify a node z on p u. Let u = (x 1,..., x d ) with x i [n], i [d]. Since p X u is ending at u and H n,d is directed, there is a first node z in p X u such

12 that starting fro z, all the nodes in p X u lie on one of the x i s (of u). Let p X z be the subpath obtained fro truncating p X u to the node z. Notice that by the definition of z, z u. Since u T (z), by lea III.12 we can find a path q z fro z to T avoiding u. In case (B), we have X T (z) because X S(u) and by the construction of z, S(u) T (z) is the set of nodes in p X u fro z to u and p X u avoids X. In case (B) when U W = d 1, the path p v concatenating p X z with q z goes fro S to T, touches W but avoids U. In case (B), when U W < d 1 and in case (A), we have X U. Therefore, the other node of U, ight belong to p v and hence in this case p v does not avoid U. To handle this last case we do as the following: If another node of U, say u 1, belongs p v, we do this way: Let t z be the subpath of p v fro S to z. Let q be the subpath of p v starting at z and ending in u 1. Assue that u 1 = (y 1,..., y d ). Siilarly as done for z, there is a first node z 1 in q such that starting fro z 1, all the nodes in q lie on one of the y i s (of u 1 ). z 1 u since q is a subpath of p v which avoids u. Also z 1 u 1 by the definition of z 1. Let q z,z1 be the subpath obtained fro truncating q to z 1. Since u 1 T (z 1 ), by lea III.12 there is a path q z1 fro z 1 to T avoiding u 1. Hence the path p v obtained fro concatenating t z with q z,z1 and then with q z1 goes fro S to T, touches v but avoids both u and u 1. If still another node of U, say u 2, belongs to p v, then we do the sae arguent (as above) until our obtained path avoids the whole U. Clai is proved. In case (ii), first we fix w 1,..., w d 1 W this way: For case (A), let the be any d 1 nodes in W. For case (B), if U W = d 1; w 1,..., w d 1 are the nodes in U W and let X = {w 1,..., w d 1 }. If U W < d 1; Let w 1,..., w i (i < d 1) be the nodes in U W. The rest of d 1 i nodes that we need are any nodes in W \ (U W ). Now we have w 1,..., w d 1 and let X = {w 1,..., w d 1 }. Now let v be the other node in W. Since u w i, i [d 1], then by lea III.12 there is a path p X u fro u to T avoiding X. oreover since X T (u), any path p u fro S to u avoids X. Hence the path p u concatenation p u of with p X u touches U and avoids W unless v p u. If v p u, then we odify p u into a new path p v touching W but avoiding U. We first identify a node z on p u. Let u = (x 1,..., x d ) with x i [n], i [d]. Since p X u is starting fro u and H n,d is directed, there is a first node z in p X u such that before z (starting fro u to z), all the nodes in p X u lie on one of the x i s (of u). Let p X z be the subpath obtained fro truncating p X u at the node z. Notice that by the definition of z, z u. Since u S(z), we can find a path q z fro S to z avoiding u by lea III.11. In case (B), we have X S(z) because X T (u) and by the construction of z, S(z) T (u) is the set of nodes in p X u fro u to z and p X u avoids X. In case (B) when U W = d 1, the path p v concatenating q z with p X z goes fro S to T, touches W but avoids U. In case (B), when U W < d 1 and in case (A), we have X U. Therefore, the other node of U, ight belong to p v and hence in this case p v does not avoid U. To handle this last case we do as the following: If another node of U, say u 1, belongs p v, we do this way: Let t z be the subpath of p v fro z to T. Let q be the subpath of p v starting at u 1 and ending in z. Assue that u 1 = (y 1,..., y d ). Siilarly as done for z, there is a first node z 1 in q such that before z 1, all the nodes in q lie on one of the y i s (of u 1 ). z 1 u since q is a subpath of p v which avoids u. Also z 1 u 1 by the definition of z 1. Let q z,z1 be the subpath obtained fro truncating q at z 1. Since u 1 S(z 1 ), there is a path q z1 fro S to z 1 avoiding u 1 by lea III.11. Hence the path p v obtained fro concatenating q z1 with q z,z1 and then with t z goes fro S to T, touches v but avoids both u and u 1. If still another node of U, say u 2, belongs to p v, then we do the sae arguent until our obtained path avoids the whole U. Clai is proved. Proof. (of Theore IV.6) Let V be the set of nodes of H n,d. We have to prove that for any U, W V with U W such that U, W d 1, then P(U) P(W ). It is sufficient to find a path p fro S to T touching U but avoiding W. We split in two cases: 1. U W = 0 2. U W < d 1; For case 1, since U W and U W = 0, then we fix u be any node in U. For case 2, we fix u be the node of U not in W. Now, it is obvious that u w, w W. First let u = (u 1,..., u d ) be a node in H n,d. We define S(u) and T (u) as below: S(u) = {v = (v 1,..., v d ) 1 v i u i, i [d]} T (u) = {v = (v 1,..., v d ) u i v i n, i [d]} According to the position of W, we have four following cases: i. W is neither in S(u) nor in T (u) ii. W S(u) iii. W T (u) iv. W S(u) d 1 and W T (u) d 1; In case i, any path p fro S to T passing through u is not touching W and proves the clai. In case ii, if W is linked to input onitors, since W d 1, then any path p fro the other input onitor passing through u to an output onitor is avoiding W and the clai is proved. If W N(u) d 1, then there is u N(u) such that u / W and w / N(u ), w W. Now, we prove that there exists a path p u fro S to u avoiding W by induction on S(u ) =. If S(u ) = 1, then S(u ) = {u } and the path p u ade by u avoids W. If S(u ) =, then there is v N(u ) S(u ). Therefore S(v) S(u ) and S(v) <. By induction, there is a path p v fro S to v avoiding W. So p u, the coposition of p v and vu, aviods W as well. Hence the coposition of p u and u u fro S to u is avoiding W. oreover, any path p fro u to T is avoiding

13 W. Therefore the coposition of p u, u u and p is touching u but avoiding W and the clai is proved. For the case iii, a syetric arguent of case ii works. For the case iv, let X = {x 1,..., x i } = W S(u) such that i d 1 and V = {v 1,..., v j } = W T (u) such that j d 1. If X and V are both (or one of the) linked to input and output onitors, since i, j d 1, then any path fro the other d 1 i input onitors passing through u to any other d 1 j output onitors is not touching W and proves the clai. Now assue that (X V ) N(u) d 1. There exists u N(u) such that u / W and w / N(u ), w W. Now by the sae arguent in the case ii, there is a path p u fro S to u avoiding W. Thus the path p fro S to u, the coposition of p u and u u, avoids W. By the syetric arguent, there is a path p fro u to T avoiding W. Hence the coposition of p and p touches U but avoids W and proves the clai. Proof. (of Theore V.6 - sketch) Upper bounds on µ(g ) iply upper bounds on µ(g). Fro µ(g ) k get U, W distinct set of nodes in G of cardinality k + 1 s.t. P(U) P(W ) = Define U = f(u) W = f(w ). If by contradiction there is a path touching nodes of only one between U and W, then since f is an ebedding (and is injective) and G is closed under transitivity, there is path in G touching nodes only in U. Contradiction. IX. EXPERIENTS OF SUBSECTION VI-B We explain soe easureents we have run to try to validate the idea that to boost axial identifiability one can approxiate a grid adding a nuber of edges in such a way that the inial degree of the graph underlying the network topologies increases with soe suitable function f(n) of the nuber nodes N of the network (we consider only f = log N and f = log N). According to Section V where we explore ebeddings another idea we experient is that of perforing soe steps (we start by only 2) of the transitive closure of the incidence atrix of the network graph. Our networks exaples are networks whose topology is publicly available. We consider the AGIS Network for North Aerica and ATT Network See Figure 2 and 1 as available on the Internet Topology Zoo. First we introduce soe notations to present the data which are collected in Table I and Table II. First we transfor the two topologies in two graphs through a nubering of the nodes. To avoid trivial case for identifiability in both networks we first perfor a reoval of the siple paths substituting the by one edge and recursively repeating this operation if needed. For each network in the Figures then we collect data in next the Tables for three instantiation of the original network: 1) the original network, G; 2) the network where we add a certain nuber of rando edges G r ; Fig. 1: ATT North Aerica Fig. 2: AGIS 3) the network obtained fro the original graph boolean squaring its incidence atrix G 2. For each of the networks we have to decide where to place the onitors (both input and output) and how any of the to add. For the second case we also have to decide how any edges to add to each node in order to increase inial degree. In these experients we follow the idea to approxiate a grid of diension d = f(n) inside the original network: hence we decide to adding rando to each node a nuber of edges in such a way to reach degree d Considering d input e d output onitors that we choose at rando In order to copare each other all the instantiations of the networks considered, we decide of choosing in cases (1) and (3) the sae nuber of input and output onitors than in the rando case. In this case however we do no choose rando the onitors but we follow the strategy of considered as onitor all the nodes with saller degrees and including always nodes with degree 1. TABLE I: P easures µ axial identifiability P n. easurent paths L length of longest path l length of shortest path + n. onitors d siulated diension U Certificate for µ W Certificate for µ