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

James MacLaurin

TL;DR

This work establishes a Large Deviation Principle for high-dimensional jump-Markov processes on sparse, disordered graphs embedded in a geometric space, showing that the LDP rate function coincides with the all-to-all connectivity case. It introduces empirical reaction fluxes and occupation measures, derives a Poisson-driven rate functional $\mathcal{G}$, and then contracts to an occupation-based rate $\mathcal{H}$ via the contraction principle, leveraging a graphon-type connectivity $\mathcal{J}$. The proof proceeds in two steps: (i) approximate the system with averaged interactions and prove an LDP for this averaged model, and (ii) transfer the LDP to the original model using Girsanov's theorem and projective-limit arguments. As an application, a stochastic SIS model on a disordered network is analyzed to obtain Euler-Lagrange equations for the most likely transition paths between metastable states, illustrating the practical computation of optimal trajectories in spatially extended Hawkes-type systems.

Abstract

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.

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

TL;DR

This work establishes a Large Deviation Principle for high-dimensional jump-Markov processes on sparse, disordered graphs embedded in a geometric space, showing that the LDP rate function coincides with the all-to-all connectivity case. It introduces empirical reaction fluxes and occupation measures, derives a Poisson-driven rate functional , and then contracts to an occupation-based rate via the contraction principle, leveraging a graphon-type connectivity . The proof proceeds in two steps: (i) approximate the system with averaged interactions and prove an LDP for this averaged model, and (ii) transfer the LDP to the original model using Girsanov's theorem and projective-limit arguments. As an application, a stochastic SIS model on a disordered network is analyzed to obtain Euler-Lagrange equations for the most likely transition paths between metastable states, illustrating the practical computation of optimal trajectories in spatially extended Hawkes-type systems.

Abstract

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 epidemiological model on a disordered networks, and determine Euler-Lagrange equations that dictate the most likely transition path between different states of the network.

Paper Structure

This paper contains 15 sections, 24 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Structure of the Paper
  3. Notation
  4. Model Outline and Main Result
  5. Geometry of the Connectivity
  6. Stochastic Transitions
  7. Main Results
  8. Contracted Rate Function
  9. Proofs
  10. Proof Outline
  11. System with Averaged Interactions
  12. Proof of the Large Deviations of the Averaged System
  13. An Application: Transition Paths For Hawkes Models in Epidemiology and Neuroscience
  14. Euler-Lagrange Equations for the Optimal Trajectory
  15. Acknowledgements

Key Result

Lemma 1

Suppose that $\lbrace J^{jk} \rbrace_{j,k \in I_N}$ assume values in $\lbrace -1, 0 , 1 \rbrace$ and that they have been sampled randomly from a distribution. Assume furthermore that they are either (i) mutually independent or (ii) independent, except that $J^{jk} = J^{kj}$. Suppose also that there Suppose that the scaling is such that for any positive constant $c >0$, Then Assumption hypothesis

Theorems & Definitions (39)

  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Corollary 5
  • proof
  • Lemma 6
  • Theorem 7
  • Lemma 8
  • proof
  • ...and 29 more