Counting Balanced Triangles on Social Networks With Uncertain Edge Signs

2026-02-15

AI Summary

This work addresses the problem of counting balanced triangles in social networks with uncertain edge signs by extending structural balance theory to probabilistic graphs. Edge signs are modeled as probabilities, and the paper introduces both exact baseline and optimized algorithms for precise counting and enumeration, alongside an efficient sampling-based approximation method. Key technical contributions include leveraging edge probability information to prune the search space and enhance sampling efficiency. Experimental results demonstrate that the proposed sampling approach achieves query speeds over two orders of magnitude faster than baseline methods across diverse real-world network topologies and probability distributions, substantially improving the scalability of balanced triangle analysis on large-scale uncertain graphs.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

On signed social networks, balanced and unbalanced triangles are a critical motif due to their role as the foundations of Structural Balance Theory. The uses for these motifs have been extensively explored in networks with known edge signs, however in the real-world graphs with ground-truth signs are near non-existent, particularly on a large-scale. In reality, edge signs are inferred via various techniques with differing levels of confidence, meaning the edge signs on these graphs should be modelled with a probability value. In this work, we adapt balanced and unbalanced triangles to a setting with uncertain edge signs and explore the problems of triangle counting and enumeration. We provide a baseline and improved method (leveraging the inherent information provided by the edge probabilities in order to reduce the search space) for fast exact counting and enumeration. We also explore approximate solutions for counting via different sampling approaches, including leveraging insights from our improved exact solution to significantly reduce the runtime of each sample resulting in upwards of two magnitudes more queries executed per second. We evaluate the efficiency of all our solutions as well as examine the effectiveness of our sampling approaches on real-world topological networks with a variety of probability distributions.

Problem

• balanced triangles
• signed social networks
• uncertain edge signs
• triangle counting
• structural balance

Innovation

• uncertain edge signs
• balanced triangles
• exact counting
• sampling approximation
• structural balance theory