# Community Correlations and Testing Independence Between Binary Graphs

arXiv: https://arxiv.org/abs/1906.03661 · HTML: https://mlsanity.com/paper/1906.03661/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jun 2019.

## Built on by (most cited first, 4 of 4)

- [Refined Graph Encoder Embedding via Self-Training and Latent Community Recovery](https://mlsanity.com/paper/2405.12797/) (2024)
- [Encoder Embedding for General Graph and Node Classification](https://mlsanity.com/paper/2405.15473/) (2024)
- [Principal Graph Encoder Embedding and Principal Community Detection](https://mlsanity.com/paper/2501.14939/) (2025)
- [Fast and Scalable Multi-Kernel Encoder Classifier](https://mlsanity.com/paper/2406.02189/) (2024)

## Abstract (arXiv)

Graph data has a unique structure that deviates from standard data assumptions, often necessitating modifications to existing methods or the development of new ones to ensure valid statistical analysis. In this paper, we explore the notion of correlation and dependence between two binary graphs. Given vertex communities, we propose community correlations to measure the edge association, which equals zero if and only if the two graphs are conditionally independent within a specific pair of communities. The set of community correlations naturally leads to the maximum community correlation, indicating conditional independence on all possible pairs of communities, and to the overall graph correlation, which equals zero if and only if the two binary graphs are unconditionally independent. We then compute the sample community correlations via graph encoder embedding, proving they converge to their respective population versions, and derive the asymptotic null distribution to enable a fast, valid, and consistent test for conditional or unconditional independence between two binary graphs. The theoretical results are validated through comprehensive simulations, and we provide two real-data examples: one using Enron email networks and another using mouse connectome graphs, to demonstrate the utility of the proposed correlation measures.