Convergence Guarantees for the DeepWalk Embedding on Block Models

TL;DR

This paper provides theoretical convergence guarantees for DeepWalk embeddings on stochastic block model graphs, demonstrating that it can reliably recover cluster structures similar to spectral methods.

Contribution

It establishes the first convergence guarantees for DeepWalk on SBMs, bridging the gap between empirical success and theoretical understanding.

Findings

DeepWalk recovers cluster structure with high probability on SBMs.

Convergence properties mirror spectral embedding results.

Theoretical analysis applies to one-dimensional embeddings.

Abstract

Graph embeddings have emerged as a powerful tool for understanding the structure of graphs. Unlike classical spectral methods, recent methods such as DeepWalk, Node2Vec, etc. are based on solving nonlinear optimization problems on the graph, using local information obtained by performing random walks. These techniques have empirically been shown to produce ''better'' embeddings than their classical counterparts. However, due to their reliance on solving a nonconvex optimization problem, obtaining theoretical guarantees on the properties of the solution has remained a challenge, even for simple classes of graphs. In this work, we show convergence properties for the DeepWalk algorithm on graphs obtained from the Stochastic Block Model (SBM). Despite being simplistic, the SBM has proved to be a classic model for analyzing the behavior of algorithms on large graphs. Our results mirror the existing ones for spectral embeddings on SBMs, showing that even in the case of one-dimensional embeddings, the output of the DeepWalk algorithm provably recovers the cluster structure with high probability.