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**1 - 3**of**3**### Swarming on Random Graphs

, 2013

"... 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 oth ..."

Abstract
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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 pc = O ( logNN). 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

, 2013

"... 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 Rd this system has a lowest energy state in which an equal number of agents occupy the vertices of th ..."

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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 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. 1