Network coherence and eigentime identity on a family of weighted fractal networks

Abstract The study on network coherence and eigentime identity has gained much interest. In this paper, the first-order network coherence is characterized by the entire mean first-passage time (EMFPT) for weight-dependent walk, while the eigentime identity is quantified by the sum of reciprocals of all nonzero normalized Laplacian eigenvalues. We construct a family of weighted fractal networks with the weight factor r  (0  r  ≤ 1). Based on the relationship between the first-order network coherence and the EMFPT, the asymptotic behavior of the first-order network coherence is obtained. The obtained results show that the scalings of first-order coherence with network size obey three laws according to the range of the weight factor. The first law is that the scaling obeys a power-law function of the network size N n with the exponent, represented by log s r , when 1 s r ≤ 1 ; The second law is that the scaling obeys ( ln N n ) 2 N n (i.e., the quotient of the square logarithm of the network size and the network size), when r = 1 s ; The third law is that the scaling obeys ln N n N n (i.e., the quotient of the logarithm of the network size and the network size), when 0 r 1 s . Thus, the scaling of the first-order coherence of weighted fractal networks decreases with the decreasing of r , when 0  r  ≤ 1. Then, all nonzero normalized Laplacian eigenvalues can be obtained by computing the roots of several small-degree polynomials defined recursively. The obtained results show that the scalings of the eigentime identity obey two laws according to the range of the weight factor. The first law is that the scaling obeys ln  N n (i.e., the logarithm of the network size), when 0  r  ≤ 1 and r ≠ 1 s ; The second law is that the scaling obeys N n ln  N n (i.e., the product of network size and its logarithm), when r = 1 s .