Absorbing states of zero-temperature Glauber dynamics in random networks

TL;DR

This paper investigates the properties and distribution of absorbing states in zero-temperature Glauber dynamics on random networks, revealing two main peaks in the distribution and their dependence on network parameters, with implications for opinion models.

Contribution

It provides a detailed analysis of the probability distribution of absorbing states in finite systems, highlighting the existence of two distinct peaks and their behavior as system size and average degree vary.

Findings

Distribution of absorbing states has two peaks influenced by average degree.

One peak is near the ground state, the other farther away, persisting as system size grows.

Implications for opinion dynamics models are discussed.

Abstract

We study zero-temperature Glauber dynamics for Ising-like spin variable models in quenched random networks with random zero-magnetization initial conditions. In particular, we focus on the absorbing states of finite systems. While it has quite often been observed that Glauber dynamics lets the system be stuck into an absorbing state distinct from its ground state in the thermodynamic limit, very little is known about the likelihood of each absorbing state. In order to explore the variety of absorbing states, we investigate the probability distribution profile of the active link density after saturation as the system size $N$ and $<k >$ vary. As a result, we find that the distribution of absorbing states can be split into two self-averaging peaks whose positions are determined by $<k>$, one slightly above the ground state and the other farther away. Moreover, we suggest that the latter peak accounts for a non-vanishing portion of samples when $N$ goes to infinity while $<k >$ stays fixed. Finally, we discuss the possible implications of our results on opinion dynamics models.