Co-clustering for directed graphs: the Stochastic co-Blockmodel and spectral algorithm Di-Sim

TL;DR

This paper introduces di-sim, a spectral algorithm for directed graph co-clustering that detects asymmetries by analyzing nodes' sending and receiving edges, validated on real networks and supported by theoretical guarantees.

Contribution

The paper presents di-sim, a novel spectral co-clustering algorithm for directed graphs that captures asymmetries and is theoretically proven to be weakly consistent under a generalized stochastic model.

Findings

di-sim effectively identifies asymmetric nodes in real networks

Theoretical guarantees show weak consistency under the degree corrected co-Blockmodel

Algorithm works without assumptions on smallest degree nodes

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 di-sim that builds on a dual measure of similarity that correspond to how a node (i) sends and (ii) receives edges. Using di-sim, 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 di-sim 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 co-Blockmodel, 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.