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Large Deviations for Marked Sparse Random Graphs with Applications to Interacting Diffusions
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DOI：10.1007/s10955-025-03565-z.png)
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Abstract
En 中文
We consider the empirical neighborhood distribution of marked sparse Erd & odblac;s-R & eacute;nyi random graphs, obtained by decorating edges and vertices of a sparse Erd & odblac;s-R & eacute;nyi random graph with i.i.d. random elements taking values on Polish spaces. We prove that the empirical neighborhood distribution of this model satisfies a large deviation principle in the framework of local weak convergence. We rely on the concept of BC-entropy introduced by Delgosha and Anantharam (2019) which is inspired on the previous work by Bordenave and Caputo (2015). Our main technical contribution is an approximation result that allows one to pass from graph with marks in discrete spaces to marks in general Polish spaces. As an application of the results developed here, we prove a large deviation principle for interacting diffusions driven by gradient evolution and defined on top of sparse Erd & odblac;s-R & eacute;nyi random graphs. In particular, our results apply for the stochastic Kuramoto model. We obtain analogous results for the sparse uniform random graph with given number of edges.
Keywords:
large deviations
Local topology
Sparse random graphs
Interacting diffusions
Stochastic Kuramoto model
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