Emergence of Collective Behaviors from Local Voronoi Topological Perception

TL;DR

A discrete time model in 2D in which individual agents are aware of their local Voronoi environment and may seek static target locations and communicate directly with their Voronoi neighbors and make decisions based on the geometry of their own Voronoi cells.

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.

Figures (8)

• Figure 0: A Voronoi diagram and dual graph. The Voronoi diagram generated by a set of points, consisting of the solid bordered regions, and its dual graph (dotted red) offer a natural communication topology for agent-based models and also give rise to many broadly applicable observables. The Voronoi (dual) topology differs from other communication networks---in particular $k$-nearest neighbor---in several respects. E.g., focusing on the encircled site, its second-nearest site is not among its Voronoi neighbors at all. Moreover, different sites generally have different numbers of Voronoi neighbors.
• Figure 1: Schematic of the influences on a generic agent at time $t$. Here we show one agent $i$ at position $\mathbf{x}_i$ as well as its Voronoi cell and Voronoi neighbors whose positions are marked with black dots. We illustrate the three components which influence $i$'s motion in the triptych above. Repulsion $\hat{\mathbf{r}}_i$ and homing $\hat{\mathbf{h}}_i$ are weighted with coefficients $\sigma_i=\sigma(\delta_i/L)$ and convex complement $1-\sigma_i=1 - \sigma(\delta_i/L)$, respectively, where $\delta_i$ is the distance to $i$’s nearest neighbor, as shown in (b) above. The relative weight of alignment $\textcolor{blue}{\mathbf{a}_i}$ is given by the parameter $\nu$.
• Figure 2: At each time step, the personal space of the $i$-th agent located at $\mathbf{x}_i$ and its Voronoi-neighboring agents (the position of a generic neighbor is labeled as $\mathbf{x}_j$). The desired direction vector $\mathbf{d}_i$ associated with the $i$-th agent determines the frontal area $F_i$ and frontal distance $\ell_i$ used to evaluate the personal-space speed $\rho$ in (\ref{['eq:speed_scl']}) and (\ref{['eq:speed_scl-2']}) for Models I and II respectively.
• Figure 3: Pinwheel (a), ring (b), and aligned orbiting cluster (c) for $n = 700$ agents under Model II. The red crosshair indicates the target point in each figure. Click the plots to run corresponding simulations.
• Figure 4: Example of the breathing regime observed under Model I. Here there are $n = 700$ agents and the alignment strength is $\nu = 8$. The curve (black) is the median radius of all agents (against time), i.e., the median distance to the center of mass of the swarm. The secondary curve (green) is the Voronoi pressure. Each is nondimensionalized with a suitable power of $L$ (although here $L = 1$). The initial spike in pressure is clipped for space but the maximum is approximately $60$. Click the plot to run a corresponding simulation.