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A model of fractal hierarchical structures that share the property of non-homogeneous weighted networks is introduced. These networks can be completely and analytically characterized in terms of the involved parameters, i.e., the size of the original graph Nk and the non-homogeneous weight scaling factors r 1, r 2, · · · rM. We also study the average weighted shortest path (AWSP), the average degree and the average node strength, taking place on the non-homogeneous hierarchical weighted networks. Moreover the AWSP is scrupulously calculated. We show that the AWSP depends on the number of copies and the sum of all non-homogeneous weight scaling factors in the infinite network order limit.

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This paper reports a weighted hierarchical network generated on the basis of self-similarity, in which each edge is assigned a different weight in the same scale. We studied two substantial properties of random walk: the first-passage time (FPT) between a hub node and a peripheral node and the FPT from a peripheral node to a local hub node over the network. Meanwhile, an analytical expression of the average sending time (AST) is deduced, which reflects the average value of FPT from a hub node to any other node. Our result shows that the AST from a hub node to any other node is related to the scale factor and the number of modules. We found that the AST grows sublinearly, linearly and superlinearly respectively with the network order, depending on the range of the scale factor. Our work may shed some light on revealing the diffusion process in hierarchical networks.