Competitive Social Mobilization in Threshold Models of Collective Action

TL;DR

This work generalizes the threshold model to competitive social mobilization across multiple movements and analyzes how environmental stability of individual dispositions—captured by quenched versus annealed thresholds—drives macroscopic outcomes. Using a threshold distribution with P(T_i ≤ k) = (k/N)^\gamma, the authors derive distinct phase transitions: a discontinuous collapse from consensus to fragmentation at \gamma_c^{(q)} ≈ 1 for quenched thresholds, and a separate discontinuous transition at \gamma_c^{(a)} ≈ 1.5 for annealed thresholds, where a single giant group can emerge (winner-takes-all) or be suppressed (pulverized phase). The quenched case yields analytic results at \gamma = 0 and a critical scaling S_{max} ∼ N^{1/2}, while the annealed case admits mean-field rate equations with exact solutions for \gamma = 0 and 1 and extensive finite-N simulations using rejection-free algorithms. Together, the findings show that raising participation costs can either foster unity or shatter collective action, depending on environmental stability, with clear implications for designing interventions in real-world social systems.

Abstract

Social mobilization often fails not for a lack of collective interest, but because of fierce competition between rival movements for the same limited pool of participants. We generalize the classic threshold model of collective behavior to analyze this competitive aggregation, exploring how populations with diverse participation thresholds navigate multiple, mutually exclusive causes. Focusing on the conditions necessary for a single consensus movement to encompass an entire population, our analysis reveals that the outcome of social competition depends critically on the stability of individual dispositions. In quenched environments where participation thresholds are fixed, increasing resistance initially allows a dominant movement to suppress its competitors; however, further resistance triggers a sudden collapse into total fragmentation as low-threshold instigators become too rare to sustain growth. Conversely, in annealed environments where opinions are fluid, higher resistance paradoxically drives a winner-takes-all consensus. In this fluid scenario, massive movements can only be avoided through a deliberate divide-and-conquer strategy. In both cases, the transitions between pulverized and massive movements are discontinuous. These findings demonstrate that the effectiveness of social control depends entirely on environmental stability: raising the cost of participation can either forge unity or shatter collective action into insignificance.

Figures (6)

• Figure 1: Mean fraction of agents in the largest group, $\rho_\infty$, for the quenched scenario as a function of the attachment exponent $\gamma$. The data points represent Monte Carlo simulation results for system sizes ranging from $N=4000$ to $N=64000$, as indicated. These quantities are measured in the final stationary state, which is reached when isolates can no longer join any existing group. The solid black curve represents a fit to the data for $N=64000$ in the region $\gamma < 0.8$, given by $\rho_\infty = 1 - a\exp[-b/(1-\gamma)]$ with $a = 6.96$ and $b=1.97$. The dashed vertical line indicates the critical value $\gamma_c^{(q)}=1$.
• Figure 2: Left panel) Mean fraction of agents in the largest group, $\rho_\infty$, for the quenched scenario as a function of the attachment exponent $\gamma$ in the critical region. The intersection of curves for different system sizes $N$ occurs in the interval $(0.99, 1)$. (Right panel) Log-log plot showing the dependence of $\rho_\infty$ on $N$ at $\gamma_c^{(q)}=1$. The solid line represents the power-law fit $\rho_\infty = a N^{-b}$ with $a=0.407$ and $b=0.505$.
• Figure 3: Mean fraction of isolates $\phi_\infty$ (left panel) and mean density of groups $\mu_\infty$ (right panel) for the quenched scenario as a function of the attachment exponent $\gamma$. The data points represent Monte Carlo simulation results for system sizes ranging from $N=4000$ to $N=64000$, as indicated. These quantities are measured in the final stationary state, which is reached when the remaining isolates cannot join any of the existing groups.
• Figure 4: Mean density of groups $\mu_\infty$ (upper panel) and the mean fraction of agents in the largest group $\rho_\infty$ (lower panel) for the annealed scenario as a function of the attachment exponent $\gamma$. The data points represent Monte Carlo simulation results for system sizes ranging from $N=4000$ to $N=1024000$ as indicated. These quantities are measured in the final stationary state, characterized by the complete exhaustion of isolated agents ($n_1=0$). The solid black curve shows the analytical prediction from the mean-field rate equations. The dashed vertical line signals the critical value, $\gamma_c^{(a)} \approx 1.499$.
• Figure 5: (Left panel) Mean fraction of agents in the largest group $\rho_\infty$ for the annealed scenario as a function of the attachment exponent $\gamma$ in the critical region. The arrows show the intersection of curves for consecutive system sizes $N$. (Right panel) The intersection values $\gamma_{N_1}$, defined where the curves for $N_1$ and $N_2=2N_1$ cross, plotted as a function of the inverse system size $1/N_1$. The solid curve represents the scaling fit $\gamma_{N_1} = \gamma_c^{(a)} + a N_1^{-b}$, yielding parameters $a = 1.569$, $b = 0.266$, and an asymptotic critical value $\gamma_c^{(a)} = 1.5$. Note the logarithmic scale on the $x$-axis.