Large Deviation Principles for Critical Transitions in Adaptive Ecological Networks

<div xmlns="http://www.tei-c.org/ns/1.0"> We develop a large deviations framework for a class of coupled stochastic differential systems evolving on adaptive networks, motivated by modeling critical transitions in ecological systems. Each network node represents a local dynamical subsystem, while the edges evolve in time according to an interaction-driven adaptive rewiring mechanism. We introduce a novel class of jointly evolving graphdiffusion processes, where the network structure and the node dynamics co-evolve under feedback, capturing ecological adaptation, extinction cascades, and restoration processes. Our main result is a uniform sample path large deviation principle for the empirical process of the joint state-structure evolution under a scaling regime where both the diffusion noise and the rate of topological rewiring vanish simultaneously. The associated quasipotential landscape exhibits nontrivial phase geometry, encoding the hierarchy of metastable ecological configurations and their basin-crossing dynamics. We show that classical Freidlin-Wentzell theory fails to capture transitions in such adaptive systems, and we introduce a new graph-aware action functional defined over the Skorokhod space of càdlàg network trajectories. As an application, we characterize rare ecological regime shifts-such as biodiversity collapse or restoration tipping-in terms of the minimal-cost adaptive paths connecting invariant graph-measure configurations. Our results provide a new mathematical foundation for the probabilistic analysis of resilience and restoration in adaptive complex systems. </div>