Spectra of weighted scale-free networks

Abstract

Much information about the structure and dynamics of a network is encoded in the eigenvalues of its transition matrix. In this paper, we present a first study on the transition matrix of a family of weight driven networks, whose degree, strength, and edge weight obey power-law distributions, as observed in diverse real networks. We analytically obtain all the eigenvalues, as well as their multiplicities. We then apply the obtained eigenvalues to derive a closed-form expression for the random target access time for biased random walks occurring on the studied weighted networks. Moreover, using the connection between the eigenvalues of the transition matrix of a network and its weighted spanning trees, we validate the obtained eigenvalues and their multiplicities. We show that the power-law weight distribution has a strong effect on the behavior of random walks.

Eigenvalues of transition weight matrix and eigentime identity of weighted network with two hub nodes

“…[10], the weight of each edge at two successive generations of this model constructed by the weight factor is invariable, while the weight of each edge at two successive generations is variable in Ref. [10]. Comparing with the two models in Ref.…”

“…The other case is that the number of nodes is not exponential [10,11]. The normalized Laplacian matrix is used to study the architectures and dynamics of the network.…”

“…Zhang et al presented a first study on the transition weight matrix of a family of weight driven networks. They applied the obtained eigenvalues to derive a closed-form expression for the random target access time for biased random walks occurring on the studied weighted networks [10]. Previous works about spectra of the transition matrix were limited to binary networks, and the influence of inhomogeneous weight distribution on the spectral properties of transition matrix still remains unknown.…”

First-Order Network Coherence and Eigentime Identity on the Weighted Cayley Networks

“…Thus, the (i, j)th element of M n is M n (i, j) = w ij (g)/s i (g), which represents the local transition probability for a walker going from node i to node j. 25 Transition matrix M n describes the biased discrete-time random walks in G n , and thus various interesting quantities related to random walks are reflected in eigenvalues of the transition matrix. For example, the sum of reciprocals of 1 minus each eigenvalue (excluding eigenvalue 1 itself) of transition matrix M n determines the eigentime identity, also called random target access time in G n .…”

“…For example, the sum of reciprocals of 1 minus each eigenvalue (excluding eigenvalue 1 itself) of transition matrix M n determines the eigentime identity, also called random target access time in G n . 25 Let H ij (n) be the MFPT from node i to node j in G n , which is the expected time for a walker starting from node i to arrive at j for the first time. The stationary distribution 26 for random walks on G n is π = (π 1 , π 2 , .…”