Percolation of a general network of networks

Abstract

Percolation theory is an approach to study the vulnerability of a system. We develop an analytical framework and analyze the percolation properties of a network composed of interdependent networks (NetONet). Typically, percolation of a single network shows that the damage in the network due to a failure is a continuous function of the size of the failure, i.e., the fraction of failed nodes. In sharp contrast, in NetONet, due to the cascading failures, the percolation transition may be discontinuous and even a single node failure may lead to an abrupt collapse of the system. We demonstrate our general framework for a NetONet composed of n classic Erdős-Rényi (ER) networks, where each network depends on the same number m of other networks, i.e., for a random regular network (RR) formed of interdependent ER networks. The dependency between nodes of different networks is taken as one-to-one correspondence, i.e., a node in one network can depend only on one node in the other network (no-feedback condition). In contrast to a treelike NetONet in which the size of the largest connected cluster (mutual component) depends on n, the loops in the RR NetONet cause the largest connected cluster to depend only on m and the topology of each network but not on n. We also analyzed the extremely vulnerable feedback condition of coupling, where the coupling between nodes of different networks is not one-to-one correspondence. In the case of NetONet formed of ER networks, percolation only exhibits two phases, a second order phase transition and collapse, and no first order percolation transition regime is found in the case of the no-feedback condition. In the case of NetONet composed of RR networks, there exists a first order phase transition when the coupling strength q (fraction of interdependency links) is large and a second order phase transition when q is small. Our insight on the resilience of coupled networks might help in designing robust interdependent systems.

Critical tipping point distinguishing two types of transitions in modular network structures

“…In particular, we consider the case of scale-free networks with average intra-and interdegree k intra , k inter , respectively, generated by [44] …”

Robustness of a partially interdependent network formed of clustered networks

“…Here we consider two cases of NON [29][30][31] composed of n interdependent networks, (i) A star-like NON and (ii) a random regular NON (see Fig. 4).…”

“…The final giant component of each network can be express as ψ ∞,i = x i g i (x i ) and the unknowns x i can be found from a system of n equations [29][30][31],…”

Mutually connected component of networks of networks with replica nodes

“…For example, if we want to understand the robustness of critical infrastructures [5] it is necessary to characterize the complex interdependencies between them, or if we aim at characterizing the function of a cell, we need to scale up the analysis of single cellular networks such as a protein interaction network and a metabolic network and study also interactions between different cellular networks. Critical phenomena in a network of networks and multilayer structures [3,4,6] show surprising new features [5,[7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In particular it has been recently shown [5,[7][8][9]] that when we consider several interdependent networks, the system as a whole might be much more fragile than single networks taken in isolation, and that the interdependencies between different networks can trigger cascading failure events of dramatic impact on the networks of networks.…”