The entire mean weighted first-passage time on infinite families of weighted tree networks

Abstract

We propose the entire mean weighted first-passage time (EMWFPT) for the first time in the literature. The EMWFPT is obtained by the sum of the reciprocals of all nonzero Laplacian eigenvalues on weighted networks. Simplified calculation of EMWFPT is the key quantity in the study of infinite families of weighted tree networks, since the weighted complex systems have become a fundamental mechanism for diverse dynamic processes. We base on the relationships between characteristic polynomials at different generations of their Laplacian matrix and Laplacian eigenvalues to compute EMWFPT. This technique of simplified calculation of EMWFPT is significant both in theory and practice. In this paper, firstly, we introduce infinite families of weighted tree networks with recursive properties. Then, we use the sum of the reciprocals of all nonzero Laplacian eigenvalues to calculate EMWFPT, which is equal to the average of MWFPTs over all pairs of nodes on infinite families of weighted networks. In order to compute EMWFPT, we try to obtain the analytical expressions for the sum of the reciprocals of all nonzero Laplacian eigenvalues. The key step here is to calculate the constant terms and the coefficients of first-order terms of characteristic polynomials. Finally, we obtain analytically the closed-form solutions to EMWFPT on the weighted tree networks and show that the leading term of EMWFPT grows superlinearly with the network size.