Large Deviations of Mean-Field Jump-Markov Processes on Structured Sparse Disordered Graphs

We prove a Large Deviation Principle for {\color{blue} jump-Markov } Processes on sparse large disordered network with disordered connectivity. The network is embedded in a geometric space, with the probability of a connection a (scaled) function of the spatial positions of the nodes. This type of model has numerous applications, including neuroscience, epidemiology and social networks. We prove that the rate function (that indicates the asymptotic likelihood of state transitions) is the same as for a network with all-to-all connectivity. We apply our results to a stochastic $SIS$ epidemiological model on a disordered networks, and determine Euler-Lagrange equations that dictate the most likely transition path between different states of the network.

💡 Research Summary

The manuscript establishes a rigorous Large Deviation Principle (LDP) for high‑dimensional jump‑Markov processes evolving on sparse, disordered graphs whose vertices are embedded in a compact geometric space. The authors assume that each vertex carries a deterministic spatial coordinate on a smooth Riemannian manifold and that the edge weights converge, in the large‑N limit, to a continuous kernel J(x,y). This “graphon” structure guarantees that the typical degree of each vertex diverges as N→∞, reproducing the mean‑field regime despite the underlying sparsity.

The stochastic dynamics are defined by Poissonian state transitions. For a node j currently in state α, the instantaneous jump rate to state β is a function fβ(xj,α,wj(t)), where wj(t) aggregates the states of neighboring nodes weighted by the kernel J. The empirical occupation measure ν̂N(t) and the empirical reaction flux μ̂Nα→β, which counts scaled numbers of α→β transitions, are shown to converge to deterministic limits described by coupled integro‑differential equations. The limiting density q_t(α,θ) satisfies a master‑type equation with mean‑field interaction terms w_t(θ)=∫J(θ,y)q_t(·,y)κ(dy). The fluxes satisfy dμ_tα→β(x,t)=fβ(x,α,w_t(x))q_t(α,x)κ(dx)dt.

The core contribution is the explicit rate functional G(μ) governing the exponential decay of probabilities of deviations of the empirical fluxes from their deterministic limit. For a flux μ possessing a density pα→β(x,t) with respect to κ⊗dt, the functional reads

 G(μ)=∑_{(α,β)∈Ξ}∫_E∫_0^∞ ℓ(pα→β(x,t), λα→β(x,t)) λα→β(x,t) dt κ(dx),

where ℓ(a,b)=a log(a/b)−a+b is the standard relative entropy for Poisson processes and λα→β(x,t)=fβ(x,α,w(x,t)) q_t(α,x) is the expected transition rate under the mean‑field limit. G is lower semicontinuous, has compact level sets, and yields the usual upper and lower bounds for closed and open sets in the space of flux measures, establishing the LDP. Notably, the functional does not depend on the microscopic edge configuration; it coincides with the rate function for an all‑to‑all (fully connected) network, demonstrating a universality of large‑deviation behavior under the graphon assumption.

To illustrate the theory, the authors apply the framework to a stochastic SIS epidemic model on the same class of graphs. The state space Γ={S,I} leads to transition rates f_I(x,S,w)=β w (infection) and f_S(x,I,w)=γ (recovery). The deterministic limit reproduces the familiar spatially extended mean‑field SIS equations, while the rate functional provides the cost of atypical epidemic trajectories. By solving the associated Euler–Lagrange equations, the most probable path for a rare outbreak (transition from low to high infection prevalence) is obtained. The optimal path depends explicitly on the spatial kernel J, revealing how network geometry shapes the shape and speed of epidemic waves even in a sparse setting.

Methodologically, the paper bridges three strands of literature: (i) large‑deviation theory for Poisson random measures, (ii) mean‑field limits on graphons, and (iii) stochastic neural‑field or epidemic models with spatially structured interactions. The authors’ assumptions are minimal: they require only that the empirical distribution of node locations converges weakly and that the edge weights satisfy uniform boundedness, a sparsity scaling φ_N, and a Lipschitz continuity of the limiting kernel. Lemma 1 shows that these conditions are satisfied by a broad class of random graph models, including power‑law, distance‑dependent, and small‑world graphs.

Overall, the work delivers a unified, mathematically rigorous description of rare events in high‑dimensional jump‑Markov systems on realistic sparse networks. The universality of the rate function suggests that, for a wide range of applications—neuroscience (spiking networks), epidemiology (structured populations), and social dynamics (opinion spread)—the probability of large deviations can be estimated using the same functional as in fully connected mean‑field models, provided the underlying graph converges to a graphon. This insight opens the door to quantitative risk assessment and optimal control strategies in complex spatial networks where exact simulation of rare events would otherwise be infeasible.