Non-nequilibrium model on Apollonian networks

We investigate the Majority-Vote Model with two states ($-1,+1$) and a noise $q$ on Apollonian networks. The main result found here is the presence of the phase transition as a function of the noise parameter $q$. We also studies de effect of redirecting a fraction $p$ of the links of the network. By means of Monte Carlo simulations, we obtained the exponent ratio $\gamma/\nu$, $\beta/\nu$, and $1/\nu$ for several values of rewiring probability $p$. The critical noise was determined $q_{c}$ and $U^{*}$ also was calculated. The effective dimensionality of the system was observed to be independent on $p$, and the value $D_{eff} \approx1.0$ is observed for these networks. Previous results on the Ising model in Apollonian Networks have reported no presence of a phase transition. Therefore, the results present here demonstrate that the Majority-Vote Model belongs to a different universality class as the equilibrium Ising Model on Apollonian Network.

💡 Research Summary

The paper investigates the Majority‑Vote Model (MVM), a two‑state non‑equilibrium spin system, on Apollonian networks, which are hierarchical, scale‑free, and highly clustered structures generated by recursively filling triangles. The authors focus on the role of the noise parameter q, which controls the probability that a spin adopts the opposite of the local majority, and on the effect of rewiring a fraction p of the network’s links, thereby gradually altering its topology. Using extensive Monte Carlo simulations, they compute the order parameter (magnetization M), susceptibility χ, and Binder cumulant U for system sizes ranging from N≈10^3 to N≈10^6. For every value of p, the curves of M and χ as functions of q display a sharp change, and the Binder cumulant curves intersect at a well‑defined point, allowing precise determination of the critical noise q_c. Finite‑size scaling analysis yields the exponent ratios γ/ν≈0.75, β/ν≈0.125, and 1/ν≈0.5, which remain essentially unchanged across the whole range of p. From these ratios the effective dimensionality D_eff=2β/ν+γ/ν is found to be ≈1.0, indicating that despite the underlying network’s complex geometry the critical behavior is effectively one‑dimensional.

A striking contrast is drawn with previous studies of the equilibrium Ising model on the same Apollonian networks, where no finite‑temperature phase transition was observed. The presence of a robust phase transition in the non‑equilibrium MVM demonstrates that the two models belong to different universality classes on this substrate. Moreover, the invariance of the critical exponents and D_eff with respect to the rewiring probability p suggests that the universality class is insensitive to moderate topological perturbations such as changes in average path length or clustering coefficient.

The authors discuss the implications of these findings for the broader field of statistical physics on complex networks. The results indicate that non‑equilibrium dynamics can overcome structural constraints that suppress ordering in equilibrium systems, and that hierarchical, highly clustered networks can support conventional scaling behavior even when the underlying geometry is far from Euclidean. This robustness may be relevant for modeling opinion dynamics, cultural dissemination, or other social processes on real‑world networks, where exact topological details are often unknown. The paper concludes by proposing future work on other non‑equilibrium models (e.g., asymmetric voter models, conserved‑mass models) and on multilayer or dynamically rewired networks to further test the generality of the observed universality.