Emergence of social hierarchies in a society with two competitive groups

TL;DR

The paper proposes a two-group extension of the Bonabeau agent-based model where only cross-group interactions occur and total fitness is conserved. The winning probability depends on normalized fitness via $P_{ij}(t)=\frac{1}{1+\exp[\eta(\hat{F}_{j}^{B}-\hat{F}_{i}^{A})]}$, and fitness exchanges are proportional to opponents' fitness, implemented through a Gillespie-based Monte Carlo on a 2D lattice without relaxation. Numerical simulations reveal a sharp egalitarian-to-hierarchical transition controlled by $\eta$, with leader emergence robust to system size and dictated by group sizes and ratios; a few leaders accumulate most of the fitness in the stationary state. The authors derive explicit scaling laws showing data collapse across parameter choices, providing analytic forms for $F_{\max}^{A}$, $F_{\max}^{A}/F_{\text{tot}}^{A}$, and $F_{\text{tot}}^{A}$ as functions of $\eta$, $N_T$, and $N_A$, and they discuss the implications for social hierarchies in urban-like settings and potential extensions to more complex interactions.

Abstract

Agent-based models describing social interactions among individuals can help to better understand emerging macroscopic patterns in societies. One of the topics which is worth tackling is the formation of different kinds of hierarchies that emerge in social spaces such as cities. Here we propose a Bonabeau-like model by adding a second group of agents. The fundamental particularity of our model is that only a pairwise interaction between agents of the opposite group is allowed. Agent fitness can thus only change by competition among the two groups, while the total fitness in the society remains constant. The main result is that for a broad range of values of the model parameters, the fitness of the agents of each group show a decay in time except for one or very few agents which capture almost all the fitness in the society. Numerical simulations also reveal a singular shift from egalitarian to hierarchical society for each group. This behaviour depends on the control parameter $η$, playing the role of the inverse of the temperature of the system. Results are invariant with regard to the system size, contingent solely on the quantity of agents within each group. Finally, scaling laws are provided thus showing a data collapse from different model parameters and they follow a shape which can be related to the presence of a phase transition in the model.

Figures (15)

• Figure 1: Flux code diagram of computer simulations. Flux code diagram of the Monte Carlo simulations with the Gillespie algorithm implemented, where $u\sim U(0,1)$ is a uniform random number between 0 and 1. A detailed description can be found in Appendix \ref{['sec:MC']}.
• Figure 2: Initial spatial distribution of agents in a regular square lattice. Initial random positions of both groups of agents with $N_\text{A}=500$ (empty circles) and $N_\text{B}=50$ (filled circles) for $L=45$. There can be more than one agent in each site.
• Figure 3: Temporal evolution of the fitnesses of group B. Time evolution of all $F_{j}^\text{B}$ for $N_\text{A}=500$ and $N_\text{B}=50$ individuals, $\eta= 5$ and $x=0.01$, randomly simulated on a $L\times L$ lattice ($L=25$). (a) The first $1\,100$ time steps. (b) Larger time window to observe the transient and stationary fluctuations of the leader while the inset shows the behaviour of the other agents of the group.
• Figure 4: Maximum fitness temporal evolution of the two groups for several systems sizes. Time evolution of the maximum fitness for $N_\text{A}=500$ and $N_\text{B}=50$ agents, $\eta=5$ and $x=0.01$, randomly simulated on several $L\times L$ square lattice sizes (largest to smallest ascending). (a) $F_\text{max}^\text{A}(t)$ and (b) $F_\text{max}^\text{B}(t)$. The vertical axis is in log scale.
• Figure 5: Fitness distribution mapping for the two groups at two distinct times. Fitness heat map of both groups of a single simulation for $N_\text{A}=500$ agents, $N_\text{B}=50$ agents, $\eta= 5$, $x=0.01$ in a $L\times L$ square lattice with $L=45$. (a) $20\,000$ Gillespie time units and group A. (b) 20 000 Gillespie time units and group B. (c) Stationary regime at $600\,000$ Gillespie time units and group A. (d) Stationary regime at $600\,000$ Gillespie time units and group B. There can be more than one agent in each site.