Asynchronous semi-anonymous dynamics over large-scale networks

TL;DR

This paper develops a mean-field framework to analyze asynchronous semi-anonymous dynamics on large-scale directed networks, approximating their evolution with differential equations, applicable to various graph structures including community and power-law networks.

Contribution

It introduces a general mean-field approach for ASD processes on large random networks, extending analysis to complex graph models like community and power-law networks.

Findings

The dynamics can be approximated by non-linear ODEs in large networks.

The framework applies to various graph ensembles, including power-law and community-structured networks.

Simulation confirms the framework's applicability to real social network data.

Abstract

We analyze a class of stochastic processes, referred to as asynchronous and semi-anonymous dynamics (ASD), over directed labeled random networks. These processes are a natural tool to describe general best-response and noisy best-response dynamics in network games where each agent, at random times governed by independent Poisson clocks, can choose among a finite set of actions. The payoff is determined by the relative popularity of different actions among neighbors, while being independent of the specific identities of neighbors. Using a mean-field approach, we prove that, under certain conditions on the network and initial node configuration, the evolution of ASD can be approximated, in the limit of large network sizes, by the solution of a system of non-linear ordinary differential equations. Our framework is very general and applies to a large class of graph ensembles for which the typical random graph locally behaves like a tree. In particular, we will focus on labeled configuration-model random graphs, a generalization of the traditional configuration model which allows different classes of nodes to be mixed together in the network, permitting us, for example, to incorporate a community structure in the system. Our analysis also applies to configuration-model graphs having a power-law degree distribution, an essential feature of many real systems. To demonstrate the power and flexibility of our framework, we consider several examples of dynamics belonging to our class of stochastic processes. Moreover, we illustrate by simulation the applicability of our analysis to realistic scenarios by running our example dynamics over a real social network graph.