Diffusion in large networks

TL;DR

This paper explores how diffusion processes behave in large, infinite networks, analyzing the effects of different mechanisms and aggregation functions on societal polarization and state evolution.

Contribution

It introduces a unified framework for diffusion in infinite networks, distinguishing between probabilistic and deterministic mechanisms and their impact on societal polarization.

Findings

Under strict aggregation, society evolves towards mixed or homogeneous states.

Boolean aggregation allows for polarization and deterministic diffusion.

Network structure influences irreducibility but not the core diffusion outcomes.

Abstract

We investigate the phenomenon of diffusion in a countably infinite society of individuals interacting with their neighbors in a network. At a given time, each individual is either active or inactive. The diffusion is driven by two characteristics: the network structure and the diffusion mechanism represented by an aggregation function. We distinguish between two diffusion mechanisms (probabilistic, deterministic) and focus on two types of aggregation functions (strict, Boolean). Under strict aggregation functions, polarization of the society cannot happen, and its state evolves towards a mixture of infinitely many active and infinitely many inactive agents, or towards a homogeneous society. Under Boolean aggregation functions, the diffusion process becomes deterministic and the contagion model of Morris (2000) becomes a particular case of our framework. Polarization can then happen. Our dynamics also allows for cycles in both cases. The network structure is not relevant for these questions, but is important for establishing irreducibility, at the price of a richness assumption: the network should contain infinitely many complex stars and have enough space for storing local configurations. Our model can be given a game-theoretic interpretation via a local coordination game, where each player would apply a best-response strategy in a random neighborhood.