Exchangeable Hyperedge Model

• Exchangeable hyperedge models are probabilistic frameworks that impose exchangeability on hyperedges or interactions rather than on vertex labels.
• The model leverages an interval hypergraph construction where hyperedges appear as intervals determined by a linear order and a compact set in the unit triangle.
• Extensions of the framework address edge, relational, and multiplicity-aware settings, facilitating advanced network inference and subgraph frequency estimation.

Searching arXiv for the specified paper and closely related exchangeable hyperedge / interval hypergraph work. An exchangeable hyperedge model is a probabilistic model for hypergraphs or interaction processes in which invariance is imposed on the sampled relations rather than only on vertex-labeled adjacency arrays. In the literature considered here, the term covers both edge- or relation-exchangeable constructions in which hyperedges are conditionally i.i.d. from a random directing law on finite subsets or multisets, and the specialized ordered setting of exchangeable interval hypergraphs, where every hyperedge is an interval in some linear order and the law is represented by a compact subset $K\subseteq \Delta:=\{(x,y):0\le x\le y\le 1\}$ together with i.i.d. latent positions $U_1,U_2,\dots$ (Crane et al., 2016, Crane et al., 2016, Gerstenberg, 2018).

1. Exchangeability regimes and formal definitions

A hypergraph is a pair $H=(V,E)$ with $E\subseteq \mathcal{P}(V)$. In the interval-hypergraph framework, hypergraphs always contain all singletons $\{j\}\in E$ for every $j\in V$, and also include the empty set $\emptyset$. An interval hypergraph on a finite vertex set $[n]:=\{1,\dots,n\}$ is a set $H\subseteq \mathcal{P}([n])$ such that $\emptyset\in H$, $U_1,U_2,\dots$0 for all $U_1,U_2,\dots$1, and there exists a linear order $U_1,U_2,\dots$2 on $U_1,U_2,\dots$3 such that every edge $U_1,U_2,\dots$4 is an interval with respect to $U_1,U_2,\dots$5: if $U_1,U_2,\dots$6 and $U_1,U_2,\dots$7, then $U_1,U_2,\dots$8. Interval hypergraphs on $U_1,U_2,\dots$9 are defined as projective sequences $H=(V,E)$0 with $H=(V,E)$1 and $H=(V,E)$2, where $H=(V,E)$3 (Gerstenberg, 2018).

In the general interaction-based setting, the statistical unit is the hyperedge or relation. Crane and Dempsey define an interaction process $H=(V,E)$4, where $H=(V,E)$5 is the set of finite multisets of a population $H=(V,E)$6, and define the induced hypergraph representation by multiplicities,

$H=(V,E)$7

An edge-labeled network $H=(V,E)$8 is edge exchangeable if $H=(V,E)$9 for all permutations $E\subseteq \mathcal{P}(V)$0 of edge labels. Relational exchangeability extends this idea to general finite relational structures by requiring invariance under permutations of relation indices rather than element labels (Crane et al., 2016, Crane et al., 2016).

A distinct but related contrast appears in statistical network analysis. Vertex-exchangeable binary adjacency models require invariance under relabeling of vertices, for example

$E\subseteq \mathcal{P}(V)$1

whereas the exchangeable hyperedge model of subgraph-frequency inference instead takes an exchangeable sequence of hyperedges $E\subseteq \mathcal{P}(V)$2 satisfying

$E\subseteq \mathcal{P}(V)$3

for any bijection $E\subseteq \mathcal{P}(V)$4, and then restricts to one de Finetti mixture component $E\subseteq \mathcal{P}(V)$5 on finite subsets of a countable vertex set $E\subseteq \mathcal{P}(V)$6 (Bhattacharya et al., 18 Aug 2025).

Framework	Exchangeability	Directing object
Exchangeable interval hypergraph	Finite permutations of vertex labels	Compact $E\subseteq \mathcal{P}(V)$7 with diagonal included
Edge exchangeable hypergraph	Permutations of edge labels	Probability measure on $E\subseteq \mathcal{P}(V)$8 or $E\subseteq \mathcal{P}(V)$9
Relational exchangeability	Permutations of relation indices	Probability vector $\{j\}\in E$0
Structured interaction models	Permutations of interactions	Paintbox $\{j\}\in E$1 or hierarchical random measures

A common misconception is that all exchangeable hyperedge models are vertex-exchangeable hypergraphons. The sources considered here describe a different organizing principle: exchangeability may be imposed on relations, on sampled interactions, or on vertices subject to an interval-order constraint, and these choices lead to different state spaces, limit objects, and inferential targets (Crane et al., 2016, Gerstenberg, 2018, Bhattacharya et al., 18 Aug 2025).

2. Exchangeable interval hypergraphs and the compact-set representation

The main representation theorem for exchangeable interval hypergraphs states that every exchangeable interval hypergraph on $\{j\}\in E$2 can be obtained by sampling from a random compact subset of the triangle

$\{j\}\in E$3

If $\{j\}\in E$4 is a random compact subset containing the full diagonal,

$\{j\}\in E$5

and $\{j\}\in E$6 are i.i.d. $\{j\}\in E$7, independent of $\{j\}\in E$8, then restricted to $\{j\}\in E$9, every non-singleton hyperedge has the form

$j\in V$0

Equivalently,

$j\in V$1

in distribution for every $j\in V$2 (Gerstenberg, 2018).

The inclusion of the diagonal $j\in V$3 for all $j\in V$4 ensures the presence of all singletons. Operationally, the construction includes all $j\in V$5 regardless of $j\in V$6. Exchangeability follows because the hyperedges are functions of the latent positions $j\in V$7, which are i.i.d. $j\in V$8, and of the compact template $j\in V$9; permuting indices only permutes an i.i.d. sample and therefore does not change the law (Gerstenberg, 2018).

For a fixed interval point $\emptyset$0 with $\emptyset$1, the edge size obeys

$\emptyset$2

with

$\emptyset$3

If $\emptyset$4 is random, or if many points of $\emptyset$5 are present, edge-size distributions become mixtures of binomials over the law of $\emptyset$6. If $\emptyset$7 contains nested or overlapping pairs $\emptyset$8, the corresponding edges overlap as subsets of $\emptyset$9, and this overlap structure reflects the geometry of $[n]:=\{1,\dots,n\}$0 (Gerstenberg, 2018).

This representation is specific to ordered discrete structures. It does not model arbitrary non-interval hypergraphs. The strength of the construction is that the state space is reduced from general hypergraphs to compact subsets of $[n]:=\{1,\dots,n\}$1 with the diagonal included, which makes boundary theory and asymptotic representation tractable (Gerstenberg, 2018).

3. Erased-interval processes, Martin boundary, and ordered subclasses

The interval-hypergraph representation is obtained through erased-interval processes (EIPs), a class of transient Markov chains $[n]:=\{1,\dots,n\}$2. Here $[n]:=\{1,\dots,n\}$3 is an interval system on $[n]:=\{1,\dots,n\}$4 with respect to the usual order $[n]:=\{1,\dots,n\}$5, $[n]:=\{1,\dots,n\}$6 is uniformly distributed and independent of the future $[n]:=\{1,\dots,n\}$7-field $[n]:=\{1,\dots,n\}$8, and

$[n]:=\{1,\dots,n\}$9

where $H\subseteq \mathcal{P}([n])$0 erases the label $H\subseteq \mathcal{P}([n])$1 from $H\subseteq \mathcal{P}([n])$2 and relabels strictly increasingly. For an interval $H\subseteq \mathcal{P}([n])$3, erasing $H\subseteq \mathcal{P}([n])$4 is defined by

$H\subseteq \mathcal{P}([n])$5

This operation is applied to all intervals in $H\subseteq \mathcal{P}([n])$6, together with $H\subseteq \mathcal{P}([n])$7 (Gerstenberg, 2018).

The compact limit space is

$H\subseteq \mathcal{P}([n])$8

equipped with the Hausdorff metric on $H\subseteq \mathcal{P}([n])$9. For $\emptyset\in H$0 and $\emptyset\in H$1, the sampling map can be written as

$\emptyset\in H$2

with the conventions $\emptyset\in H$3 and $\emptyset\in H$4. The main EIP theorem asserts that for any EIP $\emptyset\in H$5, there exists a random $\emptyset\in H$6, independent of the corresponding $\emptyset\in H$7-process, such that almost surely

$\emptyset\in H$8

Moreover, the law of an ergodic EIP is uniquely parametrized by a deterministic $\emptyset\in H$9, and the map $U_1,U_2,\dots$00 is a homeomorphism between $U_1,U_2,\dots$01 and the extreme points $U_1,U_2,\dots$02 (Gerstenberg, 2018).

Passing from interval systems to interval hypergraphs is done by random relabeling. If $U_1,U_2,\dots$03 is an EIP and $U_1,U_2,\dots$04 is the uniform permutation process associated to $U_1,U_2,\dots$05, then $U_1,U_2,\dots$06 is an exchangeable interval hypergraph, and

$U_1,U_2,\dots$07

The mapping from EIP laws to EIH laws is continuous, affine, and surjective, so all ergodic EIHs arise from deterministic $U_1,U_2,\dots$08 (Gerstenberg, 2018).

The Martin boundary gives the asymptotic description of growing interval systems. Limits of growing interval systems correspond one-to-one to compact subsets $U_1,U_2,\dots$09 with the diagonal included, and the Martin boundary of erased-interval processes is homeomorphic to $U_1,U_2,\dots$10. The same framework contains hierarchies, Schröder trees, and binary trees as ordered subclasses. Schröder trees and binary trees embed as closed subspaces $U_1,U_2,\dots$11 and $U_1,U_2,\dots$12, sampling via $U_1,U_2,\dots$13 preserves these classes, and the Martin boundary of Rémy’s tree growth chain is homeomorphic to $U_1,U_2,\dots$14, consistent with the description by Evans, Grübel, and Wakolbinger but phrased through interval systems rather than didendritic systems and real trees. At the level of law spaces, the laws of EIHs form a compact, convex Bauer simplex $U_1,U_2,\dots$15, while the laws of EIPs form a Bauer simplex $U_1,U_2,\dots$16 affinely homeomorphic to the probability measures on $U_1,U_2,\dots$17 (Gerstenberg, 2018).

4. Edge exchangeability, relational exchangeability, and rank-based hyperedge sampling

Crane and Dempsey’s formulation begins with a random edge-labeled network induced by an interaction process $U_1,U_2,\dots$18. In the blip-free case, every edge-exchangeable network admits a de Finetti mixture representation: there exists a probability measure $U_1,U_2,\dots$19 on the $U_1,U_2,\dots$20-simplex

$U_1,U_2,\dots$21

such that, conditional on $U_1,U_2,\dots$22, the hyperedges $U_1,U_2,\dots$23 are i.i.d. with

$U_1,U_2,\dots$24

and the induced edge-labeled network has law $U_1,U_2,\dots$25. In a tractable subclass, one chooses an edge-size distribution $U_1,U_2,\dots$26 and random vertex weights $U_1,U_2,\dots$27, and defines

$U_1,U_2,\dots$28

Conditional on size $U_1,U_2,\dots$29, the model samples $U_1,U_2,\dots$30 vertices i.i.d. from $U_1,U_2,\dots$31 to form a hyperedge; repeated participants are allowed because sampling is with replacement on multisets (Crane et al., 2016).

The Hollywood model is the canonical sequential specification of this family. With parameters $U_1,U_2,\dots$32, the probability that the next participant is an existing vertex $U_1,U_2,\dots$33 or a new vertex is

$U_1,U_2,\dots$34

For $U_1,U_2,\dots$35, the degree distribution satisfies

$U_1,U_2,\dots$36

so the power-law exponent is $U_1,U_2,\dots$37. If $U_1,U_2,\dots$38 is the mean hyperedge size, then

$U_1,U_2,\dots$39

and the model is almost surely sparse when $U_1,U_2,\dots$40 (Crane et al., 2016).

Janson’s analysis of edge-exchangeable random graphs studies the simple graph obtained by merging parallel edges and deleting loops. In that setting, hyperedges $U_1,U_2,\dots$41 are i.i.d. from a probability measure $U_1,U_2,\dots$42 on $U_1,U_2,\dots$43, the set of finite non-empty multisets of points in a Borel space $U_1,U_2,\dots$44, and the resulting random graph can be dense, sparse, or extremely sparse. In a rank-1 model with weights $U_1,U_2,\dots$45, one has

$U_1,U_2,\dots$46

and for $U_1,U_2,\dots$47, $U_1,U_2,\dots$48,

$U_1,U_2,\dots$49

The same paper proves a power-law tail with exponent $U_1,U_2,\dots$50 for a natural sparse regime and gives examples of dense graph-limit convergence and sparse graphon convergence on $U_1,U_2,\dots$51 (Janson, 2017).

Relational exchangeability supplies the general structure theorem. For a countable set $U_1,U_2,\dots$52 of finite relational templates, one defines the simplex

$U_1,U_2,\dots$53

draws $U_1,U_2,\dots$54 i.i.d. with $U_1,U_2,\dots$55, applies the dagger operation to make blip labels globally unique, and then quotients by vertex relabeling. The main theorem states that every relationally exchangeable random structure has law

$U_1,U_2,\dots$56

for some probability measure $U_1,U_2,\dots$57 on $U_1,U_2,\dots$58. Positive labels correspond to recurrent vertices, while non-positive labels encode blips or dust that appear at most once (Crane et al., 2016).

Compared with the interval-hypergraph representation, these frameworks do not require hyperedges to be intervals in any linear order. The interval model imposes stronger structural constraints and obtains a compact-set parameterization in $U_1,U_2,\dots$59; the edge- and relation-exchangeable models trade that ordered geometry for a more general hyperedge state space (Crane et al., 2016, Crane et al., 2016, Gerstenberg, 2018).

5. Multiplicity-aware subgraph frequencies and statistical inference

The recent inferential treatment of exchangeable hyperedge models starts from a countable vertex set $U_1,U_2,\dots$60 and i.i.d. hyperedges

$U_1,U_2,\dots$61

where $U_1,U_2,\dots$62 is an arbitrary probability law on finite subsets of $U_1,U_2,\dots$63. No Aldous–Hoover or hypergraphon latent-variable representation is imposed. Hyperedges may have any size, and repetition of hyperedges is allowed. The induced edge-colored graph $U_1,U_2,\dots$64 has vertex set

$U_1,U_2,\dots$65

and colored edge set

$U_1,U_2,\dots$66

so the multiplicity of an unordered pair $U_1,U_2,\dots$67 is $U_1,U_2,\dots$68 (Bhattacharya et al., 18 Aug 2025).

For a simple colored subgraph $U_1,U_2,\dots$69 with $U_1,U_2,\dots$70 vertices, $U_1,U_2,\dots$71 edges, and $U_1,U_2,\dots$72 colors, the colored subgraph frequency is

$U_1,U_2,\dots$73

which is an unbiased estimator of

$U_1,U_2,\dots$74

The paper distinguishes three colored triangle types: Type 1, in which all three edges come from the same hyperedge; Type 2, in which exactly two edges come from the same hyperedge; and Type 3, in which each edge comes from a different hyperedge. Type 2 triangles require an edge of multiplicity $U_1,U_2,\dots$75, so no induced subgraph can be isomorphic to Type 2, although induced subgraphs may contain a Type 2 copy (Bhattacharya et al., 18 Aug 2025).

For colorless motifs, the colorless homomorphism frequency $U_1,U_2,\dots$76 and the total number of colorless copies $U_1,U_2,\dots$77 are defined so that multiplicity is retained through color assignments and then marginalized. Among colorless counts, rainbow subgraphs, meaning subgraphs with $U_1,U_2,\dots$78 distinct colors, dominate asymptotic fluctuations of $U_1,U_2,\dots$79. By contrast, the binarized statistic

$U_1,U_2,\dots$80

collapses multiple colored edges on a vertex pair to a single edge and therefore discards multiplicity (Bhattacharya et al., 18 Aug 2025).

The asymptotic theory is U-statistic based. If $U_1,U_2,\dots$81 and the moment condition

$U_1,U_2,\dots$82

holds, then for finite positive-semidefinite covariance matrices $U_1,U_2,\dots$83, $U_1,U_2,\dots$84, and $U_1,U_2,\dots$85,

$U_1,U_2,\dots$86

$U_1,U_2,\dots$87

The proof uses U-statistics theory with kernels defined on Borel spaces and the Hájek projection, and incomplete U-statistics with $U_1,U_2,\dots$88 sampled tuples preserve asymptotic variance (Bhattacharya et al., 18 Aug 2025).

Deletion robustness is formulated through hyperdegrees $U_1,U_2,\dots$89 and the degree-filtered statistic $U_1,U_2,\dots$90, which removes nodes with $U_1,U_2,\dots$91. In finite-vertex models, if $U_1,U_2,\dots$92, then

$U_1,U_2,\dots$93

and $U_1,U_2,\dots$94 has the same asymptotic variance as the unfiltered statistic. For countably infinite vertex sets, robustness depends on the decay of node appearance probabilities $U_1,U_2,\dots$95 and on the color structure. Under $U_1,U_2,\dots$96, $U_1,U_2,\dots$97, rainbow subgraphs are the most stable class, and the paper gives explicit degree-filtering thresholds through the quantity $U_1,U_2,\dots$98 defined from $U_1,U_2,\dots$99 and $H=(V,E)$00. For colored triangles, the stated sufficient rates are $H=(V,E)$01 for Type 2 and $H=(V,E)$02 for Type 3 (Bhattacharya et al., 18 Aug 2025).

The same paper identifies a finite-vertex pathology for binarized statistics: if $H=(V,E)$03, then for any subgraph $H=(V,E)$04 there exists $H=(V,E)$05 such that

$H=(V,E)$06

Thus no non-degenerate limiting distribution exists in that regime. A positive result remains for the number of unique $H=(V,E)$07-order interactions,

$H=(V,E)$08

which satisfies a central limit theorem under $H=(V,E)$09, $H=(V,E)$10, on countably infinite vertex sets (Bhattacharya et al., 18 Aug 2025).

For computation, the recommended approach is incomplete U-statistics that sample $H=(V,E)$11 hyperedge tuples. Subsampling for covariance estimation uses subsamples of size $H=(V,E)$12 with $H=(V,E)$13 subsamples, and within each subsample the counts are again computed by incomplete U-statistics with $H=(V,E)$14 sampled tuples. In simulations with $H=(V,E)$15 for $H=(V,E)$16 and $H=(V,E)$17, coverage of $H=(V,E)$18 confidence intervals constructed via normal approximation with the subsampling covariance estimator is near nominal for $H=(V,E)$19. On coauthorship and movie-collaboration hypergraphs, Type 2 clustering coefficients and Type 2 two-star frequencies distinguish networks in ways that binarized approaches do not (Bhattacharya et al., 18 Aug 2025).

6. Hierarchical structured interaction models

A structured interaction process introduces typed roles. Let $H=(V,E)$20 be role types, $H=(V,E)$21 the population for role $H=(V,E)$22, and define an interaction as an ordered tuple of finite multisets

$H=(V,E)$23

The interaction-labeled network is exchangeable if for every finite permutation $H=(V,E)$24 of interaction indices,

$H=(V,E)$25

A blip-free exchangeable structured interaction network admits a de Finetti mixture $H=(V,E)$26, where $H=(V,E)$27 is the simplex of probability mass functions on structured hyperedges (Dempsey et al., 2019).

The hierarchical vertex components model (HVCM) specializes this framework to roles such as sender and receiver. With one sender role $H=(V,E)$28 and one receiver role $H=(V,E)$29, the paintbox probability of a hyperedge $H=(V,E)$30, where $H=(V,E)$31 and $H=(V,E)$32, is

$H=(V,E)$33

The population-level sender frequencies $H=(V,E)$34, role-specific weights $H=(V,E)$35, and sender-specific receiver distributions $H=(V,E)$36 are given Pitman–Yor or Dirichlet-process stick-breaking priors, with a global receiver base measure $H=(V,E)$37 and local receiver measures centered on $H=(V,E)$38 (Dempsey et al., 2019).

The partial-pooling interpretation is explicit: $H=(V,E)$39 Thus receiver popularity is sender-specific but shrunk toward a shared population-level distribution. In the one-sender specialization, the sender is generated by the Pitman–Yor “Hollywood” rule

$H=(V,E)$40

and receivers are then sampled sequentially from a hierarchical rule involving local and global counts $H=(V,E)$41, $H=(V,E)$42, $H=(V,E)$43, and $H=(V,E)$44 (Dempsey et al., 2019).

The model is exchangeable as a structured interaction process for all parameters in its admissible parameter space. It also yields explicit sparsity and degree-tail statements. If $H=(V,E)$45 has $H=(V,E)$46 senders, $H=(V,E)$47 follows a multinomial allocation, $H=(V,E)$48 is the average number of receivers per email for sender $H=(V,E)$49, and $H=(V,E)$50, then

$H=(V,E)$51

In particular, if $H=(V,E)$52, the sequence is almost surely sparse. Under $H=(V,E)$53 for all senders, the global receiver degree distribution satisfies

$H=(V,E)$54

so the exponent is $H=(V,E)$55 (Dempsey et al., 2019).

Posterior inference is implemented by a Gibbs algorithm with auxiliary variables. The paper gives explicit conditional updates for $H=(V,E)$56, $H=(V,E)$57, $H=(V,E)$58, $H=(V,E)$59, $H=(V,E)$60, $H=(V,E)$61, $H=(V,E)$62, $H=(V,E)$63, $H=(V,E)$64, and $H=(V,E)$65, and states that computational complexity is linear in the number of receiver events per iteration. Empirical evaluation is reported for the Enron e-mail corpus and for an arXiv coauthorship-with-subjects dataset. In the Enron study, the paper reports posterior predictive improvements over Hollywood and GGP for local receiver statistics, and in the arXiv study the multiple-sender extension with latent $H=(V,E)$66 is used for subject-overlap analysis (Dempsey et al., 2019).

7. Assumptions, identifiability, and recurring limitations

The interval-hypergraph model assumes interval structure, inclusion of all singletons and $H=(V,E)$67, existence of a linear order, and compactness of $H=(V,E)$68 together with inclusion of the full diagonal. These assumptions are structural rather than incidental: the paper states that inclusion of all singletons and $H=(V,E)$69 is essential for the filtration and almost sure representation, and non-interval hyperedges are not modeled. Identifiability is also asymmetric. For EIPs, $H=(V,E)$70 parametrizes ergodic laws bijectively via a homeomorphism. For EIHs, however, the map $H=(V,E)$71 on ergodic laws is surjective and continuous but not generally injective, so different compact sets can generate the same exchangeable hierarchy or hypergraph law (Gerstenberg, 2018).

The general edge-exchangeable framework also has identifiability subtleties. In Crane and Dempsey’s blip-free representation, the mixing measure $H=(V,E)$72 is not unique without the fuller ranked parametrization, and the supplement gives uniqueness only in a more technical representation. The same paper emphasizes that projection or thresholding destroys edge exchangeability, makes likelihood-based inference intractable after projection, and may spuriously alter sparsity statements; in the binary Hollywood specialization, projection makes the sequence sparse for all $H=(V,E)$73, whereas raw edge data require $H=(V,E)$74. It also notes that $H=(V,E)$75 converges slowly, at the Ewens-type $H=(V,E)$76 rate, and can be practically imprecise (Crane et al., 2016).

The multiplicity-aware inferential framework of subgraph frequencies imposes its own regularity conditions. It assumes i.i.d. hyperedges, the moment bound

$H=(V,E)$77

and, for degree-filtering robustness over infinite vertex sets, decay assumptions on $H=(V,E)$78. The paper states that identifiability beyond subgraph frequencies is not addressed, multiplicity distributions are not modeled parametrically, robustness beyond rainbow subclasses can be delicate, and without-multiplicity statistics may degenerate in finite-vertex models (Bhattacharya et al., 18 Aug 2025).

The hierarchical structured-interaction model is explicitly interaction-exchangeable rather than vertex-exchangeable. The paper states that with very sparse local data, $H=(V,E)$79 and $H=(V,E)$80 may be weakly identified, and that the multiple-sender latent-$H=(V,E)$81 approximation can introduce small biases. It also stresses that component labels are exchangeable and posterior comparisons should be made through functionals such as degree distributions and shared-receiver counts rather than through labels themselves (Dempsey et al., 2019).

These limitations clarify a second common misconception: exchangeability in hypergraph models does not by itself imply a universal latent-variable description or full statistical identifiability. In the interval setting, the latent object is a compact subset of $H=(V,E)$82; in edge- and relation-exchangeable settings, it is a probability law on hyperedges or relational templates; in hierarchical interaction models, it is a collection of shared and local random measures. What is shared across these constructions is the de Finetti principle, but the corresponding state spaces, ergodic parameters, and inferential equivalence classes differ substantially (Crane et al., 2016, Crane et al., 2016, Gerstenberg, 2018, Dempsey et al., 2019, Bhattacharya et al., 18 Aug 2025).