The $beta$-skeleton is a mathematical method to construct graphs from a set of points that has been widely applied in the areas of image analysis, machine learning, visual perception, and pattern recognition. In this work, we apply the $beta$-skeleton to study the cosmic web. We use this tool on observed and simulated data to identify the filamentary structures and characterize the statistical properties of the skeleton. In particular, we compare the $beta$-skeletons built from SDSS-III galaxies to those obtained from MD-PATCHY mocks, and also to mocks directly built from the Big MultiDark $N$-body simulation. We find that the $beta$-skeleton is able to reveal the underlying structures in observed and simulated samples without any parameter fine-tuning. A different degree of sparseness can be obtained by adjusting the value of $beta$; in addition, the statistical properties of the length and direction of the skeleton connections show a clear dependence on redshift space distortions (RSDs), cosmological effects and galaxy bias. We also find that the $N$-body simulation accurately reproduces the RSD effect in the data, while the MD-PATCHY mocks appear to underestimate its magnitude. Our proof-of-concept study shows that the statistical properties of the $beta$-skeleton can be used to probe cosmological parameters and galaxy evolution.
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