Propagation of chaos and large deviations in mean-field models with jumps on block-structured networks

TL;DR

This paper studies the behavior of large networks of interacting jump processes with complex block structures, establishing propagation of chaos, law of large numbers, and large deviation principles for the system's empirical measures.

Contribution

It introduces a multi-population mean-field model with block-structured interactions and proves propagation of chaos and large deviations in this setting.

Findings

Propagation of chaos established for multi-class jump processes.

Law of large numbers demonstrated in the block-structured network.

Large deviation principles derived for empirical measures.

Abstract

A system of interacting multiclass finite-state jump processes is analyzed. The model under consideration consists of a block-structured network with dynamically changing multi-colors nodes. The interaction is local and described through local empirical measures. Two levels of heterogeneity are considered: between and within the blocks where the nodes are labeled into two types. The central nodes are those connected only to the nodes of the same block whereas the peripheral nodes are connected to both the nodes of the same block and to some nodes from other blocks. The limits of such systems as the number of particles tends to infinity are investigated. Under regularity conditions on the peripheral nodes, propagation of chaos and law of large numbers are established in a multi-population setting. In particular, it is shown that, as the number of nodes goes to infinity, the behavior of the different classes of nodes can be represented by the solution of a McKean-Vlasov system. Moreover, we prove large deviation principles for the vectors of empirical measures and the empirical processes.