Physica A: Statistical Mechanics and its Applications
Clustering coefficient and community structure of bipartite networks
Introduction
In recent years, the complex networks have attracted more and more people’s attention [1], [2], [3]. Many real-world systems are depicted as complex networks to investigate their structures and functions. Examples include WWW, internet, food webs, biochemical networks, social networks, and so on [4], [5], [6], [7], [8], [9]. These researches in networks not only raised new concepts and methods, but also helped us understand complex systems.
Bipartite networks are an important kind of complex network. In fact, many real-world networks are naturally bipartite, such as the actors-films network [10], the papers-scientists network [11], [12], [13], and so on. In bipartite networks, there are two non-overlapping sets of nodes called top nodes and bottom nodes. The edges only connect a pair of vertices which belong to different sets. To investigate the properties of bipartite networks, people usually project them into classical networks which are also called one-mode networks. However, the one-mode projection of a bipartite graph, generally loses some information of the original networks, brings an inflation of the number of edges and other drawbacks which are caused by the projection [14]. Some prior works have confirmed these. In Ref. [29], Guimera et al. strikingly demonstrated that the analysis of a projection can give incorrect results in a model network. It will affect the properties including the community structures of the networks. Barber in his work [28] found pronounced differences between communities detected in a real-world bipartite network and its one-mode projection. So we should pay more attention to study the properties of the original bipartite networks, and develop some methods for detecting community structure in the original bipartite networks.
In this paper, we propose a definition of clustering coefficient for bipartite networks and develop a method for detecting communities in bipartite networks by cutting the edge with the least edge clustering coefficient. The outline of the article is as follows: In Section 2, clustering coefficient, as one important property in one-mode graphs, is defined for bipartite networks. Based on the similar consideration of standard clustering coefficient of Watts and Strogatz in one-mode networks, potential links are considered when calculating possible squares. In Section 3, we define two different edge clustering coefficients and of bipartite networks, which are based on the squares and triples respectively. Then, we develop an edge-cutting algorithm based on the clustering coefficient of links to detect the community structure of original bipartite networks. The results are different from the communities that are got by one-mode projection. They show us some detailed properties of original bipartite networks. In Section 4 we give some concluding remarks.
Section snippets
The clustering coefficient of bipartite networks
Because of the drawbacks brought by the projection, many authors try to analyze the networks by using the bipartite structure of the data. Some notions and properties, which are investigated in original bipartite networks, are also introduced, such as clustering [15], [23], overlapping [24], betweenness [23], and others [14], [15], [23], [25], [26], [28], [29].
The clustering coefficient is one of the most important properties in classical networks. For a node , the clustering coefficient
The community structure of bipartite graphs
In one-mode networks, community structure has been proved to be an important property in most social and biology networks [20], [21], [5] and it is tightly related to the functional groups of the system [6], [7], [8]. In order to detect the communities in binary networks, many algorithms have been proposed. The deep understanding in community structure will make us comprehend and analyze the characteristic of the systems better.
For bipartite networks, one-mode projection will lose some
Conclusions
Bipartite networks are an important class of networks. Many real-world networks display natural bipartite structure. One-mode projection of them will bring some drawbacks and affect the properties of the networks. So, we should develop some concepts and methods to analyze the original bipartite networks. In this paper, we discuss the node and edge clustering coefficients and propose an algorithm to detect community structures in bipartite networks.
Based on the consideration of standard
Acknowledgements
The author Peng Zhang thanks S.N. Dorogovtsev, J.F.F. Mendes for useful discussions and suggestions. This work is partially supported by 985 Projet, NSFC under the grant No.70771011 and No.70431002, and Chinese Scholarship Council.
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