A specific type of irregular ring-and-hub network structure and the average shortest distance of its rings

Ring-and-hub network structure is very common in the real world, while that neighboring rings may sometimes share nodes or line segments makes this structure irregular. In this paper, we modify Dorogovtsev-Mendes model and its subsequent models to analytically estimate the average distance between nodes on the same ring in irregular ring-and-hub networks. In order to observe the accuracy of our modified model, we develop an algorithm to generate irregular ring-and-hub networks by computer. Then, we compare the analytic estimates with the practical values on those computer-generated and real networks. The results show that our modified model actually estimates the average shortest distance of its rings when only straight and U-shape paths between the start and end points are allowed. The accuracy of estimates for innermost several rings can be acceptable.