Ergodic Limits, Relaxations, and Geometric Properties of Random Walk Node Embeddings

TL;DR

This paper analyzes the theoretical properties of random walk-based node embeddings, focusing on ergodic limits and geometric structures, and demonstrates their behavior on stochastic block models with extensive experiments.

Contribution

It introduces a theoretical framework for understanding the ergodic limits and relaxations of node embedding objectives, revealing their geometric and probabilistic properties.

Findings

Embeddings converge to Gaussian-like distributions within communities.

Optimal embeddings have rank 1 under certain relaxations.

Embeddings concentrate as network size increases.

Abstract

Random walk based node embedding algorithms learn vector representations of nodes by optimizing an objective function of node embedding vectors and skip-bigram statistics computed from random walks on the network. They have been applied to many supervised learning problems such as link prediction and node classification and have demonstrated state-of-the-art performance. Yet, their properties remain poorly understood. This paper studies properties of random walk based node embeddings in the unsupervised setting of discovering hidden block structure in the network, i.e., learning node representations whose cluster structure in Euclidean space reflects their adjacency structure within the network. We characterize the ergodic limits of the embedding objective, its generalization, and related convex relaxations to derive corresponding non-randomized versions of the node embedding objectives. We also characterize the optimal node embedding Grammians of the non-randomized objectives for the expected graph of a two-community Stochastic Block Model (SBM). We prove that the solution Grammian has rank $1$ for a suitable nuclear norm relaxation of the non-randomized objective. Comprehensive experimental results on SBM random networks reveal that our non-randomized ergodic objectives yield node embeddings whose distribution is Gaussian-like, centered at the node embeddings of the expected network within each community, and concentrate in the linear degree-scaling regime as the number of nodes increases.