Interdependent Networks: Reducing the Coupling Strength Leads to a Change from a First to Second Order Percolation Transition

Abstract

We study a system composed from two interdependent networks A and B, where a fraction of the nodes in network A depends on the nodes of network B and a fraction of the nodes in network B depends on the nodes of network A. Due to the coupling between the networks when nodes in one network fail they cause dependent nodes in the other network to also fail. This invokes an iterative cascade of failures in both networks. When a critical fraction of nodes fail the iterative process results in a percolation phase transition that completely fragments both networks. We show both analytically and numerically that reducing the coupling between the networks leads to a change from a first order percolation phase transition to a second order percolation transition at a critical point. The scaling of the percolation order parameter near the critical point is characterized by the critical exponent β = 1.Most of the research on networks has concentrated on the limited case of a single network [1][2][3][4][5] while real world systems are composed from many interdependent networks that interact with one another [6][7][8]. As a real example , consider a power-network and an Internet communication network that are coupled together. The Internet nodes depend on the power stations for electricity while the power stations depend on the Internet for control [9].We show that introducing interactions between networks is analogous to introducing interactions among molecules in the ideal gas model. Interactions among molecules lead to the replacement of the ideal gas law by the Van der Waals equation that predicts a liquid-gas first order phase transitions line ending at a critical point characterized by a second order transition ( Fig.1(a)). Similarly, interactions between networks give rise to a first order percolation phase transition line that changes to a second order transition, as the coupling strength between the networks is reduced (Fig.1(b)). At the critical point the first order line merges with the second order line, near which the order parameter (the size of giant component) scales linearly with the distance to the critical point, leading to the critical exponent β = 1.In interdependent networks, nodes from one network depend on nodes from another network. Consequently, when nodes from one network fail they cause nodes from another network to also fail. If the connections within each network are different, this may trigger a recursive process of a cascade of failures that can completely fragments both networks. Recently, Buldyrev et al [10] studied the coupling between two N node networks A and B assuming the following restrictions: (i) Each and every node in network A depends on one node from network B and vice versa. FIG. 1: (a)The van der Waals phase diagram. Along the liquid-gas equilibrium line the order parameter (density) abruptly changes from a low value in the gas phase to a high value in the liquid phase. At the critical point(Pc, Tc) the order parameter changes continuously as function of temperature if the pressur...

Networks formed from interdependent networks

“…66. It was shown 76 that, when the dependence coupling between the networks is reduced, at a critical coupling strength the percolation transition becomes second order.…”

“…In all of the above studies 73,[76][77][78] , the dependent pairs of nodes in both networks were chosen randomly. Thus when highdegree nodes in one network depend with a high probability on low-degree nodes of another network the configuration becomes vulnerable.…”

“…A generalization of the percolation theory of two fully interdependent networks 73 has been developed by Parshani et al 76 , where a more realistic case of a pair of partially interdependent networks has been studied. In this case, both interacting networks have a certain fraction of completely autonomous nodes whose function does not directly depend on the nodes of the other network.…”

“…76. This framework consists of two networks A and B with the numbers of nodes N A and N B , respectively.…”

“…Here we review the generalization of the theory of a pair of interdependent networks 73,76 to a system of n interacting networks 96 , which can be graphically represented (Fig. 5) approach for percolation of an NON system composed of n fully or partially interdependent randomly connected networks.…”