Scalable Algorithm for Finding Balanced Subgraphs with Tolerance in Signed Networks

TL;DR

This work introduces an innovative generalized balanced subgraph model that incorporates tolerance for imbalance, and proposes a region-based heuristic algorithm, tailored for this NP -hard problem, that strikes a balance between low time complexity and high-quality outcomes.

Abstract

Signed networks, characterized by edges labeled as either positive or negative, offer nuanced insights into interaction dynamics beyond the capabilities of unsigned graphs. Central to this is the task of identifying the maximum balanced subgraph, crucial for applications like polarized community detection in social networks and portfolio analysis in finance. Traditional models, however, are limited by an assumption of perfect partitioning, which fails to mirror the complexities of real-world data. Addressing this gap, we introduce an innovative generalized balanced subgraph model that incorporates tolerance for irregularities. Our proposed region-based heuristic algorithm, tailored for this NP-hard problem, strikes a balance between low time complexity and high-quality outcomes. Comparative experiments validate its superior performance against leading solutions, delivering enhanced effectiveness (notably larger subgraph sizes) and efficiency (achieving up to 100x speedup) in both traditional and generalized contexts.

Figures (6)

• Figure 1: Balanced subgraphs are found in Cloister with different tolerance, where solid edges are positive, dashed edges are negative, black edges are balanced, and red edges are imbalanced.
• Figure 2: A signed graph (a), its MBS-V and MBS-E (b), its TMBS-V and TMBS-E (c), and its $\beta$-TMBS (d) with $\beta=\frac{1}{3}$.
• Figure 3: Problem\ref{['p3']}: Comparing maximum tolerantly balanced subgraphs in vertex cardinality ($|V|$) across various tolerances ($\beta = 2^{-x/2}$), where the part corresponding to trivial $\beta$ values for this problem has been shaded.
• Figure 4: Probelm\ref{['p4']}: Comparing maximum tolerantly balanced subgraphs in edge cardinality ($|E^{+} \cup E^{-}|$) across various tolerances ($\beta = 2^{-x/2}$), where the part corresponding to trivial $\beta$ values for this problem has been shaded.
• Figure 5: Probelm\ref{['p5']}: Comparing maximum tolerantly balanced subgraphs in TBC ($\hat{\Phi}$) across various tolerances ($\beta = 2^{-x/2}$).

Theorems & Definitions (12)

• Definition 3.1: Balanced Graph
• Definition 3.2: Balanced Graph under $\beta$-Tolerance
• lemma 1
• Definition 3.3: Tolerant Balance Index (TBI)
• lemma 2
• Definition 4.1: Tolerant Balance Count (TBC)
• lemma 3
• corollary 1
• lemma 4
• lemma 5