Aging in coevolving voter models

TL;DR

This study investigates how aging affects coevolving voter models by introducing two aging mechanisms: Link Aging Model (LAM) and Node Aging Model (NAM), parameterized by aging exponents $\alpha_l$ and $\alpha_n$. Analyzing interevent-time distributions, the authors show aging induces non-Markovian dynamics and slows competing processes—link rewiring in LAM and state copying in NAM—altering the absorbing phase transition at critical plasticity $p_c$. Numerically, aging shifts the transition in opposite directions ($p_c$ increases for LAM and decreases for NAM) and, for $\alpha_l$ or $\alpha_n\ge1$, can erase the generic transition, yielding solely active or frozen phases depending on the model. The findings hold across random regular and Erdős–Rényi networks, highlighting memory effects in coevolving dynamics and suggesting avenues for studying aging in a broader class of coevolutionary systems.

Abstract

Aging, understood as the tendency to remain in a given state the longer the persistence time in that state, plays a crucial role in the dynamics of complex systems. In this paper, we explore the influence of aging on coevolution models, that is, models in which the dynamics of the states of the nodes in a complex network is coupled to the dynamics of the structure of the network. In particular we consider the coevolving voter model, and we introduce two versions of this model that include aging effects: the Link Aging Model (LAM) and the Node Aging Model (NAM). In the LAM, aging is associated with the persistence time of a link in the evolving network, while in the NAM, aging is associated with the persistence time of a node in a given state. We show that aging significantly affects the absorbing phase transition of the coevolution voter model, shifting the transition point in opposite directions for the LAM and NAM. We also show that the generic absorbing phase transition can disappear due to aging effects.

Figures (7)

• Figure 1: A schematic of a general coevolving voter model with aging is shown. The model consists of two key processes: copying and rewiring, with probabilities determined by the node age, $T_i$, and the link age, $\tau_{ij}$.
• Figure 2: Average interevent time $\langle t \rangle$ as a function of $\alpha$ is depicted, based on Eq. (5).
• Figure 3: LAM:(a) The absolute value of magnetization $|m|$, (b) the size of the largest component $S$, (c) the fraction of active links in survival runs $\rho$ as a function of $p$ for various $\alpha_l$ for the link aging model, and NAM: (d) $|m|$, (e) $S$, (f) $\rho$ for various $\alpha_n$ for the node aging model are shown. We use random regular networks with the average degree $z=4$ and $N=10^4$.
• Figure 4: The density of active links $\rho(t)$ as a function of time for (a) LAM with $\alpha_l=1$ and (b) NAM with $\alpha_n=1$ with various values of $p$. The insets show $\rho(t)$ in a log scale for (a) LAM with $p=1$ and (b) NAM with $p=0$.
• Figure 5: Phase diagram for (a) LAM and (b) NAM with active and frozen phases as functions of network plasticity $p$ and aging parameter $\alpha$ for random regular networks with $z=4$ (solid) and $8$ (dotted) and $N=10^4$. The inset in Fig. (a) shows the time $t_{max}$ to reach absorbing state for the LAM with $\alpha_l=0.4$ and $z=4$.