Line Graphs of Weighted Networks for Overlapping Communities

📝 Abstract

In this paper, we develop the idea to partition the edges of a weighted graph in order to uncover overlapping communities of its nodes. Our approach is based on the construction of different types of weighted line graphs, i.e. graphs whose nodes are the links of the original graph, that encapsulate differently the relations between the edges. Weighted line graphs are argued to provide an alternative, valuable representation of the system’s topology, and are shown to have important applications in community detection, as the usual node partition of a line graph naturally leads to an edge partition of the original graph. This identification allows us to use traditional partitioning methods in order to address the long-standing problem of the detection of overlapping communities. We apply it to the analysis of different social and geographical networks.

💡 Analysis

In this paper, we develop the idea to partition the edges of a weighted graph in order to uncover overlapping communities of its nodes. Our approach is based on the construction of different types of weighted line graphs, i.e. graphs whose nodes are the links of the original graph, that encapsulate differently the relations between the edges. Weighted line graphs are argued to provide an alternative, valuable representation of the system’s topology, and are shown to have important applications in community detection, as the usual node partition of a line graph naturally leads to an edge partition of the original graph. This identification allows us to use traditional partitioning methods in order to address the long-standing problem of the detection of overlapping communities. We apply it to the analysis of different social and geographical networks.

📄 Content

arXiv:0912.4389v2 [physics.data-an] 9 Jun 2010 EPJ manuscript No. (will be inserted by the editor) Line Graphs of Weighted Networks for Overlapping Communities T.S. Evans1,2a and R. Lambiotte1b 1 Institute for Mathematical Sciences, Imperial College London, SW7 2PG London, UK 2 Theoretical Physics, Imperial College London, SW7 2AZ, U.K. 9th June 2010 Abstract. In this paper, we develop the idea to partition the edges of a weighted graph in order to uncover overlapping communities of its nodes. Our approach is based on the construction of different types of weighted line graphs, i.e. graphs whose nodes are the links of the original graph, that encapsulate differently the relations between the edges. Weighted line graphs are argued to provide an alternative, valuable representation of the system’s topology, and are shown to have important applications in community detection, as the usual node partition of a line graph naturally leads to an edge partition of the original graph. This identification allows us to use traditional partitioning methods in order to address the long- standing problem of the detection of overlapping communities. We apply it to the analysis of different social and geographical networks. Key words. Edge partition, line graphs, community detection, overlapping communities, vertex cover PACS. 89.75.Hc Networks and genealogical trees – 89.75.Fb Structures and organization in complex systems – 05.40.Fb Random walks and Levy flights 1 Introduction In the last decade, the interdisciplinary field of complex networks has led to the development of universal tools in order to characterise and model systems as diverse as information, biological or social networks [4]. Many stud- ies focus on the properties of the vertices, e.g. studying their degree distribution or ranking them by some mea- sure. However graphs are both a set of vertices and a set of relationships between vertices — the edges. It is therefore useful sometimes to look at a network from the view point of the edges. We do this by defining ‘weighted line graphs’ for any type of graph, extending our original work on weighted line graphs for simple graphs [23]. Our weighted line graphs are topologically equivalent to the standard line graph of the literature [1,2,3]. However the weights we define play a crucial role in avoiding a bias inherent in unweighted line graphs towards high degree vertices in the original graph. Our work can be seen as providing a general framework to shift our view from a vertex centric one to an edge centric viewpoint. We illustrate our ideas in the context of community detection [5,6,7,8]. When dealing with complex networks one crucial step is the identification of communities or modules, some sort of highly connected subgraphs. It has been shown that many systems of interest are organised in a modular way and that these topological modules usually a e-mail: t.evans@imperial.ac.uk b e-mail: r.lambiotte@imperial.ac.uk correspond to functional sub-units. In a large number of situations, these building blocks themselves may be mod- ular, in which case the network is said to be hierarchical. Modularity at different scales has long been argued to be a universal property of complex systems because of the cru- cial evolutionary advantage it confers, by providing stable intermediate forms (modules) and thereby improving the system’s adaptability [9]. Multi-scale modularity is also associated to a separation of time scales for the dynamics taking place on the graph [10,11,12,13], which is essen- tial in order to ensure the persistence of diversity in the system [14]. The fundamental idea behind most community detec- tion methods is to partition the nodes of the network into modules. By doing so, each node is therefore assigned to one single module. However a vertex partition has the disadvantage of being incompatible with the existence of overlapping communities, i.e. situations where nodes be- long to several communities. This overlap is known to be present at the interface between modules, but can also be pervasive in the whole network [25]. This is the case in many social networks where individuals typically belong to several communities defined by their type of interaction, e.g. work, sport buddy, family, etc, but also in biological networks where proteins may belong to several functional categories. In those situations where the interface between the communities occurs throughout the system, a parti- tion of the nodes is questionable as it imposes undesired constraints on the community detection problem. There 2 T.S. Evans, R. Lambiotte: Line Graphs of Weighted Networks for Overlapping Communities are many different approaches to finding overlapping com- munities (for example see [18,19,20,21,22,23,24,25,26,27,28,29,30]). A popular choice is k-clique percolation, which consists in looking for connected components of cliques of size k [19]. However, this approach has several disadvantages as its outcome strongly depends on th

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