Inverse Percolation to Quantify Robustness in Multiplex Networks

Abstract

Inverse percolation is known as the problem of finding the minimum set of nodes whose elimination of their links causes the rupture of the network. Inverse percolation has been widely used in various studies of single-layer networks. However, the use and generalization of multiplex networks have been little considered. In this work, we propose a methodology based on inverse percolation to quantify the robustness of multiplex networks. Specifically, we present a modified version of the mathematical model for the multiplex-vertex separator problem (m-VSP). By solving the m-VSP, we can find nodes that cause the rupture of the mutually connected giant component (MCGC) and the large viable cluster (LVC) when their links are removed from the network. The methodology presented in this work was tested in a set of benchmark networks, and as case study, we present an analysis using a set of multiplex social networks modeled with information about the main characteristics of the best universities in the world and the universities in Mexico. The results show that the methodology presented in this work can work in different models and types of 2- and 3-layer multiplex networks without dividing the entire multiplex network into single-layer as some techniques described in the specific literature. Furthermore, thanks to the fact that the technique does not require the calculation of some structural measure or centrality metric, and it is easy to scale for networks of different sizes.

Identification of COVID-19 Spreaders Using Multiplex Networks Approach

“…On the other hand, the VSP [53] consists of finding a minimum set of C nodes that, when their links are removed from the network, produce a disconnection from the multiplex network into at least two connected components (A, B), such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$|A|$ \end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$| B|$ \end{document} are maximized. Then, the adaptation of VSP can be summarized as: Instance: A MCGC of a multiplex network \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\text {GP}=(\mathcal {G_{\alpha }}, \mathcal {C})$ \end{document} .…”

“…Finally, there are some optimization methods to solve the network percolation problem [45] , [52] , and recently, Montes-Orozco et al [53] proposed a methodology based on inverse percolation that causes a rupture of the GC in monoplex and multiplex networks.…”

Analysis and Characterization of the Spread of COVID‐19 in Mexico through Complex Networks and Optimization Approaches

“…Based on the previous idea, Montes-Orozco et al [22] present a methodology based on the percolation process and asortality, where certain nodes are eliminated (which have certain special characteristics) to generate disconnection from the network, with which robustness can be quantified. It is known that the elements that belong to a complex system play different roles; therefore, identifying the elements (nodes) that are more influential is very important in real-world issues, since these nodes determine the entire system's behavior.…”

“…As we mentioned above, we present a case study analyzing the spread of COVID-19 for 2,457 municipalities in Mexico. For these networks, each layer (network) of the coupled network represents a different month, where a node represents each municipality, and the links between the nodes are created using the Euclidean distance between the characteristics to be analyzed (for more information about the process of modeling of networks, see [22,27,29,30]), where mainly we have the following:…”

“…Specifically, for this work, we used a simulated annealing (SA) algorithm [34] developed in C language (for more information, see [30]).…”