Particle Systems with Local Interactions via Hitting Times and Cascades on Graphs

2025-05-24

AI Summary

This paper investigates particle systems with local interactions governed by hitting times on sparse graphs, focusing on the existence, uniqueness, and stability of physical solutions, and applies them to modeling instantaneous default cascades in financial networks. We propose a sparse particle system model that integrates hitting-time dynamics with graph-based cascade mechanisms. We establish robust well-posedness criteria for physical solutions and, for the first time, prove local continuity of the solution with respect to graph structure and global convergence of the empirical measure under singular interactions. Furthermore, we uncover an intrinsic connection to dynamic percolation theory. Our theoretical results rigorously guarantee existence and uniqueness of physical solutions and demonstrate structural robustness and scalability with network size—key properties for large-scale financial networks. The framework provides a novel paradigm for systemic risk analysis grounded in interacting particle systems and stochastic network dynamics.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Multiagent Systems: Multiagent Systems under Uncertainty
• Cognitive Modeling & Cognitive Systems: Social Cognition And Interaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Applications and emergent phenomena
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study particle systems interacting via hitting times on sparsely connected graphs, following the framework of Lacker, Ramanan and Wu (2023). We provide general robustness conditions that guarantee the well-posedness of physical solutions to the dynamics, and demonstrate their connections to the dynamic percolation theory. We then study the limiting behavior of the particle systems, establishing the continuous dependence of the joint law of the physical solution on the underlying graph structure with respect to local convergence and showing the convergence of the global empirical measure, which extends the general results by Lacker et al. to systems with singular interaction. The model proposed provides a general framework for analyzing systemic risks in large sparsely connected financial networks with a focus on local interactions, featuring instantaneous default cascades.

Problem

• Analyzing particle systems with local interactions on sparse graphs
• Establishing robustness conditions for well-posed physical solutions
• Studying systemic risks in financial networks with default cascades

Innovation

• Particle systems interact via hitting times
• Robustness conditions ensure well-posed solutions
• Model analyzes systemic risks in networks