Vicsek Model Meets DBSCAN: Cluster Phases in the Vicsek Model

TL;DR

This work analyzes clustering in the Vicsek model by applying DBSCAN and contrasting with Mean Shift, uncovering a phase transition in the number of clusters from $O(N)$ to $O(1)$ as noise grows at fixed interaction radius. It builds a mathematical link between the Vicsek potential and DBSCAN's cost function, and defines new order parameters, including the cluster-structure ratio and intra-cluster order, to identify multiple cluster-based phases. The study demonstrates phase diagrams in $(r_V,\eta)$, reveals system-size scaling of cluster counts, and shows that A2-type phases can exist only under specific conditions, while A2' does not occur in the Vicsek model. Together, these results bridge active-matter dynamics with density-based clustering, offering a framework to classify flocking and clustering phenomena with cluster-aware thermodynamic-like metrics.

Abstract

The Vicsek model, which was originally proposed to explain the dynamics of bird flocking, exhibits a phase transition with respect to the absolute value of the mean velocity. Although clusters of agents can be easily observed via numerical simulations of the Vicsek model, qualitative studies are lacking. We study the clustering structure of the Vicsek model by applying DBSCAN, a recently-introduced clustering algorithm, and report that the Vicsek model shows a phase transition with respect to the number of clusters: from O(N) to O(1), with N being the number of agents, when increasing the magnitude of noise for a fixed radius that specifies the interaction of the Vicsek model. We also report that the combination of the order parameter proposed by Vicsek et al. and the number of clusters defines at least four phases of the Vicsek model.

Figures (23)

• Figure 1: (Color online) Schematics of DBSCAN. (a) Red, blue, green points are core points, border points, and outliers, respectively. Red and blue points form one cluster. These definitions are consistent with setting $n_\mathrm{min}$ to any value in the set $\{2, 3, 4\}$. (b) Red and blue data points form clusters and green data points are outliers. DBSCAN is applicable to linearly nonseparable datasets. These clustering results correspond to setting $n_\mathrm{min}$ to any value in the set $\{3, 4, \cdots, 15\}$.
• Figure 2: (Color online) Schematic of the phases of the Vicsek model: (A1) non-Vicsek-ordered state with no clusters, (A2, A2') non-Vicsek-ordered state with multiple clusters, (A3) non-Vicsek-ordered state with one cluster, (B1) Vicsek-ordered state with no clusters, (B2) Vicsek-ordered state with multiple clusters, and (B3) Vicsek-ordered state with one cluster. The difference between phase A2 and phase A2' is whether agents within clusters are Vicsek-ordered or not. Note that we call a state to be non-Vicsek-ordered if the global average velocity is zero, even if each cluster may have ordered motion.
• Figure 3: (Color online) Snapshots of the Vicsek model with DBSCAN for $r_\mathrm{V} = r_\mathrm{D} = 0.4$. We set (upper) $\eta = 1.0$ and (lower) $\eta = 4.0$. We also set $n_\mathrm{ag} = 400$, $L = 10.0$, $\Delta t = 1.0$, $v_\mathrm{abs} = 0.030$, $n_\mathrm{min} = 5$, and $t = 100$. Agents are colored based on their clusters and thin black arrows do not belong to any clusters. Here, $n_\mathrm{ag}$ is the total number of agents.
• Figure 4: (Color online) Snapshots of the Vicsek model with DBSCAN for $r_\mathrm{V} = r_\mathrm{D} = 1.0$. We set (upper) $\eta = 1.0$ and (lower) $\eta = 4.0$. We also set $n_\mathrm{ag} = 400$, $L = 10.0$, $\Delta t = 1.0$, $v_\mathrm{abs} = 0.030$, $n_\mathrm{min} = 5$, and $t = 100$. Agents form a single cluster independently from the magnitude of $\eta$.
• Figure 5: (Color online) Snapshots of the Vicsek model with mean shift for $r_\mathrm{V} = r_\mathrm{m} = 0.4$. We set (upper) $\eta = 1.0$ and (lower) $\eta = 4.0$. We also set $n_\mathrm{ag} = 400$, $L = 10.0$, $\Delta t = 1.0$, $v_\mathrm{abs} = 0.030$, $n_\mathrm{min} = 5$, and $t = 100$. The threshold is set to be $10 \%$. Agents are colored based on their clusters and thin black arrows belong to clusters in which the number of agents is less than $n_\mathrm{ag}$. Here, $n_\mathrm{ag}$ is the total number of agents.