Kinetic description of swarming dynamics with topological interaction and transient leaders

TL;DR

A novel stochastic particle method is proposed to simulate the mesoscopic dynamics of birds and reduce the computational cost of identifying closer agents from quadratic to logarithmic complexity using a $k$-nearest neighbours search algorithm with a binary tree.

Abstract

In this paper, we present a model describing the collective motion of birds. The model introduces spontaneous changes in direction which are initialized by few agents, here referred as leaders, whose influence act on their nearest neighbors, in the following referred as followers. Starting at the microscopic level, we develop a kinetic model that characterizes the behaviour of large flocks with transient leadership. One significant challenge lies in managing topological interactions, as identifying nearest neighbors in extensive systems can be computationally expensive. To address this, we propose a novel stochastic particle method to simulate the mesoscopic dynamics and reduce the computational cost of identifying closer agents from quadratic to logarithmic complexity using a $k$-nearest neighbours search algorithm with a binary tree. Lastly, we conduct various numerical experiments for different scenarios to validate the algorithm's effectiveness and investigate collective dynamics in both two and three dimensions.

Figures (21)

• Figure 1: On the left the case in which agent $x$ remains or switches to follower status ($\lambda =0$), and on the right the case in which it remains or switches to leader status ($\lambda = 1$).
• Figure 2: Validation test: initial configuration and its marginals in $x$ and $v$.
• Figure 3: Validation test: comparison between the solution to the kinetic equation in \ref{['eq:boltzmann_ali']} computed by means of Asymptotic Nanbu algorithm \ref{['alg_binary']} and the exact solution in \ref{['eq:exact_sol']}. Mean (dashed line) and standard deviation (shaded area) of the velocity distribution computed over $S=100$ simulations for different values of $p$. From the left to the right $\rho^* = 0.01, 0.35, 0.75$. Markers have been added just to indicate different lines.
• Figure 4: Validation test: $L_2$- norm of the error between the solution to the kinetic equation in \ref{['eq:boltzmann_ali']} simulated by means of the Asymptotic Nanbu Algorithm \ref{['alg_binary']} (one simulation) and the exact solution in \ref{['eq:exact_sol']}. From the left to the right $\rho^* =0.01, 0.35, 1$. Markers represent the error for the different values of $\varepsilon$.
• Figure 5: Comparison between the computational costs of the exhaustive search and the $k$--NN search for different values of $N_s$ and as the subsample percentage size varies from $p=100\%$ to $p=2\%$. From the left to the right $\rho^* = 0.01,0.35,0.75$. Markers represent the computational costs relative to the different values of $N_s$.

Theorems & Definitions (4)

• Remark 1
• Remark 2: Multiple-label case and continuous limit
• Remark 3
• Remark 4