Spectral Complexity of Directed Graphs and Application to Structural Decomposition

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.

Dynamics and Robust Control of a New Realizable Chaotic Nonlinear Model

“…e eigenvalues of the matrix in (9) can be used to compute the energy of the digraph G (see [19]). To compute the energy of the proposed graph, we follow the method explained in [19] (page 6).…”

“…where |E| is the number of edges in G, w |E| is the edge weights, and SVD(M(G)) is a vector of singular values of matrix. For more details about graph energy, see [19]. Let λ 1 , λ 2 , .…”

“…It helps in understanding the structure and the complexity of the paradigm using the tools of the graph theory [18]. In directed graph theory, finding directed cycles in the graph is known as a common source of complexity [19]. ose cycles are significant in the context of engineering structures.…”

“…For instance, cycles can produce positive feedback loops [20], which drive the system to be unstable. Cycles in engineering paradigms also increase the complexity of design and analyse the context of simulation convergence [19]. ere are several studies in the literature on graph complexity criteria [19].…”

“…Cycles in engineering paradigms also increase the complexity of design and analyse the context of simulation convergence [19]. ere are several studies in the literature on graph complexity criteria [19]. Such criteria can either directly or indirectly be linked to the values of eigenvalues of the studied graph matrices.…”