Network Embedding as Matrix Factorization: Unifying DeepWalk, LINE, PTE, and node2vec

TL;DR

This paper unifies popular network embedding methods like DeepWalk, LINE, PTE, and node2vec under a matrix factorization framework, revealing their theoretical connections to graph Laplacian and introducing the NetMF method for improved embeddings.

Contribution

It provides a unified matrix factorization perspective for several network embedding algorithms and introduces NetMF, enhancing embedding quality and theoretical understanding.

Findings

DeepWalk empirically approximates a low-rank Laplacian transformation.

LINE is a special case of DeepWalk with context size one.

PTE extends LINE by joint factorization of multiple network Laplacians.

Abstract

Since the invention of word2vec, the skip-gram model has significantly advanced the research of network embedding, such as the recent emergence of the DeepWalk, LINE, PTE, and node2vec approaches. In this work, we show that all of the aforementioned models with negative sampling can be unified into the matrix factorization framework with closed forms. Our analysis and proofs reveal that: (1) DeepWalk empirically produces a low-rank transformation of a network's normalized Laplacian matrix; (2) LINE, in theory, is a special case of DeepWalk when the size of vertices' context is set to one; (3) As an extension of LINE, PTE can be viewed as the joint factorization of multiple networks' Laplacians; (4) node2vec is factorizing a matrix related to the stationary distribution and transition probability tensor of a 2nd-order random walk. We further provide the theoretical connections between skip-gram based network embedding algorithms and the theory of graph Laplacian. Finally, we present the NetMF method as well as its approximation algorithm for computing network embedding. Our method offers significant improvements over DeepWalk and LINE for conventional network mining tasks. This work lays the theoretical foundation for skip-gram based network embedding methods, leading to a better understanding of latent network representation learning.