PageRank model of opinion formation on Ulam networks

We consider a PageRank model of opinion formation on Ulam networks, generated by the intermittency map and the typical Chirikov map. The Ulam networks generated by these maps have certain similarities with such scale-free networks as the World Wide Web (WWW), showing an algebraic decay of the PageRank probability. We find that the opinion formation process on Ulam networks have certain similarities but also distinct features comparing to the WWW. We attribute these distinctions to internal differences in network structure of the Ulam and WWW networks. We also analyze the process of opinion formation in the frame of generalized Sznajd model which protects opinion of small communities.

💡 Research Summary

The paper investigates opinion dynamics on directed networks generated by the Ulam method, using two representative chaotic maps: a one‑dimensional intermittency map and the standard dissipative Chirikov map. The Ulam construction discretizes the phase space into N cells, propagates many trajectories from each cell, and records transition probabilities between cells to form a stochastic matrix S. From S the Google matrix G = αS + (1 − α)E/N is built, where α is the damping factor and E is a matrix of ones. The dominant eigenvector of G yields the PageRank vector P, which quantifies node importance. For the intermittency map (α = 1, z₁ = 2, z₂ = 0) the PageRank follows a pure power law P_j ∝ 1/j, while for the Chirikov map (α = 0.95, η = 0.99, k = 0.22) the decay exponent is ≈0.48.

On these networks the authors apply the PageRank Opinion Formation (PROF) model. Each node i carries a binary opinion σ_i = ±1 (red/blue). At each iteration the weighted sum Σ_i = a ∑_j_in P_j σ_j + b ∑_j_out P_j σ_j is computed, where a + b = 1. The parameters a and b control the relative influence of incoming versus outgoing links, allowing the model to interpolate between a “tenacious” society (large a) and a “conformist” society (large b). If Σ_i > 0 the node adopts the red opinion, otherwise blue. Simulations start either from a random assignment of opinions or from a configuration where the top‑ranked N_top nodes share the same opinion.

Key findings for the PROF dynamics on Ulam networks are: (i) convergence to a stationary state occurs after roughly 25 iterations for random initial conditions; (ii) an elite of high‑PageRank nodes can amplify its opinion, increasing the final red fraction by a factor of two to three compared with the initial fraction, but this effect is weaker than on real WWW or university networks because the Ulam graphs have a lower average degree; (iii) the critical initial red fraction f_c separating extinction from domination depends on a: for a = 0.2, f_c ≈ 0.45, while for a = 0.65, f_c ≈ 0.35. Unlike the WWW case, bistability is largely absent for a < 0.7; only for strongly tenacious societies (a > 0.7) does a small probability of minority takeover appear.

The authors then extend the model by incorporating the generalized Sznajd mechanism. At each discrete time step a node i is selected, and together with the N_g − 1 highest‑PageRank nodes that point to i they may form a group if all share the same opinion. The group’s effective PageRank P_g is the sum of its members’ PageRanks. Any node whose own PageRank is smaller than P_g and which points to a group member adopts the group’s opinion, thereby increasing P_g. This group‑based dynamics dramatically lengthens the convergence time (τ_c ≈ 10 N) and further suppresses the survival of small minority clusters. Because the Ulam networks generated from the intermittency map have a maximum indegree of four, groups larger than N_g = 5 are meaningless; varying N_g from 3 to 4 changes minority resistance by only about 2 %.

When the Chirikov‑based Ulam network is examined, the richer link structure (higher average degree) yields stronger resistance of minorities. Nevertheless, the same qualitative picture holds: the final red fraction f_f remains close to zero for initial red fractions f_i < 0.3, while for 0.45 < f_i < 0.55 there is roughly an 8 % chance that the red opinion becomes dominant. Varying α between 0.95 and 0.99 changes the PageRank decay exponent from ≈0.48 to ≈0.9, but does not induce bistability, contrary to what is observed on real social networks where a steeper decay promotes multiple stable states.

Overall, the study demonstrates that while Ulam‑generated directed graphs share the scale‑free PageRank distribution of the World Wide Web, their structural constraints—particularly low average degree, limited indegree, and the deterministic nature of the underlying map—lead to markedly different opinion dynamics. The influence of elite nodes is weaker, the system exhibits limited bistability, and minority opinions are more easily extinguished, especially under the Sznajd group rule. These results highlight the crucial role of network topology in shaping collective decision processes and suggest that models of opinion formation must account for specific structural features when applied to real‑world systems.