Geographical networks evolving with an optimal policy.
In this article we propose a growing network model based on an optimal policy involving both topological and geographical measures. In this model, at each time step, a node, having randomly assigned coordinates in a 1x1 square, is added and connected to a previously existing node i, which minimizes the quantity ri2/kialpha, where ri is the geographical distance, ki the degree, and alpha a free parameter. The degree distribution obeys a power-law form when alpha=1, and an exponential form when alpha=0. When alpha is in the interval (0, 1), the network exhibits a stretched exponential distribution. We prove that the average topological distance increases in a logarithmic scale of the network size, indicating the existence of the small-world property. Furthermore, we obtain the geographical edge length distribution, the total geographical length of all edges, and the average geographical distance of the whole network. Interestingly, we found that the total edge length will sharply increase when alpha exceeds the critical value alphac=1, and the average geographical distance has an upper bound independent of the network size. All the results are obtained analytically with some reasonable approximations, which are well verified by simulations.
Xie
Yan-Bo
YB
Department of Modern Physics and Nonlinear Science Center, University of Science and Technology of China, Hefei 230026, People's Republic of China.
Zhou
Tao
T
Bai
Wen-Jie
WJ
Chen
Guanrong
G
Xiao
Wei-Ke
WK
Wang
Bing-Hong
BH