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.  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 pressure is kept constant at the critical value, but its derivative (compressibility) diverges. This is a characteristic 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 increases the stability of network A increases.) Below the critical 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 transition where the giant component continuously approaches zero. fail, the system undergoes a first order phase transition due to the recursive process of cascading failures. However, when examining the features of real interdependent networks such as the power network and the communication network presented above, we observe that in practice not all nodes of network A depend on network B and vice versa. We therefore introduce a general model
Parshani et al. REF studied the case where only a fraction of the nodes in both networks depend on each other, that is, some nodes in each network are not connected to the other network.
Interdependent networks: Reducing the coupling strength leads to a change from a first to second order percolation transition
Physical review letters
Physical review letters
Physics Medicine