\begin{algorithmic} 
\State \textbf{Input:}  1) fixed $L_s$; 2) current positions $X(t)$ of agents at time $t$; 3) a nondecreasing transition function $\xi:\mathbb{R}^+\to [0,1]$ (e.g. $\xi(s):=1-\sigma(s)$, see \textsection \ref{sec:weights})\\
\\
set $k=0$\\
set $Y=\emptyset$
\While{$true$} 		
\State for all $\mathbf{\mathbf{y}}\in \mathcal{S}_{r}$ let $R_\mathbf{y}:=\text{dist}(\mathbf{y};X\cup Y)$ and construct
\[
f_{k}(\mathbf{y}):=\begin{cases} \xi\left(R_\mathbf{y}\right/L_s-1) & \text{if }R_\mathbf{y}\geq L_{s}\\ 0 & \text{if }R_\mathbf{y}<L_{s}\\ 
\end{cases}
\]

\If{$f_k(\mathbf{y})\equiv 0$} 
\State break (meaning $\mathcal{S}_{r}$ is saturated)
\EndIf
\State (a) normalize $f_{k}(\mathbf{y})$ so that $\int_{\mathcal{S}_{r}}f_{k}(\mathbf{y})d\mathbf{y}=1$
\State (b) generate the position $\mathbf{y}_k$ of the $k^{\text{th}}$ new agent  
\State \quad on $\mathcal{S}_{r}$ by a random sampling using the PDF $f_k(y)$
\State (c) updates: $Y\leftarrow Y\cup \{\mathbf{y}_k\}$ and $k\leftarrow k+1$

\EndWhile 


\State \textbf{Output:} $Y$ are the collected positions on $\mathcal{S}_{r}$ of the $k$ newly inserted agents to be added to $X(t)$.\\

\State \underline{NOTE}: it is possible that $k=0$ and $Y=\emptyset$, indicating that no new agent will enter at time $t$ due to $\mathcal{S}_{r}$ being overcrowded from the beginning.
\end{algorithmic} 

\begin{algorithmic}
\State \textbf{Input:} 1) fixed $L_s$; 2) current positions $X(t)$ of agents at time $t$; 3) a nondecreasing transition function $\xi:\mathbb{R}^+\to [0,1]$ (e.g. $\xi(s):=1-\sigma(s)$, see \textsection \ref{sec:weights})\\
\\
set $k=0$\\
set $Y=\emptyset$
\While{$true$}
\State for all $\mathbf{\mathbf{y}}\in \mathcal{S}_{r}$ let $R_\mathbf{y}:=\text{dist}(\mathbf{y};X\cup Y)$ and construct
\[
f_{k}(\mathbf{y}):=\begin{cases}
\xi\left(R_\mathbf{y}\right/L_s-1) & \text{if }R_\mathbf{y}\geq L_{s}\\ 0 & \text{if }R_\mathbf{y}<L_{s}\\
\end{cases}
\]
\If{$f_k(\mathbf{y})\equiv 0$}
\State break (meaning $\mathcal{S}_{r}$ is saturated)
\EndIf
\State (a) normalize $f_{k}(\mathbf{y})$ so that $\int_{\mathcal{S}_{r}}f_{k}(\mathbf{y})d\mathbf{y}=1$
\State (b) generate the position $\mathbf{y}_k$ of the $k^{\text{th}}$ new agent
\State \quad on $\mathcal{S}_{r}$ by a random sampling using the PDF $f_k(y)$
\State (c) updates: $Y\leftarrow Y\cup \{\mathbf{y}_k\}$ and $k\leftarrow k+1$
\EndWhile
\State \textbf{Output:} $Y$ are the collected positions on $\mathcal{S}_{r}$ of the $k$ newly inserted agents to be added to $X(t)$.\\
\State \underline{NOTE}: it is possible that $k=0$ and $Y=\emptyset$, indicating that no new agent will enter at time $t$ due to $\mathcal{S}_{r}$ being overcrowded from the beginning.
\end{algorithmic}
{
    "title": "emergence of collective behaviors from local voronoi topological   perception",
    "id": "2111.03448",
    "abstract": "this article addresses how diverse collective behaviors arise from simple and realistic decisions made entirely at the level of each agent's personal space in the sense of the voronoi diagram. we present a discrete time model in 2d in which individual agents are aware of their local voronoi environment and may seek static target locations. in particular, agents only communicate directly with their voronoi neighbors and make decisions based on the geometry of their own voronoi cells. with two effective control parameters, it is shown numerically to capture a wide range of collective behaviors in different scenarios. further, we show that the voronoi topology facilitates the computation of several novel observables for quantifying discrete collective behaviors. these observables are applicable to all agent-based models and to empirical data.",
    "categories": "physics.soc-ph math.ds q-bio.pe",
    "doi": "",
    "created": "2021-10-29",
    "updated": "2024-02-13",
    "authors": [
        "ivan gonzalez",
        "jack tisdell",
        "rustum choksi",
        "jean-christophe nave"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2111.03448"
}
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