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Over the years, quantifying the similarity of nodes has been a hot topic in complex networks, yet little has been known about the distributions of node-similarity. In this paper, we consider a typical measure of node-similarity called the common neighbor based similarity (CNS). By means of the generating function, we propose a general framework for calculating the CNS distributions of node sets in various complex networks. In particular, we show that for the Erd{o}s-R{e}nyi (ER) random network, the CNS distribution of node sets of any particular size obeys the Poisson law. We also connect the node-similarity distribution to the link prediction problem. We found that the performance of link prediction depends solely on the CNS distributions of the connected and unconnected node pairs in the network. Furthermore, we derive theoretical solutions of two key evaluation metrics in link prediction: i) precision and ii) area under the receiver operating characteristic curve (AUC). We show that for any link prediction method, if the similarity distributions of the connected and unconnected node pairs are identical, the AUC will be $0.5$. The theoretical solutions are elegant alternatives of the traditional experimental evaluation methods with nevertheless much lower computational cost.
Identification of communities in complex networks has become an effective means to analysis of complex systems. It has broad applications in diverse areas such as social science, engineering, biology and medicine. Finding communities of nodes and fin
Link prediction in complex network based on solely topological information is a challenging problem. In this paper, we propose a novel similarity index, which is efficient and parameter free, based on clustering ability. Here clustering ability is de
Community detection and link prediction are both of great significance in network analysis, which provide very valuable insights into topological structures of the network from different perspectives. In this paper, we propose a novel community detec
Many real networks that are inferred or collected from data are incomplete due to missing edges. Missing edges can be inherent to the dataset (Facebook friend links will never be complete) or the result of sampling (one may only have access to a port
Information entropy has been proved to be an effective tool to quantify the structural importance of complex networks. In the previous work (Xu et al, 2016 cite{xu2016}), we measure the contribution of a path in link prediction with information entro