Evolution mechanism of the local network structure recorded in distribution of distances between neighbors of each vertex

Abstract

The explosive development of research on real networks have led to the proposal of many mathematical network models explaining properties shared in the real networked systems. However, considering that different models can similarly reproduce the typical properties of networks, the estimation of the evolutionary process is more essential for appropriate network modeling than is the description of current network structures. In this paper, we propose a new measure “distribution of distances between neighbors (DDN) of each vertex” to investigate network development. We first examine several model networks, including growth models based on the attachment rule and on the local rule and their hybrid models, and show that the DDN profiles of these networks accurately reflect the differences in growth mechanisms. In particular, for the local rule model, the dependence of the mean DDN of vertices on the vertex degree records the evolution of the local network structure. The application of the same analysis to several real networks show that they share the characteristics of DDN profiles with the local rule and hybrid models. These results imply that these model networks appropriately describe the evolutionary processes of real networks. In addition, we examine the characteristics of DDN specific to fractal and non-fractal networks, and show that the presence or absence of fractality of some real networks can be explained by the influence of long-range edges on the local rule. An exception to this is the co-authorships network, which is expected to be generated by connecting several fractal sub-networks.