Cluster Paths in Data Analysis

• Cluster paths are path-centered representations that encode structural continuity and multi-scale evolution used in diverse clustering frameworks.
• They are applied in graph-based, manifold, and convex clustering to enhance partitioning accuracy and robustness, impacting fields like trajectory analysis and neural interpretability.
• Key methodologies include shortest-path formulations, angle constraints, and regularization paths, achieving improvements such as F1-scores up to 0.99 on benchmark datasets.

Cluster paths are path-centered representations used to organize cluster structure in several distinct senses. In graph-based clustering, a path is a sequence of edges whose cumulative, minimax, or learned cost determines assignment to density peaks; in manifold clustering, it may be an angle-constrained graph path, a maximum-density path between mixture components, or a bottleneck path on a simplex graph; in convex clustering and local graph diffusion, it is a regularization path indexed by a control parameter; in temporal analysis, it is an inferred evolution path through a learned cluster-transition graph; and in neural-network interpretability, it is the sequence of cluster IDs traversed by an input across layers (Pizzagalli et al., 2018, Babaeian, 2018, Ritzert et al., 19 Mar 2025, Wu et al., 27 Jan 2025, Kloster et al., 2015, Rizoiu et al., 2015, Kroeger et al., 8 Oct 2025).

1. Meanings of the term

The term is not standardized across fields. In some literatures it denotes a path used to define clusters; in others, a path of clusterings generated as a parameter changes; in still others, a path through clusters that summarizes evolution or representation learning.

Setting	Path object	Representative formulation
Graph-based assignment	Node path $\Gamma$ from a super-source to a sample	Non-decreasing shortest-path cost (Pizzagalli et al., 2018)
Intersecting manifolds	$\alpha$-constrained neighborhood-graph path	Turning-angle threshold (Babaeian, 2018)
Density hierarchy	Continuous path between mixture centers	Maximum-density bottleneck path (Ritzert et al., 19 Mar 2025)
Simplex geometry	Path on a weighted simplex graph	LAPD bottleneck metric (Chen et al., 14 Jul 2025)
Regularization	Sequence $\{U^*(\lambda_t)\}$ or $\{x(\delta)\}$	Convex clustering and seeded PageRank paths (Wu et al., 27 Jan 2025, Kloster et al., 2015)
Neural interpretability	Layer-wise sequence $\gamma(x)=(c^{(1)},\dots,c^{(L)})$	Activation-cluster path (Kroeger et al., 8 Oct 2025)

A temporal-data usage is explicit in ClusPath, where observations are partitioned into clusters $C_1,\dots,C_k$ while simultaneously learning a directed weighted graph $A=(a_{pq})$ that models transitions between successive phases under a “slow changing world” assumption (Rizoiu et al., 2015). A separate combinatorial-optimization usage studies clustering directly on a path graph with arbitrary pairwise correlation costs; in that setting the problem is polynomial-time solvable by dynamic programming, and the associated lifted multicut polytope for paths has a complete totally dual integral description (Lange et al., 2017).

This suggests a common design pattern: replacing purely local point-to-point similarity with an object that encodes continuity, bottlenecks, admissibility, or multiscale evolution.

2. Shortest-path formulations for clustering on graphs

A canonical path-based clustering construction appears in the shortest-path method for joint human-machine analysis of complex datasets. Given a point cloud, a weighted graph is built with vertices $V=\{v_1,\dots,v_N\}$ and nonnegative edge costs $w_{ij}$. Local point-density and distance-to-higher-density are defined as

$$\rho_v = \sum_{u\in V}\mathbf{1}(\|x_u-x_v\|\le r), \qquad \delta_v = \min_{u:\rho_u>\rho_v}\|x_u-x_v\|,$$

and density peaks are selected as

$\alpha$0

A fictitious source $\alpha$1 is connected to each density peak with negligible cost $\alpha$2, and a single-source shortest-path problem is solved from $\alpha$3 to all nodes (Pizzagalli et al., 2018).

The crucial constraint is that the path-cost functional must satisfy Bellman’s optimality, i.e. it must be non-decreasing on path extension. The paper gives three standard forms:

$\alpha$4

and a learned local-window cost

$\alpha$5

Because these costs are non-decreasing under extension, Dijkstra-style search remains valid. This directly addresses a failure mode of local association rules: on the “dendrites + blob” benchmark, CDP mis-assigns tips of long thin projections and DBSCAN cannot separate two touching high-density blobs, whereas minimax-path clustering corrects both artifacts. The reported F1-score is CDP $\alpha$6, DBSCAN $\alpha$7, minimax-path $\alpha$8, SVM-enhanced $\alpha$9; the corresponding Jaccard scores are CDP $\{U^*(\lambda_t)\}$0, DBSCAN $\{U^*(\lambda_t)\}$1, minimax-path $\{U^*(\lambda_t)\}$2, SVM-enhanced $\{U^*(\lambda_t)\}$3 (Pizzagalli et al., 2018).

The same work extends path costs with a trainable path classifier. Features are extracted from the last $\{U^*(\lambda_t)\}$4 edges of a candidate extension, including densities, gap sizes, and

$\{U^*(\lambda_t)\}$5

A binary SVM with RBF kernel is trained on admissible versus forbidden fragments, and the classifier score is converted into a pseudo-cost. In 4D centroid tracking of T-cells, CDP gives F1 $\{U^*(\lambda_t)\}$6, minimax only gives F1 $\{U^*(\lambda_t)\}$7, and the SVM path-classifier reaches F1 $\{U^*(\lambda_t)\}$8 with Jaccard $\{U^*(\lambda_t)\}$9 (Pizzagalli et al., 2018).

A related high-dimensional scheme first bins the data into occupied partitions, builds a nearest-neighbor partition graph, and computes shortest-path matrices such as $\{x(\delta)\}$0. The resulting path-length features support MAXPATHL clustering around local maxima and a Line-Of-Sight criterion based on convex-hull connectivity, corner-turn tests, and variance tests. The dominant cost is all-pairs shortest paths on the reduced partition graph, nominally $\{x(\delta)\}$1, but the method is designed for the regime $\{x(\delta)\}$2 (McIlhany et al., 2017).

3. Path admissibility, bottlenecks, and manifold separation

For multi-manifold clustering, path admissibility can be imposed geometrically. In the angle-constrained method, a symmetric $\{x(\delta)\}$3-nearest-neighbor graph is built on $\{x(\delta)\}$4, and a path $\{x(\delta)\}$5 is called $\{x(\delta)\}$6-constrained when every turning angle satisfies

$\{x(\delta)\}$7

Several landmarks are chosen at random, and for each landmark a modified Dijkstra algorithm marks which nodes are reachable by an $\{x(\delta)\}$8-constrained path. Each sample then receives a binary signature

$\{x(\delta)\}$9

and hierarchical clustering with complete linkage is applied to these signatures (Babaeian, 2018). Reported synthetic accuracies exceed $\gamma(x)=(c^{(1)},\dots,c^{(L)})$0 on all eight benchmark cases, including TP $\gamma(x)=(c^{(1)},\dots,c^{(L)})$1, TSI $\gamma(x)=(c^{(1)},\dots,c^{(L)})$2, FS $\gamma(x)=(c^{(1)},\dots,c^{(L)})$3, DSPR $\gamma(x)=(c^{(1)},\dots,c^{(L)})$4, and RCC $\gamma(x)=(c^{(1)},\dots,c^{(L)})$5. On COIL-20, clustering the objects $\gamma(x)=(c^{(1)},\dots,c^{(L)})$6 yields $\gamma(x)=(c^{(1)},\dots,c^{(L)})$7 accuracy and clustering all 20 objects yields $\gamma(x)=(c^{(1)},\dots,c^{(L)})$8 (Babaeian, 2018).

A density-landscape variant appears in t-NEB. Let $\gamma(x)=(c^{(1)},\dots,c^{(L)})$9 be a smooth density estimated by a Gaussian or Student’s-$C_1,\dots,C_k$0 mixture. For two cluster centers $C_1,\dots,C_k$1, the maximum-density path is

$C_1,\dots,C_k$2

equivalently

$C_1,\dots,C_k$3

After overclustering, nudged elastic band optimization estimates the bottleneck density $C_1,\dots,C_k$4 on edges of a $C_1,\dots,C_k$5-NN graph, an MST is computed on weights $C_1,\dots,C_k$6, and bottom-up merging yields a dendrogram (Ritzert et al., 19 Mar 2025). On 16D MNIST-Nd, NEB-based merging achieves $C_1,\dots,C_k$7, whereas Euclidean and dip-statistic merging are both around $C_1,\dots,C_k$8; stability across random EM seeds is pairwise ARI $C_1,\dots,C_k$9 in $A=(a_{pq})$0 (Ritzert et al., 19 Mar 2025).

A simplex-graph construction generalizes bottleneck paths to intersecting manifolds. Candidate $A=(a_{pq})$1-simplices are built from an annular graph, adjacent simplices share a $A=(a_{pq})$2-face, and edge weights are defined from the dihedral angle. The largest-angle path distance is

$A=(a_{pq})$3

Denoising removes simplices with large $A=(a_{pq})$4-nearest-neighbor LAPD, and single-linkage on the denoised simplex graph recovers manifold components (Chen et al., 14 Jul 2025). The theory gives

$A=(a_{pq})$5

under the stated sampling, curvature, and noise conditions, and the scalable implementation reaches overall complexity $A=(a_{pq})$6 when $A=(a_{pq})$7. Empirically, LAPD is reported as robust to noise, curvature, and small intersection angle, with runtime for $A=(a_{pq})$8 up to $A=(a_{pq})$9 remaining $V=\{v_1,\dots,v_N\}$0–$V=\{v_1,\dots,v_N\}$1 (Chen et al., 14 Jul 2025).

4. Trajectory and flight-path clustering

Path-based clustering is particularly natural for trajectory data. Song et al. use the U.S. Federal Aviation Administration’s Traffic Flow Management System dataset together with DV8 to categorize flight paths by agglomerative hierarchical clustering under two distance models: a spatial-based geographic distance model and a vector-based cosine similarity model (Song et al., 2023). For trajectories sampled to $V=\{v_1,\dots,v_N\}$2 points, the geographic model averages great-circle distances,

$V=\{v_1,\dots,v_N\}$3

while the directional model computes

$V=\{v_1,\dots,v_N\}$4

A silhouette-score cutoff selects the dendrogram cut, and point extraction uniformly subsamples trajectories for speed (Song et al., 2023).

The reported behavior is phase-dependent. For enroute segments, the geographic-distance model yields silhouette scores in the $V=\{v_1,\dots,v_N\}$5 range, including $V=\{v_1,\dots,v_N\}$6 for CMH$V=\{v_1,\dots,v_N\}$7ATL. For CMH$V=\{v_1,\dots,v_N\}$8PHL arrival patterns, cosine-based clustering gives a silhouette of $V=\{v_1,\dots,v_N\}$9 versus $w_{ij}$0 for geographic distance. On SFO$w_{ij}$1PIT enroute data, geographic distance gives score $w_{ij}$2, while cosine similarity gives $w_{ij}$3 (Song et al., 2023). Point extraction with $w_{ij}$4 shows that for long-haul queries, $w_{ij}$5 up to $w_{ij}$6 or $w_{ij}$7 incurs negligible silhouette loss $w_{ij}$8 while reducing runtime by $w_{ij}$9–$$\rho_v = \sum_{u\in V}\mathbf{1}(\|x_u-x_v\|\le r), \qquad \delta_v = \min_{u:\rho_u>\rho_v}\|x_u-x_v\|,$$0; for short-haul or highly curved approaches, smaller $$\rho_v = \sum_{u\in V}\mathbf{1}(\|x_u-x_v\|\le r), \qquad \delta_v = \min_{u:\rho_u>\rho_v}\|x_u-x_v\|,$$1 values preserve cluster quality. The practitioner recommendation is to segment flights into departure, enroute, and arrival portions; use cosine for departure/arrival and geographic for enroute; and expose a human-in-the-loop threshold override (Song et al., 2023).

A related trajectory hierarchy appears in scan-path generation. HPCGen decomposes raw gaze data into a multi-level tree by hierarchical K-Means, reclustering shift vectors relative to parent centroids and computing a PCA model within each cluster. New scan paths are generated by traversing the tree top-down and sampling principal-component perturbations (Fuhl, 2022). This is not a path-cost clustering algorithm, but it uses cluster hierarchies explicitly as a path-generating mechanism.

5. Regularization paths and clustering paths

In convex clustering, the path is a sequence of optimization solutions indexed by a fusion parameter. PyClustrPath formulates convex clustering as

$$\rho_v = \sum_{u\in V}\mathbf{1}(\|x_u-x_v\|\le r), \qquad \delta_v = \min_{u:\rho_u>\rho_v}\|x_u-x_v\|,$$2

For a sequence $$\rho_v = \sum_{u\in V}\mathbf{1}(\|x_u-x_v\|\le r), \qquad \delta_v = \min_{u:\rho_u>\rho_v}\|x_u-x_v\|,$$3, the clustering path is

$$\rho_v = \sum_{u\in V}\mathbf{1}(\|x_u-x_v\|\le r), \qquad \delta_v = \min_{u:\rho_u>\rho_v}\|x_u-x_v\|,$$4

which evolves from $$\rho_v = \sum_{u\in V}\mathbf{1}(\|x_u-x_v\|\le r), \qquad \delta_v = \min_{u:\rho_u>\rho_v}\|x_u-x_v\|,$$5 to the fully fused solution as $$\rho_v = \sum_{u\in V}\mathbf{1}(\|x_u-x_v\|\le r), \qquad \delta_v = \min_{u:\rho_u>\rho_v}\|x_u-x_v\|,$$6 (Wu et al., 27 Jan 2025). PyClustrPath implements ADMM, fast AMA, and semismooth-Newton augmented Lagrangian, with warm-starting across the $$\rho_v = \sum_{u\in V}\mathbf{1}(\|x_u-x_v\|\le r), \qquad \delta_v = \min_{u:\rho_u>\rho_v}\|x_u-x_v\|,$$7 sequence and GPU acceleration through PyTorch. Reported timings for $$\rho_v = \sum_{u\in V}\mathbf{1}(\|x_u-x_v\|\le r), \qquad \delta_v = \min_{u:\rho_u>\rho_v}\|x_u-x_v\|,$$8 $$\rho_v = \sum_{u\in V}\mathbf{1}(\|x_u-x_v\|\le r), \qquad \delta_v = \min_{u:\rho_u>\rho_v}\|x_u-x_v\|,$$9 values at $\alpha$00 include SSNAL-GPU at $\alpha$01 on LIBRAS-6 versus $\alpha$02 for SSNAL-CPU, $\alpha$03 on COIL-20 versus $\alpha$04, and $\alpha$05 on MNIST versus $\alpha$06 (Wu et al., 27 Jan 2025).

A graph-diffusion analogue is the seeded PageRank solution path. The seeded PageRank vector solves

$\alpha$07

and the approximate solution is controlled by a residual threshold $\alpha$08 satisfying

$\alpha$09

The same parameter acts both as an accuracy parameter and as a regularization weight in a strongly convex, $\alpha$10-norm-regularized problem (Kloster et al., 2015). The ppr-path algorithm tracks approximate solutions for all $\alpha$11 in an interval in one pass, performs a sweep-cut at each scale, and records the conductance-minimizing cluster $\alpha$12. Its runtime bound is

$\alpha$13

independent of network size (Kloster et al., 2015). On networks with up to $\alpha$14 billion edges, ppr-path visits hundreds-to-thousands of distinct $\alpha$15 values in $\alpha$16–$\alpha$17 sec, ppr-grid with $\alpha$18 grid points runs $\alpha$19–$\alpha$20 faster than $\alpha$21 independent PPR computations, and selecting the best conductance along the path yields improvements up to $\alpha$22 over a single PPR at $\alpha$23 (Kloster et al., 2015).

These regularization-path constructions differ from shortest-path clustering in mechanism, but they preserve the central idea that cluster structure is best interpreted as a trajectory across scales rather than a single flat partition.

6. Temporal evolution paths and dynamical clusters

ClusPath treats cluster paths as typical evolution paths in longitudinal data. Each observation is a triple

$\alpha$24

where $\alpha$25 is the entity, $\alpha$26 a timestamp, and $\alpha$27 a descriptive vector. The temporal-aware dissimilarity is

$\alpha$28

with weights $\alpha$29 chosen by a slider $\alpha$30 (Rizoiu et al., 2015). Clustering is semi-supervised through a soft must-link penalty between observations of the same entity, and a directed weighted cluster graph $\alpha$31 is learned simultaneously. The global objective

$\alpha$32

balances prototype fit and contiguous segments, smoothness between successive phases, and frequently observed transitions (Rizoiu et al., 2015).

The learned graph operationalizes the “slow changing world” assumption: entities should pass through slowly evolving phases, so high-inertia paths receive high weight. ClusPath also includes Pareto-based evolutionary tuning of $\alpha$33 against four antagonistic criteria: MDvar, Tvar, ShaP, and SPass. One ClusPath iteration has complexity $\alpha$34, and the evolutionary tuning cost is $\alpha$35 but is parallelizable across the population (Rizoiu et al., 2015).

A different disciplinary meaning of cluster paths appears in stellar dynamics. O’Leary, Stahler, and Ma distinguish a global expansion path and a core collapse path for gravitationally bound clusters. The collapse line in $\alpha$36 space separates the two, with the dividing $\alpha$37-value ranging “from $\alpha$38 to over $\alpha$39” for realistic starting radii. At the Sun’s Galactocentric radius, $\alpha$40, the minimum population for core collapse is $\alpha$41, and $\alpha$42 scales as $\alpha$43 (O'Leary et al., 2014). Here “path” refers to physical evolution rather than a clustering algorithm, but it illustrates that path language can describe trajectories of clusters themselves.

7. Interpretability, path graphs, and algebraic formulations

In neural-network interpretability, cluster paths are post-hoc activation summaries. Selected layers are clustered by k-means++, and a new input is encoded as

$\alpha$44

The paper defines four metrics: path complexity

$\alpha$45

weighted-path purity, decision-alignment faithfulness, and path agreement (Kroeger et al., 8 Oct 2025). On a spurious-cue CIFAR-10 task, normal-set weighted purity is $\alpha$46 and corrupted-set weighted purity is $\alpha$47, while decision-alignment faithfulness is approximately $\alpha$48 with $\alpha$49 layers and $\alpha$50. On a five-class CelebA hair-color task, path complexity is approximately $\alpha$51, faithfulness plateaus at $\alpha$52 with $\alpha$53, weighted purity is approximately $\alpha$54, and path agreement is approximately $\alpha$55 under mild noise with no drop in classification accuracy (approximately $\alpha$56) (Kroeger et al., 8 Oct 2025). On a ViT-Base/16 pretrained on ImageNet, $\alpha$57 gives theoretical $\alpha$58, but only $\alpha$59 unique paths are observed. The same work uses path rarity for OOD detection and reports AUROC $\alpha$60 on CelebA, SVHN, and CIFAR-10, with FPR@95%TPR of $\alpha$61, $\alpha$62, and $\alpha$63 respectively (Kroeger et al., 8 Oct 2025).

Path structure also appears in exact optimization and algebra. For clustering a path graph with arbitrary pairwise correlation costs, the optimum can be computed by

$\alpha$64

in total time $\alpha$65 (Lange et al., 2017). Peterson–Serhiyenko establish a bijection between tame $\alpha$66-tilings and pairs of bi-infinite paths in $\alpha$67 via

$\alpha$68

connecting path representations with Plücker friezes and Grassmannian cluster algebras (Peterson et al., 2 Apr 2025). Di Francesco–Kedem represent $\alpha$69 $\alpha$70-system cluster variables as path partition functions on weighted graphs and as boundary measurements of totally positive planar networks, yielding explicit Laurent-positivity statements for the cluster variables (0906.3421). In these settings, “cluster” refers to cluster algebras or combinatorial partitions rather than statistical clustering, but the recurrent device is the same: encode global structure through constrained paths on a graph or network.

Across these literatures, cluster paths are best understood not as a single algorithm but as a family of path-centric constructions. The invariant idea is that cluster structure is often revealed more faithfully by continuity constraints, bottleneck densities, transition graphs, or regularization trajectories than by a single local similarity rule.