Swarming on Random Graphs II

We consider an individual-based model where agents interact over a random network via firstorder dynamics that involve both attraction and repulsion. In the case of all-to-all coupling of agents in Rd this system has a lowest energy state in which an equal number of agents occupy the vertices of the d-dimensional simplex. The purpose of this paper is to study the behavior of this model when the interaction between the N agents occurs according to an Erdős-Rényi random graph G(N, p) instead of all-to-all coupling. In particular, we study the e↵ect of randomness on the stability of these simplicial solutions, and provide rigorous results to demonstrate that stability of these solutions persists for probabilities greater than Np = O(logN). In other words, only a relatively small number of interactions are required to maintain stability of the state. The results rely on basic probability arguments together with spectral properties of random graphs.