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Structural Balance in Real-World Social Networks: Incorporating Direction and Transitivity in Measuring Partial Balance

Rezvaneh Rezapour, Ly Dinh, Lan Jiang, Jana Diesner

TL;DR

The paper addresses how to measure structural balance in real-world signed directed networks by jointly incorporating transitivity and sign consistency. It introduces a triad-census–based framework that computes per-triad balance $B_{T^{(i)}}^{j}$, aggregates to $B_{T^{(i)}}$ and the network balance $B_{Avg(G)}$, and compares against an undirected baseline $B_{Avg(G')}$. Applied to ten networks with edge signs derived from NLP and surveys, the study finds an average partial balance around $0.835$, with substantial variation across relation types and notable differences between directed and undirected realizations. The results demonstrate that directionality and relation type shape the configuration of balanced triads (e.g., prevalence of $300$ and $+++$ triples) and offer refined insights into hierarchy, influence, and reciprocity in social dynamics across contexts and platforms.

Abstract

Structural balance theory predicts that triads in networks gravitate towards stable configurations. The theory has been verified for undirected graphs. Since real-world networks are often directed, we introduce a novel method for considering both transitivity and sign consistency for evaluating partial balance in signed digraphs. We test our approach on graphs constructed by using different methods for identifying edge signs: natural language processing to infer signs from underlying text data, and self-reported survey data. Our results show that for various social contexts and edge sign detection methods, partial balance of these digraphs are moderately high, ranging from 61% to 96%. Our approach not only enhances the theoretical framework of structural balance but also provides practical insights into the stability of social networks, enabling a deeper understanding of interpersonal and group dynamics across different communication platforms.

Structural Balance in Real-World Social Networks: Incorporating Direction and Transitivity in Measuring Partial Balance

TL;DR

The paper addresses how to measure structural balance in real-world signed directed networks by jointly incorporating transitivity and sign consistency. It introduces a triad-census–based framework that computes per-triad balance , aggregates to and the network balance , and compares against an undirected baseline . Applied to ten networks with edge signs derived from NLP and surveys, the study finds an average partial balance around , with substantial variation across relation types and notable differences between directed and undirected realizations. The results demonstrate that directionality and relation type shape the configuration of balanced triads (e.g., prevalence of and triples) and offer refined insights into hierarchy, influence, and reciprocity in social dynamics across contexts and platforms.

Abstract

Structural balance theory predicts that triads in networks gravitate towards stable configurations. The theory has been verified for undirected graphs. Since real-world networks are often directed, we introduce a novel method for considering both transitivity and sign consistency for evaluating partial balance in signed digraphs. We test our approach on graphs constructed by using different methods for identifying edge signs: natural language processing to infer signs from underlying text data, and self-reported survey data. Our results show that for various social contexts and edge sign detection methods, partial balance of these digraphs are moderately high, ranging from 61% to 96%. Our approach not only enhances the theoretical framework of structural balance but also provides practical insights into the stability of social networks, enabling a deeper understanding of interpersonal and group dynamics across different communication platforms.
Paper Structure (14 sections, 5 figures, 2 tables)

This paper contains 14 sections, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Balance evaluation for signed directed networks with transitivity considered
  4. Problem definition
  5. Empirical Analysis
  6. Data
  7. Balance analysis based on edge signs and transitivity
  8. Results
  9. Descriptive network measures
  10. Partial balance analysis
  11. Sign analysis of triples
  12. Utility of directed networks
  13. Discussion and Conclusion
  14. Analysis of Emails using NLP

Figures (5)

  • Figure 1: Triad census (include only triads which are transitive and balanced). Example of MAN census label - 030T: Mutual=0, Asymmetric=3, Null=0, Letter Label: T=Transitive, D=Down, U=Up
  • Figure 2: Balanced and imbalanced triples
  • Figure 3: Case study: College House C network as a directed network (left) and undirected network (right)
  • Figure 4: Case study: College House B network as a directed network (left) and undirected network (right). Edge Legends: all indicates the whole network, and V2 egonet indicates the egonet of node V2.
  • Figure 5: Examining differences in network analysis outcomes with and without considering directionality in real-world data