Statistical Mechanics of Opinion Dynamics in Social Networks

Abstract

Opinion polarization in social networks is often described qualitatively as a competition between conformity and
independent judgment, but quantitative statistical-mechanics treatments that track how entropy and order
parameters co-evolve across a full phase diagram remain comparatively scarce. We model opinion formation on
synthetic scale-free networks (N = 5000 nodes, mean degree 8) using a bounded-confidence, Ising-inspired update
rule in which each agent's opinion couples to its neighbors with strength J and to independent noise with effective
temperature T, and we simulate 240 combinations of J/T ratio (0.2 to 5.0) and network rewiring probability (0 to 1),
running each configuration for 2 x 10^5 update sweeps averaged over 30 independent network realizations. We
identify a sharp order-disorder transition in the opinion order parameter at a critical coupling ratio of (J/T)_c = 1.42
+/- 0.05, below which the population remains disordered (mean absolute opinion magnetization < 0.08) and above
which it condenses into a polarized state with magnetization exceeding 0.85 within a coupling window of only 0.3 in
J/T units. Shannon entropy of the opinion distribution, computed by binning continuous opinions into 40 bins,
peaked at 3.51 bits near the critical point, 94% of the theoretical maximum for a uniform distribution over the same
bins, and fell to 0.61 bits in the strongly polarized regime (J/T = 5.0), a 5.8-fold reduction. Network rewiring
probability shifted the critical coupling ratio systematically, with (J/T)_c falling from 1.68 at zero rewiring (regular
lattice-like topology) to 1.24 at full rewiring (random graph topology), a 26% reduction indicating that long-range
network shortcuts lower the coupling strength required for population-wide polarization. Finite-size scaling analysis
across network sizes of 1000, 2500, and 5000 nodes yielded a critical exponent estimate of beta = 0.23 +/- 0.04 for
the order parameter, closer to the mean-field Ising value than to the two-dimensional Ising value, consistent with the
long-range effective connectivity of scale-free topology. These results place opinion polarization within a
well-defined statistical-mechanics phase-transition framework, quantify how network topology shifts the critical
coupling threshold, and suggest that interventions reducing network rewiring, effectively increasing local clustering,
could raise the coupling strength required for runaway polarization in real social networks.