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ABCD: Algorithm for Balanced Component Discovery in Signed Networks

Muhieddine Shebaro, Jelena Tešić

TL;DR

A scalable state-of-the-art approach for the maximum balanced subgraph detection in the network of any size is proposed, which finds the largest balanced subgraph by considering only the top K balanced states with the lowest frustration index.

Abstract

The largest balanced element in signed graphs plays a vital role in helping researchers understand the fundamental structure of the graph, as it reveals valuable information about the complex relationships between vertices in the network. The challenge is an NP-hard problem; there is no current baseline to evaluate state-of-the-art signed graphs derived from real networks. In this paper, we propose a scalable state-of-the-art approach for the maximum balanced sub-graph detection in the network of any size. The proposed approach finds the largest balanced sub-graph by considering only the top $K$ balanced states with the lowest frustration index. We show that the ABCD method selects a subset from an extensive signed network with millions of vertices and edges, and the size of the discovered subset is double that of the state-of-the-art in a similar time frame.

ABCD: Algorithm for Balanced Component Discovery in Signed Networks

TL;DR

A scalable state-of-the-art approach for the maximum balanced subgraph detection in the network of any size is proposed, which finds the largest balanced subgraph by considering only the top K balanced states with the lowest frustration index.

Abstract

The largest balanced element in signed graphs plays a vital role in helping researchers understand the fundamental structure of the graph, as it reveals valuable information about the complex relationships between vertices in the network. The challenge is an NP-hard problem; there is no current baseline to evaluate state-of-the-art signed graphs derived from real networks. In this paper, we propose a scalable state-of-the-art approach for the maximum balanced sub-graph detection in the network of any size. The proposed approach finds the largest balanced sub-graph by considering only the top balanced states with the lowest frustration index. We show that the ABCD method selects a subset from an extensive signed network with millions of vertices and edges, and the size of the discovered subset is double that of the state-of-the-art in a similar time frame.
Paper Structure (15 sections, 2 theorems, 2 equations, 6 figures, 8 tables, 4 algorithms)

This paper contains 15 sections, 2 theorems, 2 equations, 6 figures, 8 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Definitions & Corollaries
  3. Related Work
  4. Methodology
  5. ABCD Phase 1
  6. ABCD Phase 2
  7. ABCDD: Algorithm for Balanced Component Discovery Degree
  8. ABCDH: Algorithm for Balanced Component Discovery Degree Harary
  9. ABCDS: Algorithm for Balanced Component Discovery Status
  10. Complexity Analysis
  11. Proof Of Concept
  12. TIMBAL vs. ABCD for the Konect Benchmark
  13. TIMBAL vs. ABCD for the Amazon Benchmark
  14. ABCD Ablation Study
  15. Conclusion

Key Result

Corollary 2.1

All the cycles formed by combining a path in the tree and a single edge outside the tree create a fundamental cycle basis from a spanning tree. Thus, the underlying unsigned graph $G$ with $|V|$ vertices and $|E|$ edges has precisely $|V|-|E|+1$ fundamental cycles.

Figures (6)

  • Figure 1: (a): The unbalanced signed network. Green edges are the candidate edges causing imbalance, and red vertices are the candidate vertices. (b): The maximum balanced signed sub-graph obtained after deleting one candidate vertex along each edge.
  • Figure 2: The ABCD pipeline.
  • Figure 3: Degree (black, in node) vs. Sum of Neighborhood Degrees (green, next to the node) computation. The sum of neighborhood degrees labels the red vertices connected by a positive link that are candidates for deletion based on the ABCDD criteria.
  • Figure 4: The ABCD algorithm applied to a sample signed graph with ten vertices and thirteen edges. For connectivity approximations, we compute the Harary bipartition in ABCD Phase 1 (Algorithm \ref{['alg-ABCD1']}) and compute the sum of neighborhood (green), degree (blue), and adjusted status (red) at the beginning of Phase 2 (Algorithm \ref{['alg-ABCD1']}) once for the entire graph. Orange edges are the edges of the unbalanced fundamental cycles. Green edges are the candidate edges. The result of Phase 2 for all three connectivity approximations is the sub-graph defined by black and red vertices.
  • Figure 5: ABCD and TIMBAL performance comparison for Konect benchmark in terms of subset graph fractions (left) and algorithmic timing (right).
  • ...and 1 more figures

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Corollary 2.1
  • Definition 2.6
  • Theorem 2.2