Swarming on Random Graphs II

Abstract

We consider an individual-based model where agents interact over a random network via first-order dynamics that involve both attraction and repulsion. In the case of all-to-all coupling of agents in \(\mathbb {R}^{d}\) 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 sharpen and extend a line of work initiated in [56], which studies the behavior of this model when the interaction between the \(N\) agents occurs according to an Erdős–Renyi random graph \(\mathcal {G}(N,p)\) instead of all-to-all coupling. In particular, we study the effect 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 \(Np = O ( \log N )\). 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.