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Interacting Particle Systems on Networks: joint inference of the network and the interaction kernel

Quanjun Lang, Xiong Wang, Fei Lu, Mauro Maggioni

TL;DR

This work tackles joint learning of network structure and interaction rules in interacting particle systems on graphs, formulating a nonconvex problem over the network $\mathbf{a}$ and the kernel coefficients $c$ of a parametric kernel $\Phi$. It introduces two scalable algorithms, ALS and ORALS, and establishes identifiability and stability through coercivity conditions and an exploration measure, with ORALS providing asymptotic normality results. Numerically, ALS excels in small-to-moderate sample regimes while ORALS proves reliable in large-sample settings, and both yield accurate trajectory predictions when data cover the interaction space well. The framework is demonstrated on Kuramoto-type networks, leader–follower dynamics, and multi-type kernel scenarios, highlighting practical applicability to network discovery, kernel learning, and model selection in complex multi-agent systems.

Abstract

Modeling multi-agent systems on networks is a fundamental challenge in a wide variety of disciplines. We jointly infer the weight matrix of the network and the interaction kernel, which determine respectively which agents interact with which others and the rules of such interactions from data consisting of multiple trajectories. The estimator we propose leads naturally to a non-convex optimization problem, and we investigate two approaches for its solution: one is based on the alternating least squares (ALS) algorithm; another is based on a new algorithm named operator regression with alternating least squares (ORALS). Both algorithms are scalable to large ensembles of data trajectories. We establish coercivity conditions guaranteeing identifiability and well-posedness. The ALS algorithm appears statistically efficient and robust even in the small data regime but lacks performance and convergence guarantees. The ORALS estimator is consistent and asymptotically normal under a coercivity condition. We conduct several numerical experiments ranging from Kuramoto particle systems on networks to opinion dynamics in leader-follower models.

Interacting Particle Systems on Networks: joint inference of the network and the interaction kernel

TL;DR

This work tackles joint learning of network structure and interaction rules in interacting particle systems on graphs, formulating a nonconvex problem over the network and the kernel coefficients of a parametric kernel . It introduces two scalable algorithms, ALS and ORALS, and establishes identifiability and stability through coercivity conditions and an exploration measure, with ORALS providing asymptotic normality results. Numerically, ALS excels in small-to-moderate sample regimes while ORALS proves reliable in large-sample settings, and both yield accurate trajectory predictions when data cover the interaction space well. The framework is demonstrated on Kuramoto-type networks, leader–follower dynamics, and multi-type kernel scenarios, highlighting practical applicability to network discovery, kernel learning, and model selection in complex multi-agent systems.

Abstract

Modeling multi-agent systems on networks is a fundamental challenge in a wide variety of disciplines. We jointly infer the weight matrix of the network and the interaction kernel, which determine respectively which agents interact with which others and the rules of such interactions from data consisting of multiple trajectories. The estimator we propose leads naturally to a non-convex optimization problem, and we investigate two approaches for its solution: one is based on the alternating least squares (ALS) algorithm; another is based on a new algorithm named operator regression with alternating least squares (ORALS). Both algorithms are scalable to large ensembles of data trajectories. We establish coercivity conditions guaranteeing identifiability and well-posedness. The ALS algorithm appears statistically efficient and robust even in the small data regime but lacks performance and convergence guarantees. The ORALS estimator is consistent and asymptotically normal under a coercivity condition. We conduct several numerical experiments ranging from Kuramoto particle systems on networks to opinion dynamics in leader-follower models.
Paper Structure (50 sections, 11 theorems, 140 equations, 17 figures, 5 tables, 3 algorithms)

This paper contains 50 sections, 11 theorems, 140 equations, 17 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Problem statement.
  4. Proposed estimator: scalable algorithms, identifiability, and convergence
  5. Extensions
  6. Construction and analysis of the estimator, via ALS and ORALS
  7. Two algorithms: ALS and ORALS
  8. Alternating Least Squares (ALS)
  9. Operator Regression and Alternating Least Squares (ORALS)
  10. Theoretical guarantees
  11. Exploration measure
  12. Two coercivity conditions
  13. Coercivity and invertibility of normal matrices
  14. Convergence and asymptotic normality of the ORALS estimator
  15. Trajectory prediction
  16. ...and 35 more sections

Key Result

Proposition 2.3

Let the true parameters be $\mathbf{a}^{*}\in \mathcal{M}$ and $\Phi_* \in \mathcal{H}\backslash \{0\}\subset L^2_\rho$. Assume the rank-2 joint coercivity condition holds with $c_{\mathcal{H}}>0$. Then, we have the identifiability, namely, $(\mathbf{a}^{*},\Phi_*)$ is the unique solution to $\mathc

Figures (17)

  • Figure 1: The coercivity conditions: connections with RIP conditions, identifiability, and well-conditionedness of ALS and ORALS algorithms.
  • Figure 2: Top: a typical weight matrix estimation. The first two columns show the true graph and its weight matrix. The two columns on the right show the entry-wise errors of the ALS and ORALS estimators. Bottom: Estimator of interaction kernel and trajectory prediction. The left column shows a true trajectory. The middle two columns show the true and estimated kernels with a zoom-in to show the details in a rectangular region. The fourth column presents the true (the same as in column 1) and predicted trajectories. Note that $X_3$ and $X_2$ do not converge to the same cluster, in both the true and estimated trajectory, even though they are close at time 0, since there are no edges between them in the graph.
  • Figure 3: Convergence with sample size M increasing in 100 independent experiment runs. The top row shows almost perfect rates of $M^{-1/2}$ for both algorithms for the case of noiseless data and a well-specified basis. For the case of noisy data and misspecified basis, the bottom row shows robust convergence with the errors decaying until they reach $10^{-4}$, the variance of observation noise.
  • Figure 4: Top: Estimation errors as a function of $M$ (with all other parameters fixed), for both ALS and ORALS, for a random Fourier interaction kernel with $p=16$, $N=32$, $L=2$ (left) and $L=8$ (right). In the small and medium sample regime, between the two vertical bars, ALS significantly and consistently outperforms ORALS; for large sample sizes, the two estimators have similar performance. Bottom: The performance of the ALS estimator improves not only as $M$ increases but also as $L$ increases.
  • Figure 5: The first column shows the true weight matrix $\mathbf{a}$ and a trajectory of the system with an interesting clustering pattern. In the remaining columns, we show the estimators of the interaction function with misspecified and well-specified hypothesis spaces, i.e., $\phi\notin\mathcal{H}$ (top row) and $\phi\in\mathcal{H}_\phi$ (bottom row) respectively, with M ranging in $[8, 64, 521]$. Our estimators appear robust to basis misspecification, albeit with performance worse than in the well-specified case.
  • ...and 12 more figures

Theorems & Definitions (18)

  • Definition 2.1: Exploration measure
  • Definition 2.2: Joint coercivity conditions
  • Proposition 2.3: Rank-2 Joint coercivity implies identifiability
  • Definition 2.4: Interaction kernel coercivity condition
  • Proposition 2.6
  • Theorem 2.7
  • Proposition 2.8: Trajectory prediction error
  • Proposition A.1: Interaction kernel coercivity implies joint coercivity
  • Remark A.2: Sufficient but not necessary for identifiability
  • Lemma A.3
  • ...and 8 more