Exploring the “Middle Earth” of network spectra via a Gaussian matrix function

We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. We motivate the use of this function on the basis of a dynamical process modeled by the time-dependent Schrodinger equation with a squared Hamiltonian. In particular, we study the Gaussian Estrada index - an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Such method is a generalization of the so-called "Folded Spectrum Method" used in quantum molecular sciences. Here we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. We also show that in real-world networks this index is related to the existence of important structural patters, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs

Eigenvector-based identification of bipartite subgraphs

We report our experiments in identifying large bipartite subgraphs of simple connected graphs which are based on the sign pattern of eigenvectors belonging to the extremal eigenvalues of different graph matrices: adjacency, signless Laplacian, Laplacian, and normalized Laplacian matrix. We compare the performance of these methods to a local switching algorithm based on the Erdos bound that each graph contains a bipartite subgraph with at least half of its edges. Experiments with one scale-free and three random graph models, which cover a wide range of real-world networks, show that the methods based on the eigenvectors of the normalized Laplacian and the adjacency matrix yield slightly better, but comparable results to the local switching algorithm. We also formulate two edge bipartivity indices based on the former eigenvectors, and observe that the method of iterative removal of edges with maximum bipartivity index until one obtains a bipartite subgraph, yields comparable results to the local switching algorithm, and significantly better results than an analogous method that employs the edge bipartivity index of Estrada and Gomez-Gardenes.Comment: 20 pages, 8 figure

Gaussianization of the spectra of graphs and networks : theory and applications

Matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks, such as communicability, node centrality, bipartivity and many more. Here we propose a new matrix function based on the Gaussianization of the adjacency matrix of a graph. This function gives more weight to a selected reference eigenvalue λref, which may be located in any region of the graph spectra. In particular, we study the Gaussian Estrada indices for two reference eigenvalues 0 and -1 separately. In each case, we obtain bounds for this index in simple graphs. We also obtain formulas for the Gaussian Estrada index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. Moreover, for λref = 0, we show that in real-world networks this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of evolutionary processes giving rise to them. In addition, we fold the graph spectrum at a given pair of reference eigenvalues, then exponentiate the resulting folded graph spectrum to produce the double Gaussian function of the graph adjacency matrix which give more importance to the reference eigenvalues than to the rest of the spectrum. Based on evidence from mathematical chemistry we focus here our attention on the reference eigenvalues ±1. They enclose most of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of organic molecular graphs. We prove several results for the trace of the double Gaussian adjacency matrix of simple graphs - the double Gaussian Estrada index - and we apply this index to the classification of polycyclic aromatic hydrocarbons (PAHs) as carcinogenic or non-carcinogenic. We discover that local indices based on the previously developed matrix function allow to classify correctly 100% of the PAHs analyzed. In general, folding the spectrum of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs.Matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks, such as communicability, node centrality, bipartivity and many more. Here we propose a new matrix function based on the Gaussianization of the adjacency matrix of a graph. This function gives more weight to a selected reference eigenvalue λref, which may be located in any region of the graph spectra. In particular, we study the Gaussian Estrada indices for two reference eigenvalues 0 and -1 separately. In each case, we obtain bounds for this index in simple graphs. We also obtain formulas for the Gaussian Estrada index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. Moreover, for λref = 0, we show that in real-world networks this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of evolutionary processes giving rise to them. In addition, we fold the graph spectrum at a given pair of reference eigenvalues, then exponentiate the resulting folded graph spectrum to produce the double Gaussian function of the graph adjacency matrix which give more importance to the reference eigenvalues than to the rest of the spectrum. Based on evidence from mathematical chemistry we focus here our attention on the reference eigenvalues ±1. They enclose most of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of organic molecular graphs. We prove several results for the trace of the double Gaussian adjacency matrix of simple graphs - the double Gaussian Estrada index - and we apply this index to the classification of polycyclic aromatic hydrocarbons (PAHs) as carcinogenic or non-carcinogenic. We discover that local indices based on the previously developed matrix function allow to classify correctly 100% of the PAHs analyzed. In general, folding the spectrum of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs

Spectral graph theory : from practice to theory

Graph theory is the area of mathematics that studies networks, or graphs. It arose from the need to analyse many diverse network-like structures like road networks, molecules, the Internet, social networks and electrical networks. In spectral graph theory, which is a branch of graph theory, matrices are constructed from such graphs and analysed from the point of view of their so-called eigenvalues and eigenvectors. The first practical need for studying graph eigenvalues was in quantum chemistry in the thirties, forties and fifties, specifically to describe the Hückel molecular orbital theory for unsaturated conjugated hydrocarbons. This study led to the field which nowadays is called chemical graph theory. A few years later, during the late fifties and sixties, graph eigenvalues also proved to be important in physics, particularly in the solution of the membrane vibration problem via the discrete approximation of the membrane as a graph. This paper delves into the journey of how the practical needs of quantum chemistry and vibrating membranes compelled the creation of the more abstract spectral graph theory. Important, yet basic, mathematical results stemming from spectral graph theory shall be mentioned in this paper. Later, areas of study that make full use of these mathematical results, thus benefitting greatly from spectral graph theory, shall be described. These fields of study include the P versus NP problem in the field of computational complexity, Internet search, network centrality measures and control theory.peer-reviewe

Graphs and networks theory

This chapter discusses graphs and networks theory

Edge modification criteria for enhancing the communicability of digraphs

We introduce new broadcast and receive communicability indices that can be used as global measures of how effectively information is spread in a directed network. Furthermore, we describe fast and effective criteria for the selection of edges to be added to (or deleted from) a given directed network so as to enhance these network communicability measures. Numerical experiments illustrate the effectiveness of the proposed techniques.Comment: 26 pages, 11 figures, 4 table