A Two-Time-Scale Approximation of the Dynamics Toward Consensus Among Agents in Diffusive Interactions

AbstractIn this paper we deal with the dynamics toward the consensus of a family of integrators in diffusive interaction. The diffusive process is assumed to be anisotropic so that system matrix is a non-symmetric Laplacian. Under certain assumptions that limit the degree of asymmetry, we will be able to construct a special decomposition of the Laplacian in the form of a symmetric and a non-symmetric component. We show that the symmetric part is responsible of fast dynamics which evolve orthogonally to the equilibrium manifold. The slow dynamics, instead, are proven to deploy in the direction of the equilibrium manifold.