Higher-order percolation in simplicial complexes

Many empirical systems display group interactions, that is, connections and relationships do not occur between pairs of nodes but instead are collective actions at the level of groups of nodes. Pairwise interactions are insufficient to characterize the dynamics process of real networks, such as epidemic spread, social contagion, or opinion formation. Conversely, the effect of higher-order interactions in networks has attracted extensive attention. Here we introduce a generalized theoretical model for describing higher-order networks with simplicial complexes, in which failure occurs through the synergistic effects of pairwise and higher-order interactions. In this model, removing one node causes all other nodes in the same 2-simplex to be removed. This process may happen recursively, leading to cascading processes. We develop an analytical framework for studying the robustness of simplicial complexes and give exact analytical solutions for giant components’ size and critical value. We find that when the number of triangles exceeds a fixed value, the simplicial complexes will become highly vulnerable, and phase transition undergoes a double transition. An initial phase in which a fraction of the simplicial complexes are removed discontinuously and a final phase in which the giant components disappear into simplicial complexes. Our theoretical method corresponds well with the Monte-Carlo simulation.