Elsevier

Computer Science Review

Volume 28, May 2018, Pages 92-117
Computer Science Review

Survey
The Critical Node Detection Problem in networks: A survey

https://doi.org/10.1016/j.cosrev.2018.02.002Get rights and content

Abstract

In networks, not all nodes have the same importance, and some are more important than others. The issue of finding the most important nodes in networks has been addressed extensively, particularly for nodes whose importance is related to network connectivity. These nodes are usually known as Critical Nodes. The Critical Node Detection Problem (CNDP) is the optimization problem that consists in finding the set of nodes, the deletion of which maximally degrades network connectivity according to some predefined connectivity metrics. Recently, this problem has attracted much attention, and depending on the predefined metric, different variants have been developed. In this survey, we review, classify and discuss several recent advances and results obtained for each variant, including theoretical complexity, exact solving algorithms, approximation schemes and heuristic approaches. We also prove new complexity results and induce some solving algorithms through relationships established between different variants.

Introduction

Many mechanisms and functions in networks are greatly affected by a fraction of nodes, which are usually qualified as important. A node is important if its failure or malicious behavior significantly degrades network performance. The issue of identifying the most important nodes in networks has long been the focus of an intensive amount of research. In the literature, these nodes appear under various names, depending on their role in the network, such as: most influential nodes [1], most vital nodes [2], most k-mediator nodes [3], key-player nodes [4]. With respect to network connectivity, the most important nodes are often known as Critical Nodes.

Therefore, critical nodes of a network are those the removal of which significantly degrades network connectivity. The Critical Node Detection Problem (CNDP) is the optimization problem that consists in finding the set of these nodes. In other words, the CNDP consists in finding the set of nodes whose deletion leads to achieving a certain objective related to making the network disconnected.

As a network can take different forms after disconnection, node criticality depends on how the network is disconnected once the node has been deleted, which depends, in turn, on the objective of the application considered. Thus, a node that is critical for some purposes or considerations may not be critical for others. Considering the network in Fig. 1, if we ask for the set of two nodes, the removal of which maximizes the number of connected components, then the optimal solution is to delete nodes {a,d}, which generates seven components. While if we consider the case where we aim at minimizing the set of nodes, the removal of which constrains the cardinality of each connected components to four nodes at most, the optimal solution is then to delete the nodes {b,c}. In both cases, we ask for the same thing: to delete nodes from the graph, but for different objectives. These objectives depend on the application at hand. For instance, the last case is relevant when inheriting the spreading of complex contagions1 where more than four interactions are needed to acquire the contagion. In fact, if we assume that the graph in Fig. 1 corresponds to a social network, then vaccinating critical nodes {b,c} (corresponding to individuals) allows the network to be partitioned on communities of four susceptible individuals each, at most (see Fig. 1). Hence either all individuals in the community are infected or not, there is no spread of the contagion as there is no chance of having more than four interactions with infected individuals.

Therefore, we can define critical nodes as those the deletion of which disconnects the network according to some predefined connectivity metrics, such as: maximizing the number of connected components, minimizing pairwise connectivity in the network, minimizing the largest component size, etc.

Graph connectivity has long been considered as the measure of network robustness, and assessing how well the graph is connected or, on the contrary, stating how much effort (by deleting nodes or edges) is required to disconnect it has been extensively studied in the literature. This problem is known as the edge- or vertex-connectivity problem. For a detailed review, we refer the reader to [5]. Generally, deleting nodes (or edges) from a graph in order to obtain a disconnected graph with some specific properties is one of the oldest issues in graph theory. For problems based on the deletion of edges, a variety of variants have been defined and studied in the literature, such as the graph partitioning problem [[6], [7], [8]], the minimum k-cut problem [[9], [10]], the multicut problem [[11], [12], [13]], the multiway cut problem [[14], [15], [16]], the multi-multiway cut problem [17], etc. For problems based on the deletion of nodes, which include the problem we are reviewing in this paper, we ask for deleting nodes rather than edges. Many variants have been studied in the literature, including:

  • The vertex separator problem. Given a graph G=(V,E) and an integer k, this problem aims at partitioning G into three subsets of nodes A, B and C such that |C| is minimum, A and B are disconnected and max(|A|,|B|)<k. In other words, it seeks to find the minimum set of nodes CV, the deletion of which partitions the graph into two bounded subsets A and B of at most k nodes. The problem is NP-hard even on 3-bounded degree graphs2  [18], and it has been studied on general graphs [18], planar graphs [[19], [20]], using polyhedral approach [[21], [22]], metaheuristic algorithms [23], etc.

  • The multi-terminal vertex separator problem. Given a graph G=(V,E) and k terminal nodes, this problem consists in finding a subset SV of non-terminal nodes of minimum weight, the deletion of which generates k components, each one contains exactly one terminal. This problem is also known as the vertex multi-terminal cut problem, and it is at least as hard as the edge version [15]. In the literature, different studies have been carried out considering this problem [[24], [15], [25]]

  • The vertex multicut problem. This problem is the vertex version of the multicut problem [11]. It consists in, given a graph G=(V,E), a set of s terminal-pairs and an integer k, finding a set of at most k nodes, the deletion of which disconnects the nodes in each terminal-pair. This problem was defined on two versions depending on whether the removal of terminal nodes is allowed (the restricted version) or not (the unrestricted version). The problem is NP-hard on bounded-degree trees [26], and different results were obtained for some classes of graphs, namely split, co-bipartite and permutation graphs [27], trees and complete graphs [27], interval graphs [28] and bounded treewidth graphs [[26], [28]]. Moreover the fixed-parameter complexity of the problem was explored in [29].

  • The minimum Vertex Cover problem (MVC). Given a graph G=(V,E), the MVC problem consists in finding a minimum set of nodes AV such that G[VA] is an independent set. Thus, the MVC can be stated as follows: find the minimum subset of nodes, the deletion of which results in a set of connected components of one node each, at most. This problem is a classical NP-complete problem in graph theory, and it is one of Karp’s 21 problems. It has been extensively studied in the literature [30].

All theseproblems, including the CNDP, are variants of the well-known class of problems called Node-deletion problems [[31], [32]], and many results arisen from studying these problems exist in the literature. But only recently the CNDP was reconsidered by Arulselvan et al. (2009) [33] through a detailed study of the first variant called CNP, for Critical Node Problem. Since then the problem has received much attention, and several variants have been developed, with many intensive studies have been carried out to deal with each one. The purpose of this paper is to give a structured overview of recent results that mostly appeared subsequent to the work of Arulselvan et al. (2009) [33].

Identifying critical nodes is an efficient way to analyze and apprehend the properties, structures, and functions of networks. Indeed, this facilitates network control whether the objective is to keep or to destroy it, since these nodes are those which maintain its cohesiveness and the removal of which significantly degrades its connectivity. Therefore, identifying critical nodes is of prime importance and has many applications in several domains, including computational biology [[34], [35]], network vulnerability assessment [[36], [37], [38], [39]], network immunization [[33], [40], [41]], etc. Section 7 gives more details about possible applications of the CNDP in different areas.

The three major factors that make the CNDP an important parameter are the following:

  • Almost all network applications are usually designed to be run in a connected environment. The CNDP is the problem that determines the nodes whose preserving provides such an environment, and the removal of which disrupts it. Thus, finding these nodes is very useful for studying applications before and after design.

  • The CNDP is a double-edged parameter. In fact, it can be used for offensive or defensive purposes, depending on the objective of the application at hand. For instance, on a computer network, critical nodes are those mainly immunized against virus attacks (defensive), or those primarily targeted to destroy an opponent network (offensive). Then, once identified, critical nodes may be the focus of protection and defensive monitoring for positive application purposes, or the focus of attacks and offensive attempts for negative purposes.

  • The CNDP is useful for one of the most interesting issues, namely application robustness and security analysis. Indeed, the design of network applications (such as routing protocols) is often based on the selection of a kernel set of nodes, such as the maximum independent set or dominating set, and hence application security is proportional to the criticality of the selected kernel. Based on the hypothesis that the more critical nodes we have in the kernel, the more the application is vulnerable, and conversely, the fewer critical nodes we have, the more the application is robust, the CNDP is worth taking into account when designing secured network applications.

As partitioning networks can be done by deleting nodes or links, and as the link-deletion based problems have been extensively studied in the literature, we may wonder about the importance of studying problems based on node deletion. It should be noted that there are many situations where removing nodes makes more sense than removing links. This is the case, for example, when dealing with a virus spreading in the society, where we aim at stopping its propagation. In fact, nodes in the social network are people and links are social relationships, and since removing connections between individuals is usually difficult and may not be possible, we can instead vaccinate some individuals against contaminations in such a way that the spreading will be inhibited (here, the vaccination of a node is equivalent to removing it from the network).

The specific motivations behind developing such a review are summarized below. First, the importance of the problem, as it is a fundamental problem that has major applications in many areas: hence creation of a reference material with key results is essential. Second, the large number of results arising from the studies tackling this problem: thus it is extremely useful to summarize and classify these results in the same work.

To the best of our knowledge, this is the first survey conducted on the Critical Node Detection problem, and we are aware of any prior work. However, as this problem has become quite popular and has been studied in different fields including graph theory, network analysis, network vulnerability assessment, etc., leading to a variety of publications, we do not pretend to be able to give a comprehensive survey of all methods in all fields, but rather review recent theoretical results that mostly published further to the work of Arulselvan et al. (2009) [33]. Moreover, we are primarily focused on the combinatorial aspect of the problem. In other words, our main objective is to review, classify and discuss several recent advances and results obtained in the literature for each variant, including theoretical complexity, exact algorithms, approximation schemes and heuristic algorithms, as well as deduce and prove new results by establishing relationships between different variants.

The rest of the paper is structured as follows. We start with providing the basic concepts used in this survey in Section 2. In Section 3, we give a general formulation for the CNDP and study its complexity on general graphs. Section 4 presents a detailed discussion of the general greedy approach for solving the problem. In Section 5, we classify different variants of the CNDP into two main classes, and then detail each variant by reviewing different complexity analyses and solving results presented in the literature. In Section 6, we conduct a discussion about different approaches presented for solving the CNDP and thus present some ideas for future considerations. Different applications of the problem in several areas are reviewed in Section 7. The paper ends with a conclusion.

Section snippets

Definitions and notations

In this section, we introduce the necessary terminology used in the rest of the survey. We provide some of the basic definitions of graph theory, computational complexity and approximation algorithms. Readers not familiar with these topics are invited to refer to [42].

Identifying critical nodes in networks

In this section, we first introduce a general formulation for the CNDP taking into account different variants of the problem. We then discuss its complexity.

Solving CNDP

To solve hard problems, one of the most popular and effective approaches is to use greedy algorithms. These algorithms iteratively make the locally optimal choice with the hope of finding a global optimum. They often yield good solutions within a reasonable time. In what follows, we present a general greedy algorithm for solving the CNDP.

Variants of CNDP

In this section, we present the variants of the CNDP, those that we rate as among the top fundamental variants of this problem. First, we begin by classifying them into two main classes, before reviewing, for each one, the recent important combinatorial results, as well as the algorithmic aspects proposed for solving it in different graph classes, if any.

Discussion and future research directions

From this review, we can easily note that the CNDP is generally NP-complete and that almost all its variants remain NP-complete even on particular graph classes. Also, proofs of NP-completeness have been thoroughly detailed. However, the development of solving algorithms, which is even more important, has attracted less attention, except for the first variant, namely the CNP. We can also note that the development of improved ILP formulations has drawn a large attention, where a variety of ILP

Applications of CNDP

The CNDP has many real-world applications in a number of fields, including network risk management [75], network vulnerability assessment [[36], [63], [37]], social network analysis [[134], [135]], biological molecule studies [[34], [35]] and network immunization [[33], [136]], network communication [[137], [138], [139]]. In this section, we present a variety of applications, considered in the literature, for which critical nodes of the corresponding network have been identified to tackle the

Conclusion

This survey deals with one of the most important issues, namely the problem of finding critical nodes in networks. Critical nodes are those the deletion of which disconnects the network according to some predefined connectivity metrics. This problem has attracted much attention in recent years, and many variants have been defined in the literature depending on the connectivity metric to optimize once the nodes have been removed. The problem is generally NP-complete even on some particular graph

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