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Tractable description of hydrodynamic limits of a class of interacting jump processes on sparse graphs

Juniper Cocomello, Michel Davydov, Kavita Ramanan

TL;DR

This work derives a tractable hydrodynamic description for a broad class of interacting pure-jump processes on sparse graphs by exploiting local-field limits on unimodular Galton–Watson trees. The authors establish a time-m marginal 2-MRF structure and a Markovian projection of the local-field dynamics, yielding an autonomous Markov local-field process governed by a finite system of ODEs for the root-neighborhood law $ \mathfrak{p}_t $. They prove well-posedness of the ODE and show that, for configuration-model graphs, the empirical neighborhood distribution converges to the corresponding limit on the UGW tree, providing principled approximations for complex dynamics such as seizure propagation and nonlinear voter-type interactions. The framework unifies local convergence, conditional independence, and Markovian mimicking to produce scalable, accurate descriptions for sparse networks with non-pairwise interactions, with potential applications in neuroscience, epidemiology, and networked systems. The paper also illustrates practical implementations via simulations, highlighting improved tractability over full trajectory-based analyses while preserving essential space-time marginals.

Abstract

We consider dynamics of the empirical measure of vertex neighborhood states of Markov interacting jump processes on sparse random graphs, in a suitable asymptotic limit as the graph size goes to infinity. Under the assumption of a certain acyclic structure on single-particle transitions, we provide a tractable autonomous description of the evolution of this hydrodynamic limit in terms of a finite coupled system of ordinary differential equations. Key ingredients of the proof include a characterization of the hydrodynamic limit of the neighborhood empirical measure in terms of a certain local-field equation, well-posedness of its Markovian projection, and a Markov random field property of the time-marginals, which may be of independent interest. We also show how our results lead to principled approximations for classes of interacting jump processes and illustrate its efficacy via simulations on several examples, including an idealized model of seizure spread in the brain.

Tractable description of hydrodynamic limits of a class of interacting jump processes on sparse graphs

TL;DR

This work derives a tractable hydrodynamic description for a broad class of interacting pure-jump processes on sparse graphs by exploiting local-field limits on unimodular Galton–Watson trees. The authors establish a time-m marginal 2-MRF structure and a Markovian projection of the local-field dynamics, yielding an autonomous Markov local-field process governed by a finite system of ODEs for the root-neighborhood law . They prove well-posedness of the ODE and show that, for configuration-model graphs, the empirical neighborhood distribution converges to the corresponding limit on the UGW tree, providing principled approximations for complex dynamics such as seizure propagation and nonlinear voter-type interactions. The framework unifies local convergence, conditional independence, and Markovian mimicking to produce scalable, accurate descriptions for sparse networks with non-pairwise interactions, with potential applications in neuroscience, epidemiology, and networked systems. The paper also illustrates practical implementations via simulations, highlighting improved tractability over full trajectory-based analyses while preserving essential space-time marginals.

Abstract

We consider dynamics of the empirical measure of vertex neighborhood states of Markov interacting jump processes on sparse random graphs, in a suitable asymptotic limit as the graph size goes to infinity. Under the assumption of a certain acyclic structure on single-particle transitions, we provide a tractable autonomous description of the evolution of this hydrodynamic limit in terms of a finite coupled system of ordinary differential equations. Key ingredients of the proof include a characterization of the hydrodynamic limit of the neighborhood empirical measure in terms of a certain local-field equation, well-posedness of its Markovian projection, and a Markov random field property of the time-marginals, which may be of independent interest. We also show how our results lead to principled approximations for classes of interacting jump processes and illustrate its efficacy via simulations on several examples, including an idealized model of seizure spread in the brain.

Paper Structure

This paper contains 33 sections, 22 theorems, 114 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Interacting particle systems on graphs
  3. Comparison with alternative approaches
  4. Organization of the article
  5. Notation
  6. Main result
  7. Model description and basic assumptions
  8. A class of IPS
  9. Main result: tractable description of space-time marginal dynamics
  10. A limit theorem formulation of the main result
  11. Ramifications and simulations
  12. Algorithm
  13. Examples
  14. Seizure propagation dynamics
  15. The entrenched majority voter model
  16. ...and 18 more sections

Key Result

Proposition 2.3

Under Assumptions Ass:A_1, Ass:B_1 and Ass:A_2, the SDE eq:IPS_SDE is strongly well-posed, in the sense that there exists at least one weak solution, and the SDE is pathwise unique.

Figures (3)

  • Figure 1: State space representation for some example dynamics
  • Figure 2: Fraction of individuals in states S and I in the seizure propagation model from Section \ref{['subsubsec:neural_SIR']} on the random 3-regular graph with the number of vertices equal to $n=50$ (a), $200$ (b), $400$ (c). We compare our ODE (dashed lines) with simulations (dotted lines) averaged over 500 runs.
  • Figure 3: Fraction of individuals in each state of the entrenched majority voter model from Section \ref{['subsubsec:voter_model']} on the random $2-$regular graph with 200 vertices. We compare our ODE, simulations (500 realizations), and the mean-field approximation.

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2: Semi-global Markov Random Field
  • Proposition 2.3: GanRam2024*Theorem 4.3
  • Definition 2.4: Transition Graph
  • Remark 2.5
  • Proposition 2.6
  • Corollary 2.7
  • Theorem 2.8
  • Remark 2.9
  • Definition 2.10: Configuration model
  • ...and 48 more