Parameter estimation in interacting particle systems on dynamic random networks

TL;DR

This work tackles parameter estimation for interacting particle systems on dynamic random graphs with one-way feedback, when only partial observations of the network are available through edge counts $S(t)$. A moments-based approach derives $\mathbb{E}[S(t)]$, $\mathbb{E}[S^2(t)]$, and $\mathbb{E}[(S(t+1)-S(t))^2]$ in terms of the parameters $(\pi_+,\pi_-,\alpha)$, enabling separate estimation of $\pi_+$ and $\pi_-$ from the first two moments and of $\alpha$ from the one-step increment moment. The estimators are proven to be consistent and asymptotically normal, with numerical experiments across opinion dynamics, communication networks, and neural systems demonstrating robustness to moderate noise and heterogeneity, while also highlighting identifiability considerations under symmetric vs asymmetric link functions. The framework provides a data-efficient tool for learning co-evolving network dynamics from partial observations and lays groundwork for extensions to fully coupled vertex–edge dynamics in complex adaptive systems.

Abstract

In this paper we consider a class of interacting particle systems on dynamic random networks, in which the joint dynamics of vertices and edges acts as one-way feedback, i.e., edges appear and disappear over time depending on the state of the two connected vertices, while the vertex dynamics does not depend on the edge process. Our goal is to estimate the underlying dynamics from partial information of the process, specifically from snapshots of the total number of edges present. We showcase the effectiveness of our inference method through various numerical results.

Figures (6)

• Figure 3.1: Example of transition matrix when $n=3$, $\alpha=0.3$, $\pi_+=0.9$, $\pi_-=0.4$ and $f$ as in \ref{['eq:fdef2']}.
• Figure 4.1: Estimation of the parameters $\pi_+$, $\pi_-$, $\alpha$ when $n = 10$, using the methodology described in Section \ref{['sec:methodology']}, with $f(\pi_+, \pi_-) = (\pi_+ + \pi_-)/2$.
• Figure 4.2: Estimation of the parameters $\pi_+$, $\pi_-$, $\alpha$ when $n = 10$, using the methodology described in Section \ref{['sec:methodology']}, with $f(\pi_+,\pi_-)=\pi_+\pi_-/(\pi_++\pi_-)$.
• Figure 4.3: Estimation of the parameters $\pi_+$, $\pi_-$, $\alpha$ when $n = 5$, using the methodology described in Section \ref{['sec:methodology']}, with $f(\pi_+,\pi_-)=\pi_+ \pi_- / (\pi_+ + \pi_-)$.
• Figure 4.4: Estimation of the parameters $\pi_+$, $\pi_-$, $\alpha$ when $n = 20$, using the methodology described in Section \ref{['sec:methodology']}, with $f(\pi_+,\pi_-)=\pi_+ \pi_- / (\pi_+ + \pi_-)$.