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تسجيل مستخدم جديدGeographical networks evolving with optimal policy
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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 new node, having randomly assigned coordinates in a $1 times 1$ square, is added and connected to a previously existing node $i$, which minimizes the quantity $r_i^2/k_i^alpha$, where $r_i$ is the geographical distance, $k_i$ 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 $alpha_c=1$, 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.
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