Community detection is of considerable importance for understanding both the structure and function of complex networks. In this paper, we introduced the general procedure of the community detection algorithms using global and local structural information, where the edge betweenness and the local similarity measures respectively based on local random walk dynamics and local cyclic structures were used. The algorithms were tested on artificial and real-world networks. The results clearly show that all the algorithms have excellent performance in the tests and the local similarity measure based on local random walk dynamics is superior to that based on local cyclic structures.

The bound states of four bosons in the quantum $\beta$-Fermi–Pasta–Ulam model are investigated and some interesting results are presented using the number conserving approximation combined with the number state method. We find that the relative magnitude of anharmonic coefficient has a significant effect on forming localized energy in the model, and the wave number plays an important role in forming different bound states. The signature of the quantum breather is also set up by the square of the amplitudes of the corresponding eigenvectors in real space.

By revisiting the preferential attachment (PA) mechanism for generating a classical scale-free network, we propose a class of novel preferential attachment similarity indices for predicting future links in evolving networks. Extensive experiments on 14 real-life networks show that these new indices can provide more accurate prediction than the traditional one. Due to the improved prediction accuracy and low computational complexity, these proposed preferential attachment indices can be helpful for providing both instructions for mining unknown links and new insights to understand the underlying mechanisms that drive the network evolution.

Many real-world networks such as the protein–protein interaction networks and metabolic networks often display nontrivial correlations between degrees of vertices connected by edges. Here, we analyse the statistical methods used usually to describe the degree correlation in the networks, and analytically give linear relation in the degree correlation. It provides a simple and interesting perspective on the analysis of the degree correlation in networks, which is usefully complementary to the existing methods for degree correlation in networks. Especially, the slope in the linear relation corresponds exactly to the degree correlation coefficient in networks, meaning that it can not only characterize the level of degree correlation in networks, but also reflects the speed that the average nearest neighbours’ degree varies with the vertex degree. Finally, we applied our results to several real-world networks, validating the conclusions of the linear analysis of degree correlation. We hope that the work in this paper can be helpful for further understanding the degree correlation in complex networks.