Coclustering for directed graphs: the Stochastic coBlockmodel and spectral algorithm DiSim
Abstract
Directed graphs have asymmetric connections, yet the current graph clustering methodologies cannot identify the potentially global structure of these asymmetries. We give a spectral algorithm called disim that builds on a dual measure of similarity that correspond to how a node (i) sends and (ii) receives edges. Using disim, we analyze the global asymmetries in the networks of Enron emails, political blogs, and the c elegans neural connectome. In each example, a small subset of nodes have persistent asymmetries; these nodes send edges with one cluster, but receive edges with another cluster. Previous approaches would have assigned these asymmetric nodes to only one cluster, failing to identify their sending/receiving asymmetries. Regularization and "projection" are two steps of disim that are essential for spectral clustering algorithms to work in practice. The theoretical results show that these steps make the algorithm weakly consistent under the degree corrected Stochastic coBlockmodel, a model that generalizes the Stochastic Blockmodel to allow for both (i) degree heterogeneity and (ii) the global asymmetries that we intend to detect. The theoretical results make no assumptions on the smallest degree nodes. Instead, the theorem requires that the average degree grows sufficiently fast and that the weak consistency only applies to the subset of the nodes with sufficiently large leverage scores. The results results also apply to bipartite graphs.
 Publication:

arXiv eprints
 Pub Date:
 April 2012
 arXiv:
 arXiv:1204.2296
 Bibcode:
 2012arXiv1204.2296R
 Keywords:

 Statistics  Machine Learning;
 Mathematics  Statistics Theory