Exact eigenvalue spectrum of a class of fractal scale-free networks

Abstract

-The eigenvalue spectrum of the transition matrix of a network encodes important information about its structural and dynamical properties. We study the transition matrix of a family of fractal scale-free networks and analytically determine all the eigenvalues and their degeneracies. We then use these eigenvalues to evaluate the closed-form solution to the eigentime for random walks on the networks under consideration. Through the connection between the spectrum of transition matrix and the number of spanning trees, we corroborate the obtained eigenvalues and their multiplicities.

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.…”

“…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].…”

“…An interesting finding in nonlinear dynamics is that chaotic attractors are often accompanied by fractal structures [13]. In the field of complex networks, fractal properties and self-similarities are shared by many network systems [9][10][11][12]14,15], which motivates us to explore how the fractal structure affects the dynamical processes that take place on complex networks. Prototypical approaches include transportation and diffusion [16][17][18][19].…”