Interdependent networks: Reducing the coupling strength leads to a change from a first to second order percolation transition

arXiv:1004.3989v1 [physics.data-an] 22 Apr 2010 Interdependent networks: Reducing the coupling strength leads to a change from a first to second order percolation transition Roni Parshani,1 Sergey V. Buldyrev,2, 3 and Shlomo Havlin1 1Minerva Center & Department of Physics, Bar-Ilan University, Ramat Gan, Israel 2Center for Polymer Studies and Dept. of Physics, Boston Univ., Boston, MA 02215 USA 3Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA (Dated: October 22, 2018) 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 [1–5] while real world systems are composed from many interdependent net- works that interact with one another [6–8]. As a real example , consider a power-network and an Internet com- munication network that are coupled together. The In- ternet nodes depend on the power stations for electricity while the power stations depend on the Internet for con- trol [9]. We show that introducing interactions between net- works 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 be- tween 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 crit- ical 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 frag- ments both networks. Recently, Buldyrev et al [10] stud- ied 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. (ii) If node Ai depends on node Bi then node Bi depends on node Ai. They show that for such a model when a critical fraction of the nodes in one network 0 0.2 0.4 0.6 0.8 1 T/Tc (Reduced Temperature) 0 0.2 0.4 0.6 0.8 1 P/Pc (Reduced Pressure) First Order Critical Point Gas Liquid (a) 0.1 0.2 0.3 0.4 0.5 1-p (fraction of removed nodes in network A) 0 0.2 0.4 0.6 0.8 1 1-qA (fraction of independent nodes in network A) First Order Second Order Critical Point Giant component in network B=0 Giant component in network B >0 (b) 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 temper- ature if the pressure is kept constant at the critical value, but its derivative (compressibility) diverges. This is a characteris- tic of the second order phase transition. (b) The percolation phase transition for two interdependent networks as obtained from the numerical solution of system (7) for qB = 1 and a = b = 3. Here 1 −p, the fraction of removed nodes from network A, plays the role of temperature. (As 1−p increases, the disorder increases.) The fraction 1 −qA of independent nodes in network A plays the role of pressure. (As 1 −qA in- creases the stability of network A increases.) Below the criti- cal point, the system undergoes a first order phase transition at which, β∞, the fraction of nodes in the giant component of network B abruptly changes from a finite value to zero. As we approach the critical point, β∞→0. Above the critical point, the system undergoes a second order tran

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