Statistical mechanics of multiplex networks: Entropy and overlap

Abstract

There is growing interest in multiplex networks where individual nodes take part in several layers of networks simultaneously. This is the case for example in social networks where each individual node has different kind of social ties or transportation systems where each location is connected to another location by different types of transport. Many of these multiplex are characterized by a significant overlap of the links in different layers. In this paper we introduce a statistical mechanics framework to describe multiplex ensembles. A multiplex is a system formed by N nodes and M layers of interactions where each node belongs to the M layers at the same time. Each layer α is formed by a network G α . Here we introduce the concept of correlated multiplex ensembles in which the existence of a link in one layer is correlated with the existence of a link in another layer. This implies that a typical multiplex of the ensemble can have a significant overlap of the links in the different layers. Moreover we characterize microcanonical and canonical multiplex ensembles satisfying respectively hard and soft constraints and we discuss how to construct multiplex in these ensembles. Finally we provide the expression for the entropy of these ensembles that can be useful to address different inference problems involving multiplexes.

Message passing theory for percolation models on multiplex networks with link overlap

“…The matrix element a [ whether node i is connected to node j in layer α (a [α] ij = 1) or not (a [α] ij = 0). In a multiplex network, we define the total overlap O [α,α ] [37] of the links in layer α and layer α as the total number of pairs of nodes connected both in layer α and layer α ; i.e.,…”

“…IV we report the message passing theory for calculating the MCGC in multiplex networks without link overlap, using the formalism of Ref. [37]. In Sec.…”

“…Numerous multilayer networks have a significant link overlap [4,7,9,37], which explains the need to explore the percolation transition on this type of correlated multilayer structures [38]. Recently, two approaches were used to describe the transition in duplex networks (i.e.,, networks formed by M = 2 layers) with link overlap.…”

Structural reducibility of multilayer networks

“…Similarly, in biological systems basic constituents such as proteins can have physical, co-localization, genetic or many other types of interactions. Recently, it has been shown that retaining such multi-dimensional information 7 and modelling the structure of interdependent and multilayer systems respectively through interdependent 8 and multilayer networks [9][10][11][12] reveals new nontrivial structural properties [13][14][15][16][17][18][19][20] and relevant emergent physical phenomena [21][22][23][24][25][26][27] .…”

Multi-Universality and Localized Attacks in Spatially Embedded Networks

“…Therefore, there is a need of modeling these systems as interdependent for understanding their structure, function and robustness [19][20][21][22][23][24][25][26]. Studies on spatially embedded interdependent networks found that in many cases they are significantly more vulnerable then non-embedded systems [27].…”