Counting Balanced Triangles on Social Networks With Uncertain Edge Signs

TL;DR

This work provides 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 and explores approximate solutions for counting via different sampling approaches.

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.

Figures (10)

• Figure 1: An example of balanced and unbalanced triangles.
• Figure 2: An example of a signed graph with uncertain edges. The number associated with the edge is the probability the edge is positive.
• Figure 3: An example of uncertain balanced, unbalanced and unclassified triangles for $t = 0.7$. The numerical value of each edge corresponds to $P_+(e)$
• Figure 4: An example of different ordering options for the neighbourhood list of node $Z$. (a) is the baseline of sorting by Node Order, (b) is sorted by descending sign probability and (c) is sorted by Absolute Edge Order. Boxes in red and blue are edges which are likely to be positive and negative, respectively, and satisfy the threshold pruning requirement.
• Figure 5: Comparing the runtime of the Baseline with the Improved solution for all networks, for $t = 0.8$

Theorems & Definitions (22)

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