TITLE

Swarming on Random Graphs

AUTHOR(S)
Brecht, James; Kolokolnikov, Theodore; Bertozzi, Andrea; Sun, Hui
PUB. DATE
April 2013
SOURCE
Journal of Statistical Physics;Apr2013, Vol. 151 Issue 1/2, p150
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
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 $\mathcal{G}(N,p)$. 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 $p_{c}=O(\frac{\log N}{N})$. 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.
ACCESSION #
86449479

 

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