We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e., a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erdős–Rényi graphs with edge probability pn, n is the number of vertices, such that lim n→∞pnn= ∞. The purpose of this note is twofold: (1) to establish this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker–Planck PDE (or, equivalently, by a nonlinear diffusion process) in the n= ∞ limit; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with n large but finite, for example the values of n that can be reached in simulations or that correspond to the typical number of interacting units in a biological system.
A Note on Dynamical Models on Random Graphs and Fokker–Planck Equations
Giacomin G.;
2016
Abstract
We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e., a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erdős–Rényi graphs with edge probability pn, n is the number of vertices, such that lim n→∞pnn= ∞. The purpose of this note is twofold: (1) to establish this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker–Planck PDE (or, equivalently, by a nonlinear diffusion process) in the n= ∞ limit; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with n large but finite, for example the values of n that can be reached in simulations or that correspond to the typical number of interacting units in a biological system.
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