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Non-Euclidean Mixture Model for Social Network Embedding

Roshni G. Iyer, Yewen Wang, Wei Wang, Yizhou Sun

TL;DR

Experiments on public datasets show NMM-GNN significantly outperforms state-of-the-art baselines on social network generation and classification tasks, demonstrating its ability to better explain how the social network is formed.

Abstract

It is largely agreed that social network links are formed due to either homophily or social influence. Inspired by this, we aim at understanding the generation of links via providing a novel embedding-based graph formation model. Different from existing graph representation learning, where link generation probabilities are defined as a simple function of the corresponding node embeddings, we model the link generation as a mixture model of the two factors. In addition, we model the homophily factor in spherical space and the influence factor in hyperbolic space to accommodate the fact that (1) homophily results in cycles and (2) influence results in hierarchies in networks. We also design a special projection to align these two spaces. We call this model Non-Euclidean Mixture Model, i.e., NMM. We further integrate NMM with our non-Euclidean graph variational autoencoder (VAE) framework, NMM-GNN. NMM-GNN learns embeddings through a unified framework which uses non-Euclidean GNN encoders, non-Euclidean Gaussian priors, a non-Euclidean decoder, and a novel space unification loss component to unify distinct non-Euclidean geometric spaces. Experiments on public datasets show NMM-GNN significantly outperforms state-of-the-art baselines on social network generation and classification tasks, demonstrating its ability to better explain how the social network is formed.

Non-Euclidean Mixture Model for Social Network Embedding

TL;DR

Experiments on public datasets show NMM-GNN significantly outperforms state-of-the-art baselines on social network generation and classification tasks, demonstrating its ability to better explain how the social network is formed.

Abstract

It is largely agreed that social network links are formed due to either homophily or social influence. Inspired by this, we aim at understanding the generation of links via providing a novel embedding-based graph formation model. Different from existing graph representation learning, where link generation probabilities are defined as a simple function of the corresponding node embeddings, we model the link generation as a mixture model of the two factors. In addition, we model the homophily factor in spherical space and the influence factor in hyperbolic space to accommodate the fact that (1) homophily results in cycles and (2) influence results in hierarchies in networks. We also design a special projection to align these two spaces. We call this model Non-Euclidean Mixture Model, i.e., NMM. We further integrate NMM with our non-Euclidean graph variational autoencoder (VAE) framework, NMM-GNN. NMM-GNN learns embeddings through a unified framework which uses non-Euclidean GNN encoders, non-Euclidean Gaussian priors, a non-Euclidean decoder, and a novel space unification loss component to unify distinct non-Euclidean geometric spaces. Experiments on public datasets show NMM-GNN significantly outperforms state-of-the-art baselines on social network generation and classification tasks, demonstrating its ability to better explain how the social network is formed.

Paper Structure

This paper contains 49 sections, 4 equations, 2 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Preliminary and Related Work
  3. Social Network Embedding
  4. Non-Euclidean Geometry for Graphs
  5. Methodology
  6. Overview
  7. Modeling via Non-Euclidean Mixture Model (NMM)
  8. Modeling for homophily
  9. Representation of homophily regulated nodes.
  10. Link prediction using homophily based distribution.
  11. Modeling for social influence
  12. Representation of social influence regulated nodes.
  13. Link prediction using social influence based distribution.
  14. Non-Euclidean Mixture Model
  15. Link Prediction Using Mixed Space Distribution.
  16. ...and 34 more sections

Figures (2)

  • Figure 1: Model Architecture. (a) Architecture overview of NMM-GNN, a non-Euclidean Mixture Model with non-Euclidean VAE framework. (b) Illustration of space unification loss of NMM-GNN . $\boldsymbol{z}_i^{H}$ from the hyperbolic space is projected to its corresponding point in the spherical space, $\boldsymbol{z}_i^{S^{'}}$, such that its geodesic distance is ensured to be close to $v_{i}$'s existing spherical space representation, $\boldsymbol{z}_i^{S}$.
  • Figure 2: Ablation studies. (a) Quality of mixture model, where $\boldsymbol{\mathrm{NMM_{hom}}}$ and $\boldsymbol{\mathrm{NMM_{rank}}}$ are homophily-only and social influence-only deconstructed NMM components. (b) Inductive reasoning for NMM-GNN and RaRE on LiveJournal. % nodes$\{10, 30, 50, 70, 90\}$ are sampled ensuring no overlap with test. (c) Ablation study on quality of using space unification loss (SUL) component.