Complex Network Structure of Flocks in the Standard Vicsek Model

Abstract

In flocking models, the collective motion of self-driven individuals leads to the formation of complex spatiotemporal patterns. The Standard Vicsek Model (SVM) considers individuals that tend to adopt the direction of movement of their neighbors under the influence of noise. By performing an extensive complex network characterization of the structure of SVM flocks, we show that flocks are highly clustered, assortative, and non-hierarchical networks with short-tailed degree distributions. Moreover, we also find that the SVM dynamics leads to the formation of complex structures with an effective dimension higher than that of the space where the actual displacements take place. Furthermore, we show that these structures are capable of sustaining mean-field-like orientationally ordered states when the displacements are suppressed, thus suggesting a linkage between the onset of order and the enhanced dimensionality of SVM flocks.

Appendix

Here we derive Eq. (10) for the mean clustering coefficient of a vertex in the bulk of a large cluster. Large flocks generally consist of a core that contains most of the particles and links between them distributed in a highly uniform fashion. Indeed, the near-uniform distribution of nodes and links within flock cores is not only spatial but it also manifests itself in the network’s structure, as for instance shown by the short-tailed degree distributions in Fig. 4. Based on these observations and for the sake of simplicity, we assume that particles in the bulk are distributed homogeneously with a constant density ρ _in.

Let us recall that the clustering coefficient for node i with k _i links is defined as C _i =2n _i /(k _i (k _i −1)), where n _i is the number of links between the k _i neighbors of i. Since particles are distributed uniformly within the interaction radius R ₀=1, it follows straightforwardly that k _i =πρ _in−1. In order to evaluate n _i , let us focus our attention on one of the neighbor nodes of i, which we call node j. The number of nodes that are neighbors of i and j simultaneously is, on average, given by

$$ n_{ij}=\rho_{\mathrm{in}}A_{ij}-2, $$

(14)

where A _ij is the area of the intersection between the interaction circles centered at i and j. A _ij , which depends only on the distance r between i and j, can be expressed as

$$ A_{ij}(r)= 2\int_{-\sqrt{1-\frac{r^{2}}{4}}}^{\sqrt{1-\frac {r^{2}}{4}}}\biggl( \sqrt{1-x^{2}}-\frac{r}{2}\biggr) dx. $$

(15)

Therefore, the number of links between the k _i neighbors of i is obtained by replacing Eq. (15) in Eq. (14) and integrating ρ _in n _ij /2 over the unit circle (notice that we divide by 2 because we are dealing with undirected links, so we must count each pair of neighbor nodes just once), i.e.

$$ n_i=\pi\rho_{\mathrm{in}}\int_0^1 rn_{ij}(r)dr. $$

(16)

Solving the integrals and replacing in the definition of the clustering coefficient, we finally arrive at

$$ C_\infty= \frac{[(4\pi-3\sqrt{3})\rho_{\mathrm{in}}-8]\pi\rho_{\mathrm{in}}}{4(\pi\rho _{\mathrm{in}}-1)(\pi\rho_{\mathrm{in}}-2)}, $$

(17)

which provides an analytic solution for the mean clustering coefficient of particles in the bulk of a large cluster.