[Paper Review] Large Deviations of Non-Stochastic Interacting Particles on Sparse Random Graphs

James MacLaurin|arXiv (Cornell University)|Oct 27, 2020

Complex Network Analysis Techniques37 references4 citations

TL;DR

This paper establishes a large deviations principle for non-stochastic interacting particle systems on sparse random graphs, where randomness arises solely from the graph topology and initial conditions. By introducing a nested empirical measure and using a push-forward argument via a continuous dynamics map, it proves that the large deviations rate function for the system's empirical measure is derived from the initial condition's rate function, valid even when the average degree grows arbitrarily slowly.

ABSTRACT

This paper concerns the large deviations of a system of interacting particles on a random graph. There is no stochasticity, and the only sources of disorder are the random graph connections, and the initial condition. The average number of afferent edges on any particular vertex must diverge to infinity as $N o \infty$, but can do so at an arbitrarily slow rate. These results are thus accurate for both sparse and dense random graphs. A particular application to sparse Erdos-Renyi graphs is provided. The theorem is proved by pushing forward a Large Deviation Principle for a `nested empirical measure' generated by the initial conditions to the dynamics. The nested empirical measure can be thought of as the density of the density of edge connections: the associated weak topology is more coarse than the topology generated by the graph cut norm, and thus there is a broader range of application.

Motivation & Objective

• To develop a large deviations framework for interacting particle systems without time-varying stochasticity, where disorder arises only from random graph structure and initial conditions.
• To extend large deviations theory to sparse random graphs, including those with arbitrarily slow-growing average degree.
• To address the challenge of non-ergodic or heterogeneous network topologies where standard graphon or cut-norm methods fail due to lack of regularity.
• To establish a rigorous connection between the large deviations of initial conditions and the resulting dynamics via a continuous push-forward map.
• To provide a theoretical foundation for understanding how disordered network structure alone can generate complex collective behaviors, such as neural avalanches or transient states.

Proposed method

• Introduce a 'nested empirical measure' that encodes the conditional distribution of afferent connections to each node, enabling a finer topology than the standard graph cut norm.
• Define a dynamics map Ψ that pushes forward the initial condition's empirical measure to the path-dynamics measure of the particle system.
• Prove the continuity of the push-forward map Ψ under a weak topology on the space of probability measures over continuous paths.
• Use the inverse contraction principle and Varadhan's contraction principle to transfer the large deviations principle from the initial condition space to the dynamical system space.
• Establish exponential tightness and compactness in the topology induced by the nested empirical measure to ensure convergence of approximating sequences.
• Leverage weak convergence techniques and construct a Cauchy sequence of approximating measures to define the limit of the push-forward map in the absence of uniform continuity.

Experimental results

Research questions

• RQ1 Can a large deviations principle be established for non-stochastic interacting particle systems on sparse random graphs where no time-dependent noise is present?
• RQ2 How does the large deviations behavior of the system depend on the initial condition and graph topology when the average degree diverges slowly?
• RQ3 What topological structure on the space of empirical measures is necessary to ensure continuity of the dynamics map in sparse graph settings?
• RQ4 To what extent can the graph disorder alone, without noise, generate rare but significant collective behaviors in particle systems?
• RQ5 Can the push-forward method be adapted to systems with heterogeneous connectivity, including nodes with O(1) connections, under a refined empirical measure topology?

Key findings

• The large deviations rate function for the empirical measure of the particle system is the push-forward of the rate function from the initial condition's empirical measure via a continuous dynamics map Ψ.
• The method applies to both sparse and dense random graphs, including Erdos-Renyi graphs, as long as the average number of incoming edges per node diverges with N, even at an arbitrarily slow rate.
• The use of a nested empirical measure allows for a coarser topology than the graph cut norm, enabling broader applicability despite the lack of regularity in sparse graphs.
• The dynamics map Ψ is continuous under the weak topology on path-space, which is essential for applying Varadhan's contraction principle.
• The proof establishes that the limit of the push-forward map exists and is unique, even when the graph structure is highly heterogeneous.
• The framework successfully captures rare events in systems driven purely by topological disorder, such as transient neural states or avalanches, without relying on intrinsic noise.