Large Deviations for Markovian Graphon Processes and Associated Dynamical Systems on Networks

TL;DR

This paper develops large deviation principles for Markovian graphon processes and their associated dynamical systems, providing explicit solutions for rate functions and analyzing their behavior over long time horizons.

Contribution

It introduces large deviation principles for time-evolving Markovian networks with graphon-based descriptions, extending static network results to dynamic settings.

Findings

Established laws of large numbers for graphon processes.

Derived large deviation principles with explicit rate functions.

Applied results to node-valent dynamical systems on evolving networks.

Abstract

We consider temporal models of rapidly changing Markovian networks modulated by time-evolving spatially dependent kernels that define rates for edge formation and dissolution. Alternatively, these can be viewed as Markovian networks with $O(1)$ jump rates viewed over a long time horizon. In the regimes we consider, the window averages of graphon valued processes over suitable time intervals are natural state descriptors for the system. Under appropriate conditions on the jump-rate kernels, we establish laws of large numbers and large deviation principles(LDP) for the graphon processes averaged over a suitable time window, both in the weak topology and with respect to the cut norm in the associated graphon space. Although the problem setting and analysis are more involved than for the well-studied static random network model, the variational problem associated with the rate function admits an explicit solution, yielding an equally tractable, though different, expression for the rate function, similar to the static case. Using these results, we then establish the LDP for node-valent dynamical systems driven by the underlying evolving network.