Evidence of a robust universality class in the critical behavior of self-propelled agents: Metric versus topological interactions

The nature of the interactions among self-propelled agents (SPA), i.e., topological versus metric or a combination of both types, is a relevant open question in the field of self-organization phenomena. We studied the critical behavior of a Vicsek-like system of SPA given by a group of agents moving at constant speed and interacting among themselves under the action of a topological rule: each agent aligns itself with the average direction of its seven nearest neighbors, independent of the distance, under the influence of some noise. Based on both stationary and dynamic measurements, we provide strong evidence that both types of interactions are manifestations of the same phenomenon, which defines a robust universality class. Also, the cluster size distribution evaluated at the critical point shows a power-law behavior, and the exponent corresponding to the topological model is in excellent agreement with that of the metric one, further reinforcing our claim. Furthermore, we found that with topological interactions the average distance of influence between agents undergoes large fluctuations that diverge at the critical noise, thus providing clues about a mechanism that could be implemented by the agents to change their moving strategy.

Figure 1

Stationary measurements of (a) the order parameter and (b) susceptibility, corresponding to four system sizes.

Figure 2

Stationary measurements of the Binder cumulant, corresponding to four system sizes. (a) At the critical noise ${\eta }_c=0.03975\left(25\right)$, the cumulant has the same value for all system sizes. The variable $ln(2/3-U)$ was chosen to depict smooth curves. (b) Overview of the Binder cumulant for the whole range of the order-disorder transition.

Figure 3

(a) Scaling plot of the peak position of the susceptibility vs system size, according to ${\eta }_{\max }\left(L\right)={\eta }_c+\mathrm{const}{L}^{-1/1.34}$. The extrapolation to $L\to \infty$ gives ${\eta }_c=0.0397\pm 0.0001$. (b) Log-log plot of the maximum value of the susceptibility vs $L^{\gamma /\nu }$, which allows us to determine $\gamma /\nu =1.51\left(3\right)$ (solid line). The dashed line corresponds to $\gamma /\nu =1.45\left(3\right)$, which was already reported for the metric model [6].

Figure 4

Dynamic measurement for a system of 65 536 agents. (a) The rise in the order of the system from a randomly disordered initial state gives $\gamma /z\nu =1.008\left(9\right)$. (b) The second-order Binder cumulant provides $d/z=1.55\left(3\right)$. (c) The logarithmic derivative of $\varphi$ gives $1/z\nu =0.58\left(6\right)$. (d) The time evolution of the order parameter gives $\beta /z\nu =0.265\left(5\right)$.

Figure 5

Scaling plots corresponding to the stationary measurements already shown in Figs. 1 and 2. (a) Scaling of the order parameter according to $\varphi L^{\beta /\nu }$ vs $\left(\right|\eta -{\eta }_c\left|L^{1/\nu }\right)/{\rho }^{1/2}$. (b) Susceptibility, namely, log-log plots of $\chi L^{\gamma /\nu }$ vs $\left(\right|\eta -{\eta }_c\left|L^{1/\nu }\right)/{\rho }^{1/2}$. (c) Binder cumulant versus $(\eta -{\eta }_c)L^{1/\nu }$.

Figure 6

Snapshot of a typical configuration close to the critical point for $N=32000$. An incipient traveling band (right-hand side) is formed by correlated clusters. Agents belonging to the same cluster are labeled with the same color.

Figure 7

Cluster size distribution $p\left(m\right)$ vs the cluster size $m$. A power-law fitting (solid line) gives the exponent $\zeta =1.3\left(1\right)$.

Figure 8

(a) Mean radius $\langle r\rangle$ required by an agent for a topological interaction with seven neighbors near the critical point ${\eta }_c=0.03975$. Note that $\langle r\rangle$ has a maximum and a minimum located on both sides of the critical noise (vertical dashed line), which are more pronounced for larger system sizes, as indicated by the arrow. The inset shows the dependence of $\langle r\rangle$ on the noise for the whole domain explored. (b) Plots of the fluctuations of $\langle r\rangle$ vs the noise. The top and bottom insets show the scaling behavior of the maximum fluctuation and its position, respectively. Further details are given in the text.