Interacting diffusions on sparse graphs: hydrodynamics from local weak limits

2018/12/31 by Oliveira, Roberto I., Reis, Guilherme H., Stolerman, Lucas M. · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1812.11924

Abstract

We prove limit theorems for systems of interacting diffusions on sparse graphs. For example, we deduce a hydrodynamic limit and the propagation of chaos property for the stochastic Kuramoto model with interactions determined by Erdős-Rényi graphs with constant mean degree. The limiting object is related to a potentially infinite system of SDEs defined over a Galton-Watson tree. Our theorems apply more generally, when the sequence of graphs ("decorated" with edge and vertex parameters) converges in the local weak sense. Our main technical result is a locality estimate bounding the influence of far-away diffusions on one another. We also numerically explore the emergence of synchronization phenomena on Galton-Watson random trees, observing rich phase transitions from synchronized to desynchronized activity among nodes at different distances from the root.

Cited by

• Marginal dynamics of probabilistic cellular automata on trees

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• Local weak convergence for sparse networks of interacting processes
• Hydrodynamic Limits of non-Markovian Interacting Particle Systems on Sparse Graphs
• Weakly interacting particle systems on inhomogeneous random graphs
• Interacting particle systems on sparse W-random graphs
• Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees