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 Erdös-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.
Highlights
The results of this paper were inspired by a concrete problem
Define the Erdos-Rényi random graph Gn = G(n, p(n)) as the random graph with vertex set [n] := {1, . . . , n} where two vertices are adjacent with probability p(n), independently of all other pairs
We consider the stochastic Kuramoto model [16] over each realization of the graph Gn, which is defined as a system of interacting diffusions indexed
Summary
The results of this paper were inspired by a concrete problem. Let n ∈ N and 0 < p(n) ≤ 1. Write i ∼(n) j if i, j ∈ [n] are adjacent and let d(in) denote the degree of i in Gn. We consider the stochastic Kuramoto model [16] over each realization of the graph Gn, which is defined as a system of interacting diffusions indexed. Interacting diffusions on sparse graphs by i ∈ [n], solutions of the following system of Itô Stochastic Differential Equations (SDEs) in time interval [0, T ]: dθi(n)(t) =. We explain how we performed the numerical simulations for the stochastic Kuramoto model on Galton-Watson (GW) trees. We assume that {Wtj }t≥0 are independent brownian motions for each node j, while the noise intensity is given by ε > 0. We chose two different models for generating the GW trees: 2. We chose two different models for generating the GW trees: 2. D-Regular model: The root node has C children, while the other ones have exactly C − 1 children.
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