Nonlinear q-voter model with deadlocks on the Watts-Strogatz graph

We study the nonlinear $q$-voter model with deadlocks on a Watts-Strogats graph. Using Monte Carlo simulations, we obtain so called exit probability and exit time. We determine how network properties, such as randomness or density of links influence exit properties of a model.

💡 Research Summary

The paper investigates a special case of the nonlinear q‑voter model in which the stochastic “noise” parameter ε is set to zero, creating so‑called dead‑lock configurations that prevent any opinion change unless a unanimous q‑panel is present. The authors embed this model on Watts–Strogatz (WS) small‑world networks, which are characterized by two tunable parameters: the average degree k (the number of nearest neighbours each node has in the underlying regular lattice) and the rewiring probability β (the fraction of edges that are randomly rewired, controlling the amount of randomness). By varying k and β the study spans from a one‑dimensional ring (k = 1, β = 0) through increasingly dense regular lattices (larger k) to fully random graphs (β ≈ 1).

Monte‑Carlo simulations are used to measure two key observables as functions of the initial fraction p of +1 spins: (i) the exit probability E(p), i.e. the probability that the system finally reaches the absorbing state where all spins are +1, and (ii) the exit time T, the average number of elementary updates required to reach either of the two absorbing states. For a complete graph (k = N/2 − 1, β = 1) the dynamics reduces to a mean‑field description, yielding a step‑function E(p) that jumps from 0 to 1 at p = ½. In contrast, on regular lattices (β = 0) the exit probability is a smooth S‑shaped curve; its steepness grows with k, reflecting that denser connectivity makes consensus easier to achieve.

The exit time shows a complementary trend: larger k dramatically shortens T, indicating that higher local connectivity accelerates the spread of a unanimous opinion. Introducing a modest amount of randomness (β = 0.01–0.05) also reduces T, especially for small k, because rewired shortcuts create shorter paths across the network. However, for very small k the effect of β on T is more pronounced, leading to a pronounced separation between the k = 1 case and higher‑k cases.

A central empirical finding is that the entire family of exit‑probability curves can be fitted by a simple functional form
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