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

TLDR

Since the invention of word2vec, the skip‑gram model has significantly advanced network embedding research, giving rise to DeepWalk, LINE, PTE, and node2vec. The study aims to unify these skip‑gram based network embedding models into a single matrix factorization framework. This unification is achieved by showing that each model with negative sampling can be expressed as a closed‑form matrix factorization. The authors prove that DeepWalk, LINE, PTE, and node2vec are all special cases of matrix factorization, relate them to graph Laplacians, and demonstrate that the proposed NetMF method outperforms DeepWalk and LINE on conventional network mining tasks, thereby establishing a theoretical foundation for skip‑gram based embedding.

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.