Kinetic Models for Topological Nearest-Neighbor Interactions

Abstract

We consider systems of agents interacting through topological interactions. These have been shown to play an important part in animal and human behavior. Precisely, the system consists of a finite number of particles characterized by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of its closest neighbor, the leader. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit kinetic equation is a non-standard spatial diffusion equation for the particle distribution function. We also study the case wherein the particles interact with their K closest neighbors and show that the corresponding kinetic equation is the same. Finally, we prove that these models can be seen as a singular limit of the smooth rank-based model previously studied in Blanchet and Degond (J Stat Phys 163:41-60, 2016). The proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments.

Optimized leaders strategies for crowd evacuation in unknown environments with multiple exits

“…• Rigorous derivation of the mean-field limit( 13) from ( 1) is a challenging task due to the strong irregularities induced by the behavior of topological-type interactions. We refer to [42] for possible regularization in the case of Cucker-Smale type dynamics, and to [13,30] for alignment driven by jump-type processes. • Alternative derivation of mesoscopic models in presence of diffusion has been obtained in [4], where the authors derived a Fokker-Planck equation of the original microscopic system via quasi-invariant scaling of binary Boltzmann interactions.…”

“…For the numerical solution of the mean-field followers dynamics in (13) we employ mean-field Monte-Carlo methods (MFMCs) generalizing the approaches proposed in [7,49]. These methods fall in the class of fast algorithms developed for interacting particle systems such as direct simulation Monte-Carlo methods (DSMCs), and they are strictly related to more recent class of algorithms named Random Batch Methods (RBMs) [45].…”

“…constrained to the evolution of microscopic (1) or mean-field system (13). We observe that the minimization task for evacuation time or total mass can be extremly difficult, due to the strong irregularity and the presence of many local minima.…”

Topologically-based fractional diffusion and emergent dynamics with short-range interactions

“…n }, over the increased interference in communication with agents farther away, {x j | d N (x i , x j ) 1}. The net effect of probing low density neighborhoods using such singular kernels is communication dictated by the number of nearest agents rather than geometric proximity, [31,6,7]. Letting N → ∞ recovers the topological distance (1.11c) in the continuum setup, d N (x, y) N →∞ −→ d ρ (x, y).…”

Propagation of chaos for topological interactions

“…In [17] mean-field kinetic and fluid models for topological mean-field interactions are formally derived. Recently, [2] and [3] have formally derived kinetic models for jump processes ruled by topological interactions. In the former, the number of particles interacting with a given particle is unbounded in the large particle number limit, while in the latter, particles only interact with a fixed finite number of closest neighbors.…”