Asymmetric Transitivity Preserving Graph Embedding

TLDR

Graph embedding maps graphs to vector spaces while preserving structure, yet existing methods fail to maintain asymmetric transitivity—a key directed‑graph property that reflects the likelihood of an edge given a directed path and aids in capturing structure and recovering missing edges. The authors aim to preserve asymmetric transitivity by approximating high‑order proximity measures. They introduce HOPE, a scalable embedding framework that derives a general formulation for multiple high‑order proximities, efficiently approximates them, and provides a theoretical RMSE bound. Experiments on synthetic and three real datasets show HOPE achieves superior approximation of high‑order proximities and outperforms existing methods in reconstruction, link prediction, and vertex recommendation.

Abstract

Graph embedding algorithms embed a graph into a vector space where the structure and the inherent properties of the graph are preserved. The existing graph embedding methods cannot preserve the asymmetric transitivity well, which is a critical property of directed graphs. Asymmetric transitivity depicts the correlation among directed edges, that is, if there is a directed path from u to v, then there is likely a directed edge from u to v. Asymmetric transitivity can help in capturing structures of graphs and recovering from partially observed graphs. To tackle this challenge, we propose the idea of preserving asymmetric transitivity by approximating high-order proximity which are based on asymmetric transitivity. In particular, we develop a novel graph embedding algorithm, High-Order Proximity preserved Embedding (HOPE for short), which is scalable to preserve high-order proximities of large scale graphs and capable of capturing the asymmetric transitivity. More specifically, we first derive a general formulation that cover multiple popular high-order proximity measurements, then propose a scalable embedding algorithm to approximate the high-order proximity measurements based on their general formulation. Moreover, we provide a theoretical upper bound on the RMSE (Root Mean Squared Error) of the approximation. Our empirical experiments on a synthetic dataset and three real-world datasets demonstrate that HOPE can approximate the high-order proximities significantly better than the state-of-art algorithms and outperform the state-of-art algorithms in tasks of reconstruction, link prediction and vertex recommendation.