Vertex-Connectivity Measures for Node Failure Identification in Boolean Network Tomography

TL;DR

This paper investigates how vertex connectivity influences node failure detection in undirected graphs using Boolean Network Tomography, providing bounds for specific network classes and analyzing random network models.

Contribution

It establishes tight bounds on maximal identifiability for Line of Sight networks and explores probabilistic bounds in Erdős-Rényi and Random Regular graphs.

Findings

Tight bounds on maximal identifiability for Line of Sight networks

Weaker bounds for arbitrary networks

Probabilistic tradeoff between monitors and identifiability in random networks

Abstract

In this paper we study the node failure identification problem in undirected graphs by means of Boolean Network Tomography. We argue that vertex connectivity plays a central role. We show tight bounds on the maximal identifiability in a particular class of graphs, the Line of Sight networks. We prove slightly weaker bounds on arbitrary networks. Finally we initiate the study of maximal identifiability in random networks. We focus on two models: the classical Erd\H{o}s-R\'enyi model, and that of Random Regular graphs. The framework proposed in the paper allows a probabilistic analysis of the identifiability in random networks giving a tradeoff between the number of monitors to place and the maximal identifiability.

Figures

Equations

$$\underset{p\in \mathbb{P}}{\bigwedge }\left(\underset{v\in p}{\bigvee }{x}_v\equiv b_p\right)$$

$$\kappa (G)\le \delta (G).$$

$$SE(u)=\{(l,k):l\ge i\vee k\ge j\}\text{mbox}andNW(u)=\{(l,k):l\le i\vee k\le j\}.$$

$$\mu (G)\ge \kappa (G)-2.$$

$$\kappa (G(n,p))=\delta (G(n,p))=np-o(n).$$

$$O(n^{2k})\times (1-(1-(1-\frac{\gamma }{n}{)}^q{)}^2)\le O(n^{2k})\times 2(1-\frac{\gamma }{n}{)}^q\le O(n^{2k})\times 2{\mathrm{e}}^{-\frac{\gamma }{n}q},$$

$$SE(u)=\{(y_1,...,y_d):y_1\ge x_1\vee ...\vee y_d\ge x_d\},\text{mbox}and$$

$$NW(u)=\{(y_1,...,y_d):y_1\le x_1\vee ...\vee y_d\le x_d\}.$$

Full text

1 Introduction

A

central issue in communication networks is to ensure that the structure works reliably. To this end it is of the utmost importance to discover as quickly as possible those components that develop some sort of failure. Network Tomography is a family of distributed failure detection algorithms based on the spreading of end-to-end measurements [7, 22] rather than directly measuring individual network components. Typically a network $G=(V,E)$ is given as a graph along with a collection of paths $\mathbb{P}$ in it and the goal is to take measurements along such paths to infer properties of the given network. Quoting from [11] “A key advantage of tomographic methods is that they require no participation from network elements other than the usual forwarding of packets. This distinguishes them from well-known tools such as traceroute and ping, that require ICMP responses to function. In some networks, ICMP response has been restricted by administrators, presumably to prevent probing from external sources. Another feature of tomography is that probing and the recovery of probe data may be embedded within transport protocols, thus co-opting suitably enabled hosts to form impromptu measurement infrastructures”. The

approach is strongly related to group testing [10] where, in general, one is interested in making statements about individuals in a population by taking group measurements. The main concern is to do so with the minimum number of tests. In our setting, the connectivity structure of the network constrains the set of feasible tests. Graph-constrained group testing has been studied before, starting with [6]. We are interested in using structural graph-theoretic properties to make statements about the quality of the testing process.

Research

in Network Tomography is vast. The seminal works of Vardi [22], and Coates [7], or more recent surveys like [5] each have more that 500 citations, according to Google scholar. Methods and algorithms vary dramatically depending on the network property of interest, or the measurements one has to rely on. Boolean Network Tomography (BNT) aims to identify corrupted components in a network using boolean measurements (i.e. assuming that elementary network components can be in one of two states: “working” or “not-working”). Introduced in [11, 13], the paradigm has recently attracted a lot of interest because of its simplicity. In this work we use BNT to identify failing nodes. Assume to have a set $\mathbb{P}$ of measurement paths over a node set $V$. We would like to know the state $x_{v}$ (with $x_{v}=0$ corresponding to “$v$ in working order” and $x_{v}=1$ corresponding to “$v$ in a faulty state”) of each node $v\in V$. The localization of the failing nodes in $\mathbb{P}$ is captured by the solutions of the system:

[TABLE]

where $b_{p}$ models the (boolean) state of the path $p\in\mathbb{P}$. Of course, systems of this form may have several solutions and therefore, in general, the availability of a collection of end-to-end measurements does not necessarily lead to the unique identification of the failing nodes. We

will investigate properties of the underlying network that facilitate the solution of this problem. In particular, we follow the approach initiated by Ma et. al. [18] based on the notion of maximal identifiability (see Section 2 for a precise definition). The metric aims to capture the maximal number of simultaneously failing nodes that can be uniquely identified in a network by means of measurement along a given path system. It turns out that the network maximal identifiability is an interesting combinatorial measure and several studies [1, 14, 18, 19] have investigated variants of this measure in connection with various types of path systems. However, it seems difficult to come up with simple graph-theoretic properties that affect the given network identifiability. We contend that working with the collection of simple paths between two disjoint sets of vertices $S$ and $T$ enables us to make good progress on this issue. More specifically

we show that the proposed approach provides an almost tight characterization of the maximal identifiability in augmented hypergrids (see definition in Section 2) and more general Line-of-Sight (LoS) networks. LoS networks were introduced by Frieze et al. in [12] and have been widely studied (see for instance [9, 8, 20, 21]) as models for communication patterns in a geometric environment containing obstacles. Like grids, LoS networks can be embedded in a finite cube of $\mathbb{Z}^{d}$, for some positive integer $d$. But LoS networks generalize grids in that edges are allowed between nodes that are not necessarily next to each other in the network embedding.

Using the network vertex-connectivity, $\kappa(G)$, (i.e. the size of the minimal set of nodes disconnecting the graph) we are able to prove the following:

Theorem 1.1

Let $n$ be a positive integer, and $d$, and $\omega$ be fixed positive integers, independent of $n$.

The maximal identifiability of an augmented hypergrid ${\cal H}_{n,d,\omega}$ on $n^{d}$ vertices with range parameter $\omega$. is between $\kappa({\cal H}_{n,d,\omega})-1$ and $\kappa({\cal H}_{n,d,\omega})$. 2. 2.

Let $G=(V,E)$ be an arbitrary $d$-dimensional LoS network with range parameter $\omega$. Then the maximal identifiability of $G$ is between $\kappa(G)-2$ and $\kappa(G)$.

The result on

LoS networks immediately suggests the related question about general graphs. In this work we prove upper and lower bounds on the maximal identifiability of any network $G$. The following statement summarizes our findings:

Theorem 1.2

Let $G=(V,E)$ be an arbitrary graph. Then the maximal identifiability of $G$ is at least $\lfloor\kappa(G)/2\rfloor-2$ and at most $\kappa(G)$.

In both Theorem 1.1 and 1.2, the upper bound is proved by showing that there are sets of $\kappa(G)+1$ vertices that cannot be identified. The lower bounds which require the construction of paths separating large sets of nodes in the graph, is based on a well-known relationship between $\kappa(G)$ and the existence of collections of vertex-disjoint paths between certain sets of nodes in $G$. In fact a much higher lower bound can be proved for graphs with low connectivity (see Theorem 4.4 in Section 4.1). The result, which implies the aforementioned result for arbitrary LoS networks, applies to many topologies studies in relation to communication problems including various types of grids, butterflies, hypercubes, and sensor networks.

Finally,

we look at random networks (Erdős-Rényi and Random Regular Graphs). In these structures we are able to show a trade-off between the success probability of the relevant path construction processes and the size of the sets $S$ and $T$ defining the path set $\mathbb{P}$. Random graphs also give us alternative constructions of networks with large identifiability.

The rest of the paper is organized as follows. After

a section devoted to preliminaries and important definitions, we have a section that focuses on Theorem 1.1.1. Section 4 focuses on arbitrary graphs. First we look at the upper bound in Theorem 1.1.2. Then an additional lower bound is proved for graphs with low connectivity, which implies the lower bound in Theorem 1.1.2. Finally Section 5 is dedicated to the analysis of the maximal identifiability in random graphs. First we look at Erdős-Rényi graphs, then random regular graphs.

2 Preliminaries

Sets, Graphs, Paths, and Connectivity.

If $U$ and $W$ are sets, $U\triangle W=(U\setminus W)\cup(W\setminus U)$ is the symmetric difference between $U$ and $W$. Graphs (we will use the terms network and graph interchangeably) in this paper will be undirected, simple and loop-less. A path (of length $k$) in a graph $G=(V,E)$ from a node $u$ to a node $v$ is a sequence of nodes $p=u_{1},u_{2},\ldots,u_{k+1}$ such that $u_{1}=u$, $u_{k+1}=v$ and $\{u_{i}u_{i+1}\}\in E$ for all $i\in[k]$. The path $p$ is simple of no two $u_{i}$ and $u_{j}$ in $p$ are the same. Any sub-sequence $u_{x},\ldots,u_{x+y}$ ($x\in\{1,\ldots,k+1\}$, $y\in\{0,\dots,k+1-x\}$) is said to be contained in $p$, and dually we say that $p$ contains the sequence or passes through it. We say that path $p$ and $q$ intersect if they contain a common sub-sequence. The intersection of a path $p$ and an arbitrary set of nodes $W$ is the set of elements of $W$ that are contained in $p$. For a node $u$ in $G$, $N(u)$ is the set of neighbourhood of $u$, i.e. $\{v\in V\;|\;\{u,v\}\in E\}$. The degree of $u$, $\deg(u)$, is the cardinality of $N(u)$, and let $\delta(G)=\min_{u\in V}\deg(u)$ be the minimum degree of $G$.

In

what follows $\kappa(G)$ denotes the vertex-connectivity of the given graph $G=(V,E)$, namely $\kappa(G)$ is the size of the minimal subset $K$ of $V$, such that removing $K$ from $G$ disconnects $G$. In particular it is well-known (see for example [15], Theorem 5.1, pag 43) that

[TABLE]

It will also be convenient to work with sets of vertices disconnecting particular parts of $G$. If $S,T\subseteq V$, then $\kappa_{ST}(G)$ is the size of the smallest vertex separator of $S$ and $T$ in $G$, i.e. the smallest set of vertices whose removal disconnects $S$ and $T$ (set $\kappa_{ST}(G)=\infty$ if $S\cap T\neq\emptyset$ or there are $s\in S$ and $t\in T$ such that $\{s,t\}\in E$). Notice that $\kappa_{ST}(G)\geq\kappa(G)$.

Grids and LoS networks.

For positive integers $d$, and $n\geq 2$, let $\mathbb{Z}^{d}_{n}$ be the $d$-dimensional cube $\{1,\ldots,n\}^{d}$. We say that distinct points $P_{1}$ and $P_{2}$ in a cube share a line of sight if their coordinates differ in a single place. A graph $G=(V,E)$ is said to be a Line of Sight (LoS) network of size $n$, dimension $d$, and range parameter $\omega$ if there exists an embedding $f_{G}:V\rightarrow\mathbb{Z}^{d}_{n}$ such that $\{u,v\}\in E$ if and only if $f_{G}(u)$ and $f_{G}(v)$ share a line of sight and $|f_{G}(u)-f_{G}(v)|<\omega$. In the rest of the paper a LoS network $G$ is always given along with some embedding $f_{G}$ in $\mathbb{Z}^{d}_{n}$ for some $d$ and $n$, and with slight abus de langage we will often refer to the vertices of $G$, $u,v\in V$ in terms of their corresponding points $f_{G}(u),f_{G}(v),\ldots$ in $\mathbb{Z}^{d}_{n}$, and in fact the embedding $f_{G}$ will not be mentioned explicitly.

Note that $d$-dimensional hypergrids, $\mathcal{H}_{n,d}$, as defined in [14] are particular LoS networks with $\omega=2$ and all possible $n^{d}$ vertices. In the forthcoming sections we will study augmented hypergrids $\mathcal{H}_{n,d,\omega}$ (or simply $\mathcal{H}_{n,\omega}$ in the 2-dimensional case), namely $d$-dimensional LoS networks with range parameter $\omega>2$ containing all possible $n^{d}$ nodes.

Paths, Monitors and Identifiability.

In BNT

one takes measurements along paths, and the quality of the monitoring scheme depends on the choice of such paths. Let $\mathbb{P}$ be a set of paths over some node set $V$. For a node $v\in V$, let $\mathbb{P}(v)$ be the set of paths in $\mathbb{P}$ passing through $v$. For a set of nodes $U$, $\mathbb{P}(U)=\bigcup_{u\in U}\mathbb{P}(u)$. Hence if $U\subseteq V$, $\mathbb{P}(U)\subseteq\mathbb{P}(V)$. Crucially, we identify two disjoint sets of vertices $S$ and $T$, and assume that $\mathbb{P}$ is the set of all simple paths in $G$ with one end-point in $S$ and the other one in $T$. This is similar to the CSP probing scheme analyzed in [17], but the scheme in that paper does not assume $S\cap T=\emptyset$.

Traditionally

in Network Tomography all measurements originate and end at special monitoring stations that are connected to the structure under observation. For any tomographic process to have any chance of succeeding one has to assume that such monitors are infallible. It is therefore customary to assume that the monitors are external to the given network, but connected to it through a designated set of nodes. $S\cup T$ is such set in our case. We call the pair $(S,T)$ a monitor placement. In

this settings, two sets of vertice $U$ and $W$ are separable if $\mathbb{P}(U)\triangle\mathbb{P}(W)\not=\emptyset$. A set of vertices $N$ is $k$-identifiable (with respect to the probing scheme $(\mathbb{P},S,T)$) if and only if any $U,W\subseteq N$, with $U\triangle W\not=\emptyset$ and $|U|,|W|\leq k$, $U$ are separable. The maximal identifiability of $N$ with respect to $(\mathbb{P},S,T)$, $\mu(N,\mathbb{P},S,T)$, is the largest $k$ such that $N$ is $k$-identifiable. For a graph $G=(V,E)$, we write $\mu(G,\mathbb{P},S,T)$ to indicate the maximal identifiability of the set of nodes in $V$ which are used in at least in a path of $\mathbb{P}$. In what follow we usually omit the dependency of $\mu$ on the probing scheme $(\mathbb{P},S,T)$ when this is clear from the context.

Note

that $k$-identifiability is monotone: if $G$ is $k$-identifiable then it is $k^{\prime}$-identifiable for any $k^{\prime}<k$. This implies that to prove that $\mu(N)\leq k-1$ it is sufficient to show that $N$ is not $k$-identifiable. By the definition given above this boils down to showing the existence of two distinct node sets $U$ and $W$ in $N$ of cardinality at most $k$ that are not separable.

Conversely, if we want to prove that $\mu(N)\geq k$ for some $k$, then it is enough to argue that all distinct node sets $U$ and $W$ of cardinality $|U|,|W|\leq k$ are separable. To prove this we have to show that for any two distinct node sets $U$ and $W$ of cardinality at most $k$ there exists a path in $\mathbb{P}$ intersecting exactly one between $U$ and $W$.

3 Failure Identifiability in Augmented Hypergrids

Let $\omega>2$ be an integer. In this section we analyze the maximal identifiability of augmented hypergrids. To maximize clarity, we provide full details for the special case of $\mathcal{H}_{n,\omega}$, the 2-dimensional augmented hypergrid, and then state the general result, leaving its proof to the Appendix. In [14]

two of us showed that $\mu(G)\leq\delta(G)$ for any $(\mathbb{P},S,T)$. In $\mathcal{H}_{n,\omega}$ each node $u$ has $\omega-1$ edges for each one of the possible directions (north, south, east, west). Hence the minimal degree in $\mathcal{H}_{n,\omega}$ is reached at the corner nodes and it is $2(\omega-1)$. Thus $\mu(\mathcal{H}_{n,\omega})\leq 2(\omega-1)$ for any $(\mathbb{P},S,T)$. In the remainder of this section we pair this up with a tight lower bound for a specific monitor placement. Note that these results readily imply Theorem 1.1.1 as in augmented hypegrids the vertex connectivity is actually equal to the network’s minimum degree.

We

say that nodes with coordinates $(1,j)$ in $\mathcal{H}_{n,\omega}$, for some $j\in\{1,\ldots,n\}$, are in the north border of $\mathcal{H}_{n,\omega}$. Analogously we can define south, west and east borders of $\mathcal{H}_{n,\omega}$. A canonical monitor placement for $\mathcal{H}_{n,\omega}$ is a pair $(S,T)$, such that $|S|=|T|=2\omega-1$ and nodes in $S$ are chosen among the west and north borders of $\mathcal{H}_{n,\omega}$ and $T$ among the the south and east borders of $\mathcal{H}_{n,\omega}$.

Given a node $u$ of $\mathcal{H}_{n,\omega}$, identified as a pair $(i,j)\in\mathbb{Z}^{2}_{n}$, we define:

[TABLE]

We are now ready to state the main result of this section.

Theorem 3.1

Let $n,\omega\in\mathbb{N}$, $n\geq 2$ and $\omega>2$. Let $(S,T)$ be the canonical monitor placement for $\mathcal{H}_{n,\omega}$. Then $\mu(\mathcal{H}_{n,\omega})\geq 2(\omega-1)-1$.

Proof

We have to prove that given two node sets $U$, and $W$ of cardinality at most $2(\omega-1)-1$, with $U\triangle W\not=\emptyset$ we can build an $S$-$T$ path touching exactly one of them. Given a node $u\in U\setminus W$, let $S(u)=NW(u)$, the nodes in the North-West region of $u$ and let $T(u)=SE(u)$, the nodes in South-East region of $u$. Notice that $(1,1)\in S(u)$ and $(n,n)\in T(u)$ and $S(u)\cap T(u)=\emptyset$. Since $|S|>2(\omega-1)$ and $|W|\leq 2(\omega-1)$, there is a node in $s\in S\setminus W$. Assume that $s=(1,1)$ (if $s\not=(1,1)$ is similar and give even better results). Similarly for $T$, assume that $(n,n)\not\in W$. Consider the following definition:

Definition 1

Given a node $u\in\mathcal{H}_{n,\omega}$, and a set of nodes $W$ in $\mathcal{H}_{n,\omega}$, we say that a direction $X$ (north, south, west, east) is $W$-saturated on $u$ if moving from $u$ on direction $X$ there is, right after $u$, a consecutive block of $\omega-1$ nodes in $W$.

The following claim define two disjoint paths $i_{u}$ in $S(u)$ from $s$ to $u$ and $o_{u}$ in $T(u)$ form $u$ to $t$ not touching $W$. Their concatenation hence defines a path joining $S$ to $T$ passing through $u$ and not touching $W$. ∎

Claim

Let $u\in S(u)$. There is a path $i_{u}$ in $S(u)$ from $(1,1)$ to $u$ not touching $W$. There is a path $o_{u}$ in $T(u)$ from $u$ to $t$ not touching $W$.

Proof

We prove the first one since they are the same. By induction on $S(u)$. If $|S(u)|=1$, then $u=s$ and we have done. If $|S(u)|>1$. Since $|W|\leq 2(\omega-1)-1$, and since a direction is $W$-saturated only if a block of $\omega-1$ consecutive elements of $W$ appear after $u$ on that direction, then there is at a least a direction $X$ between North and West which is not $W$-saturated. Hence there is a node $u^{\prime}\in S(u)\setminus W$ on direction $X$ from $u$ at distance less than $\omega$. Hence there is an edge $\{u^{\prime},u\}\in\mathcal{H}_{n,\omega}$. Since $S(u^{\prime})\subset S(u)$ the inductive hypothesis give us a path $i_{u^{\prime}}$ as required. Hence the path $i_{u}=i_{u^{\prime}},u$ is as required. ∎

Theorem

3.1 easily generalizes to $d$-dimensional augmented hypergrids.

Theorem 3.2

Let $d,n,\omega\in\mathbb{N}$, $d,n\geq 2$ and $\omega>2$. Let $(S,T)$ be the canonical monitor placement for $\mathcal{H}_{n,d,\omega}$. Then $\mu(\mathcal{H}_{n,d,\omega})\geq d(\omega-1)-1$.

4 General Topologies

We now look at the maximal identifiability in arbitrary networks. Theorem 1.2 stated in Section 1 will be a consequence of two independent results. In

[14] it was proved that $\mu(G)\leq\delta(G)$, for any monitor placement $(S,T)$. Here we show that $\mu(G)$ can be upper bound in terms of $\kappa_{ST}$, the size of the minimal node set separating $S$ from $T$.

Theorem 4.1

Let $G=(V,E)$ be a graph and $(S,T)$ be a monitor placement. Then $\mu(G)\leq\kappa_{ST}(G)$.

Proof

If there is no vertex set in $G$ separating $S$ and $T$, $\kappa_{ST}(G)=\infty$ and the result is trivial. Let $K$ be the set witnessing the minimal separability of $S$ from $T$ in $G$. Hence $|K|=\kappa_{ST}(G)$. Let $N(K)$ be the set of nodes neighbours of nodes in $K$ and notice this cannot be empty since $K$ is disconnecting $G$. Pick one $w\in N(K)$ and define $U:=K$ and $W:=U\cup\{w\}$. Clearly $\mathbb{P}(U)\subseteq\mathbb{P}(W)$. To see the opposite inclusion assume that there exists a path from $S$ to $T$ passing from $w$ but not touching $U=K$. Then $K$ is not separating $S$ from $T$ in $G$. Contradiction. ∎

Note that, while in general $\kappa_{ST}(G)$ may be larger than $\delta(G)$, if $S$ and $T$ are separated by a set of $\kappa(G)$ vertices then, by inequality (2), the bound in Theorem 4.1 is at least as good as the minimum degree bound proved earlier by the first two authors [14]. This implies the upper bound in Theorem 1.2.

Moving

to lower bounds, in this section we prove the following:

Theorem 4.2

Let $G=(V,E)$ and $(S,T)$ be a monitor placement for $G$. Then $\mu(G)\geq\min(\kappa(G),|S|,|T|)-2$.

The lower bound in Theorem 1.2 can be derived easily from Theorem 4.2. Let $K$ be a vertex separator in $G$ of size $\kappa(G)$, set $S^{K}$ to be the first $\lfloor\kappa(G)/2\rfloor$ elements of $K$ and $T^{K}=K\setminus S^{K}$. By Theorem 4.2 the maximal identifiability of $G$ is at least $|S^{K}|-2=\lfloor\kappa(G)/2\rfloor-2$.

The proof of Theorem 4.2 uses Menger’s Theorem, a well-known result in graph theory (see [15, Theorem 5.10, p. 48] for its proof).

Theorem 4.3

*(Menger’s Theorem)

Let $G=(V,E)$ be a connected graph. Then $\kappa(G)\geq k$ if and only if each pair of nodes in $V$ is connected by at least $k$ node-disjoint paths in $G$.*

Menger’s Theorem is central to the following Lemma which is used in the proof of Theorem 4.2.

Lemma 1

Let $G=(V,E)$. Let $W\subseteq V$ such that $|W|\leq\kappa(G)-2$. Then any pair of vertices in $V\setminus W$ is connected by at least two vertex-disjoint simple paths not touching $W$.

Proof

By Menger’s Theorem, for any pair of nodes $u$ and $v$ in $V\setminus W$ there are at least $\kappa(G)$ vertex-disjoint paths from $u$ to $v$ in $G$. Call $\mathbb{P}$ the set of such paths. Since $|W|\leq\kappa(G)-2$, then the nodes of $W$ can be in at most $\kappa(G)-2$ of paths in $\mathbb{P}$. Hence there are at least two paths in $\mathbb{P}$ not touching $W$. ∎

Proof of Theorem 4.2.

Let $G=(V,E)$ be an undirected connected graph and $(S,T)$ be a monitor placement in $G$. Note that without loss of generality that $\min(\kappa(G),|S|,|T|)>2$ (for otherwise there is nothing to prove).

Assume first that $|S|\geq\kappa(G)$ and $|T|\geq\kappa(G)$. We claim that

[TABLE]

We show that for any distinct non-empty subsets $U$ and $W$ of $V$ of size at most $\kappa(G)-2$, there is a path in $\mathbb{P}$ touching exactly one between $U$ and $W$. Given such $U$ and $W$, fix a node $u\in U\triangle W$ and wlog $u\in U$. Since $|W|\leq\kappa(G)-2$ and $|S|\geq\kappa(G)$ there is at least a node in $s\in S\setminus W$. By the Claim above applied to nodes $s$ and $u$ and to the set $W$, there are two vertex-disjoint simple paths $\pi^{s}_{1},\pi^{s}_{2}$ from $s$ to $u$ not touching $W$. The same reasoning applied to $T$, guarantees the existence of a node $t\in T\setminus W$ and two vertex-disjoint paths $\pi^{t}_{1},\pi^{t}_{2}$ from $u$ to $t$ not touching $W$. If at least one between $\pi^{s}_{1}$, and $\pi^{s}_{2}$ only intersects one of $\pi^{t}_{1}$, and $\pi^{t}_{2}$ at $u$ then the concatenation of such paths is a (longer) simple path from $s$ to $t$ passing through $u$ and not touching $W$. Otherwise the concatenation of one between $\pi^{s}_{1}$, and $\pi^{s}_{2}$ with one between $\pi^{t}_{1}$, and $\pi^{t}_{2}$ is a non simple path. In what follow we show that the subgraph of $G$ induced by the four paths does contain a simple path from $s$ to $t$ passing through $u$ and not touching $W$. In the construction below we exploit the fact that $\pi^{s}_{1}$, and $\pi^{s}_{2}$ (resp. $\pi^{t}_{1}$, and $\pi^{t}_{2}$) are simple and vertex disjoint. Let $p$ be a path from $s$ to $u$. Define an order on the nodes of $p$ as follows: $v\prec w$ if going from $v$ to $u$ we pass though $w$. For $i,j\in\{1,2\}$, let $Z_{ij}$ be the set of nodes in $\pi^{s}_{i}\cap\pi^{t}_{j}$. Notice that $Z_{ij}=Z_{ji}$. Nodes in $Z_{ij}$ can be ordered according to $\prec$. So let $z_{ij}$ be the minimal node in $Z_{ij}$ wrt $\prec$. Wlog let us say that $z_{1j}\prec z_{2j}$. Observe hence that the subpath $\pi^{s}_{1}[s\ldots z_{1j}]$ of $\pi^{s}_{1}$ going from $s$ to $z_{1j}$, before $z_{1j}$ is intersecting neither $\pi^{t}_{1}$ nor $\pi^{t}_{2}$. Hence the concatenation of the following three disjoint paths defines a simple path from $s$ to $t$ passing through $u$ avoiding $W$, hence a path in $\mathbb{P}$ with the required properties:

$\pi^{s}_{1}[s\ldots z_{1j}]$, going form $s$ to $z_{1j}$; 2. 2.

$\pi^{t}_{j}[z_{1j}\ldots u]$ a sub path of $\pi^{t}_{j}$ going from $u$ to $z_{1j}$ and traversed in the other direction; 3. 3.

$\pi^{t}_{j\!\!\!\mod 2+1}$, the other path connecting $u$ to $t$.

Now assume that least one $|S|$ or $|T|$ is less than $\kappa(G)$. Let $r=\min(|S|,|T|)-2$. As before we prove that for all distinct non-empty $U$ and $W$ subsets of $V$ of size at most $r$, there is a simple $S-T$ path in $G$, hence in $\mathbb{P}$, touching exactly one between $U$ and $W$. Let $u\in U\triangle W$ and wlog $u\in U$. Notice that $r+2=\min(|S|,|T|)$, then both $|S|\geq r+2$ and $|T|\geq r+2$. Since $|W|\leq r$, as before there are $s\in S\setminus W$ and $t\in T\setminus W$. Furthermore, since $\kappa(G)\geq\min(|S|,|T|)$, then by previous observation on $|S|$ and $|T|$, $\kappa(G)\geq r+2$ and, since $|W|\leq r$, then $\kappa(G)-|W|\geq 2$, that is $|W|\leq\kappa(G)-2$. As in the previous case we can apply the Claim above once to $s$, $u$ and $W$ getting the vertex-disjoint paths $\pi^{s}_{1}$ and $\pi^{s}_{2}$ from $s$ to $u$, and once to $t$, $u$ and $W$ getting the vertex-disjoint paths $\pi^{t}_{1}$ and $\pi^{t}_{2}$ from $t$ to $u$. The proof then follows by the same steps as in the previous case. We then have proved that if $|S|$ or $|T|$ are $\leq\kappa(G)$, then $\mu(G)\geq\min(|S|,|T|)-2$ and the proof of Theorem 4.2 is complete. ∎

4.1 Improved Bounds for Networks with Low Connectivity

We

complete this section investigating a different way to relate the graph vertex connectivity to $\mu(G)$. It is easy to see that, in general, the bounds in Theorem 1.2 are not very tight, particularly when $\kappa(G)$ is large. However, if $\kappa(G)$ is small, we can do better. Theorem 4.4 below in particular applies to LoS network with constant range parameter, and readily gives the lower bound promised in Theorem 1.1.2.

Theorem 4.4

Let $G=(V,E)$, and $\kappa(G)\leq\frac{|V|}{3}$. There exists a monitor placement for $G$ such that $\kappa(G)-2\leq\mu(G)\leq\kappa(G)$.

Proof

Assume $\kappa(G)=k$, and let $K$ be a minimal vertex separator in $G$. Let $G^{K}_{i}=(V^{K}_{i},E^{K}_{i})$, $i\in[r_{K}]$ be the $r_{K}\geq 2$ connected components remaining in $G$ after removing $K$. Since $k\leq\frac{n}{3}$, then $2k\leq n-k$. Since $|V\setminus K|$ has $n-k$ nodes there are sufficient nodes in $V\setminus K$ to define $(S,T)$ with $|S|=|T|=k$ in such a way that the smallest among the $V_{i}^{K}$’s contains only element from $S$. This can be done as follows: if the smallest $V_{i}^{K}$’s has less than $k$ nodes, say $k-\ell$, then assign all its nodes to $S$. Use other components $G_{j}^{K}$’s (that will have more than $k+\ell$ nodes) to assign $\ell$ nodes to $S$ and $k$ other nodes to $T$. If the smallest $V_{i}^{K}$ has more than $k$ nodes, choose $k$ among them and put them in $S$. Choose $k$ nodes in other components and assign them to $T$.

We now prove that $\mu(G)\leq\kappa(G)$. Let $G_{i}^{K}$ be the component where all the $S$-nodes are assigned. Let $w$ be a node in $V^{K}_{i}\cap N(K)$. This node has to exists since $G$ was connected and the removal of $K$ is disconnecting $G^{K}_{i}$ from $K$. Fix $U=K$ and $W=K\cup\{w\}$. We will show that $\mathbb{P}(U)=\mathbb{P}(W)$. It sufficient to prove that $\mathbb{P}(\{w\})\subseteq\mathbb{P}(K)$, since clearly $\mathbb{P}(U)\subseteq\mathbb{P}(W)$. Observe that no $S-T$ path $p$ in $G$ can live entirely inside $G^{K}_{i}$, i.e. have all of its nodes in $V_{i}^{K}$. This is because at least one end-point (that in $T$) it is necessarily missing in any path entirely living only in $G^{K}_{i}$. Hence a path touching $w$ is either entering or leaving $G^{K}_{i}$. But outside of $G^{K}_{i}$ $w$ is connected only to $K$, since otherwise $K$ would not be a minimal vertex separator. Hence it must be necessary that $\mathbb{P}(\{w\})\subseteq\mathbb{P}(K)$. We have found $U,W$ of size $\leq\kappa(G)$ such that $\mathbb{P}(U)=\mathbb{P}(W)$. The upper bounds follows. The lower bound follows form Theorem 4.2 noticing that $|S|=|T|=\kappa$. ∎

Arbitrary LoS networks have minimum degree, and hence also vertex connectivity at most $2d(\omega-1)$. The next corollary follows directly from Theorem 4.4.

Corollary 1

Let $G$ be an arbitrary LoS network, with fixed range parameter $\omega$. Then $\mu(G)\geq\kappa(G)-2$.

5 Random Networks and Tradeoffs

The main aim of this work is to characterize the identifiability in terms of the vertex connectivity. In this section we prove that tight results are possible in random graphs. Also we show an interesting trade-off between the success probability of the various random processes and the size of the sets $S$ and $T$. Finally, random graphs give us constructions of networks with large identifiability.

5.1 Sub-Linear Separability in Erdős-Rényi Graphs

We start our investigation of the identifiability of node failures in random graphs by looking at the binomial model $G(n,p)$, for fixed $p\leq 1/2$ (in this section only we follow the traditional random graph jargon and use $p$ to denote the graph edge probability rather than a generic path). The following equalities, which hold with probability approaching one as $n$ tends to infinity high probability (that is with high probability (w.h.p.)), are folklore:

[TABLE]

(see [4]). Here we describe a simple method which can be used to separate sets of vertices of sublinear size.

We assume, for now, that $S$ and $T$ are each formed by $\gamma=\gamma(n)$ nodes with $\kappa(G(n,p))\leq\gamma<n/2$. Let $M=S\cup T$.

Let $U$ and $W$ be two arbitrary subsets of $V\setminus M$ of size $k$. The probability that $U$ and $W$ are separable is at least the probability that an element $v$ of $U\Delta W$ (w.l.o.g. assume $v\in U\setminus W$) is directly connected to a node in $S$ and to a node in $T$. This event has probability $(1-(1-p)^{\gamma})^{2}.$ Hence the probability that $U$ and $W$ cannot be separated is at most $1-(1-(1-p)^{\gamma})^{2}=2(1-p)^{\gamma}-(1-p)^{2\gamma}$ and therefore the probability that some pair of sets $U$ and $W$ of size $k$ (not intersecting $M$) fail is at most $2{n-2\gamma\choose k}^{2}(1-p)^{\gamma}.$

Theorem 5.1

For fixed $p$ with $p\leq 1/2$, under the assumptions above about the way monitors are placed in $G(n,p)$, the probability that $G(n,p)$ is not $k$-vertex separable is at most $2k{n\choose k}^{2}{\rm e}^{(2k-\gamma)p}.$

Proof

The argument above works if both $U$ and $W$ contain no vertex in $M$. The presence of elements of vertices in $M$ in $U$ or $W$ may affect the analysis in two ways. First $v$ could be in $M$ (say $v\in S$). In this case $U$ and $W$ are separable if $v$ is directly connected to a vertex in $T$. This happens with probability $(1-(1-p)^{\gamma})>(1-(1-p)^{\gamma})^{2}.$ Second, $M$ might contain some elements of $U$ and $W$ different from $v$. In the worst case when $v$ is trying to connect to $M$, it must avoid at most $2k$ element of such set. There is at most $\sum_{h\leq k}{n\choose h}^{2}\leq k{n\choose k}^{2}$ pairs of $U$ and $W$ of size at most $k$. Thus the probability that $G(n,p)$ fails to be $k$-vertex separable is at most $2k{n\choose k}^{2}(1-p)^{\gamma-2k}.$ and the result follows as $1-p\leq{\rm e}^{-p}$. ∎

5.2 Linear Separability in Erdős-Rényi Graphs

The argument above cannot be pushed all the way up to $\kappa(G(n,p))$. When trying to separate vertex sets containing $\Omega(n)$ vertices the problem is that these sets can form a large part of $M$ and the existence of direct links from $v$ to $S\setminus W$ and $T\setminus W$ is not guaranteed with sufficiently high probability. However a different argument allow us to prove the following:

Theorem 5.2

For fixed $p$, $\mu(G(n,p)=\kappa(G(n,p))$ w.h.p.

Full details of the proof are left to the final version of this paper, but here is an informal explanation. The upper bound follows immediately from (3) and Theorem 4.1. For the lower bound we claim that the chance that two sets of size at most $np$ are not vertex separable is small. To see this pick two sets $U$ and $W$, and remove, say, $W$. $G(n,p)\setminus W$ is still a random graph on at least $n-np$ vertices and constant edge probability. Results in [3] imply that $G(n,p)\setminus W$ has a Hamilton path starting at some $s\in S$ and ending at some $t\in T$ with probability at least $1-o(2^{-n})$ (and in fact there is a fast polynomial time algorithm that finds one). Such Hamilton path, by definition, contains a path from $S$ to $T$ passing through $v\not\in W$, for every possible choice of $v$. This proves, w.h.p., the separability of sets of size up to $\kappa(G(n,p))$. Past such value the construction in Theorem 4.1 applies.

5.3 Random Regular Graphs

A standard way to model random graphs with fixed vertex degrees is Bollobas’ configuration model [2]. There’s $n$ buckets, each with $r$ free points. A random pairing of these free points has a constant probability of not containing any pair containing two points from the same bucket or two pairs containing points from just two buckets. These configurations are in one-to-one correspondence with $r$-regular $n$-vertex simple graphs. Denote by ${\cal C}_{n,r}$ the set of all configurations $C(n,r)$ on $n$ buckets each containing $r$ points, and let $G(r$-reg$)$ be a random $r$-regular graph.

As before assume $|S|=|T|=\gamma$. The main result of this section is the following:

Theorem 5.3

Let $r\geq 3$ be a fixed integer. $\mu(G(r$-reg$)=r$ w.h.p.

The upper bound follows from Theorem 4.1 and the weil-known fact that random $r$-regular graphs are $r$-connected w.h.p. The lower bound is a consequence of the following:

Lemma 2

Let $r\geq 3$ be a fixed integer. Two sets $U$ and $W$ with $U,W\subseteq V(G(r$-reg$))$ and $\max(|U|,|W|)\leq k$ are separable w.h.p. if $k=r-o(1)$.

Proof

In what follows we often use graph-theoretic terms, but we actually work with a random configuration $C(n,r)$. Let $U$ and $W$ be two sets of $k$ buckets. For simplicity assume that (the vertices corresponding to the elements of) both $U$ and $W$ are subsets of $V\setminus M$. The probability that $U$ and $W$ can be separated is at least the probability that a (say) random element $v$ of $U\triangle W$ (w.l.o.g. $v\in U\setminus W$) is connected to $S$ by a path of length at most $\ell_{s}$ and to $T$ by a path of length at most $\ell_{t}$, neither of which “touch” $W$. Fig. 3 provides a simple example of the event under consideration. The desired paths can be found using algorithm PathFinder below that builds the paths and $C(n,r)$ at the same time.

PathFinder$(v,\ell_{s},\ell_{t},W)$

SimplePaths$(v,\ell_{s},\ell_{t},W)$.

Starting from $v$, build a simple path $p^{s}$ of length $\ell_{s}$ that avoids $W$. Similarly, starting from $v$, build a simple path $p^{t}$ of length $\ell_{t}$ that avoids $W$.

RandomShooting$(p^{s},p^{t})$.

Pair up all un-matched points in $p^{s}$ and $p^{t}$.

Complete the configuration $C(n,r)$ by pairing up all remaining points.

Sub-algorithm SimplePaths can complete its constructions by pairing points starting from elements of the bucket $v$ then choosing a random un-matched point in a bucket $u$, then picking any other point $u$ and then again a random un-matched point and so on, essentially simulating two random walks RWs and RWt on the set of buckets. Note that the process may fail if at any point we re-visit a previously visited bucket or if we hit $W$ or even $M$. However the following can be proved easily.

Claim

RWs and RWt succeed w.h.p. provided $\ell_{s},\ell_{t}\in o(n)$.

As to RandomShooting, the process succeeds if we manage to hit an element of $S$ from $p^{s}$ and an element of $T$ from $p^{t}$.

Claim

RandomShooting$(q_{s},q_{t},S,T)$ succeeds w.h.p. if $\ell_{s},\ell_{t}\in\omega(1)$.

Any un-matched point in $p^{s}$ or $p^{t}$ after SimplePaths is complete is called useful. Path $p^{s}$ (resp. $p^{t}$) contains $q_{s}=(r-2)\ell_{s}+1$ (resp $q_{t}=(r-2)\ell_{t}+1$) useful points. During the execution of RandomShooting a single useful point “hits” its target set, say $S$, with probability proportional to the cardinality of $S$. Hence the probability that none of the $q_{s}$ useful points hits $S$ is $(1-\frac{\gamma}{n})^{q_{s}}$ and the overall success probability is $(1-(1-\frac{\gamma}{n})^{q_{s}})(1-(1-\frac{\gamma}{n})^{q_{t}}).$

Back to the proof of Lemma 2 Set $\ell_{s}=\ell_{t}=\ell$ and $q$ the common value of $q_{s}$ an $q_{t}$. The argument above implies that the success probability for $U$ and $W$ is asymptotically approximately $(1-(1-\frac{\gamma}{n})^{q})^{2}$ and the rest of the argument (and its conclusion) is very similar to the $G(n,p)$ case (the final bound is slightly weaker, though). The chance that a random $r$-regular graph is not $k$-vertex separable is at most

[TABLE]

which goes to zero as $n^{-C}$ provided $k\leq(r-o(1))\frac{\gamma\ell}{n\log n}$. The constraints on $\ell$ from the claims above imply that that parameter can be traded-off agains $\gamma$ to achieve optimal identifiability. ∎

APPENDIX

Lemma 3

Let $d\in\mathbb{N}^{+}$ and $n,\omega\in\mathbb{N}$, $n\geq 2$ and $\omega>2$. $\delta(\mathcal{H}_{n,d,\omega})=d(\omega-1)$.

Proof

Each node $u\in\mathcal{H}_{n,d,\omega}$ has $\omega-1$ edges for each one of the possible directions the node is linked to. There can be at most $2d$ direction. Thus an internal node has degree $2d(\omega-1)$, the nodes on a border have degree $(d+t)(\omega-1)$ for some $t\in\{0,\ldots d-1\}$ and, in particular, the nodes on the corners of the grid have degree $d(\omega-1)$. Hence the minimal degree in $\mathcal{H}_{n,d,\omega}$ is reached at the corner nodes and it is $d(\omega-1)$.

By Lemma IV.4 of [14] we have $\mu(\mathcal{H}_{n,d,\omega})\leq d(\omega-1)$. In the remainder of this section we provide a tight lower bound.

Given a node $u\in\mathcal{H}_{n,d,\omega}$, identified as $(x_{1},...,x_{d})\in\mathbb{Z}^{d}_{n}$, we define:

[TABLE]

[TABLE]

Note that $\mathcal{H}_{n,d,\omega}$ has many more edges than the simple hypergrid $\mathcal{H}_{n,d}$ (studied in [14]). For non-degenerate monitor placement $(S,T)$, we place $d\omega-1$ input monitors on the west and north borders of $\mathcal{H}_{n,d,\omega}$ and $d\omega-1$ output monitors on the south and east borders of $\mathcal{H}_{n,d,\omega}$.

Definition 2

Given a node $u\in\mathcal{H}_{n,d,\omega}$ and $W$ a set of nodes in $\mathcal{H}_{n,d,\omega}$. We say that a direction $X$ ($2d$ directions) is $W$-saturated on $u$ if moving from $u$ on direction $X$ there is, right after $u$, a consecutive block of $\omega-1$ nodes in $W$.

Proof of Theorem 3.2.

We have to prove that for two node sets $U$, and $W$ of cardinality at most $d(\omega-1)-1$, with $U\triangle W\not=\emptyset$ we can build a path from $S$ to $T$ touching exactly one of them. Given a node $u\in U\setminus W$, let $S(u)=NW(u)$, the nodes in the North-West of $u$ and let $T(u)=SE(u)$, the nodes in South-East of $u$. Notice that $(\overbrace{1\times\cdots\times 1}^{d})\in S(u)$ and $(\overbrace{n\times\cdots\times n}^{d})\in T(u)$ and $S(u)\cap T(u)=\emptyset$. Since $|S|=d\omega-1>d(\omega-1)$ and $|W|\leq d(\omega-1)$, there is a node in $s\in S\setminus W$. Assume that $s=(1,...,1)$ (if $s\not=(1,...,1)$ is similar). Similarly for $T$, assume that $t=(n,...,n)\not\in W$. Now let $u\in S(u)$. We show that there is a path $p_{u}$ in $S(u)$ from $s=(1,...,1)$ to $u$ not touching $W$. There is also a path $q_{u}$ in $T(u)$ from $u$ to $t=(n,...,n)$ not touching $W$. We prove the first one since they are the same. By induction on $S(u)$. If $|S(u)|=1$, then $u=s$. Take the path $u$ itself and we have done. If $|S(u)|>1$, since $|W|\leq d(\omega-1)-1$, and since a direction is $W$-saturated only if a block of $\omega-1$ consecutive elements of $W$ appear after $u$ on that direction, then there is at least a direction $X$ between North and West which is not $W$-saturated. Hence there is a node $u^{\prime}\in S(u)\setminus W$ on direction $X$ from $u$ at distance less than $\omega$. Thus there is an edge $\{u^{\prime},u\}\in\mathcal{H}_{n,d,\omega}$. Since $S(u^{\prime})\subset S(u)$, by the inductive hypothesis we have a path $p_{u^{\prime}}$ as required. Hence the path $p_{u}=p_{u^{\prime}},\{u^{\prime},u\}$ is as required. Now we have found two disjoint paths $p_{u}$ in $S(u)$ from $s$ to $u$ and $q_{u}$ in $T(u)$ form $u$ to $t$ not touching $W$. Their concatenation gives us a path from $S$ to $T$ passing from $u$ and not touching $W$ and proves the theorem. ∎