Principal Graph Encoder Embedding and Principal Community Detection

TL;DR

This paper introduces a novel method for detecting principal communities and embedding vertices in graphs, combining community importance ranking with vertex embedding to improve visualization and classification.

Contribution

The paper proposes a new principal graph encoder embedding method that simultaneously detects principal communities and produces vertex embeddings, with theoretical guarantees and practical advantages.

Findings

Accurately detects ground-truth principal communities in simulations

Enhances embedding visualization and vertex classification

Demonstrates robustness to label noise and scalability on real graphs

Abstract

In this paper, we introduce the concept of principal communities and propose a principal graph encoder embedding method that concurrently detects these communities and achieves vertex embedding. Given a graph adjacency matrix with vertex labels, the method computes a sample community score for each community, ranking them to measure community importance and estimate a set of principal communities. The method then produces a vertex embedding by retaining only the dimensions corresponding to these principal communities. Theoretically, we define the population version of the encoder embedding and the community score based on a random Bernoulli graph distribution. We prove that the population principal graph encoder embedding preserves the conditional density of the vertex labels and that the population community score successfully distinguishes the principal communities. We conduct a variety of simulations to demonstrate the finite-sample accuracy in detecting ground-truth principal communities, as well as the advantages in embedding visualization and subsequent vertex classification. The method is further applied to a set of real-world graphs, showcasing its numerical advantages, including robustness to label noise and computational scalability.