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.

翻译：图嵌入已成为理解图结构的强大工具。与经典谱方法不同，DeepWalk、Node2Vec等近期方法基于解决图上的非线性优化问题，利用通过随机游走获取的局部信息。经验表明，这些技术能够产生比经典方法"更优"的嵌入。然而，由于这些方法依赖于求解非凸优化问题，即使在简单图类上，获得解的性质理论保证仍具挑战性。本工作证明了DeepWalk算法在随机块模型（SBM）生成图上的收敛性质。尽管SBM是简化模型，但已被证明是分析大型图算法行为的经典模型。我们的结果与现有SBM谱嵌入的研究结论相呼应，表明即使在一维嵌入情况下，DeepWalk算法的输出也能以高概率可证明地恢复聚类结构。