Spectral Complexity of Directed Graphs and Application to Structural Decomposition

TL;DR

This paper introduces a spectral complexity measure for directed graphs that captures cycle structures and applies it to structural decomposition, aiding in understanding system interdependencies and stability.

Contribution

It proposes a novel spectral complexity metric based on the spectrum of the recurrence matrix and Wasserstein distance, along with a spectral decomposition technique for directed graphs.

Findings

Spectral complexity increases with the average degree in random graphs.

The metric effectively captures directed cycles and interdependencies.

Application to an aircraft system demonstrates practical utility.

Abstract

We introduce a new measure of complexity (called spectral complexity) for directed graphs. We start with splitting of the directed graph into its recurrent and non-recurrent parts. We define the spectral complexity metric in terms of the spectrum of the recurrence matrix (associated with the reccurent part of the graph) and the Wasserstein distance. We show that the total complexity of the graph can then be defined in terms of the spectral complexity, complexities of individual components and edge weights. The essential property of the spectral complexity metric is that it accounts for directed cycles in the graph. In engineered and software systems, such cycles give rise to sub-system interdependencies and increase risk for unintended consequences through positive feedback loops, instabilities, and infinite execution loops in software. In addition, we present a structural decomposition technique that identifies such cycles using a spectral technique. We show that this decomposition complements the well-known spectral decomposition analysis based on the Fiedler vector. We provide several examples of computation of spectral and total complexities, including the demonstration that the complexity increases monotonically with the average degree of a random graph. We also provide an example of spectral complexity computation for the architecture of a realistic fixed wing aircraft system.