Navigability of interconnected networks under random failures

Abstract

Assessing the navigability of interconnected networks (transporting information, people, or goods) under eventual random failures is of utmost importance to design and protect critical infrastructures. Random walks are a good proxy to determine this navigability, specifically the coverage time of random walks, which is a measure of the dynamical functionality of the network. Here, we introduce the theoretical tools required to describe random walks in interconnected networks accounting for structure and dynamics inherent to real systems. We develop an analytical approach for the covering time of random walks in interconnected networks and compare it with extensive Monte Carlo simulations. Generally speaking, interconnected networks are more resilient to random failures than their individual layers per se, and we are able to quantify this effect. As an application--which we illustrate by considering the public transport of London--we show how the efficiency in exploring the multiplex critically depends on layers' topology, interconnection strengths, and walk strategy. Our findings are corroborated by data-driven simulations, where the empirical distribution of check-ins and checksout is considered and passengers travel along fastest paths in a network affected by real disruptions. These findings are fundamental for further development of searching and navigability strategies in real interconnected systems.N etwork theory has been revealed to be a perfect instrument to model the structure of complex systems and the dynamical process in which they are involved. However, the classical approach does not take into account the possibility that agents can be networked in different ways, and with different intensity, on multiple layers simultaneously. As an example, in the case of social sciences the same user might choose to subscribe to two or more online social networks and to build different social relationships with different users on each social platform (e.g., LinkedIn for the network of professional contacts, Facebook for the network of friends, etc.). Another example is represented by transportation networks in a city: the network of bus stops, the first layer, is different from the tube network, the second layer, but people make use of both by combining paths to move from one place to another within the city. The cases where one vertex is not present in the full multiplex can be accounted for by including it as an isolated vertex in the layers where it is missing, without altering either topological or dynamical properties of the interconnected network.The existence of such multiple connections on different layers invites a generalization of the theory of complex networks to cope with multilayer interconnected networked systems. More specifically, very recent studies focused on a particular type of multilayer network, the multiplex, where each agent participates in different layers simultaneously, like our previous example in the case of online social networks. Indeed, the actors (vertices) in every layer are t...

Structural reducibility of multilayer networks

“…As an example, we have applied it also to social 17 and economical systems, coauthorship networks 36 , metropolitan transportation networks 24 and continental air transportation systems 20 (see Table 1). A particularly interesting case is that of the FAO (Food and Agriculture Organization of the United Nations) worldwide food import/export network, an economic network in which layers represent products, nodes are countries and edges at each layer represent import/export relationships of a specific food product among countries.…”

“…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] .…”

Exact coupling threshold for structural transition reveals diversified behaviors in interconnected networks

“…Interconnected networks are mathematical representations of systems where two or more simple networks, possibly with different functionalities, are coupled to each other. The omnipresence of interconnected networks has spurred a variety of research [4][5][6][7], with particular interest in dynamical processes such as percolation [8,9], epidemic spreading [10][11][12][13], and diffusion [14,15].…”

An Ensemble Perspective on Multi-layer Networks

“…As mentioned above diffusion dynamics on networks can be studied analytically using transition matrices of random walkers [8,9]. The multi-layer structure of a network can explicitly be incorporated in a random walk model by constructing a so-called supra-transition matrix [3,12] similar to the supra-adjacency matrix used in [9,10,11].…”

“…The multi-layer structure of a network can explicitly be incorporated in a random walk model by constructing a so-called supra-transition matrix [3,12] similar to the supra-adjacency matrix used in [9,10,11]. The supra-adjacency matrix of a multi-layer network G can be defined in a block-matrix structure as…”