Characterizing the hierarchical structure of filaments I. On the origin of the length-mass (L-M) scaling relation

Aims. Filamentary structures in molecular clouds are widely recognized as fundamental components of star formation. Observations report a length-mass relation of the form $L\propto M^{\alpha }$ with $\alpha \simeq 0.5$. We characterize this relation in simulated molecular clouds and investigate the influence of hierarchical fragmentation and spatial resolution. Methods. We apply a two-step filament identification method to high-resolution 3D hydrodynamical and magnetohydrodynamical simulations from the SILCC-Zoom project, combining the Rolling Hough Transform with dendrogram-based segmentation for hierarchical decomposition. We analyze the resulting structures across multiple spatial resolutions. Results. The identified structures span column densities of $\sim {10}^{21}$–$\sim {10}^{23}{\mathrm{c}\mathrm{m}}^{-2}$. Across multiple resolutions and, separately, the full hierarchical decomposition at the highest resolution, the ensemble exhibits a sub-linear $L-M$ relation close to $L\propto M^{0.5}$. When individual structures are traced across resolutions, the slopes remain sub-linear ($\alpha \simeq 0.45$–$0.70$), consistent with the geometric prediction $L\propto M^{0.5}$ of a fragmentation-driven random walk. In contrast, decomposing individual structures at the highest resolution yields a wide range of slopes ($\alpha \simeq 0.2$–$3$), peaking at $\alpha <1$. A simple toy model shows that this diversity arises from differences in the internal column-density distribution and segmentation geometry. Conclusions. Taken together, these results indicate that the average sub-linear length-mass relation with $\alpha \simeq 0.5$ is a robust property of the hierarchical structure and fragmentation occurring in molecular clouds.