Weighted Total Persistence in TDA

• Weighted Total Persistence is a family of aggregate functionals that weight persistence diagram intervals by their lifetimes or endpoint-dependent kernels to capture geometric features.
• Its degree-p formulation and analytic bounds convert geometric properties into quantitative persistence moments, ensuring stability and robust control in TDA.
• The concept bridges applications from minimal spanning acycles to persistence-weighted kernels, enabling both theoretical insights and practical algorithmic implementations.

Searching arXiv for the cited papers to ground the article in current metadata. arXiv search query: (Skraba et al., 2017) Randomly Weighted d-complexes Minimal Spanning Acycles and Persistence Diagrams Weighted total persistence is a family of aggregate functionals on persistence diagrams that summarize barcode geometry by weighting intervals through their lifetimes or endpoint-dependent kernels. In the degree-$p$ form, it is the sum of persistence lengths $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$ with $\mathrm{pers}(x)=d-b$; in randomly weighted $d$-complexes with births in $H_{d-1}$ at $0$, it becomes $\sum_i D_i^p$ and, for $p=1$, the lifetime sum $L_{d-1}$; in persistent magnitude, it is replaced by the signed exponential weighting $\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$; and in weighted persistence diagrams on metric measure spaces, it is the interval-moment functional $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$0 (Kusano et al., 2016, Skraba et al., 2017, Govc et al., 2019, Gülen et al., 16 Apr 2025). These formulations suggest that weighted total persistence is not a single invariant but a class of persistence summaries whose weights may be polynomial, bounded monotone, exponential, or measure-theoretic.

1. Definitions and principal formulations

A persistence diagram $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$1 is a multiset of points $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$2 with $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$3, together with the diagonal $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$4 counted with infinite multiplicity. The persistence of a point $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$5 is $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$6 (Kusano et al., 2016).

Formulation	Formula	Source
Degree-$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$7 total persistence	$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$8	(Kusano et al., 2016)
Thresholded total persistence	$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$9	(Kusano et al., 2016)
Birth-zero lifetime moments	$\mathrm{pers}(x)=d-b$0	(Skraba et al., 2017)
General barcode moment	$\mathrm{pers}(x)=d-b$1	(Skraba et al., 2017)
Persistent magnitude	$\mathrm{pers}(x)=d-b$2	(Govc et al., 2019)
Weighted-diagram moment	$\mathrm{pers}(x)=d-b$3	(Gülen et al., 16 Apr 2025)

The literature represented by these papers does not impose a single universal definition. One paper explicitly defines degree-$\mathrm{pers}(x)=d-b$4 total persistence, one identifies weighted total persistence with sums of powers of death times when births are zero, one introduces a signed endpoint-weighted variant through persistent magnitude, and one places the construction on weighted diagrams endowed with a probability measure on intervals (Kusano et al., 2016, Skraba et al., 2017, Govc et al., 2019, Gülen et al., 16 Apr 2025). A plausible implication is that “weighted total persistence” is best understood as a design pattern for persistence summaries rather than as a uniquely standardized term.

2. Degree-$\mathrm{pers}(x)=d-b$5 total persistence and analytic bounds

In the sublevel-set formulation of persistent homology, for a tame Lipschitz function $\mathrm{pers}(x)=d-b$6 on a triangulable compact metric space, the thresholded degree-$\mathrm{pers}(x)=d-b$7 total persistence is

$\mathrm{pers}(x)=d-b$8

for $\mathrm{pers}(x)=d-b$9, where $d$0. The full degree-$d$1 total persistence is $d$2 (Kusano et al., 2016).

The analytic importance of this definition is that it admits explicit upper bounds in standard TDA settings. If $d$3 is triangulable and compact, $d$4 is tame and Lipschitz with Lipschitz constant $d$5, and $d$6 is the minimum number of simplices needed in a triangulation with mesh at most $d$7, then

$d$8

When $d$9 and $H_{d-1}$0, the estimate $H_{d-1}$1 yields

$H_{d-1}$2

For a finite subset $H_{d-1}$3, the function $H_{d-1}$4 satisfies $H_{d-1}$5 and $H_{d-1}$6, so

$H_{d-1}$7

These inequalities are central because they convert geometric control of the underlying space into quantitative control of persistence moments (Kusano et al., 2016).

The same paper also proves monotonicity in the persistence exponent: if $H_{d-1}$8 and $H_{d-1}$9 is bounded, then $0$0 is also bounded. Equivalently, if $0$1, then $0$2, hence

$0$3

This places weighted total persistence in the same hierarchy of $0$4-type summaries that underlies many stability and approximation arguments (Kusano et al., 2016).

3. Minimal spanning acycles, death times, and lifetime sums

For a finite simplicial complex $0$5, a weighted $0$6-complex is a pair $0$7 in which $0$8 is monotone in the sense that $0$9 whenever $\sum_i D_i^p$0. The associated filtration is the sublevel filtration $\sum_i D_i^p$1. Within this framework, a subset $\sum_i D_i^p$2 is spanning if $\sum_i D_i^p$3, an acycle if $\sum_i D_i^p$4, and a spanning acycle if both hold. The minimal spanning acycle $\sum_i D_i^p$5 minimizes $\sum_i D_i^p$6 among spanning acycles (Skraba et al., 2017).

The central structural result is the equivalence between minimal spanning acycle weights and persistence death times. If $\sum_i D_i^p$7, $\sum_i D_i^p$8 is the multiset of death times in $\sum_i D_i^p$9, $p=1$0 is the multiset of birth times in $p=1$1, and $p=1$2 is a $p=1$3-minimal spanning acycle, then

$p=1$4

This identification is obtained by comparing the Incremental Persistence Algorithm with the simplicial Kruskal algorithm: a $p=1$5-face is added to the MSA exactly when it is negative, and negative faces are exactly the faces whose weights are recorded as death times in $p=1$6 (Skraba et al., 2017).

A further identity, attributed there as the Hiraoka–Shirai identity, links lifetime sums to spanning acycle weights:

$p=1$7

In the mean-field random model in which all lower-dimensional faces have weight $p=1$8, births in $p=1$9 occur at $L_{d-1}$0 and $L_{d-1}$1. Consequently,

$L_{d-1}$2

and, more generally,

$L_{d-1}$3

In this regime, weighted total persistence in degree $L_{d-1}$4 is literally the total weight, or the moment sequence of weights, of the $L_{d-1}$5-dimensional minimal spanning acycle (Skraba et al., 2017).

4. Stability and random asymptotics in weighted $L_{d-1}$6-complexes

The stability theorem for weighted $L_{d-1}$7-complexes compares two monotone weight functions $L_{d-1}$8 and $L_{d-1}$9 on the same finite complex. If $\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$0 and $\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$1 are the birth and death times under $\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$2, and $\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$3 and $\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$4 are the corresponding sets under $\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$5, then for any $\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$6,

$\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$7

with the $\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$8 case interpreted as a supremum. Via the equivalence between MSA weights and death times, the same bound applies to the multiset of MSA $\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$9-face weights (Skraba et al., 2017).

In the mean-field model, where $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$00 for $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$01 and $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$02 on $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$03, this stability estimate yields continuity of total persistence moments under perturbations of the $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$04-face weights. The paper derives

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$05

and an accompanying expectation bound. For $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$06, the inequality simplifies to

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$07

so $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$08 is $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$09-Lipschitz with respect to the $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$10 change in $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$11-face weights up to optimal matching of death sets (Skraba et al., 2017).

The random model also supports a detailed extremal theory. If $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$12 is Lipschitz continuous and $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$13 in probability, then the scaled point processes of extremal nearest-face distances, extremal death times in $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$14, and extremal weights in the $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$15-MSA all converge vaguely in distribution to a Poisson point process on $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$16 with intensity $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$17. In particular,

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$18

This completely characterizes the extremal points of persistence diagrams and MSAs in the model (Skraba et al., 2017).

The same stability mechanism transfers classical random spanning-tree asymptotics to noisy higher-dimensional settings. If $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$19 is uniform on $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$20 and

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$21

then

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$22

For $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$23 and $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$24, this recovers Frieze’s $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$25 limit for random minimal spanning trees under suitable noise; for higher $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$26, it gives the corresponding asymptotics for random minimal spanning acycles and therefore for weighted total persistence in $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$27 (Skraba et al., 2017).

5. Persistence-weighted kernels and bounded monotone weighting

In the kernel approach of the persistence weighted Gaussian kernel, weighted total persistence appears both as a direct summary and as a control quantity for diagram embeddings. The weight assigned to a point $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$28 is

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$29

which is increasing in persistence. Near-diagonal points with small persistence receive small weight, while large-persistence points receive larger weight, and the saturation of $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$30 prevents unbounded influence (Kusano et al., 2016).

The weighted atomic measure associated with a diagram is

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$31

and the Gaussian RKHS embedding is

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$32

This yields the linear kernel

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$33

the RKHS distance $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$34, and the nonlinear Gaussian kernel on the RKHS

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$35

The analytic role of total persistence is explicit in the stability estimate

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$36

with

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$37

Thus the continuity of the weighted embedding is governed by degree-$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$38 and degree-$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$39 total persistence (Kusano et al., 2016).

For finite point clouds $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$40, the data-level theorem states that if $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$41, then

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$42

where the constant is independent of $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$43 and $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$44. The reason for the condition $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$45 is that uniform control of $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$46 and $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$47 over diagrams arising from finite subsets in $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$48 requires $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$49 (Kusano et al., 2016).

The method also has an explicit computational approximation via random Fourier features. Exact Gram-matrix computation among $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$50 diagrams of size at most $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$51 costs $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$52 kernel evaluations, while the random-feature approximation reduces this to $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$53 for $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$54 sampled frequencies. The same paper reports that, in a synthesized XOR-like classification problem, PWGK with the RKHS Gaussian kernel achieved approximately $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$55 test accuracy, compared to approximately $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$56 for PSSK and approximately $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$57 for an unweighted Gaussian embedding; in oxide-glass change-point detection it identified a change point converging to $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$58; and in two protein classification tasks it achieved $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$59 and $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$60 cross-validation accuracies (Kusano et al., 2016).

6. Persistent magnitude as signed exponential-weighted total persistence

Persistent magnitude defines a different weighting paradigm for barcodes. For a finitely presented graded persistence module $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$61 with half-open bars $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$62 appearing in degrees $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$63, the persistent magnitude is

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$64

with the convention $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$65. Because

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$66

each bar contributes an exponentially weighted length, and the sign $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$67 inserts Euler-characteristic cancellation across homological degrees (Govc et al., 2019).

The magnitude function introduces a scaling parameter:

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$68

This admits two equivalent interpretations. First, in terms of the associated graded functor,

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$69

Second, in terms of the Laplace transform, if $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$70 is the Euler characteristic curve, then

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$71

A discrete breakpoint formula is

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$72

where $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$73 list the distinct barcode endpoints (Govc et al., 2019).

These formulations imply strong formal properties. Persistent magnitude is additive with respect to short exact sequences:

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$74

It is also compatible with tensor products:

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$75

In settings with a Künneth short exact sequence for a graded persistent homology theory, this yields multiplicativity of magnitude under products (Govc et al., 2019).

Relative to standard total persistence,

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$76

persistent magnitude is a weighted total persistence with kernel $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$77 and with graded signs. It therefore differs from the unsigned polynomial moment $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$78 in two essential ways: it depends on endpoints through $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$79 rather than only on bar length, and it is alternating across homological degree. The paper further shows that the magnitude of a finite metric space is precisely the persistent magnitude of its blurred magnitude homology, providing a bridge between classical magnitude and persistent homology (Govc et al., 2019).

7. Weighted persistence diagrams on metric measure spaces

A more recent framework defines weighted persistence diagrams functorially from finite metric measure spaces $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$80, where $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$81 is a fully supported probability measure. The weighted Vietoris–Rips filtration is built from the finite index set

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$82

together with the global distribution of distances

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$83

The filtration is

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$84

From a weighted filtration $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$85, the weighted persistence diagram is

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$86

with $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$87 (Gülen et al., 16 Apr 2025).

In this setting, weighted total persistence is defined by

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$88

where $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$89. If multiplicities are also incorporated, the alternative definition is

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$90

For weighted Vietoris–Rips diagrams, the moment has the explicit probabilistic interpretation

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$91

because flipping does not change $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$92. Hence weighted total persistence captures the $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$93-th absolute moment of pairwise distance differences drawn from the global distance distribution (Gülen et al., 16 Apr 2025).

Differences between weighted diagrams are measured by a $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$94-edit distance with an optimal-transport-like reformulation. The main stability theorem states that for all $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$95,

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$96

From this, one obtains a robustness bound for weighted total persistence:

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$97

and therefore

$\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$98

where $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$99 bounds the index range (Gülen et al., 16 Apr 2025).

This framework also sharpens the distinction between topology and weighting. The paper shows that two $\mathrm{pers}(x)=d-b$00-point ultrametric spaces can have identical Vietoris–Rips persistence diagrams in all degrees yet different weighted diagrams because their $\mathrm{pers}(x)=d-b$01 differ. It also proves that for $\mathrm{pers}(x)=d-b$02, the weighted edit distance is most discriminative, while for finite $\mathrm{pers}(x)=d-b$03 it may become insensitive to diagram multiplicities and depend primarily on the weight distribution, although interval ordering can still distinguish cases where the flipped-GDD Gromov–Wasserstein distance is zero (Gülen et al., 16 Apr 2025). A concrete toy calculation illustrates the moment interpretation: if $\mathrm{pers}(x)=d-b$04 is uniform on $\mathrm{pers}(x)=d-b$05, then

$\mathrm{pers}(x)=d-b$06

These are exactly the weighted total persistence values of the associated weighted Vietoris–Rips diagram (Gülen et al., 16 Apr 2025).

Taken together, these developments place weighted total persistence at the intersection of persistent homology, random topology, kernel methods, categorical invariants, and metric-measure stability theory. The common theme is aggregation of barcode information under a prescribed weighting rule, but the rule itself varies substantially across settings: polynomial lifetime moments, bounded persistence weights, exponential endpoint kernels, and probability measures on intervals each produce distinct invariants with distinct formal and statistical properties (Kusano et al., 2016, Skraba et al., 2017, Govc et al., 2019, Gülen et al., 16 Apr 2025).