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 nonrecurrent 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 subsystem 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.
Highlights
Given that complex engineering systems are constructed by composing various subsystems and components that interact with one another, it is common practice in modern engineering design to consider the directed interconnectivity graph as a representation of the underlying system [1]
Inspired by the above argument, we develop a class of complexity metrics based on the algebraic properties of a matrix that represents the underlying directed graph
The use of the “counting” of eigenvaues with θj = 0 in the second term of F makes the spectral complexity measure have some features of discrete metrics, as the following example shows: Example 5 (Spectral complexity in a class of recurrent 2-graphs) We consider graphs with 2 elements that have both a self loop and an edge connecting them to the other element, with uniform probabilities as shown in figure 1
Summary
Given that complex engineering systems are constructed by composing various subsystems and components that interact with one another, it is common practice in modern engineering design to consider the directed interconnectivity graph as a representation of the underlying system [1]. Based on the above spectral complexity approach, we develop a novel graph decomposition technique that is based on cyclic interaction between subsystems and does not resort to symmetrization of the underlying matrices. This approach facilitates the identification of strongly interacting subsystems that can be used for design and analysis of complex systems. One computes the eigenvector corresponding to the smallest non-zero eigenvalue of the Laplacian matrix This eigenvector is known as the Fiedler vector [6] and is related to the minimum cut in undirected graphs [26, 27].
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