Quantifying sunspot group nesting with density-based unsupervised clustering

Abstract: Sunspot groups often emerge in spatial-temporal clusters, known as nests or complexes of activity. Quantifying how frequently such nesting occurs is important for understanding the organisation and recurrence of solar magnetic fields. We introduce an automated approach to identify nests in the longitude-time domain and to measure the fraction of sunspot groups that belong to them. The method combines a smooth representation of emergence patterns with a density-based clustering procedure, validated using synthetic solar-like cycles and corrected for variations in data density. We apply this method to 151 years of sunspot-group observations from the Royal Greenwich Observatory Photoheliographic Results (RGO, 1874-1976) and Kislovodsk Mountain Astronomical Station (KMAS, 1955-2025) catalogues. Across all cycles and latitude bands, the mean nesting degree is $\langle D\rangle = 0.61 \pm 0.12$, implying that about 60 percent all sunspot groups emerge within nests. Nesting is strongest at mid-latitudes (10$^\circ$-20$^\circ$), and results from the two independent datasets agree in the period of overlap. The identified nests range from compact clusters to long-lived, drifting structures, offering new quantitative constraints on the persistence and organisation of solar magnetic activity.