标题： 复杂网络中壳层的结构 显示英文标题

标题： Structure of shells in complex networks

摘要： 在网络中，我们定义壳层$\ell$为相对于给定节点距离为$\ell$的节点集合，并定义$r_\ell$为壳层$\ell$外的节点的比例。 在传输过程中，信息或疾病通常从一个随机节点扩散，并逐层到达各个壳层。 因此，理解壳层结构对于研究网络的传输特性至关重要。 对于具有给定度分布的随机连接网络，我们解析地推导出位于壳层$\ell$外的节点的度分布和平均度作为$r_\ell$的函数。 此外，我们发现 $r_\ell$ 遵循迭代函数形式 $r_\ell=\phi(r_{\ell-1})$，其中 $\phi$ 是用网络原始度分布的生成函数表示的。 我们的结果可以解释在壳层中节点数 $B_\ell$ 的幂律分布，这些壳层的 $\ell$ 大于网络直径 $d$，即所有节点对之间的平均距离。 对于现实世界网络，理论预测$r_\ell$与经验结果$r_\ell$存在偏差。 我们引入了一个网络相关函数$c(r_\ell)\equiv r_{\ell+1}/\phi(r_\ell)$来表征网络中的相关性，其中$r_{\ell+1}$是经验值，$\phi(r_\ell)$是理论预测值。 $c(r_\ell)=1$表示经验结果与理论的完美一致。 我们将$c(r_\ell)$应用于多个模型和现实世界网络。 我们发现网络分为两个不同的类别：(i) 一个具有$c(r_\ell)>1$的{\it 连接不良}网络类，与具有相同度分布的随机连接网络相比，它们的平均距离更大；以及 (ii) 一个具有$c(r_\ell)<1$的{\it 联系广泛的}网络类。 显示英文摘要

摘要： In a network, we define shell $\ell$ as the set of nodes at distance $\ell$ with respect to a given node and define $r_\ell$ as the fraction of nodes outside shell $\ell$. In a transport process, information or disease usually diffuses from a random node and reach nodes shell after shell. Thus, understanding the shell structure is crucial for the study of the transport property of networks. For a randomly connected network with given degree distribution, we derive analytically the degree distribution and average degree of the nodes residing outside shell $\ell$ as a function of $r_\ell$. Further, we find that $r_\ell$ follows an iterative functional form $r_\ell=\phi(r_{\ell-1})$, where $\phi$ is expressed in terms of the generating function of the original degree distribution of the network. Our results can explain the power-law distribution of the number of nodes $B_\ell$ found in shells with $\ell$ larger than the network diameter $d$, which is the average distance between all pairs of nodes. For real world networks the theoretical prediction of $r_\ell$ deviates from the empirical $r_\ell$. We introduce a network correlation function $c(r_\ell)\equiv r_{\ell+1}/\phi(r_\ell)$ to characterize the correlations in the network, where $r_{\ell+1}$ is the empirical value and $\phi(r_\ell)$ is the theoretical prediction. $c(r_\ell)=1$ indicates perfect agreement between empirical results and theory. We apply $c(r_\ell)$ to several model and real world networks. We find that the networks fall into two distinct classes: (i) a class of {\it poorly-connected} networks with $c(r_\ell)>1$, which have larger average distances compared with randomly connected networks with the same degree distributions; and (ii) a class of {\it well-connected} networks with $c(r_\ell)<1$.