Optimal percolation on multiplex networks

Abstract

Optimal percolation is the problem of finding the minimal set of nodes whose removal from a network fragments the system into non-extensive disconnected clusters. The solution to this problem is important for strategies of immunization in disease spreading, and influence maximization in opinion dynamics. Optimal percolation has received considerable attention in the context of isolated networks. However, its generalization to multiplex networks has not yet been considered. Here we show that approximating the solution of the optimal percolation problem on a multiplex network with solutions valid for single-layer networks extracted from the multiplex may have serious consequences in the characterization of the true robustness of the system. We reach this conclusion by extending many of the methods for finding approximate solutions of the optimal percolation problem from single-layer to multiplex networks, and performing a systematic analysis on synthetic and real-world multiplex networks.

Controlling the uncertain response of real multiplex networks to random damage

“…Is it possible to safeguard some nodes so that the sufficient condition that the network will be in the functional state is met? Please note that the latter question is different from the one defined in optimal percolation, where the goal is to dismantle a system rather than preserving its cohesiveness [33,37].…”

“…In order to investigate whether the top-ranked nodes according to ∆s have some relation with the nodes identified in solutions to the optimal percolation problem, we performed systematic comparisons between the top 5 nodes according to ∆s and various methods used in optimal percolation [37]. We find that the top-ranked nodes correspond with good accuracy to the nodes in the optimal structural set detected by Simulated Annealing optimization [37]. High correlation (measured using Kendall τ ) is also found with sets determined using other state-ofthe-art techniques (see Table II).…”

“…In this section we describe a number of state of the art algorithms [37,38] to detect the optimal structural node set or to rank the nodes according to their likelihood to be found in the optimal structural node set. These algorithms include the duplex network version of the algorithms: Simulated Annealing (SA), High Degree (HD), High Degree Adaptive (HDA) and Collective Influence (Cl).…”

Asymmetry in interdependence makes a multilayer system more robust against cascading failures

“…It was found that an effective way to destroy the giant component of a single network, i.e., destruction of the 2-core, does not carry over to multilayer networks. The methods of effective multiplex degree [56] and optimal percolation [57] were articulated for multilayer networks to find the minimal damage set that destroys the largest mutually connected component.…”

Optimal shattering of complex networks

“…In [10] belief propagation is used to find efficient shattering sets. In [11] optimal attacks on multiplex networks are studied using simulated annealing methods.…”