Eigentime identity of the weighted scale-free triangulation networks for weight-dependent walk

The eigenvalues of the normalized Laplacian matrix of a network provide information on its structural properties and some relevant dynamical aspects, in particular for weight-dependent walk. In order to get the eigentime identity for weight-dependent walk, we need to obtain the eigenvalues and their multiplicities of the Laplacian matrix. Firstly, the model of the weighted scale-free triangulation networks is constructed. Then, the eigenvalues and their multiplicities of transition weight matrix are presented, after the recursive relationship of those eigenvalues at two successive generations are given. Consequently, the Laplacian spectrum is obtained. Finally, the analytical expression of the eigentime identity, indicating that the eigentime identity grows sublinearly with the network order, is deduced.