随机网络中最短路长分布的解析结果
Analytical results for the distribution of shortest path lengths in random networks
摘要：
本文提出了两种互补的分析方法来计算Erdos-Rényi网络中最短路径长度的分布，这两种方法都是基于参考节点周围的壳层和从参考节点出发的路径的递归方程，其结果与广泛的网络规模和连通性的数值模拟结果一致。还获得了分布的平均值和标准差。在平均度随网络大小按$N^α$缩放的情况下，分布在渐近极限处变得非常窄，即几乎所有节点对都是等距的，彼此之间的距离为$d=\lfloor 1/α\rfloor$。计算度为$m$的节点与网络其余部分之间的最短路径长度的分布。它的平均值被证明是$m$的单调递减函数，提供了一个有趣的局部属性和网络的全局属性之间的关系。这里提出的方法可以应用到更一般的网络类。
英文摘要：
We present two complementary analytical approaches for calculating the distribution of shortest path lengths in Erdos-Rényi networks, based on recursion equations for the shells around a reference node and for the paths originating from it. The results are in agreement with numerical simulations for a broad range of network sizes and connectivities. The average and standard deviation of the distribution are also obtained. In the case that the mean degree scales as $N^α$ with the network size, the distribution becomes extremely narrow in the asymptotic limit, namely almost all pairs of nodes are equidistant, at distance $d=\lfloor 1/α\rfloor$ from each other. The distribution of shortest path lengths between nodes of degree $m$ and the rest of the network is calculated. Its average is shown to be a monotonically decreasing function of $m$, providing an interesting relation between a local property and a global property of the network. The methodology presented here can be applied to more general classes of networks.
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