通过节点复制增长的有向随机网络的入度和出度分布的分析结果
Analytical results for the in-degree and out-degree distributions of directed random networks that grow by node duplication
摘要：
我们提出了精确的结果的度分布在一个有向网络模型，增长的节点重复（ND）。这些模型在研究基因调控网络和科学引文网络的结构和增长动力学方面是有用的。从初始种子网络开始，在每个时间步长处，选择随机节点（母节点）进行复制。它的子节点被添加到网络中，并以概率p复制母节点的每个传出链路。此外，子节点形成到母节点本身的有向链路。我们得到了在时间t的入度分布$P_t(K_{in}=k)$和出度分布$P_t(K_{out}=k)$的分析结果。入度遵循移动幂律，因此网络是渐近无标度的。相反，出度分布是窄的，并且在稀疏网络极限中收敛到泊松分布，而在密集网络极限中收敛到高斯分布。这种宽的入度分布和窄的出度分布之间的区别在科学引文网络等经验网络中很常见。使用这个，我们计算平均度$\langle K_{in}\rangle_t=\langle K_{out}\rangle_t$，它收敛到$1/(1-p)$在大网络的限制，为整个范围的$0<p<1$。这与相应的无向网络相反，无向网络在$p=1/2$处表现出相变，使得对于$p>1/2$，平均度在大网络极限中发散。我们还提出了从一个随机节点的上游和下游节点的数量分布的分析结果。平均值$\langle N_{up}\rangle_t=\langle N_{down}\rangle_t$与网络规模成比例关系，这意味着只有一小部分节点对通过有向路径连接，不像无向ND情况，由单个组件组成，因此不是小世界网络。
英文摘要：
We present exact results for the degree distribution in a directed network model that grows by node duplication (ND). Such models are useful in the study of the structure and growth dynamics of gene regulatory networks and scientific citation networks. Starting from an initial seed network, at each time step a random node, a mother node, is selected for duplication. Its daughter node is added to the network and duplicates each outgoing link of the mother node with probability p. In addition, the daughter node forms a directed link to the mother node itself. We obtain analytical results for the in-degree distribution $P_t(K_{in}=k)$, and for the out-degree distribution $P_t(K_{out}=k)$ at time t. The in-degrees follow a shifted power-law, so the network is asymptotically scale free. In contrast, the out-degree distribution is narrow, and converges to a Poisson distribution in the sparse network limit and to a Gaussian distribution in the dense network limit. Such distinction between a broad in-degree distribution and a narrow out-degree distribution is common in empirical networks such as scientific citation networks. Using this we calculate the mean degree $\langle K_{in}\rangle_t=\langle K_{out}\rangle_t$, which converges to $1/(1-p)$ in the large network limit, for the whole range of $0<p<1$. This is in contrast to the corresponding undirected network, which exhibits a phase transition at $p=1/2$ such that for $p>1/2$ the mean degree diverges in the large network limit. We also present analytical results for the distribution of the number of upstream and downstream nodes from a random node. The mean values $\langle N_{up}\rangle_t=\langle N_{down}\rangle_t$ scale logarithmically with the network size, implying that only a small fraction of pairs of nodes are connected by directed paths, unlike the undirected ND case that consists of a single component, hence not a small-world network.
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