Swarming on Random Graphs

Abstract

We consider a compromise model in one dimension in which pairs of agents interact through first-order dynamics that involve both attraction and repulsion. In the case of all-to-all coupling of agents, this system has a lowest energy state in which half of the agents agree upon one value and the other half agree upon a different value. The purpose of this paper is to study the behavior of this compromise model when the interaction between the N agents occurs according to an Erdős-Rényi random graph G (N, p). We study the effect of changing p on the stability of the compromised state, and derive both rigorous and asymptotic results suggesting that the stability is preserved for probabilities greater than p c = O( log N N ). In other words, relatively few interactions are needed to preserve stability of the state. The results rely on basic probability arguments and the theory of eigenvalues of random matrices.

Swarming on Random Graphs II

“…Our interest lies in determining when the stability condition (4) holds for the more general linear system (6). Our main result is a proof of the following theorem given in section 3, which settles a conjecture originally posed in [49]. We then demonstrate in section 4 how to extend the analysis in order to prove similar results for higher dimensional simplex configurations.…”

“…Nevertheless, a standard technique easily adapts to the present situation and allows us to handle this lack of independence. If np  (1 ✏) log n then a previous result [49] already implies instability asymptotically almost surely. We therefore may as well assume that (1 ✏) log n  np  c 0 ()(1 ✏) log n, so that lemma 3.2 applies and there exists a c 1 > 0 su ciently small so that d ii c 1 np with probability at least c 0 n 1+✏/2 for any given diagonal entry.…”

“…Specifically, we proved theorems that yield stability or instability for two types of simplex equilibria as the number of particles becomes infinite. For the one-dimensional simplex equilibria we rigorously established a conjecture first formulated in [49], i.e. an explicit formula for the critical probability above which stability holds asymptotically almost surely.…”

“…Using a connectivity-based argument similar to that used in [49], we may conclude that stability asymptotically almost surely necessitates np c log n for some constant c > 0 as well. In other words, the stability threshold for the two-dimensional simplex configuration exhibits the same critical scaling np / log n as the one-dimensional case.…”

“…that the particles lie in equilibrium regardless of their interaction structure, necessarily reduces our study to the class of so-called simplex configurations. These simplex configurations generalize the one-dimensional simplex equilibrium or "compromise solution" studied in [49], so named due to its similarity with the classical consensus-type algorithms in control theory. This is the particular choice of equilibrium where equal numbers n = N/2 of particles occupy both vertices of the one dimensional simplex, so that x 1 = · · · = x n = 0 and x n+1 = · · · = x N = p 2R up to a reordering of the particle indices.…”