On Directed Graphs With Real Laplacian Spectra

TL;DR

This paper explores the topological features of directed graphs that determine whether their Laplacian spectra are real or complex, providing conditions for designing graphs with desired spectral properties to improve dynamical system performance.

Contribution

It derives new sufficient conditions for digraphs to have real Laplacian spectra and identifies key topological factors influencing spectral types, extending analysis to multilayer digraphs.

Findings

Real spectra linked to absence of digon sign-asymmetric interactions

Complex spectra associated with directed cycles

Strategies for topology redesign to control spectral properties

Abstract

It is reported that dynamical systems over digraphs have superior performance in terms of system damping and tolerance to time delays if the underlying graph Laplacian has a purely real spectrum. This paper investigates the topological conditions under which digraphs possess real or complex Laplacian spectra. We derive sufficient conditions for digraphs, which possibly contain self-loops and negative-weighted edges, to have real Laplacian spectra. The established conditions generally imply that a real Laplacian spectrum is linked to the absence of the so-called digon sign-asymmetric interactions and non-strong connectivity in any subgraph of the digraph. Then, two classes of digraphs with complex Laplacian spectra are identified, which imply that the occurrence of directed cycles is a major factor to cause complex Laplacian eigenvalues. Moreover, we extend our analysis to multilayer digraphs, where strategies for preserving real/complex spectra from graph interconnection are proposed. Numerical experiments demonstrate that the obtained results can effectively guide the redesign of digraph topologies for a better performance.