Symmetrizations for clustering directed graphs

Graph clustering has generally concerned itself with clustering undirected graphs; however the graphs from a number of important domains are essentially directed, e.g. networks of web pages, research papers and Twitter users. This paper investigates various ways of symmetrizing a directed graph into an undirected graph so that previous work on clustering undirected graphs may subsequently be leveraged. Recent work on clustering directed graphs has looked at generalizing objective functions such as conductance to directed graphs and minimizing such objective functions using spectral methods. We show that more meaningful clusters (as measured by an external ground truth criterion) can be obtained by symmetrizing the graph using measures that capture in- and out-link similarity, such as bibliographic coupling and co-citation strength. However, direct application of these similarity measures to modern large-scale power-law networks is problematic because of the presence of hub nodes, which become connected to the vast majority of the network in the transformed undirected graph. We carefully analyze this problem and propose a Degree-discounted similarity measure which is much more suitable for large-scale networks. We show extensive empirical validation.