Measuring Partial Balance in Signed Networks (1509.04037v6)

Abstract: Is the enemy of an enemy necessarily a friend? If not, to what extent does this tend to hold? Such questions were formulated in terms of signed (social) networks and necessary and sufficient conditions for a network to be "balanced" were obtained around 1960. Since then the idea that signed networks tend over time to become more balanced has been widely used in several application areas. However, investigation of this hypothesis has been complicated by the lack of a standard measure of partial balance, since complete balance is almost never achieved in practice. We formalize the concept of a measure of partial balance, discuss various measures, compare the measures on synthetic datasets, and investigate their axiomatic properties. The synthetic data involves Erd\H{o}s-R\'enyi and specially structured random graphs. We show that some measures behave better than others in terms of axioms and ability to differentiate between graphs. We also use well-known datasets from the sociology and biology literature, such as Read's New Guinean tribes, gene regulatory networks related to two organisms, and a network involving senate bill co-sponsorship. Our results show that substantially different levels of partial balance is observed under cycle-based, eigenvalue-based, and frustration-based measures. We make some recommendations for measures to be used in future work.

• The paper systematically evaluates cycle-based, eigenvalue-based, and frustration-based measures to quantify partial balance in signed networks.
• The paper introduces an axiomatic framework to assess measure invariance and sensitivity, guiding proper metric selection for different network structures.
• The paper demonstrates that real-world signed networks exhibit statistically significant partial balance, validating insights from structural balance theory.

Analyzing Partial Balance in Signed Networks

This paper explores the intricacies of measuring partial balance in signed networks, a concept rooted in social balance theory first introduced by Heider, and later formalized by Cartwright and Harary. While total balance implies that the network is free from cycles with an odd number of negative edges, achieving such balance is practically impossible in real-world networks, thereby necessitating the development of measures to quantify partial balance.

Key Contributions:

The paper systematically evaluates various measures of partial balance in signed networks and examines their performance on synthetic datasets, as well as several real-world datasets from sociology and biology literature.

Main Findings:

1. Measure Evaluation:
   • Cycle-Based Measures: These include the degree of balance $D(G)$, weighted degree of balance $C(G)$, and relative $k$-balance $D_k(G)$. Among these, the triangle index $T(G)$, representing the fraction of balanced triangles, provides meaningful results for networks with a substantial number of triangles, yet struggles with sensitivity and tends towards clustering around 0.5 in graphs with a significant proportion of negative edges.
   • Eigenvalue-Based Measures: The paper explores algebraic conflict, normalized by using the smallest eigenvalue of the signed Laplacian matrix. This approach is promising for connected graphs, providing a distinction between balance levels.
   • Frustration-Based Measure: The frustration index $F(G)$, calculated as the minimum number of edges whose removal renders the graph balanced, is shown to be one of the most reliable measures, despite computational challenges in large graphs.
2. Axiomatic Evaluation:
   • The authors propose an axiomatic framework to assess these measures, considering properties like invariance under switching and the ability to increase with additional balanced structures. This evaluation helps identify measures that might distort the interpretation of network balance, such as walk-based measures which suffer from double-counting issues.
3. Practical Implications:
   • The results underscore the need for careful measure selection based on the context of paper and network structure. The authors recommend using the frustration index for its robustness, or alternatively, algebraic conflict or $D_k(G)$ depending on the graph's characteristics.
4. Real-World Networks:
   • For several real-world signed networks, including social interaction and gene regulatory networks, the paper confirms that these networks exhibit partial balance beyond random expectations. This suggests a level of conformity with structural balance theory, contrary to analyses that might present conflicting results due to inappropriate measure choice.

Conclusion and Future Work:

The paper highlights the critical role of measure selection in understanding the dynamics and structural properties of signed networks. Future developments in the computational efficiency of measures like the frustration index, and exploration of their applicability to directed networks, will further enhance the utility of these analyses in complex network scenarios. The paper also paves the way for applications in link prediction and clustering, offering insights that could be leveraged across fields as diverse as sociology and biology.