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

Abstract

The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to weight-dependent walk. In this paper, we first present a study on the transition weight matrix of a weighted network. To get the eigentime identity for weight-dependent walk and weighted counting of spanning trees, we need to obtain all the eigenvalues and their multiplicities of the transition weight matrix. Then we obtain the recursive relationship of its eigenvalues at two successive generations of transition weight matrix. By substituting, we can obtain the relationship of normalized Laplacian matrix’s eigenvalues at two successive generations. Using the relationship and Vietas formulas, we obtain the scalings of the eigentime identity. Afterwards, we classify normalized Laplacian matrix’s eigenvalues and compute the product of all nonzero normalized Laplacian eigenvalues by the product recursive relationship. The product is used to obtain weighted counting of spanning trees. Finally, by weighted counting of spanning trees, we validate the obtained eigenvalues and their multiplicities. The obtained results show that the weight factor has a strong effect on the behavior of weight-dependent walks.

Scaling of average receiving time and average shortest path on claw network

“…Complex networks have been applied to many interdisciplinary fields since they show many superiority, such as communication network, airport network, biological network and so on. As a part of complex networks, deterministic networks are widely studied including the structure of network [1][2][3], spectra of the network [4][5][6][7], the applications of spectra [8][9][10] and so on.Furthermore, some dynamical processes are also hot fields in complex networks, like random walks [11][12][13], coherence problem [14,15], epidemic spreading [16], Lotka-Volterra model [17], evolutionary game [18,19], and so on.…”