AstroLink: Hierarchical Astrophysical Clustering

• AstroLink is a general-purpose astrophysical clustering algorithm that identifies statistically significant, hierarchically nested structures as overdensities in point-cloud data.
• It employs an ordered-density plot to reduce high-dimensional data into a 2D representation, enabling clear morphological diagnostics like distinguishing streams from shells.
• Featuring a single significance threshold and adaptive feature scaling, AstroLink robustly handles spatial, kinematic, and chemical data for comprehensive structure extraction.

Searching arXiv for AstroLink papers to ground the article in current research. AstroLink is a general-purpose astrophysical clustering algorithm for extracting hierarchical, arbitrarily shaped structures from point-cloud data in arbitrary feature spaces. It was developed by improving CluSTAR-ND so that clusters are identified as statistically significant overdensities rather than as ad hoc density features, and so that the implicit high-dimensional clustering structure is represented by an ordered-density plot that reduces the problem to a two-dimensional summary (Oliver et al., 2023). In subsequent work, AstroLink has been used not only to recover hierarchical substructure in simulated galactic haloes, but also to classify tidal debris morphologies such as stream-like and shell-like remnants, and to serve as the hard-clustering stage in the composite pipeline $\texttt{FuzzyCat} \circ \texttt{AstroLink}$ for phase-temporal analysis of galaxy formation and evolution (Ekanayaka et al., 30 Jul 2025).

1. Origins and design objectives

AstroLink was introduced as an “efficient and versatile clustering algorithm designed to hierarchically classify astrophysically-relevant structures from both synthetic and observational data sets” (Oliver et al., 2023). Its immediate predecessor was CluSTAR-ND, a hierarchical galaxy/(sub)halo finder, but AstroLink was redesigned to address two limitations emphasized in that earlier framework: the need for a more intuitive and statistically principled cluster-selection mechanism, and the need for greater adaptability in data with high or fluctuating noise levels (Oliver et al., 2023).

A central design decision was to replace the three cluster-extraction parameters of CluSTAR-ND with a single parameter, $S$, defined as the lower statistical significance threshold of clusters. This threshold can be estimated automatically through a dynamical model-fitting process, thereby making cluster extraction less brittle and more interpretable (Oliver et al., 2023). The redesign also removed the optional Friends-of-Friends root-level halo-finding step, allowing AstroLink to be applied seamlessly beyond purely spatial halo-finding tasks to phase space, chemical-abundance spaces, and mixed observational feature spaces (Oliver et al., 2023).

The algorithm is explicitly framed as a general-purpose astrophysical clustering method rather than a narrow halo finder. Later work describes it as a structure extractor that can operate on “any point-based data set whose geometry lives in some feature space,” provided that a Euclidean or Mahalanobis notion of neighborhood similarity is sensible (Oliver et al., 2024, Ekanayaka et al., 30 Jul 2025). This broader scope is essential to its subsequent use in 6D stellar phase-space decomposition and in morphology-sensitive analyses of tidal debris.

2. Algorithmic structure

AstroLink’s workflow consists of optional feature rescaling, local-density estimation, agglomerative grouping, statistical cluster identification, and optional hierarchy correction (Oliver et al., 2023). In formulations used across the AstroLink literature, if adaptive = 1 then each feature is rescaled to unit variance so that no coordinate dominates due to units alone (Oliver et al., 2023, Ekanayaka et al., 30 Jul 2025). This implies that AstroLink effectively calculates Mahalanobis-like distances on the original feature space while retaining a computationally simple Euclidean implementation after rescaling (Oliver et al., 2023).

Local density is estimated from the $k_{\mathrm{den}}$ nearest neighbours, with default $k_{\mathrm{den}} = 20$ (Oliver et al., 2023, Oliver et al., 2024, Ekanayaka et al., 30 Jul 2025). In the 2023 formulation, the estimator is written as

$$\rho_i \propto \frac{1}{h_i^d} \sum_{j \in N_{k_\mathrm{den}}} K\left(\frac{s(x_i, x_j)}{h_i}\right),$$

where $h_i$ is the maximum neighbour distance inside the $k_{\mathrm{den}}$ neighbourhood and $K(u) \propto (1-u^2)$ is an Epanechnikov kernel with compact support, vanishing for $u>1$ (Oliver et al., 2023). The density is then transformed to a scaled log-density,

$$\log\hat{\rho}_i \coloneq \frac{\log(\rho_i/\rho_\mathrm{min})}{\log(\rho_\mathrm{max}/\rho_\mathrm{min})},$$

mapping local densities into $S$0 so that noisy fluctuations become comparable across data sets (Oliver et al., 2023). A related description in the later phase-temporal study characterizes this step as using a multivariate Epanechnikov kernel together with a balloon estimator on the point’s $S$1-neighbourhood (Oliver et al., 2024).

The aggregation stage constructs a local-density-weighted $S$2-nearest-neighbour graph, with $S$3 data-driven by default (Oliver et al., 2023, Oliver et al., 2024). In the original AstroLink paper, graph edges are weighted by

$S$4

and the aggregation process is described as resembling Kruskal’s minimum spanning tree algorithm interwoven with additional hierarchy-tracking steps (Oliver et al., 2023). The default choice of $S$5 is

$S$6

where $S$7 is the dimensionality of the feature space (Oliver et al., 2023). Across implementations, points are linked in descending order of $S$8, generating a hierarchy of groups and an ordered-density curve or ordered-density plot (Ekanayaka et al., 30 Jul 2025).

The resulting output is not merely a flat partition. AstroLink returns a hierarchy of clusters, reflecting the multiscale nesting of overdensities in the input point cloud (Oliver et al., 2024). This hierarchical output is particularly relevant to astrophysical systems, whose structures are themselves nested across scales.

3. Ordered-density representation and statistical significance

The ordered-density plot is one of AstroLink’s defining innovations. It is obtained by composing the ordered list of points with the scaled log-density values, yielding a two-dimensional representation of the clustering structure (Oliver et al., 2023). The AstroLink paper describes this plot as reducing “the complexity of the $S$9-dimensional feature space” to “the simplicity of a $k_{\mathrm{den}}$0-dimensional dendrogram,” with peaks corresponding to overdensities and smaller embedded peaks often corresponding to stochastic clumping rather than physically meaningful substructure (Oliver et al., 2023).

AstroLink’s cluster-selection mechanism is based on prominence. For a group $k_{\mathrm{den}}$1, the prominence is defined as

$k_{\mathrm{den}}$2

where the terms encode the group’s maximum density, the boundary or saddle density where it merges upward in the hierarchy, and an internal noise correction derived from the direct children of the group (Oliver et al., 2023). The paper characterizes this quantity as effectively a logarithmic signal-to-noise ratio.

Rather than fixing a hard density threshold, AstroLink fits a descriptive model $k_{\mathrm{den}}$3 to the prominence distribution of groups from $k_{\mathrm{den}}$4 on a case-by-case basis (Oliver et al., 2023). The model is written as

$k_{\mathrm{den}}$5

where $k_{\mathrm{den}}$6 is the transition point between noise-like and cluster-like prominences (Oliver et al., 2023). The best-performing noise model among several candidates was reported to be a Beta distribution, while the cluster component was modeled with a Uniform distribution to avoid distortion by rare but highly significant clusters (Oliver et al., 2023).

Statistical significance is then defined by mapping group prominences through the fitted Beta distribution into a Gaussian significance scale:

$k_{\mathrm{den}}$7

A group is labeled a cluster when $k_{\mathrm{den}}$8, where $k_{\mathrm{den}}$9 is the lower significance threshold in units of $k_{\mathrm{den}} = 20$0 (Oliver et al., 2023). If S = auto, AstroLink sets

$k_{\mathrm{den}} = 20$1

so that the model-inferred transition point between the noise and cluster regimes determines the default threshold (Oliver et al., 2023).

If h_style = 1, which is the default, AstroLink applies a hierarchy correction that can incorporate additional outlier overdensities from $k_{\mathrm{den}} = 20$2 while simplifying redundant clusters-within-clusters that differ by only a small number of points (Oliver et al., 2023, Ekanayaka et al., 30 Jul 2025). A plausible implication is that AstroLink’s hierarchy is intentionally shallower and more interpretable than a raw density tree, even if that can penalize certain fine-grained label-comparison metrics.

4. Benchmarking, robustness, and relation to neighboring methods

AstroLink belongs to the same broad family as DBSCAN, OPTICS, and HDBSCAN, but it is presented as a more statistically robust astrophysical variant that identifies statistical outliers from noisy density fluctuations rather than simply reporting all dense regions as clusters (Oliver et al., 2024). Relative to classical halo finders such as AHF, AstroLink is not restricted to self-bound, spherical, or fixed-overdensity structures, which is precisely why it can recover tidally disrupted or transient structures such as streams, shells, and infalling groups (Oliver et al., 2024).

The 2023 AstroLink paper evaluates the method on synthetic stellar haloes generated with Galaxia from 11 $k_{\mathrm{den}} = 20$3CDM stellar haloes from Bullock et al. (2005) and 6 artificial stellar haloes from Johnston et al. (2008), using spatial coordinates $k_{\mathrm{den}} = 20$4, velocities $k_{\mathrm{den}} = 20$5, and chemical abundances $k_{\mathrm{den}} = 20$6 in various feature-space combinations (Oliver et al., 2023). Runtime measurements report that AstroLink and CluSTAR-ND both scale as $k_{\mathrm{den}} = 20$7, whereas Halo-OPTICS scales as $k_{\mathrm{den}} = 20$8 (Oliver et al., 2023). Quantitatively, AstroLink is reported to be about $k_{\mathrm{den}} = 20$9 faster than CluSTAR-ND and $$\rho_i \propto \frac{1}{h_i^d} \sum_{j \in N_{k_\mathrm{den}}} K\left(\frac{s(x_i, x_j)}{h_i}\right),$$0–$$\rho_i \propto \frac{1}{h_i^d} \sum_{j \in N_{k_\mathrm{den}}} K\left(\frac{s(x_i, x_j)}{h_i}\right),$$1 faster than Halo-OPTICS on the tested synthetic data sets, while also using less memory than CluSTAR-ND (Oliver et al., 2023).

Clustering quality was assessed using NRAMI. The paper reports that AstroLink outperforms CluSTAR-ND in about 75% of comparisons and Halo-OPTICS in about 50% of them, with broadly comparable NRAMI performance to CluSTAR-ND for $$\rho_i \propto \frac{1}{h_i^d} \sum_{j \in N_{k_\mathrm{den}}} K\left(\frac{s(x_i, x_j)}{h_i}\right),$$2 around $$\rho_i \propto \frac{1}{h_i^d} \sum_{j \in N_{k_\mathrm{den}}} K\left(\frac{s(x_i, x_j)}{h_i}\right),$$3–$$\rho_i \propto \frac{1}{h_i^d} \sum_{j \in N_{k_\mathrm{den}}} K\left(\frac{s(x_i, x_j)}{h_i}\right),$$4; the default S = auto often falls in this interval (Oliver et al., 2023). AstroLink is also reported to be robust over roughly $$\rho_i \propto \frac{1}{h_i^d} \sum_{j \in N_{k_\mathrm{den}}} K\left(\frac{s(x_i, x_j)}{h_i}\right),$$5 and $$\rho_i \propto \frac{1}{h_i^d} \sum_{j \in N_{k_\mathrm{den}}} K\left(\frac{s(x_i, x_j)}{h_i}\right),$$6 in the later stream–shell classification study, with smaller values tending to be noise-dominated and larger values able to miss real structures (Ekanayaka et al., 30 Jul 2025).

The feature-space experiments further indicate that spatial features recover compact clusters well, while kinematic and chemical features are better for stream-like structures; combined feature spaces recover multiple cluster types simultaneously (Oliver et al., 2023). The addition of chemical abundances improves recovery of phase-mixed structures because chemistry preserves formation history even when spatial coherence is reduced (Oliver et al., 2023). This suggests that AstroLink’s generality is not merely formal: its performance depends materially on the chosen feature space and on whether that space retains the relevant astrophysical signal.

5. Morphological interpretation: streams, shells, and halo structure

A major extension of AstroLink is the reinterpretation of its ordered-density output as a morphology diagnostic rather than solely a clustering device. In “Streams and Shells Decoded: A Density-Driven Approach to Stellar Clustering in Galactic Halos with AstroLink” (Ekanayaka et al., 30 Jul 2025), the ordered-density plot is used to distinguish stream-like from shell-like tidal remnants in simulated galactic haloes. In this use case, any AstroLink-identifiable structure appears as a contiguous peak in the ordered-density plot, and the shape of that peak is treated as an imprint of the underlying density profile in feature space (Ekanayaka et al., 30 Jul 2025).

The simulations in that study comprise eight tidal remnants generated by disrupting Plummer-sphere subhalos in a static Milky Way-like potential using GADGET-4, with the analysis carried out in 6D phase space $$\rho_i \propto \frac{1}{h_i^d} \sum_{j \in N_{k_\mathrm{den}}} K\left(\frac{s(x_i, x_j)}{h_i}\right),$$7 (Ekanayaka et al., 30 Jul 2025). With default settings, AstroLink recovers all simulated tidal structures, while also revealing hierarchical subdivision within some of them; notably, no two disrupted subhalos are merged into a single AstroLink cluster in the authors’ sample (Ekanayaka et al., 30 Jul 2025). This supports the method’s ability to resolve distinct tidal remnants even when internal overdensities induce subclustering.

The central classification result is that the curvature of the ordered-density plot separates stream-like from shell-like remnants: stream-like cases tend to have positive curvature, whereas shell-like systems show negative curvature (Ekanayaka et al., 30 Jul 2025). The paper also states this in terms of filtered concavity. After taking a slice of the ordered-density plot—illustratively from about 4% to 25% of the cluster extent—the profile is Fourier-filtered with a cutoff at $$\rho_i \propto \frac{1}{h_i^d} \sum_{j \in N_{k_\mathrm{den}}} K\left(\frac{s(x_i, x_j)}{h_i}\right),$$8, position-averaged, and fit with a quadratic; positive concavity is associated with streams and negative concavity with shells (Ekanayaka et al., 30 Jul 2025). The slicing window is described as arbitrary but chosen to avoid noise and secondary features.

The physical interpretation offered is that shells exhibit a gradual density decline in ordered-density space because their particles are more spread out angularly despite radial concentration near a shell edge, whereas streams show a denser core plus arms followed by a sharper drop into low-density territory, yielding a more sharply peaked ordered-density profile (Ekanayaka et al., 30 Jul 2025). AstroLink-based classification shows strong agreement with more standard stream/shell criteria based on orbital or physical morphology, but it does not require prior knowledge of the host potential or a manually specified orbital model (Ekanayaka et al., 30 Jul 2025). The authors further note that the absence of line-of-sight velocities does not strongly damage the approach: the essential dynamical signature is retained in 5D, although some observer orientations can weaken the concavity signal (Ekanayaka et al., 30 Jul 2025).

This line of work situates AstroLink within the broader study of galactic archaeology and dark matter. The paper argues that because tidal remnants are sensitive probes of the host dark matter potential, their identification and classification provide routes to reconstructing galactic evolution and probing dark matter structure formation (Ekanayaka et al., 30 Jul 2025). A plausible implication is that AstroLink’s value is not confined to segmentation; it also lies in converting cluster geometry into physically interpretable diagnostics.

6. Composition with FuzzyCat and applications to galaxy formation

AstroLink also serves as the first stage in the composite pipeline

$$\rho_i \propto \frac{1}{h_i^d} \sum_{j \in N_{k_\mathrm{den}}} K\left(\frac{s(x_i, x_j)}{h_i}\right),$$9

introduced for galaxy formation and evolution via phase-temporal clustering (Oliver et al., 2024). In that framework, AstroLink provides the hard clusterings on each realization of a changing data set, while FuzzyCat propagates the variability across realizations into a fuzzy hierarchy of stable fuzzy clusters with membership functions (Oliver et al., 2024).

The division of labor is explicit. AstroLink performs per-realization clustering on the raw feature-space point cloud, while FuzzyCat operates on the resulting catalogue of clusterings rather than on the raw data directly (Oliver et al., 2024). FuzzyCat works in Jaccard space, using pairwise cluster overlaps measured by the Jaccard index and retaining clusters that satisfy internal similarity, external dissimilarity, and stability requirements across realizations (Oliver et al., 2024). The stated defaults for FuzzyCat include

$h_i$0

together with a minStability hyperparameter that was adjusted in the galaxy-evolution application so that fuzzy clusters persist for at least $h_i$1 Myr (Oliver et al., 2024).

In the principal application, AstroLink is run on the 6D position-velocity phase space of stellar particles in NIHAO-UHD simulated galaxies, and FuzzyCat is applied across snapshots to track stable structures over time (Oliver et al., 2024). The resulting pipeline extracts dwarf galaxies, infalling groups, stellar streams and their progenitors, stellar shells, galactic bulges, and star-forming regions (Oliver et al., 2024). The paper emphasizes that this reveals a richer and more complete internal structure of galaxies than traditional halo finders such as AHF, which recover mainly self-bound groups and systematically miss transient or tidally disrupted structures (Oliver et al., 2024).

The broader methodological claim is that the pipeline avoids strong assumptions about the data, the change process, the number of structure types, the importance of specific structure classes, or extensive user input in the form of hand-tuned hyperparameters (Oliver et al., 2024). This suggests a context-agnostic strategy for astrophysical structure extraction in evolving systems, with AstroLink providing the statistically calibrated structural backbone.

7. Scientific significance, limits, and interpretation

AstroLink’s scientific significance lies in its attempt to make hierarchical clustering in astrophysics both statistically explicit and operationally tractable. It is designed to find meaningful hierarchical and arbitrarily shaped astrophysical clusters from both synthetic and observational data, to distinguish them from stochastic clumping, and to do so with a single significance control parameter whose default can be estimated from the data (Oliver et al., 2023). This makes it well suited to the hierarchical structure-formation context in which galaxies contain satellites, streams, shells, bulges, and nested substructure.

A recurrent misconception in astrophysical clustering is that all dense regions should be treated as clusters, or that the relevant structures must be self-bound. AstroLink directly contests both assumptions. Its prominence-based significance test is designed precisely to avoid labeling every density fluctuation as structure, and its successful recovery of streams, shells, and infalling groups shows that self-boundness is not a necessary criterion for astrophysical relevance (Oliver et al., 2023, Oliver et al., 2024). Another potential misconception is that ordered-density plots are merely visualization devices; the stream–shell study shows that, within AstroLink, they can be used as morphology diagnostics whose curvature or concavity carries dynamical information (Ekanayaka et al., 30 Jul 2025).

The method nevertheless has limits that are explicit in the cited work. Results depend on the choice of feature space, and some observer orientations can weaken the concavity signal in reduced-dimensional stream–shell classification (Ekanayaka et al., 30 Jul 2025). Parameter choices that are too small can become noise-dominated, while overly large significance thresholds can suppress real structures (Ekanayaka et al., 30 Jul 2025). AstroLink’s shallower hierarchy may also reduce scores under metrics that reward finer label granularity, even when the resulting hierarchy is more interpretable (Oliver et al., 2023). These are not contradictions of the algorithm’s goals; rather, they indicate that AstroLink is a deliberately regularized hierarchy extractor rather than a maximal structure enumerator.

Within current research, AstroLink occupies a position at the intersection of density-based clustering, hierarchical astrophysical structure finding, and morphology-sensitive galactic archaeology. Its initial contribution was to provide a statistically calibrated alternative to parameter-heavy hierarchical halo finders (Oliver et al., 2023). Its later development shows that the same framework can support phase-temporal fuzzy clustering in galaxy evolution and density-driven morphological classification of tidal debris (Oliver et al., 2024, Ekanayaka et al., 30 Jul 2025). This suggests that AstroLink is best understood not as a single-purpose cluster finder, but as a general hierarchical representation of astrophysical point-cloud structure whose ordered-density geometry can itself become an object of scientific interpretation.