Interdependent networks with identical degrees of mutually dependent nodes

Sergey V. Buldyrev, Nathaniel W. Shere, and Gabriel A. Cwilich

Phys. Rev. E 83, 016112 – Published 27 January 2011

Abstract

We study a problem of failure of two interdependent networks in the case of identical 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 (CCNs). 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 that belong to the mutual giant component remain functional. We assume that initially a $1 - p$ fraction of nodes are randomly removed because of an attack or failure and find analytically, for an arbitrary $P (k)$ , the fraction of nodes $μ (p)$ that belong to the mutual giant component. We find that the system undergoes a percolation transition at a certain fraction $p = p_{c}$ , which is always smaller than $p_{c}$ 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.

Received 16 September 2010

DOI:https://doi.org/10.1103/PhysRevE.83.016112

Authors & Affiliations

Sergey V. Buldyrev, Nathaniel W. Shere, and Gabriel A. Cwilich

Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA

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Vol. 83, Iss. 1 — January 2011

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Figure 1
Graphical solution of Eq. (36) for various special cases of CCNs. (a) ER networks with average degree $〈 k 〉 = 3$ . One can see that the monotonically increasing curves representing the right-hand side of Eq. (36) for different $p$ have zero slopes at $t = 1$ . The relevant solutions for $t$ are given by the lower intersection points of these curves and a straight line $y = t$ representing the left-hand side of Eq. (36). For $p = 1$ , this solution $t = 0.0602$ is indicated by a vertical straight line. The intersection of this vertical line with the plot of Eq. (35) (dot-dash line) gives the mutual giant component $μ = 0.931$ . The critical $p = p_{c} = 0.649 9451$ corresponds to a sudden disappearance of the nontrivial solution. (b) RR networks with $〈 k 〉 = 3$ . Note that for $p = 1$ the nontrivial solution is $t = 0$ , which means that $μ = 1$ . The value of $p_{c} = 0.758 751$ is grater than the $p_{c}$ for ER networks with the same average degree shown in panel (a). (c) Analogous plot for SF networks with $λ = 2.5$ . It shows that the slope of the curves is infinite for $t \to 1$ . One can see that in this case the nontrivial solution exists for any $p > 0$ . However, as $p \to 0$ , the nontrivial solution $t \to 1$ , and, accordingly, $μ \to 0$ indicating the second-order transition at $p = p_{c} = 0$ . (d) The marginal case of $λ = 3$ . The slopes of the curves for $t \to 1$ are finite. This means that there is a critical $p = p_{c} > 0$ at which the slope of the curve becomes equal to 1 at $t \to 1$ . For the displayed case of $k_{\min} = 1$ , Eq. (44) yields $p_{c} = 0.593 284 56$ . The nontrivial solution smoothly approaches 1 as $p \to p_{c}$ . This again implies $μ \to 0$ (second-order transition).Reuse & Permissions
Figure 2
Graphical solution of Eq. (37) for ER networks with different degree $〈 k 〉$ illustrating the method of finding $p_{c}$ . The bold dashed curve corresponds to $〈 k 〉 = 3$ studied in Fig. 1a. As $〈 k 〉$ decreases below 1.706, the nontrivial solution corresponding to $p ⩽ 1$ disappears. We also show the behavior of the analogous equation (45) for $〈 k 〉 = 1.706$ for RCNs. In agreement with proposition (3) this curve is always below the curve with the same average degree for CCNs studied here.Reuse & Permissions
Figure 3
The values of $p_{c}$ versus $〈 k 〉$ for several degree distributions of increasing broadness, namely RR, ER, uniform, and SF with $λ = 3$ . We define the uniform distribution as follows: $P (k) = 1 / (2 〈 k 〉 + 1)$ for $k = 0, 1, \dots, 2 〈 k 〉$ and $P (k) = 0$ for $k > 2 〈 k 〉$ . For SF distribution we use Eqs. (43) and (44), while for other distributions we numerically solve Eq. (38) and use Eq. (37) to find $p_{c}$ . One can see that $p_{c}$ decreases(and hence the robustness increases) with the broadness.Reuse & Permissions

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