On the stochastic robustness of complete clustering predictability for a first-order consensus model

We study stochastic noise response on the complete cluster predictability of a first-order stochastic consensus model on the real line with an odd, bounded, Lipschitz, and monotone coupling function. Emergence of multiclusters is one of the novel dynamical features of the deterministic consensus model, and it has been extensively studied in collective dynamics community. Recently, the second author and his collaborators proposed a complete cluster predictability of the first-order nonlinear consensus model that can be derived from the Cucker–Smale flocking model on the real line, and they verified that the proposed clustering algorithm explicitly determines the number of asymptotic clusters and their cluster velocities only in terms of the initial data, coupling strength and supremum of coupling function. In this work, we show that the same cluster algorithm does work for the complete clustering predictability problem when the deterministic consensus model is stochastically perturbed in a multiplicative way.