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

TL;DR

This paper introduces a novel class of models for emergent flocking behavior based on topological, short-range, and anisotropic communication protocols, with rigorous proofs of flocking and regularity.

Contribution

It develops a new framework for emergent dynamics using topological neighborhoods and proves flocking and regularity results, including a novel analysis for local fractional elliptic operators.

Findings

Proves flocking behavior in models with short-range topological communication.

Establishes global regularity for one-dimensional models.

Develops new analysis techniques for non-symmetric singular kernels.

Abstract

We introduce a new class of models for emergent dynamics. It is based on a new communication protocol which incorporates two main features: short-range kernels which restrict the communication to local geometric balls, and anisotropic communication kernels, adapted to the local density in these balls, which form topological neighborhoods. We prove flocking behavior -- the emergence of global alignment for regular, non-vacuous solutions of the $n$-dimensional models based on short-range topological communication. Moreover, global regularity (and hence unconditional flocking) of the one-dimensional model is proved via an application of a De Giorgi-type method. To handle the non-symmetric singular kernels that arise with our topological communication, we develop a new analysis for local fractional elliptic operators, interesting for its own sake, encountered in the construction of our class of models.