The effect of spatiality on multiplex networks

Abstract

Many multiplex networks are embedded in space, with links more likely to exist between nearby nodes than distant nodes. For example, interdependent infrastructure networks can be represented as multiplex networks, where each layer has links among nearby nodes. Here, we model the effect of spatiality on the robustness of a multiplex network embedded in 2-dimensional space, where links in each layer are of variable but constrained length. Based on empirical measurements of real-world networks, we adopt exponentially distributed link lengths with characteristic length ζ. By changing ζ, we modulate the strength of the spatial embedding. When ζ → ∞, all link lengths are equally likely, and the spatiality does not affect the topology. However, when ζ → 0 only short links are allowed, and the topology is overwhelmingly determined by the spatial embedding. We find that, though longer links strengthen a single-layer network, they make a multi-layer network more vulnerable. We further find that when ζ is longer than a certain critical value, ζc, abrupt, discontinuous transitions take place, while for ζ < ζc the transition is continuous, indicating that the risk of abrupt collapse can be eliminated if the typical link length is shorter than ζc.

Multi-Universality and Localized Attacks in Spatially Embedded Networks

“…(1). This results in a mean-field-like behavior because the space is irrelevant, similar to d ≥ 6 [1]. However, for r > ζ the links are much shorter than the measurement scale and the underlying two-dimensional lattice structure determines the behavior.…”

“…Therefore, it is important to understand the spatial network topology discussed above in the case of interacting networks. Though random failures in multiplex networks of this topology have been studied [1], localized attacks-which occur frequently in real world due to natural disasters or terrorist attacks-have not been studied on multiplex embedded structures.…”

“…The normalized value of the data is 3.7 km (power) and 1.0 km (rail), and the characteristic length is 4.8 km (power) and 1.2 km (rail) if measured as the mean or 3.3 km (power) and 2.0 km (rail) if measured as the inverse slope of the fit. After [1].…”

“…Here we study a recently developed spatial embedded network model, whose aim is to give a better description of real-world space embedded networks and of critical phenomena which takes place in them [1]. We focus on two main aspects of this topology: (i) the unique percolation critical behavior in the single-layer case, and (ii) the vulnerability to localized attacks in the multiplex case.…”

“…Here ζ is a parameter determining the characteristic link length and thereby the strength of the embeddingthe smaller ζ is, the stronger the embedding is. Real network realizations supporting this model are shown in Fig.1 [1]. For the multiplex case, we generate two (or more) sets of links with the same set of nodes.…”