Elucidation of characteristics of networks where every node has its own lifetime

Abstract

Growth is regarded as an important mechanism for explaining the structures of real networks. However, when the increase in the number of nodes is suppressed owing to their lifetime, the growth property alone is not sufficient to explain even fundamental network properties, such as the scale-free property. In this paper, we propose a network model that considers the lifetime of nodes and the excess addition of local internal links as a mechanism that supports network structures. By investigating the model network, we aimed to elucidate the network characteristics supported by local interactions between nodes via their common neighbors even when the rates of node addition and deletion were balanced. We found that the stationary state of the number of nodes is characterized by a scale-free property with the power-law exponent $\gamma \simeq 1$ and localization of the peaks at $l=2$ in the distance distributions of neighboring nodes (DDN) as the node degree $k$ increases. The specific behavior of the DDN explains the very slow decrease in the clustering strength $C\left(k\right)$ with $k$ compared with the normal behavior $C\left(k\right)\sim k^{-1}$ and the accelerated growth of the neighborhood graph of each node. Moreover, we showed that some real networks share local structures similar to those of the model network. These findings suggest that the same mechanism as that of the proposed model plays an essential role in supporting the local structures of some real networks.