Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Dissecting molecular network structures using a network subgraph approach.

Abstract: Biological processes are based on molecular networks, which exhibit biological functions through interactions of genetic elements or proteins. This study presents a graph-based method to characterize molecular networks by decomposing the networks into directed multigraphs: network subgraphs. Spectral graph theory, reciprocity and complexity measures were used to quantify the network subgraphs. Graph energy, reciprocity and cyclomatic complexity can optimally specify network subgraphs with some degree of degeneracy. Seventy-one molecular networks were analyzed from three network types: cancer networks, signal transduction networks, and cellular processes. Molecular networks are built from a finite number of subgraph patterns and subgraphs with large graph energies are not present, which implies a graph energy cutoff. In addition, certain subgraph patterns are absent from the three network types. Thus, the Shannon entropy of the subgraph frequency distribution is not maximal. Furthermore, frequently-observed subgraphs are irreducible graphs. These novel findings warrant further investigation and may lead to important applications. Finally, we observed that cancer-related cellular processes are enriched with subgraph-associated driver genes. Our study provides a systematic approach for dissecting biological networks and supports the conclusion that there are organizational principles underlying molecular networks.

A Note on the Laplacian Eigenvalues

Abstract: This note determines the maximum spectral radius for the Laplacian matrix of a graph with e edges and n vertices.

Spectral Sparse Representation for Clustering: Evolved from PCA, K-means, Laplacian Eigenmap, and Ratio Cut.

Abstract: Dimensionality reduction, cluster analysis, and sparse representation are basic components in machine learning However, their relationships have not yet been fully investigated In this paper, we find that the spectral graph theory underlies a series of these elementary methods and can unify them into a complete framework The methods include PCA, K-means, Laplacian eigenmap (LE), ratio cut (Rcut), and a new sparse representation method developed by us, called spectral sparse representation (SSR) Further, extended relations to conventional over-complete sparse representations (eg, method of optimal directions, KSVD), manifold learning (eg, kernel PCA, multidimensional scaling, Isomap, locally linear embedding), and subspace clustering (eg, sparse subspace clustering, low-rank representation) are incorporated We show that, under an ideal condition from the spectral graph theory, PCA, K-means, LE, and Rcut are unified together And when the condition is relaxed, the unification evolves to SSR, which lies in the intermediate between PCA/LE and K-mean/Rcut An efficient algorithm, NSCrt, is developed to solve the sparse codes of SSR SSR combines merits of both sides: its sparse codes reduce dimensionality of data meanwhile revealing cluster structure For its inherent relation to cluster analysis, the codes of SSR can be directly used for clustering Scut, a clustering approach derived from SSR reaches the state-of-the-art performance in the spectral clustering family The one-shot solution obtained by Scut is comparable to the optimal result of K-means that are run many times Experiments on various data sets demonstrate the properties and strengths of SSR, NSCrt, and Scut

Laplacian Energy of Digraphs and a Minimum Laplacian Energy Algorithm

Abstract: In spectral graph theory, the Laplacian energy of undirected graphs has been studied extensively. However, there has been little work yet for digraphs. Recently, Perera and Mizoguchi (2010) introduced the directed Laplacian matrix L=D−A and directed Laplacian energy LE(G)=∑i=1nλi2 using the second spectral moment of L for a digraph G with n vertices, where D is the diagonal out-degree matrix, and A=(aij) with aij=1 whenever there is an arc (i,j ) from the vertex i to the vertex j and 0 otherwise. They studied the directed Laplacian energies of two special families of digraphs (simple digraphs and symmetric digraphs). In this paper, we extend the study of Laplacian energy for digraphs which allow both simple and symmetric arcs. We present lower and upper bounds for the Laplacian energy for such digraphs and also characterize the extremal graphs that attain the lower and upper bounds. We also present a polynomial algorithm to find an optimal orientation of a simple undirected graph such that the resulting oriented graph has the minimum Laplacian energy among all orientations. This solves an open problem proposed by Perera and Mizoguchi at 2010.

A Spectral Representation of Power Systems with Applications to Adaptive Grid Partitioning and Cascading Failure Localization.

Abstract: Transmission line failures in power systems propagate and cascade non-locally. This well-known yet counter-intuitive feature makes it even more challenging to optimally and reliably operate these complex networks. In this work we present a comprehensive framework based on spectral graph theory that fully and rigorously captures how multiple simultaneous line failures propagate, distinguishing between non-cut and cut set outages. Using this spectral representation of power systems, we identify the crucial graph sub-structure that ensures line failure localization -- the network bridge-block decomposition. Leveraging this theory, we propose an adaptive network topology reconfiguration paradigm that uses a two-stage algorithm where the first stage aims to identify optimal clusters using the notion of network modularity and the second stage refines the clusters by means of optimal line switching actions. Our proposed methodology is illustrated using extensive numerical examples on standard IEEE networks and we discussed several extensions and variants of the proposed algorithm.