Linear response to leadership, effective temperature, and decision making in flocks

Large collections of autonomously moving agents, such as animals or micro-organisms, are able to flock coherently in space even in the absence of a central control mechanism. While the direction of the flock resulting from this critical behavior is random, this can be controlled by a small subset of informed individuals acting as leaders of the group. In this article we use the Vicsek model to investigate how flocks respond to leadership and make decisions. Using a combination of numerical simulations and continuous modeling we demonstrate that flocks display a linear response to leadership that can be cast in the framework of the fluctuation-dissipation theorem, identifying an effective temperature reflecting how promptly the flock reacts to the initiative of the leaders. The linear response to leadership also holds in the presence of two groups of informed individuals with competing interests, indicating that the flock's behavioral decision is determined by both the number of leaders and their degree of influence.

Figure 1

(a) Example of how the perturbation is applied to the flock. In this sketch $N_l=4$ leaders out of $N=12$ individuals turn by an angle ${\varphi }_l$, thus changing their direction of motion from the black to the red dashed line. (b) Snapshot of a typical simulation. Shown is the trajectory of one of the perturbed particles (red), a normal unperturbed particle (black), and a tracer particle inserted into the system that has $\eta =0$ (blue). This represents 200 consecutive time steps from a simulation of $N=1600$ particles with $\rho =16, \eta =0.25, N_l=50$, and ${\varphi }_l=0.1$.

Figure 2

(a) Probability distribution function (PDF) of the discrete curvature $\kappa$ defined in Eq. (2) for various effective torque and noise values. The inset shows the PDF of $(\kappa -{\langle \kappa \rangle }_{\tau })/\eta$ normalized by the noise standard deviation $\eta$. The data collapse on the same Gaussian distribution. (b) The mean curvature ${\langle \kappa \rangle }_{\tau }$, of the trajectory of the flock is linear with the torque per individual $\tau /N$ (the black line shows ${\langle \kappa \rangle }_{\tau }\sim \tau /N$). (c) Conversely, the total mean-square curvature $L^2{\langle {\kappa }^2\rangle }_{\tau }$ remains constant for small $\tau$ values. (d) The rotational diffusion coefficient $D_r$, defined in Eq. (3), follows a universal power-law relationship $D_r\sim {\eta }^2$ at high density.

Figure 3

(a) and (c) Induced mean and (b) and (d) standard deviation curvature in the motion of the flock due to competing subsets of informed individuals. Here $N_l$ and $N_r$ individuals give the flock a positive and a negative curvature, respectively, with $N_i=N_l+N_r$. (a) and (b) When the competing subsets have equal and opposite influence (${\varphi }_l=-{\varphi }_r$) the resulting curvature is of the same sign as the largest informed subset and the inclusion of uninformed individuals reduces the standard deviation of the curvature. (c) and (d) When ${\varphi }_r=-2{\varphi }_l$ the resulting curvature is distorted and the relative sizes of the informed subsets are no longer sufficient to predict the curvature. The black lines in (a) and (c) show ${\langle \kappa \rangle }_{\tau }=-0.04,0,+0.04$ and in (b) and (d) show ${\langle {\kappa }^2\rangle }_{\tau }=0.1$; all simulations here are done with $N=900, \rho =4, v_0=0.1$, and ${\varphi }_l=0.1$.

Figure 4

(a) Resulting curvature of flocks with $N_l-N_r$ fixed and varying values of ${\varphi }_r$. As is clear here, the resulting curvature is not just the function of the relative sizes of the informed subsets; rather it is the total torque. (b) The ratio between the resulting curvature and the total torque is constant across all simulations, indicating that this is the only important parameter here. All simulations here are done with $N=900, \rho =4, v_0=0.1, {\varphi }_l=0.1$, and $N_l-N_r=50$.

Figure 5

(a) Flock angular displacement $\mathrm{\Delta }\mathrm{\Theta }$, with $\mathrm{\Theta }$ defined from $\mathit{p}=(cos\mathrm{\Theta },sin\mathrm{\Theta })$, as a function of time. (b) Mean-square angular displacement versus time illustrating the long-time diffusive behavior. (c) Probability distribution function for the discrete curvature $\kappa$. All results here are for systems of $N=400$ particles with $\rho =4$ and the data were taken over $320\times {10}^4$ time steps after a $150\times {10}^3$-step relaxation period.

Figure 6

(a) Mean curvature and (b) mean-square curvature versus torque. (c) Rotational diffusion coefficient versus noise. All these quantities depend exclusively on the torque $\tau ={\varphi }_l{N}_l$, regardless of whether it is adjusted by varying the number of leaders $N_l$ or their degree of influence ${\varphi }_l$. (a) and (b) both show the results for $N=900$ and $\eta =0.1$. The black and red points represent simulations where $N_l=10$ and ${\varphi }_l=0.1$, respectively. All results here are for particles with $\rho =4$ and $v_0=0.1$ and the data were taken over $150\times {10}^3$ time steps after a ${10}^4$-step relaxation period.

Figure 7

(a) Mean curvature and (b) mean-square curvature versus torque. (c) Rotational diffusion coefficient versus noise. All these quantities are independent on the particle speed $v_0$. (a) and (b) both show the results for $N=900$ and $\eta =0.1$. All results here are for systems with $\rho =4$ and the data were taken over $150\times {10}^3$ time steps after a ${10}^4$-step relaxation period.

Figure 8

(a) Mean curvature and (b) mean-square curvature in the presence of two competing subsets of leaders: $N_l$ with influence ${\varphi }_l$ and $N_r$ with influence ${\varphi }_r$; here we have set ${\varphi }_r=-1.5{\varphi }_l$. When the curvature is normalized by the total torque applied to the system all simulations give the same response, hence the flock is acting like a thermal bath with a linear response to leadership. All results here are for systems with $N=900, v_0=0.1$, and $\rho =4$ and the data were taken over $150\times {10}^3$ time steps after a ${10}^4$-step relaxation period.