Principal Graph Encoder Embedding and Principal Community Detection

TL;DR

This work tackles simultaneous graph embedding and principled detection of principal communities when vertex labels are partially available. It extends the graph encoder embedding to a principal version (P-GEE) by introducing a sample community score that isolates principal communities and reduces embedding dimensions to those coordinates, while preserving the conditional density of vertex labels and maintaining Bayes-optimal classification under a random Bernoulli graph model ($Y|A \equiv Y|Z \equiv Y|Z^{D}$). Theoretical results show that the population score distinguishes principal from redundant communities, and simulations demonstrate consistency in detecting true principals and in improving vertex classification; real-data experiments confirm scalability and robustness to label noise, outperforming several baselines. Practically, P-GEE offers interpretable embedding coordinates linked to edge connectivity to principal communities and faster downstream inference due to reduced dimensionality, with code and data available on GitHub for reproducibility. These contributions advance scalable, interpretable graph embedding with principled community selection and strong empirical performance.

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.

Figures (3)

• Figure 1: This flowchart illustrates the input, output, and intermediate steps of the principal graph encoder embedding.
• Figure 2: This figure visualizes the sample adjacency matrix for each model at $n=5000$ and $K=20$, the sample community scores for $k \in [1,20]$, and the principal graph encoder embedding.
• Figure 3: This figure displays the average sample community score, the average principal community detection accuracy, and the average vertex classification error using the embedding, based on $100$ Monte-Carlo replicates with increasing $n$. P-GEE denotes the principal graph encoder embedding, and GEE denotes the original graph encoder embedding.

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Definition 5
• Theorem 2
• Theorem 1
• proof
• Theorem 1