Mean field systems on networks, with singular interaction through hitting times

TL;DR

This paper studies advanced mean field particle systems with singular interactions on networks, revealing new phenomena like fragility times and how strategic connection adjustments prevent systemic failures.

Contribution

It introduces a general framework for particle systems with singular interactions, characterizes fragility times explicitly, and analyzes a strategic network game showing system regularization.

Findings

Characterization of times of fragility in terms of network and process dynamics.

Demonstration that strategic behavior prevents systemic fragility.

Development of mathematical tools including a fixed-point theorem and max-plus algebra application.

Abstract

Building on the line of work [DIRT15a], [DIRT15b], [NS17a], [DT17], [HLS18], [HS18] we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous connection structures, and (iii) analyze a game in which the particles determine their connections strategically. Hereby, we uncover two completely new phenomena. First, we characterize the "times of fragility" of such systems (e.g., the times when a macroscopic part of the population defaults or gets infected simultaneously, or when the neuron cells "synchronize") explicitly in terms of the dynamics of the driving processes, the current distribution of the particles' values, and the topology of the underlying network (represented by its Perron-Frobenius eigenvalue). Second, we use such systems to describe a dynamic credit-network game and show that, in equilibrium, the system regularizes: i.e., the times of fragility never occur, as the particles avoid them by adjusting their connections strategically. Two auxiliary mathematical results, useful in their own right, are uncovered during our investigation: a generalization of Schauder's fixed-point theorem for the Skorokhod space with the M1 topology, and the application of the max-plus algebra to the equilibrium version of the network flow problem.