Normalized Topological Similarity (NTS)

• Normalized Topological Similarity (NTS) is a topology-aware measure that compares hierarchical merge orders derived from paired dissimilarity matrices.
• It leverages minimum spanning trees and Spearman’s rank correlation to compute a bounded score in [-1,1], ensuring robustness against scaling variations.
• NTS’s variants, NTS-E and NTS-M, enable nuanced analysis of hierarchical connectivity in neural representations and support standardized cross-scenario benchmarking.

Searching arXiv for recent and foundational papers on Normalized Topological Similarity and closely related constructs. Normalized Topological Similarity (NTS) is a normalized, topology-aware similarity measure for comparing two paired neural representations. In its explicit formulation, NTS measures the rank correlation of hierarchical merge orders derived from the $0$-dimensional Vietoris–Rips, single-linkage, and minimum spanning tree (MST) structure of two dissimilarity matrices, yielding a scale-invariant score in $[-1,1]$ intended for cross-scenario benchmarking (Wang et al., 4 Jun 2026). The term is not fully standardized across the broader literature: several earlier works define closely related normalized topological comparisons under different names or within different pipelines rather than as a single standalone coefficient.

1. Concept and terminological scope

In the strict sense, NTS denotes the measure introduced in "Symmetric Divergence and Normalized Similarity: A Unified Topological Framework for Representation Analysis" (Wang et al., 4 Jun 2026). That work presents NTS as the normalized counterpart to RTD- and SRTD-style topological divergences. Its stated purpose is to replace unbounded, sample-dependent scalar summaries based on sums of barcode lengths with a bounded similarity based on the rank correlation of hierarchical merge orders.

Across adjacent literatures, however, the same phrase is often only approximately applicable. "Inter-layer Information Similarity Assessment of Deep Neural Networks Via Topological Similarity and Persistence Analysis of Data Neighbour Dynamics" defines Nearest Neighbour Topological Similarity (NNTS) rather than NTS, using intersection-over-union of $k$-nearest-neighbour sets across layers (Hryniowski et al., 2020). "Topological Similarity Index and Loss Function for Blood Vessel Segmentation" introduces a path-feasibility-based topological similarity index $m(\mathbf{P},\mathbf{Y})$, also normalized to $[0,1]$, but specialized to binary vessel masks (Araújo et al., 2021). "Topological Representational Similarity Analysis in Brains and Beyond" does not define a metric called NTS at all; instead, it introduces tRSA, geo-topological transforms, representational geo-topological matrices (RGTMs), and representational geodesic-distance matrices (RGDMs), which collectively function as a normalized, topology-sensitive representational comparison pipeline (Lin, 2024). This suggests that NTS is best understood as a family of normalized topology-sensitive comparison strategies, with a particularly precise modern instantiation in (Wang et al., 4 Jun 2026).

2. Formal definition

Let

$w,\tilde w \in \mathbb{R}^{n\times n}$

be pairwise dissimilarity matrices for two paired representations on the same $n$ samples. In the NTS formulation, these matrices are assumed symmetric, nonnegative, and to have zero diagonal (Wang et al., 4 Jun 2026).

Let $T_w$ and $T_{\tilde w}$ be minimum spanning trees of the complete weighted graphs induced by $w$ and $[-1,1]$0, with edge sets $[-1,1]$1 and $[-1,1]$2. The core pair set is

$[-1,1]$3

Since each MST has $[-1,1]$4 edges, $[-1,1]$5.

NTS is defined through Spearman’s rank correlation. For vectors $[-1,1]$6, with rank vectors $[-1,1]$7 and $[-1,1]$8, the paper gives

$[-1,1]$9

The construction has two variants.

For NTS-E, one uses raw dissimilarities on the core pair set. For each core edge $k$0, define

$k$1

Then

$k$2

For NTS-M, one uses merge times induced by the hierarchy. For any pair $k$3, the merge time under $k$4 is

$k$5

The MST characterization is

$k$6

For each core edge $k$7, define

$k$8

Then

$k$9

The paper explicitly states that NTS compares the hierarchical connectivity structure induced by the two representations, namely their $m(\mathbf{P},\mathbf{Y})$0-dimensional Vietoris–Rips merge structure (Wang et al., 4 Jun 2026).

3. Topological basis and computational pipeline

NTS is built on the equivalence among $m(\mathbf{P},\mathbf{Y})$1-dimensional Vietoris–Rips filtrations, single-linkage clustering, Kruskal-style merge sequences, and MSTs (Wang et al., 4 Jun 2026). At threshold $m(\mathbf{P},\mathbf{Y})$2, points $m(\mathbf{P},\mathbf{Y})$3 and $m(\mathbf{P},\mathbf{Y})$4 are connected when $m(\mathbf{P},\mathbf{Y})$5. As $m(\mathbf{P},\mathbf{Y})$6 increases, connected components merge. In dimension $m(\mathbf{P},\mathbf{Y})$7, the relevant topological information is therefore the sequence and ordering of those merges.

The computation pipeline is concrete. Starting from paired representations $m(\mathbf{P},\mathbf{Y})$8 and $m(\mathbf{P},\mathbf{Y})$9, one computes dissimilarity matrices

$[0,1]$0

with Euclidean distance or cosine dissimilarity given as examples. One then computes $[0,1]$1 and $[0,1]$2, forms $[0,1]$3, constructs aligned vectors either of raw edge weights (NTS-E) or merge times (NTS-M), and returns their Spearman correlation.

This design is intentionally sparse. The paper contrasts NTS with representational similarity analysis based on full vectorization of all pairwise distances: rather than using all $[0,1]$4 pairs, NTS restricts comparison to the topology-aware core set $[0,1]$5 induced by the two MSTs (Wang et al., 4 Jun 2026). In this sense, it is not simply “RSA on distances,” but a rank-correlation comparator applied to a summary of the $[0,1]$6-dimensional topological hierarchy.

The two variants differ in what they preserve. NTS-E is described as stricter because it matches the rank structure of the underlying edge weights on $[0,1]$7, whereas NTS-M requires agreement only in the induced merge-event ordering. The paper provides pseudocode for both variants and specifies deterministic tie breaking for MST computation together with deterministic rank handling, such as mid-ranks, for Spearman’s $[0,1]$8 (Wang et al., 4 Jun 2026).

For complexity, the paper gives an explicit bound only for NTS-E: $[0,1]$9 where $w,\tilde w \in \mathbb{R}^{n\times n}$0 is the inverse Ackermann factor arising from union-find operations. The stated dominant costs are pairwise distance calculation and MST construction (Wang et al., 4 Jun 2026).

4. Properties and interpretation

The central normalization property is immediate from the definition: because NTS is a Spearman correlation, it is bounded in

$w,\tilde w \in \mathbb{R}^{n\times n}$1

The measure is symmetric, since both Spearman correlation and the core set $w,\tilde w \in \mathbb{R}^{n\times n}$2 are symmetric in the two inputs (Wang et al., 4 Jun 2026).

The paper states a precise characterization of perfect NTS-M. Assuming nonzero rank variance,

$w,\tilde w \in \mathbb{R}^{n\times n}$3

if and only if the values

$w,\tilde w \in \mathbb{R}^{n\times n}$4

induce the same weak order on $w,\tilde w \in \mathbb{R}^{n\times n}$5. The notion of weak order includes ties: for all $w,\tilde w \in \mathbb{R}^{n\times n}$6,

$w,\tilde w \in \mathbb{R}^{n\times n}$7

A second theorem states that, again under nonzero rank variance,

$w,\tilde w \in \mathbb{R}^{n\times n}$8

but not conversely. The interpretation is that exact agreement on the rank ordering of core edge weights is sufficient for exact agreement on the induced merge hierarchy, whereas multiple edge-order configurations may collapse to the same path-maximum merge ordering (Wang et al., 4 Jun 2026).

The paper repeatedly emphasizes the interpretive advantage of rank correlation. Because NTS keeps only relative order, it reduces sensitivity to global rescalings and to monotone distortions of dissimilarities. In practical terms, values near $w,\tilde w \in \mathbb{R}^{n\times n}$9 indicate strong agreement in hierarchical organization, values near $n$0 indicate little monotone relationship, and values near $n$1 indicate inverse ordering of merge structure. This is the sense in which NTS is presented as “scale-invariant” and suitable for cross-scenario benchmarking (Wang et al., 4 Jun 2026).

5. Relation to adjacent normalized topological measures

Several related measures share the general aim of normalized topology-sensitive comparison, but they operate on different objects and use different normalization principles.

Construct	Compared structure	Normalization principle
NTS (Wang et al., 4 Jun 2026)	MST-core edge or merge-time hierarchy of paired representations	Spearman rank correlation in $n$2
NNTS (Hryniowski et al., 2020)	Sample-wise $n$3-nearest-neighbour sets across DNN layers	Intersection-over-union averaged over samples, in $n$4
Topological similarity index (Araújo et al., 2021)	Path feasibility between prediction and ground-truth vessel masks	Symmetric geometric mean of topological recall and precision, in $n$5
tRSA descriptors (Lin, 2024)	RGTMs and RGDMs derived from transformed RDMs	GT transform to $n$6 plus rank/quantile normalization
MRS similarity (Agarwal et al., 2020)	Matched nodes in multi-resolution Reeb spaces	Normalized local node similarity and multiresolution average in $n$7

The nearest conceptual predecessor within deep representation analysis is NNTS. There, the sample-wise score is

$n$8

and the global score is

$n$9

so the normalization arises from per-sample intersection-over-union and averaging over samples (Hryniowski et al., 2020). NTS, by contrast, does not compare neighbourhood sets directly; it compares the ranked order of hierarchical merge events.

In segmentation, the topological similarity index

$T_w$0

defines topology through path feasibility in vascular trees rather than through persistent merges or MSTs (Araújo et al., 2021). The multivariate topology literature offers yet another variant: in multi-resolution Reeb-space comparison, the final similarity

$T_w$1

is also explicitly bounded in $T_w$2 (Agarwal et al., 2020).

A distinct but closely related representational line is tRSA. Starting from an RDM with entries

$T_w$3

it applies the piecewise linear geo-topological transform

$T_w$4

to obtain RGTMs, optionally followed by RGDM construction via shortest paths. There, normalization is effected by mapping distances into $T_w$5, defining $T_w$6 and $T_w$7 as quantiles of each RDM, and often replacing raw dissimilarities by their ranks within each RDM before transformation (Lin, 2024). This suggests a broader methodological pattern: normalized topological comparison is often achieved not by a single universal metric, but by rank normalization, saturation, or probability normalization applied to topology-sensitive summaries.

Another nearby development normalizes topological indices rather than defining a pairwise similarity. For branched macromolecules represented as trees, the transform

$T_w$8

produces a normalized phase space $T_w$9 for cross-size comparison (Vaupotič et al., 2024). This is not itself NTS, but it is directly relevant to the wider problem of scale-correct topological comparison.

6. Empirical findings and practical use

The empirical evidence for NTS comes primarily from synthetic and deep learning experiments in (Wang et al., 4 Jun 2026). In the synthetic Clusters experiment, NTS and SRTD-family measures track increasing structural dissimilarity as a single Gaussian cluster is split into $T_{\tilde w}$0 clusters arranged on a circle, whereas CKA is reported as largely insensitive and RTD-lite shows an anomalous inverted trend. In a UMAP experiment on MNIST, NTS and SRTD-lite show smooth, monotonic responses as n_neighbors varies, while CKA fails to reflect the gradual structural evolution.

In CNN analysis, the paper studies an 8-layer TinyCNN on CIFAR-10. NTS and CKA both exhibit a near-diagonal similarity pattern, but NTS and SRTD-lite additionally detect a sharp break at the final pooling layer. The figure caption interprets this as a transition from local feature extraction to global aggregation and states that this functional shift is missed by geometric analysis (Wang et al., 4 Jun 2026). By contrast, RTD-style divergences are described as unsuitable for fine-grained cross-layer benchmarking because their unnormalized nature leads to irregular inversions.

The LLM results are particularly important for the intended benchmarking role of NTS. For intra-model layer similarity, NTS reveals consistent hierarchical fingerprints within model families such as Qwen, InternLM, Baichuan, and Llama. For inter-model comparison, the paper reports that CKA often saturates, whereas NTS is more broadly distributed and therefore more discriminative. A highlighted case is DeepSeek-R1-Ds versus Qwen2.5-Math-7B: CKA gives unexpectedly low similarity, while NTS-E recovers high structural similarity, aligning better with the known distillation relation (Wang et al., 4 Jun 2026).

The paper also gives concrete preprocessing guidance. For inter-model LLM comparisons, Z-score normalization across the feature dimension is described as crucial for reliable NTS results. This is a practical rather than definitional requirement, but it bears directly on reproducibility and on the interpretation of cross-model comparisons (Wang et al., 4 Jun 2026).

7. Limitations, misconceptions, and scope conditions

NTS is not a generic full-topology metric. The explicit NTS construction in (Wang et al., 4 Jun 2026) is based only on $T_{\tilde w}$1-dimensional features, specifically the MST-encoded merge hierarchy. It therefore does not compare higher-dimensional persistent homology, such as loops or voids. A plausible implication is that NTS is best viewed as a normalized summary of hierarchical connectivity rather than as a complete persistent-homology distance.

The measure is also described as non-differentiable and therefore analysis-only in its current form (Wang et al., 4 Jun 2026). When barcode-level diagnosis or an optimization objective is needed, the paper positions SRTD and SRTD-lite as preferable. NTS is instead the tool for stable, standardized scalar benchmarking.

A common misconception is to treat “NTS” as a universally standardized term across topological machine learning. The record is more heterogeneous. Some papers use related names such as NNTS (Hryniowski et al., 2020); some define normalized topological similarity in domain-specific settings such as vessel trees or multi-field Reeb spaces (Araújo et al., 2021); and some, like tRSA, offer a normalized topology-sensitive comparison pipeline without introducing a scalar called NTS (Lin, 2024). This suggests that the phrase has both a narrow meaning—its specific rank-correlation construction in (Wang et al., 4 Jun 2026)—and a broader descriptive use for normalized topology-aware comparison.

Finally, because NTS is a global summary, it lacks the local explanatory detail of barcode-based diagnostics (Wang et al., 4 Jun 2026). It tells whether two representations agree in hierarchical merge order, but not which exact intervals, samples, or structural motifs account for disagreement. That trade-off between normalization and diagnosis is central to its place within the contemporary topological toolkit.