Nonlocal network dynamics via fractional graph Laplacians

TL;DR

This paper introduces a fractional Laplacian for directed networks, enabling nonlocal dynamics and random walks with jumps, with applications to consensus in multi-agent systems.

Contribution

It develops a novel fractional Laplacian for directed graphs and analyzes its properties, expanding the tools for studying nonlocal network dynamics.

Findings

Fractional Laplacian enables nonlocal network analysis.

Random walks with arbitrary jump lengths are possible.

Applications include consensus in multi-agent systems.

Abstract

We introduce nonlocal dynamics on directed networks through the construction of a fractional version of a nonsymmetric Laplacian for weighted directed graphs. Furthermore, we provide an analytic treatment of fractional dynamics for both directed and undirected graphs, showing the possibility of exploring the network employing random walks with jumps of arbitrary length. We also provide some examples of the applicability of the proposed dynamics, including consensus over multi-agent systems described by directed networks.