Graph Spectra for Complex Networks

Abstract

Analyzing the behavior of complex networks is an important element in the design of new man-made structures such as communication systems and biologically engineered molecules. Because any complex network can be represented by a graph, and therefore in turn by a matrix, graph theory has become a powerful tool in the investigation of network performance. This self-contained 2010 book provides a concise introduction to the theory of graph spectra and its applications to the study of complex networks. Covering a range of types of graphs and topics important to the analysis of complex systems, this guide provides the mathematical foundation needed to understand and apply spectral insight to real-world systems. In particular, the general properties of both the adjacency and Laplacian spectrum of graphs are derived and applied to complex networks. An ideal resource for researchers and students in communications networking as well as in physics and mathematics.

Dynamic state determination of a software-defined network via dual basis representation

“…The full mesh is intended to model a service provider's core network, and the outlying nodes are the access network onto which the customers are connected. The Laplacian matrix is a well-documented graph theory matrix representation of networks [9]. The Laplacian matrix is a combination of the adjacency matrix, A , which describes how nodes are connected, and the degree matrix, D , which describes how many links connect each node to the network [7].…”

“…The Laplacian matrix is positive semi-definite; therefore, all eigenvalues are real, and there will always be at least one zero eigenvalue [9]. The eigenvalues are typically ordered from smallest to largest by…”

“…This maximization only minimizes the difference between the largest and smallest eigenvalues. In an unweighted graph, the largest eigenvalue is bound by [9] n n λ ≤ (5) Eqn. (5) indicates that the maximum eigenvalue is bounded by the number of nodes, and not links or link weights.…”

“…The right eigenvector's dual is the left eigenvector which can be demonstrated by rearranging (9) to result in ( )…”

A Concurrent Object-Oriented Approach to the Eigenproblem Treatment in Shared Memory Multicore Environments

“…Thus, it appears in problems such as the quantum mechanical treatment of time independent systems [1], in the principal components analysis (PCA) [2], in the specific application of PCA to face recognition (eigenfaces) or in the computation of the eigenvalues of a graph in spectral graph theory applied to complex networks [3]. The eigenproblem implies, in practical terms, the computation of the eigenvalues and eigenvectors of a matrix [4].…”