Opinion formation at Ising social networks

TL;DR

This study investigates how fixed elite opinions compete with a crowd on an Ising social network derived from scientific collaborations. It introduces the Generalized INOF (GINOF) model, where non-fixed spins start with random opinions and a weak amplitude influence $W_i$ that self-ascends to 1 upon surpassing a conviction threshold $Z_c$, while spins flip only if the majority influence $Z_i$ exceeds $Z_c$. The results demonstrate a phase transition from elite-dominated to crowd-dominated outcomes as the crowd influence $W$ grows, with a critical value $W_{cr}\approx Z_c/\kappa$ (here $\kappa\approx 4.8$ and $Z_c=0.1$). Seed placement and Erdős-barrage effects can markedly alter the final opinion distribution $p(f_r)$, highlighting the sensitivity of collective outcomes to initial conditions. This work provides a framework for understanding how fixed elites lose political or social sway when crowd influence strengthens, with implications for opinion dynamics on undirected networks.

Abstract

We study the process of opinion formation in an Ising social network of scientific collaborations. The network is undirected. An Ising spin is associated with each network node being oriented up (red) or down (blue). Certain nodes carry fixed, opposite opinions whose influence propagates over the other spins, which are flipped according to the majority-influence opinion of neighbors of a given spin during the asynchronous Monte Carlo process. The amplitude influence of each spin is self-consistently adapted, and a flip occurs only if this majority influence exceeds a certain conviction threshold. All non-fixed spins are initially randomly distributed, with half of them oriented up and half down. Such a system can be viewed as a model of elite influence, coming from the fixed spins, on the opinions of the crowd of non-fixed spins. We show that a phase transition occurs as the amplitude influence of the crowd spins increases: the dominant opinion shifts from that of the elite nodes to a phase in which the crowd spins' opinion becomes dominant and the elite can no longer impose their views.

Figures (7)

• Figure 1: Evolution of the fraction of red nodes $f_r$ for $N_r = 500$ random pathway realisations. An initial condition has one red fixed node (Newman) and one blue fixed node (Barabasi); they remain fixed during an asynchronous Monte Carlo evolution based on the relation (\ref{['eqzc']}); all other nodes are initially white ($\sigma_j=0$ in (\ref{['eqzc']})). Here $x$-axis represents time time $\tau$ of Monte Carlo process, where each unit of $\tau$ marks one complete update of all nodes/spins following the INOF/GINOF model (here $Z_c=0; W=0$); steady-state configurations are reached at $\tau=20$ (or earlier).
• Figure 2: Probability distribution $p(f_r)$ of red node fractions; the histogram of $f_r$ values is obtained with 50 cells $1 \leq m \leq 50$ with normalization $\sum_m f_r(m) =1$, average red value is $<f_r> = 0.638$. Here there are $N_r=10^5$ pathway realizations. Fixed nodes are Newman (red) and Barabasi (blue), all other nodes are white (spin zero). Initially all non-fixed nodes are white for the INOF model [or for the GINOF model at $W=0; Z_c=0$]. Vertical dashed line marks the average red value $<f_r>$.
• Figure 3: Same as Fig. \ref{['fig2']}, but with initial state of node Sole being blue; $<f_r> = 0.326$
• Figure 4: Dependence of the red fraction of nodes $f_r(K)$ on PageRank index $K$ for the case of Fig. \ref{['fig2']} ($K$ is obtained at damping factor $\alpha=0.85$).
• Figure 5: Same as Fig. \ref{['fig2']} but for the GINOF model at $Z_c=0$, $W=0.005$; here $N_r = 10^5$.