# Stochastic Simulations of Casual Groups

## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. The Gillespie Algorithm

## 4. Analytical Approximation for the Equilibrium Regime

#### 4.1. Case $\alpha =0$

#### 4.2. Case $\alpha =1$

#### 4.3. The Limit $\alpha \to -\infty $

#### 4.4. General $\alpha \le 1$

## 5. Equilibrium Regime for $\alpha >1$

## 6. Discussion

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Stochastic simulations of casual groups with attractiveness exponents $\alpha =-1,0,1$ and 2, as indicated. (

**Left**) Mean density of isolates as function of time t. The dashed horizontal line is the exact result (11) for the equilibrium regime. (

**Right**) Mean group size as function of time t. The dashed horizontal lines are the equilibrium approximate analytical results for $\alpha \le 1$ presented in Section 4. The other parameters are $\lambda =1.5$, $\mu =1$ and $N=100$.

**Figure 2.**Stochastic simulations of casual groups for $\alpha =0$ and population sizes $N=5,10,20$ and 100, as indicated. (

**Left**) Mean density of isolates as function of time t. The dashed horizontal line is the exact result (11) for the equilibrium regime. (

**Right**) Mean group size as function of time t. The dashed horizontal line is the approximate analytical result for the equilibrium regime presented in Section 4. The other parameters are $\lambda =1.5$ and $\mu =1$.

**Figure 3.**Probability of observing a group of size $i=1,2,3$ and 4 in the equilibrium regime for $\alpha =0$. The population sizes are $N=10$ and 100, as indicated, and the disaggregation rate is $\mu =1$. The solid curves that perfectly fit the simulation data for $N=100$ are given by the truncated Poisson distribution (23).

**Figure 4.**Probability of observing a group of size $i=1,2,3$ and 4 in the equilibrium regime for $\alpha =1$. The population sizes are $N=10$ and 100, as indicated, and the disaggregation rate is $\mu =1$. The solid curves that perfectly fit the simulation data for $N=100$ are given by the logarithmic distribution (28).

**Figure 5.**Probability of observing a group of size $i=1$ and 2 in the equilibrium regime for $\alpha =-{10}^{4}$. The population sizes are $N=10$ and 100, as indicated, and the disaggregation rate is $\mu =1$. The solid curves that perfectly fit the simulation data for $N=100$ are given by Equations (31) and (32).

**Figure 6.**Mean-field approximation for the equilibrium regime for attractiveness exponents (top to bottom) $\alpha =1,0.5,0,-1$ and $\alpha \to -\infty $. (

**Left**) Mean group size ${m}_{\alpha}$. (

**Right**) Variance of the group size ${\sigma}_{\alpha}^{2}$.

**Figure 7.**Stochastic simulations in the equilibrium regime for $\alpha =2$ and population sizes $N=50,100,200$ and 400, as indicated. (

**Left**) Mean group size ${m}_{2}$. (

**Middle**) Variance of the group size ${\sigma}_{2}^{2}$. (

**Right**) Scaled variance ${\sigma}_{2}^{2}/N$. The solid lines are the predictions of Equations (36) and (37) for $\alpha \to \infty $. The disaggregation rate is $\mu =1$.

**Figure 8.**Distribution of group sizes in the equilibrium regime for $\alpha =2$, $\kappa =2$, $\mu =1$ and $N=100,200,400$ and 800, as indicated. The dashed vertical line indicates the relative size of the large group for $\alpha \to \infty $, viz., $1-{\langle {n}_{1}\rangle}_{eq}/N=2/3$. The lines connecting the symbols are guides to the eye.

**Figure 9.**Stochastic simulations in the equilibrium regime for $\kappa =2$ and population sizes $N=100$, $200,400$ and 800, as indicated. (

**Left**) Mean group size ${m}_{\alpha}$. (

**Right**) Scaled variance ${\sigma}_{2}^{2}/N$. The predictions of Equations (36) and (37) for $\alpha \to \infty $ are ${m}_{\infty}=3$ and ${\sigma}_{\infty}^{2}/N=4/3$ (dashed horizontal line in the right panel). The disaggregation rate is $\mu =1$. The lines connecting the symbols are guides to the eye.

**Figure 10.**Distribution of group sizes in the equilibrium regime for $\kappa =2$ and $N=100,200,400$ and 800, as indicated. (

**Left**) $\alpha =1.10$. The straight line is the fitting ${p}_{i}=0.003exp(-0.18i)$ of the data for $N=800$. (

**Right**) $\alpha =1.15$. The disaggregation rate is $\mu =1$. The lines connecting the symbols are guides to the eye.

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Fontanari, J.F.
Stochastic Simulations of Casual Groups. *Mathematics* **2023**, *11*, 2152.
https://doi.org/10.3390/math11092152

**AMA Style**

Fontanari JF.
Stochastic Simulations of Casual Groups. *Mathematics*. 2023; 11(9):2152.
https://doi.org/10.3390/math11092152

**Chicago/Turabian Style**

Fontanari, José F.
2023. "Stochastic Simulations of Casual Groups" *Mathematics* 11, no. 9: 2152.
https://doi.org/10.3390/math11092152