Eigenvalue spectrum of transition matrix of dual Sierpinski gaskets and its applications

Abstract

The eigenvalue spectrum of the transition matrix of a network contains much information about its structural properties and is related to the behavior of various dynamical processes performed on it. In this paper, we study the eigenvalues of the transition matrix of the dual Sierpinski gaskets embedded in d-dimensional Euclidean spaces. We obtain all the eigenvalues, as well as their corresponding degeneracies, by making use of the spectral decimation technique. We then apply the obtained eigenvalues to determine the exact number of spanning trees in the studied fractals and derive an explicit formula of the eigentime identity for random walks taking place on them.

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

“…Thus, for the studied scale-free network, the random target access time grows linearly with the number of nodes, which is the minimal scaling for random walks on graphs [59] and is in sharp contrast to those previously obtained for other networks [28][29][30][33][34][35]59], where the random target access timeF scales with the network size N asF ∼ N θ (θ > 1) or F ∼ N ln N .…”

“…Thus, we have obtained exact simple expressions for all eigenvalues of the transition matrix P g . We note that analytical solutions to eigenvalues of transition matrix for other deterministic networks are [28][29][30][33][34][35] invariably of the forms having radicals or trigonometric functions.…”

“…More specifically, for T g , the number of different eigenvalues N g (λ) is 2g + 1, which has a logarithmic dependence on the network size N g , that is, N g (λ) ∼ ln N g . This should be compared with that for other previously studied deterministic networks, including Sierpinski gaskets [30], fractal trees [34], and polymer networks [35], for which the number of different eigenvalues behaves linearly with the network size.…”

“…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]. However, these previously studied networks cannot simultaneously mimic some common and distinctive properties of real-world systems [36][37][38], including scale-free behavior [39] and the small-world effect [40] characterized by a small average distance and a high clustering coefficient.…”

Controllability of fractal networks: An analytical approach

“…The development of computer science offers a number of tools for exploring fractal behaviors numerically and mathematically [1,2]. Based on the underlying self-similarity, a variety of iteration models have been proposed to reproduce fractal properties [3][4][5][6][7][8][9][10][11][12]. An interesting finding in nonlinear dynamics is that chaotic attractors are often accompanied by fractal structures [13].…”