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

Abstract

We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. 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. 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 Erdős-Rï¿oenyi random graphs as well as for the Barabï¿oesi-Albert graphs. We also 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 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. 2The spectrum of a network-the set of its eigenvalues-provides important information about the structural and dynamical properties of the corresponding system. Most of the functions used to study network spectra give more weight to the largest modular eigenvalues. Then, the information contained in the eigenvalues close to the centre of the spectra, i.e, those close to zero, has remained totally unexplored in the study of graph spectra. Here we study a Gaussian matrix function that gives more weights to the eigenvalues closest to the centre of the spectrum of a network. Using this function we extract important structural information hidden in the spectra of networks, such as emergence of complete bipartite subgraphs (bicliques) which appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. These bicliques are also ubiquitous in random networks generated by preferential attachment mechanisms, such as the Barabï¿oesi-Albert model. In this work we provide a series of analytical results that pave the way for further analysis and uses of this Gaussian matrix function to understand network structure and dynamics.3

On Generalized Distance Gaussian Estrada Index of Graphs

“…Estrada index H(G) [26] has the merit of encoding the information hidden in the eigenvalues close to zero which are overlooked in other Estrada indices. It has also played an essential role in quantum mechanics [27].…”

“…In this sense, information that is hidden in the smaller eigenvalues, which are particularly useful, for example, in the context of molecular orbital theory [25], has been overlooked. Estrada et al [26] recently proposed to scope this bit of information by using a Gaussian matrix function, which gives rise to the Gaussian Estrada index, H(G). H(G) can be defined as…”

Lower Bounds for Gaussian Estrada Index of Graphs

“…For example, electron transfers from the highest occupied molecular orbital of one molecule to the lowest unoccupied molecular orbital of another molecule play a vital part in several organic chemical reactions; see [20] for a survey. As such, Estrada et al, [21] recently propose to extract key structural information hidden in the eigenvalues in proximity to zero in the spectra of networks by using a Gaussian matrix function. This novel method leads to the Gaussian Estrada index, H(G), characterized as follows:…”

“…In fact, unlike EE which gives more weight to the large eigenvalues, H stresses those close to zero. As shown via numerical simulations in [21], H is able to distinguish between the dynamics of a particle hopping over a bipartite network from the one hopping over a non-bipartite network. This is impossible for EE as the large eigenvalues are usually not correlated with the emergence of bipartite structure.…”

“…The Gaussian Estrada index of some simple graphs including complete graphs, paths, cycles, and Erdős-Rényi random graphs as well as BA random networks has been studied in [21]. Signify the star graph on n vertices by K 1,n−1 .…”