Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yucheng Guo, Qinxin Yan

Abstract

We introduce a family of particle systems on sparse graphs where local interactions occur via hitting times, providing a dynamic and tractable model for default cascades in large sparsely-connected financial networks. Building on the framework of Lacker, Ramanan and Wu (2023), we extend convergence theory to systems with singular interactions, capturing the abrupt and discontinuous nature of systemic events. We establish conditions for well-posedness through a minimality principle and connect fragility to dynamic percolation thresholds. Our analysis demonstrates continuity of the joint law of defaults with respect to local graph convergence, establishes convergence of empirical distributions, and characterizes the default time distribution in tree-like networks. This framework offers a rigorous and flexible foundation for modeling systemic risk in evolving financial systems, featuring continuous-time dynamics, heterogeneous and local interactions, and instantaneous default cascades.

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

Abstract

We introduce a family of particle systems on sparse graphs where local interactions occur via hitting times, providing a dynamic and tractable model for default cascades in large sparsely-connected financial networks. Building on the framework of Lacker, Ramanan and Wu (2023), we extend convergence theory to systems with singular interactions, capturing the abrupt and discontinuous nature of systemic events. We establish conditions for well-posedness through a minimality principle and connect fragility to dynamic percolation thresholds. Our analysis demonstrates continuity of the joint law of defaults with respect to local graph convergence, establishes convergence of empirical distributions, and characterizes the default time distribution in tree-like networks. This framework offers a rigorous and flexible foundation for modeling systemic risk in evolving financial systems, featuring continuous-time dynamics, heterogeneous and local interactions, and instantaneous default cascades.

Paper Structure

This paper contains 28 sections, 24 theorems, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related literature.
  3. Preliminaries and Notations
  4. Elementary notations
  5. Local convergence of marked graphs
  6. Directed graphs
  7. Rooted graphs, isomorphism and the space $\mathcal{G}_*$
  8. Marked graphs and the space $\mathcal{G}_*[\mathcal{Y}]$
  9. Compactness in $\mathcal{G}_*[\mathcal{Y}]$
  10. Graph convergence in the local weak sense
  11. Main Results and Proofs
  12. Physical and minimal solutions
  13. Fragility, robustness and default cascades
  14. Default trees and recovery of locality
  15. Convergence of physical solutions
  16. ...and 13 more sections

Key Result

Theorem 1.10

For each problem configuration $(G,c,x,Z)\in\mathcal{G}_*[\mathbb{R}\times\mathcal{D}]$ that drives $\delta$-robust systems, there exists a unique physical solution to the equations eq:1.1. Moreover, this solution coincides with the minimal solution.

Figures (1)

  • Figure :

Theorems & Definitions (62)

  • Remark 1.1
  • Example 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Example 1.6
  • Definition 1.7
  • Remark 1.9
  • Theorem 1.10: Well-Posedness
  • Theorem 1.11: Convergence of Physical Solutions
  • ...and 52 more