In this paper, we consider the problem of exploring structural regularities of networks by dividing the nodes of a network into groups such that the members of each group have similar patterns of connections to other groups. Specifically, we propose
a general statistical model to describe network structure. In this model, group is viewed as hidden or unobserved quantity and it is learned by fitting the observed network data using the expectation-maximization algorithm. Compared with existing models, the most prominent strength of our model is the high flexibility. This strength enables it to possess the advantages of existing models and overcomes their shortcomings in a unified way. As a result, not only broad types of structure can be detected without prior knowledge of what type of intrinsic regularities exist in the network, but also the type of identified structure can be directly learned from data. Moreover, by differentiating outgoing edges from incoming edges, our model can detect several types of structural regularities beyond competing models. Tests on a number of real world and artificial networks demonstrate that our model outperforms the state-of-the-art model at shedding light on the structural features of networks, including the overlapping community structure, multipartite structure and several other types of structure which are beyond the capability of existing models.
Spectral analysis has been successfully applied at the detection of community structure of networks, respectively being based on the adjacency matrix, the standard Laplacian matrix, the normalized Laplacian matrix, the modularity matrix, the correlat
ion matrix and several other variants of these matrices. However, the comparison between these spectral methods is less reported. More importantly, it is still unclear which matrix is more appropriate for the detection of community structure. This paper answers the question through evaluating the effectiveness of these five matrices against the benchmark networks with heterogeneous distributions of node degree and community size. Test results demonstrate that the normalized Laplacian matrix and the correlation matrix significantly outperform the other three matrices at identifying the community structure of networks. This indicates that it is crucial to take into account the heterogeneous distribution of node degree when using spectral analysis for the detection of community structure. In addition, to our surprise, the modularity matrix exhibits very similar performance to the adjacency matrix, which indicates that the modularity matrix does not gain desired benefits from using the configuration model as reference network with the consideration of the node degree heterogeneity.
Empirical studies show that real world networks often exhibit multiple scales of topological descriptions. However, it is still an open problem how to identify the intrinsic multiple scales of networks. In this article, we consider detecting the mult
i-scale community structure of network from the perspective of dimension reduction. According to this perspective, a covariance matrix of network is defined to uncover the multi-scale community structure through the translation and rotation transformations. It is proved that the covariance matrix is the unbiased version of the well-known modularity matrix. We then point out that the translation and rotation transformations fail to deal with the heterogeneous network, which is very common in nature and society. To address this problem, a correlation matrix is proposed through introducing the rescaling transformation into the covariance matrix. Extensive tests on real world and artificial networks demonstrate that the correlation matrix significantly outperforms the covariance matrix, identically the modularity matrix, as regards identifying the multi-scale community structure of network. This work provides a novel perspective to the identification of community structure and thus various dimension reduction methods might be used for the identification of community structure. Through introducing the correlation matrix, we further conclude that the rescaling transformation is crucial to identify the multi-scale community structure of network, as well as the translation and rotation transformations.
As two main focuses of the study of complex networks, the community structure and the dynamics on networks have both attracted much attention in various scientific fields. However, it is still an open question how the community structure is associate
d with the dynamics on complex networks. In this paper, through investigating the diffusion process taking place on networks, we demonstrate that the intrinsic community structure of networks can be revealed by the stable local equilibrium states of the diffusion process. Furthermore, we show that such community structure can be directly identified through the optimization of the conductance of network, which measures how easily the diffusion occurs among different communities. Tests on benchmark networks indicate that the conductance optimization method significantly outperforms the modularity optimization methods at identifying the community structure of networks. Applications on real world networks also demonstrate the effectiveness of the conductance optimization method. This work provides insights into the multiple topological scales of complex networks, and the obtained community structure can naturally reflect the diffusion capability of the underlying network.
It has been shown that the communities of complex networks often overlap with each other. However, there is no effective method to quantify the overlapping community structure. In this paper, we propose a metric to address this problem. Instead of as
suming that one node can only belong to one community, our metric assumes that a maximal clique only belongs to one community. In this way, the overlaps between communities are allowed. To identify the overlapping community structure, we construct a maximal clique network from the original network, and prove that the optimization of our metric on the original network is equivalent to the optimization of Newmans modularity on the maximal clique network. Thus the overlapping community structure can be identified through partitioning the maximal clique network using any modularity optimization method. The effectiveness of our metric is demonstrated by extensive tests on both the artificial networks and the real world networks with known community structure. The application to the word association network also reproduces excellent results.
Document networks are characteristic in that a document node, e.g. a webpage or an article, carries meaningful content. Properties of document networks are not only affected by topological connectivity between nodes, but also strongly influenced by t
he semantic relation between content of the nodes. We observe that document networks have a large number of triangles and a high value of clustering coefficient. And there is a strong correlation between the probability of formation of a triangle and the content similarity among the three nodes involved. We propose the degree-similarity product (DSP) model which well reproduces these properties. The model achieves this by using a preferential attachment mechanism which favours the linkage between nodes that are both popular and similar. This work is a step forward towards a better understanding of the structure and evolution of document networks.
