Network dismantling

Abstract

We study the network dismantling problem, which consists of determining a minimal set of vertices in which removal leaves the network broken into connected components of subextensive size. For a large class of random graphs, this problem is tightly connected to the decycling problem (the removal of vertices, leaving the graph acyclic). Exploiting this connection and recent works on epidemic spreading, we present precise predictions for the minimal size of a dismantling set in a large random graph with a prescribed (lighttailed) degree distribution. Building on the statistical mechanics perspective, we propose a three-stage Min-Sum algorithm for efficiently dismantling networks, including heavy-tailed ones for which the dismantling and decycling problems are not equivalent. We also provide additional insights into the dismantling problem, concluding that it is an intrinsically collective problem and that optimal dismantling sets cannot be viewed as a collection of individually well-performing nodes.graph fragmentation | message passing | percolation | random graphs | influence maximization A network (a graph G in the discrete mathematics language) is a set V of N entities called nodes (or vertices), along with a set E of edges connecting some pairs of nodes. In a simplified way, networks are used to describe numerous systems in very diverse fields, ranging from social sciences to information technology or biological systems (reviews are in refs. 1 and 2). Several crucial questions in the context of network studies concern the modifications of the properties of a graph when a subset S of its nodes is selected and treated in a specific way. For instance, how much does the size of the largest connected component of the graph decrease if the vertices in S (along with their adjacent edges) are removed? Do the cycles survive this removal? What is the outcome of the epidemic spreading if the vertices in S are initially contaminated, constituting the seed of the epidemic? On the contrary, what is the influence of a vaccination of nodes in S preventing them from transmitting the epidemic? It is relatively easy to answer these questions when the set S is chosen randomly, with each vertex being selected with some probability independently. Classical percolation theory is nothing but the study of the connected components of a graph in which some vertices have been removed in this way.A much more interesting case is when the set S can be chosen in some optimal way. Indeed, in all applications sketched above, it is reasonable to assign some cost to the inclusion of a vertex in S: vaccination has a socioeconomic price, incentives must be paid to customers to convince them to adopt a new product in a viral marketing campaign, and incapacitating a computer during a cyber attack requires resources. Thus, one faces a combinatorial optimization problem: the minimization of the cost of S under a constraint on its effect on the graph. These problems thus exhibit both static and dynamic features, the former referring to the combinatori...

Immunization and Targeted Destruction of Networks using Explosive Percolation

“…However, it is not suitable to keep a small SðqÞ below q c . This is due to the fact that below q c it leads to big jumps in SðqÞ when two large clusters join (similar jumps were seen in [13,34,35]). As a result of the merging process, many nodes (at the interface between the two clusters) suddenly become harmless without being treated as such.…”

“…Overall, it gives the smallest values of SðqÞ (although other strategies may locally perform better for specific q-values). Moreover, it gave in all cases by far the lowest values of q c compared to all other strategies, except for the very recent message passing algorithms of [34,35]. Following the mainstream in network studies, we focused on SðqÞ, which corresponds to outbreaks starting in GðqÞ.…”

Fast and simple decycling and dismantling of networks

“…These are two fundamental network-optimization problems with a wide range of applications, related to optimal vaccination and surveillance, information spreading, viral marketing, and identification of influential nodes. Considerable research efforts have been devoted to the network decycling and dismantling problems recently12345678.…”

“…Both the decycling and the dismantling problems belong to the class of NP-hard problems69, meaning that it is rather hopeless to look for algorithms to solve them exactly in polynomial time. However, finding the best possible approximate solutions for as large classes of networks as possible is an open and actively investigated direction.…”

Optimal deployment of resources for maximizing impact in spreading processes

“…It is quite natural that algorithms based exclusively on topological characteristics appear to have variable performance depending on particular network instances and dynamical models used [23,24]. Another line of work consists in studying the NP-complete problem of network dismantling [25][26][27]: the underlying reasoning is that removal of nodes breaking the giant component to small pieces is likely to prevent the global percolation of the contagion. The localization of an optimal immunization set has been addressed using a belief propagation algorithm built on top of percolation-like equations for SIR (Susceptible, Infected, Recovered) and SIS (Susceptible, InarXiv:1608.08278v1 [cs.SI] 29 Aug 2016 2 fected, Survived) models [28], based on cavity method techniques developed previously for deterministic threshold models [29,30].…”