Eigentime identity of the weighted (m, n)-flower networks

The eigentime identity for random walks on the weighted networks is the expected time for a walker going from a node to another node. Eigentime identity can be studied by the sum of reciprocals of all nonzero Laplacian eigenvalues on the weighted networks. In this paper, we study the weighted (m, n)-flower networks with the weight factor r. We divide the set of the nonzero Laplacian eigenvalues into three subsets according to the obtained characteristic polynomial. Then we obtain the analytic expression of the eigentime identity Ht+1 of the weighted (m, n)-flower networks by using the characteristic polynomial of Laplacian and recurrent structure of Markov spectrum. We take m = 3; n = 2 as example, and show that the leading term of the eigentime identity on the weighted (3, 2)-flower networks obey superlinearly, linearly with the network size.