Original Article
Measuring the robustness of a network using minimal vertex covers

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Abstract

We define two quantities associated to each of the vertices of a simple graph, based on the collection of minimal vertex covers of the graph. They are called covering degree and covering index. We use them to describe new strategies for measuring the robustness of a network. We study the correlation between the defined quantities and other quantities used in the context of network attacks. Using the attack strategies associated to these quantities we study their effect on the connectedness of several network models. We also consider the complexity of the computation of the defined quantities and use a computational commutative algebra approach for their actual computation.

Introduction

The robustness of networks is an important issue that has been studied in a variety of situations, see for instance [1] and the review [9] and references therein. The attack vulnerability of a network is often investigated by measuring the decrease of the network's performance due to removal of nodes or connections. Since there are many different ways to define the performance of a network, depending on the features of interest, there are many different ways to evaluate their robustness or vulnerability. Usual requirements are measures of connectedness of the network, but there are others. The study of attacks on networks in order to measure their robustness or vulnerability has different applications. Often, such study is done during the design phase of a network, in order to use the available resources to construct a network that is most reliable, robust, or easy to protect. In epidemiology, the goal is to identify good vaccination strategies to prevent the spreading of a disease in the population modelled by the network. Other applications appear in electrical or computer networks [3], or in protection against criminal organizations [6]. In all these applications the procedure is usually to identify nodes or connections that are most critical to the performance of the network, in order to either protect them (allocating more resources, for instance) or to attack them. An attack can consist on vaccination of a specific individual in an epidemiology network. This kind of node (resp. connection) selection based attack is called a targeted node (resp. connection) attack, as opposed to random attacks. An important result in this area was given in [11] concerning the drastic difference of the vulnerability of scale-free networks to random and targeted attacks.

We propose two node attack strategies based on the collection of minimal vertex covers of the graph associated to the network. These strategies take into account the structure of the whole network to help the identification of nodes with an important role. For any given node of the network we define two quantities, covering degree and covering index, that count the number of minimal (resp. minimum) vertex covers of the network's graph in which the given node is present. Our strategies consist on attacking nodes with highest covering degree (resp. covering index). The definition of these quantities and all the necessary concepts from graph theory are given in Section 2, where we also describe how these quantities are related to each other and to other quantities widely used in network attack strategies, vertex degree and betweenness centrality, cf. [6].

In Section 3 we analyse the effect of our attack strategies on several model networks, namely the Erdös-Renyi random graph model, the Watts–Strogatz small world model and the Albert–Barabasi preferential attachment model. We find good results of our strategies with respect to decreasing the network's connectivity, in particular for the covering index attack on Watts–Strogatz networks.

Finally, in Section 4 we address the computational cost of our strategies, which is very high since it is based on the list of all minimal vertex covers of a graph, which is equivalent to finding all maximal cliques of the complementary graph. This problem is known to be NP-hard and the size of the output is exponential in terms of the number of nodes. We also describe a computational commutative algebra approach to this problem, which yields simple yet efficient algorithms for our purposes. The experiments carried out using graph theory software and computational commutative algebra software show that the algebraic approach can be useful in this context.

First of all, we describe the network models that will be used to test our attack strategies.

  • 1

    Erdös-Renyi random graphs. Erdös-Renyi graphs are models for generating random graphs. There are two different Erdös-Renyi models cf. [10], [9]. We use here the model in which the graph is constructed by connecting nodes randomly. The graph has n vertices and each edge (i, j) is included in the graph with probability pi,j independent from every other edge. In the classical description of this model pi,j = p for all (i, j) and the resulting graph is denoted G(n, p). We will use for our tests ten instances of G(60, 0.1).

  • 2

    Watts–Strogatz small world model. The Watts–Strogatz graph is a model for the small world effect [14]. The graph is constructed as follows. First we create a ring with n nodes and connect each node with its k nearest neighbours. Then we replace with probability p each edge (i, j) of the ring with a new edge (i, j′) for some j′ chosen uniformly random. We denote this graph by WS(n, k, p). We will use ten instances of WS(60, 6, 0.5).

  • 3

    Albert–Barabasi preferential attachment model. The Albert–Barabasi graph AB(n, m) is a scale free network in which a graph of n nodes is formed by attaching new nodes one by one, each with m edges that are preferably attached to existing nodes with higher vertex degree [1]. We use ten instances of AB(60, 5) for our experiments.

Section snippets

Covering degree and covering index of a vertex

We view a network as a simple graph G(V, E) where V is its set of vertices and E its set of edges. An edge will be denoted by {i, j} meaning that it joins the vertices i and j. In a graph theoretical context we speak of vertices and edges while in a network theory context we will speak of nodes and connections.

We define two quantities associated with each vertex in a graph that are based on the list of all minimal vertex covers of the graph. Although very similar in the definition, we will see

Experiments

We now measure how the removal of vertices according to the described quantities affects the performance of the network in terms of its connectedness. We build basically two strategies for each quantity: (1) sorting the nodes in decreasing order according to the quantity and then deleting them in that order and (2) for each quantity deleting the node with highest quantity and then recalculating the criteria in the resulting smaller network, again deleting the node with the highest quantity on

Algorithms and computations

The strategies proposed in Section 2 rely on the computation of the list of all minimal vertex covers of a graph, which is an NP-hard problem equivalent to the listing of all maximal cliques of a graph, one of Karp's 21 problems [7]. A set of vertices of a graph is a vertex cover if and only if its complement is an independent set of vertices, and a set of vertices is independent if and only if its elements are a clique in the complement graph. The computation of the size of a minimum vertex

Conclusions

We have defined two quantities associated to each vertex of a simple graph, covering degree and covering index, based on the collection of minimal vertex covers of the graph. Motivated by the study of the robustness of networks, we have studied the correlation of these two quantities between them and with other quantities used in network attacks, namely vertex degree and betweenness centrality. In experiments on several well known network models we have found that covering index is very weakly

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