Critical nodes for distance‐based connectivity and related problems in graphs

Abstract

This study considers a class of critical node detection problems that involves minimization of a distancebased connectivity measure of a given unweighted graph via the removal of a subset of nodes (referred to as critical nodes) subject to a budgetary constraint. The distance-based connectivity measure of a graph is assumed to be a function of the actual pairwise distances between nodes in the remaining graph (e.g., graph efficiency, Harary index, characteristic path length, residual closeness) rather than simply whether nodes are connected or not, a typical assumption in the literature. We derive linear integer programming (IP) formulations, along with additional enhancements, aimed at improving the performance of standard solvers. For handling larger instances, we develop an effective exact algorithm that iteratively solves a series of simpler IPs to obtain an optimal solution for the original problem. The edge-weighted generalization is also considered, which results in some interesting implications for distance-based clique relaxations, namely, s-clubs. Finally, we conduct extensive computational experiments with real-world and randomly generated network instances under various settings that reveal interesting insights and demonstrate the advantages and limitations of the proposed approach. In particular, one important conclusion of our work is that vulnerability of real-world networks to targeted attacks can be significantly more pronounced than what can be estimated by centrality-based heuristic methods commonly used in the literature.

Combined Heuristic Attack Strategy on Complex Networks

“…Another example is cascading failures (blackouts) of electric power transmission network, where the goal is to prevent a total breakdown of power network by inhibiting some power transmission points or lines. Similar approach involves, for example, immunization of disease carriers [24,25] or critical node detection [26][27][28], where the sequence of deletion order is not taken into account, and only the optimum subset of nodes selected for deletion is important. Lately, a more computationally demanding approach to network disintegration and attack, using stochastic and evolutionary optimization, is applied on smaller scale networks, providing better results than traditional approaches [29][30][31][32].…”

A survey of strategies for communication networks to protect against large-scale natural disasters

“…For a given graph with associated node costs and a given cost budget C, Veremyev et al [9] consider a set of critical nodes as a subset of all nodes whose total cost is not higher than C and whose removal maximally degrades the connectivity of the graph. In [9], the degradation aim is the maximization of a distance-based connectivity metric, which takes into account not only the pairwise connectivity but also the shortest path distance penalties between node pairs that remain connected.…”

“…In [9], the degradation aim is the maximization of a distance-based connectivity metric, which takes into account not only the pairwise connectivity but also the shortest path distance penalties between node pairs that remain connected. The paper proposes a general ILP model (that can be adapted to the different distance-based metrics by proper parameter definition) and an alternative exact algorithm that iteratively solves a series of simpler ILP models.…”

Correction to: Finding a maximum k-club using the k-clique formulation and canonical hypercube cuts

“…This observation is a reminder that even smaller instances of this NP-hard problem can be extremely challenging to solve using this approach. Although the original instance is easy for DBC to solve, when we consider the subgraph induced by N 3 G [5], the difficulty level is dramatically different. We verified this observation independently with the DBC code by adding the constraint x 5 = 1 to the original instance that solved in under 1 s, which also resulted in suboptimal termination after an hour.…”

“…of the integer programming (IP) formulations, referred to as F1 [4] and F2 [5], serve as our comparison. All four approaches (DBC, ITDBC, F1, and F2) are implemented in C++, and Gurobi TM Optimizer 7.0.2 [2] is used to solve the IP formulations.…”