Line Graphs of Weighted Networks for Overlapping Communities

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.Comment: 8 Pages. New title and text revisions to emphasise differences from earlier paper

Random line graphs and a linear law for assortativity

For a fixed number N of nodes, the number of links L in the line graph H(N,L) can only appear in consecutive intervals, called a band of L. We prove that some consecutive integers can never represent the number of links L in H(N,L), and they are called a bandgap of L. We give the exact expressions of bands and bandgaps of L. We propose a model which can randomly generate simple graphs which are line graphs of other simple graphs. The essence of our model is to merge step by step a pair of nodes in cliques, which we use to construct line graphs. Obeying necessary rules to ensure that the resulting graphs are line graphs, two nodes to be merged are randomly chosen at each step. If the cliques are all of the same size, the assortativity of the line graphs in each step are close to 0, and the assortativity of the corresponding root graphs increases linearly from ?1 to 0 with the steps of the nodal merging process. If we dope the constructing elements of the line graphs—the cliques of the same size—with a relatively smaller number of cliques of different size, the characteristics of the assortativity of the line graphs is completely altered. We also generate line graphs with the cliques whose sizes follow a binomial distribution. The corresponding root graphs, with binomial degree distributions, zero assortativity, and semicircle eigenvalue distributions, are equivalent to Erd?s-Rényi random graphs.Electrical Engineering, Mathematics and Computer Scienc