The Impact of Social Attractiveness on Casual Group Formation: Power-Law Group Sizes and Suppressed Percolation

TL;DR

The paper investigates how heterogeneous social appeal drives casual group formation in a 2D spatial setting, contrasting an attractiveness-driven interaction model with a null movement model akin to a Random Geometric Graph. Using extensive simulations to reach equilibrium, it shows that the average degree scales linearly with system size, leading to compact, celebrity-centered clusters and a suppression of percolation, while the group-size distribution follows a density-independent power law $P(n) \propto n^{-2.5}$. In stark contrast, the null model yields exponential or bimodal $P(n)$ and exhibits a standard percolation transition near $\rho_c \approx 1.44$. These findings highlight the crucial role of individual attractiveness in shaping spatial social aggregation and percolation properties, with broad implications for modeling face-to-face interactions and the emergence of casual groups.

Abstract

The dynamics of casual group formation has long been a subject of interest in social sciences. While early stochastic models offered foundational insights into group size distributions, they often simplified individual behaviors and lacked mechanisms for heterogeneous social appeal. Here, we re-examine the attractiveness-driven interaction model, an agent-based framework where point-like agents move randomly in a 2D arena and exhibit varied social appeal, leading them to pause near highly attractive celebrity peers. We compare this model to a null model where the agents are continuously in movement, which resembles a Random Geometric Graph. Our extensive simulations reveal significant structural and dynamic differences: unlike the null model, the attractiveness-driven model's average degree increases linearly with system size for fixed density, resulting in more compact groups and the suppression of a percolation transition. Crucially, while the null model's group size distribution is either exponentially decaying or bimodal, the attractiveness-driven model robustly exhibits a power-law distribution, $P(n) \propto n^{-2.5}$, with an exponent independent of density. These findings, obtained through computationally intensive simulations due to long equilibration times, offer a thorough quantitative characterization of this model, highlighting the critical role of individual attractiveness in shaping social aggregation in physical space.

Figures (13)

• Figure 1: Spatial distribution of active agents at $t=10^5$. Snapshots illustrate the spatial distribution of active agents (red symbols) for the null model (left panel) and the attractiveness-driven interaction model (right panel). Active agents belonging to the largest group are highlighted in blue. In the null model, the largest cluster contains $4890$ agents, compared to $1226$ agents in the attractiveness-driven interaction model. Simulation parameters are $L=50$ and $N=5000$.
• Figure 2: Pair correlation function $g (r)$ at $t=10^5$. The pair correlation function is shown for the null model (left panel) and the attractiveness-driven interaction model (right panel). Results are averaged over 100 independent simulation runs. The thin black line represents the analytical result for uniformly randomly distributed agents, $g (r) = 2\pi r/L^2$, which is valid for $r <L/2$. The vertical dashed line indicates $r=d=1$. Simulation parameters are $L=50$ and $N=5000$.
• Figure 3: Time evolution of average degree. Average degree $k$ as a function of time $t$ for the null model (left panel) and the attractiveness-driven interaction model (right panel). Results are shown for $\rho=0.6$ and varying linear sizes $L=25, 50$, and $100$. Data points (symbols) represent averages obtained from $10^3$ independent simulation runs. In the left panel, the horizontal dashed line indicates the analytical prediction $k = \pi \rho/2$. In the right panel, the solid colored curves correspond to the exponential fit function given by eq. (\ref{['fitk']}).
• Figure 4: Equilibrium average degree as a function of density. Equilibrium average degree $k_\infty$ for the null model (left panel) and the attractiveness-driven interaction model (right panel) as functions of density $\rho$. Results are shown for linear sizes $L=25, 50$, and $100$. Data points (symbols) represent averages obtained from $10^3$ independent simulation runs. In the left panel, the solid black line denotes $k = \pi \rho$, while the dashed black line indicates $k = \pi \rho/2$. Lines connecting the symbols serve as visual guides.
• Figure 5: Equilibrium fraction of active agents in the largest group. Equilibrium fraction of active agents in the largest group $s_\infty$ for the null model (left panel) and the attractiveness-driven interaction model (right panel) as function of density $\rho$. Results are shown for linear sizes $L=25, 50$, and $100$. Data points (symbols) represent averages obtained from $10^3$ independent simulation runs. In the left panel, the vertical dashed line indicates the critical density $\rho_c \approx 1.44$ of the Random Geometric Graph (RGG). Lines connecting the symbols serve as visual guides.