Multiple Hybrid Phase Transition: Bootstrap Percolation on Complex Networks with Communities

Bootstrap percolation is a well-known model to study the spreading of rumors, new products or innovations on social networks. The empirical studies show that community structure is ubiquitous among various social networks. Thus, studying the bootstrap percolation on the complex networks with communities can bring us new and important insights of the spreading dynamics on social networks. It attracts a lot of scientists’ attentions recently. In this letter, we study the bootstrap percolation on Erd\H{o}s-R'{e}nyi networks with communities and observed second order, hybrid (both second and first order) and multiple hybrid phase transitions, which is rare in natural system. Moreover, we have analytically solved this system and obtained the phase diagram, which is further justified well by the corresponding simulations.

💡 Research Summary

The paper investigates bootstrap percolation on Erdős‑Rényi (ER) random graphs that contain two distinct communities, aiming to understand how community structure influences the emergence of phase transitions in spreading processes. In bootstrap percolation each node is either active (state 1) or inactive (state 0). Initially a fraction f_i of nodes in community i (i = 1, 2) are set active. At each discrete step any inactive node becomes permanently active if it has at least k active neighbours, regardless of whether those neighbours are in the same community or the other one. The dynamics proceeds until no further activations are possible, defining the final (equilibrium) state.

The authors first formulate the problem in terms of probabilities Z_{ij}, the chance that a node reached by following a randomly chosen edge from community i to community j is active in the final state. By enumerating all possible configurations of inner‑degree (edges to the same community) and outer‑degree (edges to the other community) and using the degree distributions P_i(i,j), they derive exact self‑consistency equations for Z_{ij}. For ER graphs the degree distribution is Poisson, which simplifies the expressions dramatically: Z_{11}=Z_{21}=S_1 and Z_{22}=Z_{12}=S_2, where S_i denotes the overall fraction of active nodes in community i at equilibrium.

The resulting coupled equations are
S_1 = f_1 + (1−f_1) ∑_{r≥k}