The asymptotic formula on average weighted path length for scale-free modular network

In this paper, we discuss the average path length for a class of scale-free modular networks with deterministic growth. To facilitate the analysis, we define the sum of distances from all nodes to the nearest hub nodes and the nearest peripheral nodes. For the unweighted network, we find that whether the scale-free modular network is single-hub or multiple-hub, the average path length grows logarithmically with the increase of nodes number. For the weighted network, we deduce that when the network iteration t tends to infinity, the average weighted shortest path length is bounded, and the result is independent of the connection method of network.