Laplacian spectra of a class of small-world networks and their applications

Abstract

One of the most crucial domains of interdisciplinary research is the relationship between the dynamics and structural characteristics. In this paper, we introduce a family of small-world networks, parameterized through a variable d controlling the scale of graph completeness or of network clustering. We study the Laplacian eigenvalues of these networks, which are determined through analytic recursive equations. This allows us to analyze the spectra in depth and to determine the corresponding spectral dimension. Based on these results, we consider the networks in the framework of generalized Gaussian structures, whose physical behavior is exemplified on the relaxation dynamics and on the fluorescence depolarization under quasiresonant energy transfer. Although the networks have the same number of nodes (beads) and edges (springs) as the dual Sierpinski gaskets, they display rather different dynamic behavior.

Eigenvalues for the transition matrix of a small-world scale-free network: Explicit expressions and applications

“…In recent years, extensive attention has been focused on the study of spectra for various matrices of complex networks [14][15][16], such as the adjacency matrix [17][18][19], the Laplacian matrix [20][21][22][23][24], the modularity matrix [25,26], and the nonback-tracking matrix [27]. In the context of the transition matrix, a conscientious effort has been devoted to determine the eigenvalues for some classic fractals [28][29][30][31] and treelike networks [32][33][34][35].…”

Temperature dependent network stability in simple alcohols and pure water: The evolution of Laplace spectra

“…Here S, Sc are two non-empty subsets of G, vol(S) is the sum of the edges, and E(S,Sc) is the sum of the edges within S and Sc. This inequality is related to the minimum number of edges such that, when removed, cause the graph to become disconnected ('non-percolated' in the world of Hbonded networks) [11][12][13][14]. h(G) (or a similar derived quantity like 2*nhb) can therefore serve as possible metrics to measure the 'distance' from the percolation transition.…”

“…A large number of concepts have been introduced for classifying network structures, for example degree distribution, path length, clustering, percolation, the small world property, spectral density of graph, ring structures, etc. During the last years it has become clear that the Laplacian eigenvalues and eigenvectors play an important role in revealing the multiple aspects of the characteristics of network structure and dynamics, like spanning trees, resistance distances and community structures [11][12][13][14][15][16][17][18][19][20][21][22].There are several example of application of network science concepts in chemistry and physical chemistry, too [23][24][25][26][27][28][29][30][31][32][33]. Hydrogen-bond (HB) connectivity and strength and directionality influence the anomalous properties of water and other H-bonded liquids and liquid mixtures.…”

Optimal network modification for spectral radius dependent phase transitions

“…For example, the emergence of synchronization in networks has been found to be linked to the eigenvalues of the networks Laplacian matrix [1,13,14]. As such, efficient ways to approximate this spectrum, in the case of small world networks, were given via mean field theory [15] or by recursive relations using the characteristic polynomial [16]. In another case, the spectra of the graph Laplacian was used to optimally reconstruct brain networks [17].…”