Social dynamics with peer support on heterogeneous networks: The "mafia model"

Human behavior often exhibit a scheme in which individuals adopt indifferent, neutral, or radical positions on a given topic. The mechanisms leading to community formation are strongly related with social pressure and the topology of the contact network. Here, we discuss an approach to model social behavior which accounts for the protection by alike peers proportional to their relative abundance in the closest neighborhood. We explore the ensuing non-linear dynamics emphasizing the role of the specific structure of the social network, modeled by scale-free graphs. We find that both coexistence of opinions and consensus on the default position are possible stationary states of the model. In particular, we show how these states critically depend on the heterogeneity of the social network and the specific distribution of external control elements.

💡 Research Summary

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The paper introduces a novel opinion‑dynamics framework, dubbed the “mafia model,” to study how individuals adopt one of three possible states—neutral (empty site), citizen (default opinion), or mafioso (radical minority)—within a social network. Unlike classic voter or Abrams‑Strogatz models, the transition rates in this model are defined as the product of two factors: (i) a pressure term proportional to the fraction of opposite‑type neighbors, and (ii) a support term that attenuates this pressure according to the fraction of like‑type neighbors. Concretely, the rate for a citizen to become a mafioso is w_cm = s_m m (1 − c), while the reverse rate is w_mc = s_c c (1 − m), where c, m, and φ denote the local fractions of citizens, mafiosi, and empty sites, respectively, and s_m, s_c are persuasiveness parameters. This multiplicative form makes the effective non‑linearity depend on the local composition: when empty sites dominate, the leading term is linear; when one opinion dominates, the non‑linear term becomes dominant, a feature absent in earlier models where the exponent is fixed.

The dynamics proceeds through three stochastic processes: (i) birth of a citizen at an empty node with rate b, (ii) death of any occupied node with rate d, and (iii) strategy changes governed by the above rates. The model assumes a “one‑to‑all” interaction scheme: an agent evaluates the entire composition of its immediate neighborhood before deciding, rather than interacting pairwise as in the traditional voter model. This leads to a discrete set of possible neighborhood configurations, invalidating simple mean‑field approximations.

First, the authors analyze a well‑mixed population (complete graph) using a standard mean‑field approach. By nondimensionalizing time with the inverse death rate, they obtain a set of ordinary differential equations for the global fractions c(t), m(t), and φ(t). The empty‑site fraction reaches a steady state determined solely by birth and death rates, while the citizen and mafioso fractions are coupled through the composition‑dependent rates. The analysis reveals two possible stationary regimes: (a) consensus on the neutral/default opinion (c ≈ 1, m ≈ 0) and (b) coexistence of citizens and mafiosi at non‑zero fractions. The boundary between these regimes depends on the persuasiveness parameters and the relative birth/death rates.

To explore the impact of network topology, extensive agent‑based simulations are performed on scale‑free networks generated by the Barabási‑Albert algorithm (degree exponent ≈ 3). The heterogeneous degree distribution dramatically enlarges the coexistence region compared with the well‑mixed case. High‑degree hub nodes experience strong mixed pressures, which destabilize the absorbing state where mafiosi vanish. Consequently, even for parameter values that would lead to consensus in a complete graph, the scale‑free structure sustains a persistent minority of mafiosi. The authors also discuss an extended mean‑field theory (Appendix A) that accounts for the finite set of possible neighborhood compositions; this theory quantitatively reproduces the simulation results and highlights why the simple mean‑field fails for one‑to‑all interactions on heterogeneous graphs.

A further innovation is the inclusion of external control elements, interpreted as police or regulatory agents, placed on a fraction p of the edges. These elements simultaneously (i) reduce the pressure exerted by mafiosi on citizens by a factor (1 − p) and (ii) add an extra persuasive term s_p p that encourages mafiosi to revert to citizenship. The modified transition rates become w_cm = s_m m (1 − c)(1 − p) and w_mc = (s_c c + s_p p)(1 − m). Simulations show that heterogeneous placement of control edges induces a micro‑phase separation: regions with high p become protected zones where citizens dominate, while low‑p regions allow mafiosi to persist. This phenomenon mirrors real‑world scenarios where targeted policing creates pockets of low crime amid broader areas of higher criminal activity.

Overall, the paper makes three substantive contributions: (1) it proposes a biologically plausible, composition‑dependent transition mechanism that bridges linear and non‑linear opinion dynamics; (2) it demonstrates that network heterogeneity, especially in scale‑free graphs, expands the parameter space supporting opinion coexistence and undermines absorbing states; (3) it shows that external, edge‑based control can generate spatially heterogeneous steady states, offering a theoretical framework for policy interventions in complex social systems. The work thus advances our understanding of how local peer support, network structure, and targeted regulation jointly shape collective behavior.