Six-sigma approach-based visibility graphs

Ferenczi, Martin and Kummer, Alex and Abonyi, János (2026) Six-sigma approach-based visibility graphs. Scientific Reports, 16 (1). ISSN 2045-2322

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Abstract

Visibility graphs transform time series into complex networks where nodes represent time indices and edges encode temporal visibility relationships. While these graphs preserve structural properties of time series, the semantic interpretation of derived network characteristics remains challenging, limiting interpretability and raising questions about what information visibility graphs actually encode. This study enhances visibility graph interpretability using Statistical Process Control (SPC) as an external lens. We construct zone-labeled horizontal visibility graphs (HVGs) that preserve ordinal visibility properties while labeling nodes according to SPC zone classifications (A, B, C zones and control limit violations). Following Six Sigma practices, we quantize the vertical axis into control chart zones and establish explicit correspondences between SPC patterns and graph signatures, including degree distributions, edge weights, motifs, and local communities. The zone-labeled HVG framework successfully maps SPC-charted time series to interpretable network representations. Long runs and trends correspond to skewed in/out degrees and elongated paths, alternation patterns show elevated local clustering, while limit excursions manifest as hubs or articulation points partitioning the graph. This integration provides structured subgraph-level explanations for process alarms, demonstrating that visibility graphs inherently carry actionable information aligned with classical SPC rules, enabling explainable anomaly detection in industrial processes and quality control applications.

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