Percolation transitions in partially edge-coupled interdependent networks with different group size distributions

In many systems, from brain neural networks to epidemic transmission networks, pairwise interactions are insufficient to express complex relationships. Nodes sometimes cooperate and form groups to increase their robustness to risks, and each such group can be considered a “supernode†. Furthermore, previous studies of cascading failures in interdependent networks have typically concentrated on node coupling connections; however, in many realistic scenarios, interactions occur between the edges connecting nodes rather than between the nodes themselves. Networks of this type are called edge-coupled interdependent networks. To better reflect complex networks in the real world, in this paper, we construct a theoretical model of a two-layer partially edge-coupled interdependent network with groups, where all nodes in the same group are functionally dependent on each other. We identify several types of phase transitions, namely, discontinuous, hybrid and continuous, which are determined by the strength of the dependency and the distribution of the supernodes. We first apply our developed mathematical framework to ErdsRnyi and scale-free partially edge-coupled interdependent networks with equally sized groups to analytically and numerically calculate the phase transition thresholds and the critical dependency strengths that distinguish different types of transitions. We then investigate the influence of the group size distribution on cascading failures by presenting examples of two different heterogeneous group size distributions. Our theoretical predictions and numerical findings are in close agreement, demonstrating that decreasing the dependency strength and increasing group size heterogeneity can increase the robustness of interdependent networks. Our results have significant implications for the design and optimization of network security and fill a knowledge gap in the robustness of partially edge-coupled interdependent networks with different group size distributions.