[Paper Review] Joint effect of ageing and multilayer structure prevents ordering in the voter model

Oriol Artime, Juan Fernández-Gracia+2 more|LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas)|Mar 17, 2017

Opinion Dynamics and Social Influence5,607 citations

TL;DR

The study shows that ageing combined with a multilayer (multiplex) network can trap the ageing voter model in non-consensus states, with a critical multiplexity q* separating conditions that lead to global ordering from those that induce dynamical traps. The trapping is tied to spontaneous symmetry breaking across layers and persists in various topologies, depending on q and network structure.

ABSTRACT

The voter model rules are simple, with agents copying the state of a random neighbor, but they lead to non-trivial dynamics. Besides opinion processes, the model has also applications for catalysis and species competition. Inspired by the temporal inhomogeneities found in human interactions, one can introduce ageing in the agents: the probability to update decreases with the time elapsed since the last change. This modified dynamics induces an approach to consensus via coarsening in complex networks. Additionally, multilayer networks produce profound changes in the dynamics of models. In this work, we investigate how a multilayer structure affects the dynamics of an ageing voter model. The system is studied as a function of the fraction of nodes sharing states across layers (multiplexity parameter q ). We find that the dynamics of the system suffers a notable change at an intermediate value q*. Above it, the voter model always orders to an absorbing configuration. While, below, a fraction of the realizations falls into dynamical traps associated to a spontaneous symmetry breaking in which the majority opinion in the different layers takes opposite signs and that due to the ageing indefinitely delay the arrival at the absorbing state.

Motivation & Objective

• Investigate how ageing, where update probability decreases with time since last change, interacts with multilayer network structure in the voter model.
• Determine whether multilayer coupling alters the ordering process and leads to dynamical traps instead of global consensus.
• Identify the existence and value of a critical multiplexity q* separating trapped dynamics from guaranteed ordering.
• Characterize how network topology (complete graphs, Poissonian and scale-free networks) and interlayer connections influence outcomes.

Proposed method

• Extend the endogenous update voter model to two layers with layer-specific activation rates p_i^l = b_l/τ_i.
• Introduce multiplexity q as the fraction of nodes sharing states across layers (common nodes).
• Analyze dynamics under RAU and endogenous updates, with random layer update order per Monte Carlo step.
• Derive effective activation probability P_act for a node in a full two-layer multiplex and discuss its asymptotic behavior.
• Compute and interpret plateaus in the average fraction of active links 〈ρ(t)〉 and the inter-event time distributions C(Δt).
• Explore various topologies: complete graphs, Poissonian and scale-free random networks, with different interlayer connection schemes.

Experimental results

Research questions

• RQ1 Does ageing in a two-layer voter model produce different ordering dynamics compared to a monolayer?
• RQ2 Is there a finite multiplexity threshold q* above which the system always orders by coarsening and below which dynamical traps occur?
• RQ3 How do network topology and interlayer connection patterns affect the likelihood of trapping and the value of q*?
• RQ4 What is the role of common nodes in sustaining trapped configurations and how do plateaus in 〈ρ(t)〉 arise?
• RQ5 How do mixed update rules (RAU vs endogenous) interact in a multiplex setting to influence ordering?

Key findings

• A finite-q regime exists where dynamical traps occur, with a q* between 0.25 and 0.35 on the tested sizes; for q>q*, the system always orders by coarsening.
• Dynamical traps arise from spontaneous symmetry breaking across layers and are sustained by ageing and interlayer coupling through common nodes.
• Plateaus in the fraction of active links 〈ρ(t)〉 emerge in the trapped regime, with predicted values ρ_low ≈ 2/N, ρ_up ≈ 2q(1−q), and ρ_half ≈ q(1−q/2) in the large-N limit, matching simulations.
• Trapping probability n_trap/n_tot decreases with q and remains finite below q*, with q* decreasing as system size grows.
• The phenomenon persists across complete graphs and Poissonian networks, and is modulated by degree heterogeneity and interlayer connection schemes, especially when hubs are correlated across layers (high-high linking).
• With mixed updates (one layer RAU, one endogenous), the dynamics show distinct ordering behaviors depending on q and the update combination, including fast exponential-like decay when fully shared (q=1).