Population-Level Balance in Signed Networks

Abstract

Statistical network models are useful for understanding the underlying formation mechanism and characteristics of complex networks. However, statistical models for signed networks have been largely unexplored. In signed networks, there exist both positive (e.g., like, trust) and negative (e.g., dislike, distrust) edges, which are commonly seen in real-world scenarios. The positive and negative edges in signed networks lead to unique structural patterns, which pose challenges for statistical modeling. In this article, we introduce a statistically principled latent space approach for modeling signed networks and accommodating the well-known balance theory, that is, “the enemy of my enemy is my friend” and “the friend of my friend is my friend.” The proposed approach treats both edges and their signs as random variables, and characterizes the balance theory with a novel and natural notion of population-level balance. This approach guides us towards building a class of balanced inner-product models, and toward developing scalable algorithms via projected gradient descent to estimate the latent variables. We also establish non-asymptotic error rates for the estimates, which are further verified through simulation studies. In addition, we apply the proposed approach to an international relation network, which provides an informative and interpretable model-based visualization of countries during World War II. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

KEYWORDS:

• Balance theory
• Latent space models
• Projected gradient descent
• Signed networks

Supplementary Materials

The supplementary material contains the detailed description of the separate estimation method in Algorithms S1 and S2 and their initialization algorithms, the projected gradient descent algorithm for the joint estimation, the generalization of population-level balance to any $\mathcal{l}$-loops as well as general weighted networks, the extension of Theorem 1 to the infinitely exchangeable graphon model for signed networks, all proofs, additional simulation results, and a sensitivity analysis on the application to international relation network data. The code is publicly available on GitHub at https://github.com/weijtang/SignedLSM.

Acknowledgments

The authors thank the editor, the associate editor, and referees for their constructive comments and suggestions, which led to significant improvements in the article.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

The research of Zhu was supported by NSF DMS-2210439.