Interdependent networks with identical degrees of mutually dependent nodes

Abstract

We study a problem of failure of two interdependent networks in the case of correlated degrees of mutually dependent nodes. We assume that both networks (A and B) have the same number of nodes N connected by the bidirectional dependency links establishing a one-to-one correspondence between the nodes of the two networks in a such a way that the mutually dependent nodes have the same number of connectivity links, i.e. their degrees coincide. This implies that both networks have the same degree distribution P (k). We call such networks correspondently coupled networks (CCN). We assume that the nodes in each network are randomly connected. We define the mutually connected clusters and the mutual giant component as in earlier works on randomly coupled interdependent networks and assume that only the nodes which belong to the mutual giant component remain functional. We assume that initially a 1 − p fraction of nodes are randomly removed due to an attack or failure and find analytically, for an arbitrary P (k), the fraction of nodes µ(p) which belong to the mutual giant component. We find that the system undergoes a percolation transition at certain fraction p = p c which is always smaller than the p c for randomly coupled networks with the same P (k). We also find that the system undergoes a first order transition at p c > 0 if P (k) has a finite second moment. For the case of scale free networks with 2 < λ ≤ 3, the transition becomes a second order transition. Moreover, if λ < 3 we find p c = 0 as in percolation of a single network. For λ = 3 we find an exact analytical expression for p c > 0. Finally, we find that the robustness of CCN increases with the broadness of their degree distribution.

Networks formed from interdependent networks

“…This explanation is corroborated by the analytical proof in ref. 82, which shows that if the degrees of the interdependent nodes coincide, then a network with a broader degree distribution will become more robust than a network with a narrower degree distribution, that is, the behaviour characteristic of non-interacting networks is restored. Ref.…”

“…The case in which all pairs of interdependent nodes in both networks have the same degree was solved analytically in ref. 82.…”

“…INSIGHT | PROGRESS ARTICLE there are three methods to achieve this goal: increase the fraction of autonomous nodes 76 , particularly nodes with high degree 95 ; design the dependence links such that they connect the nodes with similar degrees 79,82 and protect the high-degree nodes against attack 95 .…”

Network robustness of multiplex networks with interlayer degree correlations

“…The result for the MP case was also obtained earlier in Refs. [23,24]. The rescaled size of the giant mutual component, M/M (0) where M (0) is the size of the giant mutual component with f = 0, for the MP (MN) coupling is larger (smaller) than those for the others for any removal fraction f (Fig.…”

“…Recently, the effect of such interlayer degree correlation was addressed for connectivity of multiplex networks [12]. Furthermore, a few studies demonstrated that interdependent networks with higher interlayer degree correlation [23,24] or more assortative layers [25] are more robust under random * kgoh@korea.ac.kr damage. However, there is still lack of unified understanding of various robustness properties of multiplex networks due to the role of interlayer degree correlations.…”