V² REVOLVER Pruning

• V² REVOLVER pruning is a void identification method that uses Voronoi tessellations to estimate galaxy densities and conservatively groups adjacent low-density cells.
• It contrasts with algorithms like VoidFinder by preserving wall-like structures within voids, which moderates environmental contrasts in observed galaxy properties.
• The method integrates nonparametric Bayesian analysis with Pólya tree priors to sensitively assess distributional differences and reveal subtle trends in galaxy evolution.

V² REVOLVER pruning is a methodology designed to identify cosmic voids in large-scale galaxy surveys by leveraging Voronoi tessellations of the observed galaxy density field. The approach centers on a conservative “pruning” algorithm that distinguishes void regions by grouping together adjacent low-density zones while intentionally retaining certain wall-like structures within void boundaries. This partitioning method, in combination with robust nonparametric Bayesian statistical techniques, enables detailed investigations into how environmental factors shape galaxy evolution, providing a nuanced contrast to traditional void classification algorithms such as VoidFinder.

1. Algorithmic Foundations of V² REVOLVER Pruning

V² REVOLVER void identification initiates by mapping the galaxy density field via a Voronoi tessellation. Each observed galaxy is assigned a unique Voronoi cell, with the volume of the cell serving as a direct estimator for local galaxy density: lower-density environments correspond to larger Voronoi cell volumes. Adjacent low-density cells are clustered into zones. Unlike approaches that combine multiple low-density zones into a singular, maximally sized void, the REVOLVER pruning step is distinguished by its conservative treatment, frequently considering each low-density zone independently and restricting excessive merging.

To ensure that void boundaries correspond strictly to the surveyed volume, Voronoi cells extending beyond survey edges are assigned an infinite density, effectively excluding them from void assignment. This boundary treatment guarantees the fidelity of the void catalog to the region covered in the galaxy survey.

A concise schematic of the algorithm is as follows:

Step	Methodology	Purpose
Tessellation	Voronoi	Proxy for local galaxy density
Zone grouping	Adjacent low-density zones identified	Basis for candidate voids
Pruning	Minimal merging of low-density zones	Retains wall-like structures in void interiors
Edge constraint	Infinite density for out-of-bounds cells	Maintains strict survey-constrained void definitions

2. Comparative Context: VoidFinder and V² REVOLVER

A principal contrast in environmental studies arises between the use of the V² REVOLVER method and traditional algorithms such as VoidFinder. VoidFinder employs a sphere-growing mechanism based on third-nearest neighbor distances, tending to produce voids bounded by nearly spherical maximal spheres. The resultant void populations commonly reflect classical environmental signatures: galaxies within such voids are fainter, less massive, exhibit bluer colors, and display elevated star formation rates (SFR) compared to their denser, wall-bound counterparts.

In contrast, V² REVOLVER's conservative pruning frequently preserves wall-like features in the interior of voids and merges low-density zones in a way that results in larger and sometimes noisier void regions. This structural difference in void construction has significant statistical consequences: under the V² REVOLVER pruning, the observed contrasts between galaxies in voids and walls are diminished, with some statistical tests favoring the null hypothesis for properties such as the specific star formation rate (sSFR). As such, the choice of void-finding algorithm exerts a substantive influence on the inferred environmental dependence of galaxy properties.

3. Bayesian Nonparametric Statistical Analysis with Pólya Tree Priors

To quantitatively assess differences between galaxy populations in distinct environments, the methodology incorporates a Bayesian nonparametric testing framework. This framework circumvents the restrictive distributional assumptions of parametric approaches and improves upon the sensitivity of classical frequentist tests. Central to this framework is the use of Pólya tree priors for flexibly modeling the (unknown) probability distributions of galaxy observables.

The Pólya tree construction proceeds by recursive bisection of the sample space $\Omega$ (e.g., $\Omega = [0,1)$), with each division parameterized by independent Beta-distributed probabilities. For a subset $B$ identified along a path $$\boldsymbol{\epsilon}_k = (\epsilon_1, ..., \epsilon_k)$$:

$$P(B_{\boldsymbol{\epsilon}_k}) = \prod_{i=1}^k \theta_{\boldsymbol{\epsilon}_{i-1}}^{(1-\epsilon_i)} (1-\theta_{\boldsymbol{\epsilon}_{i-1}})^{\epsilon_i}$$

Upon observing data, the Beta variables can be marginalized using the Beta–Binomial identity, yielding a likelihood for the data given the tree partition $\Pi$ and parameters $A$:

$$\Pr(y\mid\Pi,A) = \prod_j \left[ \frac{\Gamma(\alpha_{j0}+\alpha_{j1})}{\Gamma(\alpha_{j0})\Gamma(\alpha_{j1})} \frac{\Gamma(\alpha_{j0}+n_{j0})\Gamma(\alpha_{j1}+n_{j1})}{\Gamma(\alpha_{j0}+\alpha_{j1}+n_{j0}+n_{j1})} \right]$$

Comparisons between samples are made by computing a Bayes factor,

$$B_{01} = \frac{\Pr(y^{(1,2)}\mid H_0)}{\Pr(y^{(1)},y^{(2)}\mid H_1)} = \prod_{j} b_j,$$

which directly quantifies evidence for the null (identical distributions) versus the alternative (distinct distributions). Unlike the Kolmogorov-Smirnov test, which yields a binary reject/not-reject outcome, this Bayesian procedure assigns quantitative support to each hypothesis and remains robust even as sample sizes become large.

4. Performance and Sensitivity Characteristics

The interplay between void detection algorithms and statistical methodology is critical. The Kolmogorov-Smirnov (KS) test, while widely adopted, exhibits excessive sensitivity on large datasets, resulting in frequent type I errors even for minimal differences in distribution. Parametric Bayesian tests benefit from computational efficiency when their assumptions are met but are vulnerable to outlier influence and model misspecification.

The nonparametric Bayesian Pólya tree test provides enhanced sensitivity and is model-agnostic, amplifying statistically significant differences in robust scenarios. As demonstrated in the SDSS DR7 dataset, log Bayes factors computed for VoidFinder-classified voids become strongly negative for stellar mass and luminosity, signifying pronounced environmental effects. However, for V² REVOLVER-classified voids—whose boundaries frequently encompass wall-like structures—the differences detected by the test may be negligible, with Bayes factors occasionally aligning with the null hypothesis. This outcome highlights the dependency of the statistical signal on the quality and definition of environmental partitioning.

5. Implications for Galaxy Evolution

Empirical interpretations of galaxy evolution are tightly linked to void classification strategies. Under VoidFinder, void galaxies manifest clear environmental signatures: lower mass, reduced luminosity, and higher star formation rates. When V² REVOLVER pruning is employed, and voids are defined more conservatively, these contrasts are diminished. The statistical analysis indicates that wall-like structures within V² REVOLVER voids blend the environmental differences, potentially masking subtle evolutionary trends attributable to low-density environments.

A plausible implication is that the boundaries of void regions, as set by the chosen algorithm, are not only methodological artifacts but exert a scientific influence on the interpretation of environmental effects on galactic evolution.

6. Methodological Integration: Statistical and Void Classification Considerations

The findings emphasize that both the statistical testing regime and the void identification algorithm must be considered jointly for robust cosmic environmental studies. Overly aggressive or conservative treatment in void finding can respectively exaggerate or understate environmental contrasts among galaxies. Similarly, selecting a statistical framework that is both sufficiently sensitive (as with Pólya tree-based nonparametric Bayes factors) and robust to distributional ambiguity is essential for calibrated inference.

7. Summary and Outlook

V² REVOLVER pruning represents a conservative approach to void definition that often leads to voids containing internal wall-like features. In the context of contemporary galaxy surveys and environmental studies, this method contrasts with sphere-based methods like VoidFinder by producing larger, less environmentally discrete void regions. The choice of statistical methods for downstream analysis is critical: nonparametric Bayesian frameworks equipped with Pólya tree priors provide sensitive and calibrated measures of distributional differences but may report no significant contrasts under conservative void partitions.

These methodological nuances collectively underscore the necessity of aligning void classification strategies with appropriately sensitive statistical tools to yield reliable insights into the environmental processes governing galaxy evolution.