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REVIEW 3 major objections 4 minor 1 cited by

Latent Space Modelling of Hypergraph Data

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proposes a latent space model for non-simplicial hypergraphs, built from per-order Cech skeletons with Bernoulli noise, whose likelihood is cheap to evaluate and whose posterior can be sampled with MCMC under Bookstein…

desk verdict A genuine new latent-space model for non-simplicial hypergraphs with a cheap likelihood, but the MCMC ignores the model's ordering constraint and the abstract oversells; still worth refereeing after fixes. read the letter →

arxiv 1909.00472 v3 pith:7Q2B6YBH submitted 2019-09-01 stat.ME

Kathryn Turnbull , Simón Lunagómez , Christopher Nemeth , Edoardo Airoldi This is my paper · ORCID
classification stat.ME MSC 05C6562F1560D05
keywords HypergraphsLatentSpaceNetworksSimplicialcomplexBayesianInferenceStatisticalNetworkAnalysisNon-simplicialCech
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces a latent space model for hypergraph data, extending the classic latent distance model for graphs. It claims that a hypergraph can be generated by placing nodes in Euclidean space, drawing a ball of radius r_k around each node, and declaring a hyperedge of order k whenever the balls of its k nodes have a common intersection; taking the union of these order-k skeletons across k and then flipping hyperedge indicators with small Bernoulli noise yields a model that covers non-simplicial hypergraphs. The payoff is a likelihood that can be evaluated by comparing only observed and geometry-induced hyperedges, a posterior sampleable by MCMC, and Bookstein coordinates that remove the non-identifiability of translation, rotation, and scaling. A careful reader would care because it gives hypergraphs the interpretable geometric visualization and transitive-structure control that latent space models give graphs, while escaping the simplicial restriction of nerve-based random geometric hypergraphs.

What carries the argument

The key object is the non-simplicial random geometric hypergraph (nsRGH): the union over k of the order-k skeletons of Cech complexes with per-order radii r_2 < r_3 < ... < r_K, where each Cech complex records all subsets of nodes whose radius-r_k balls have a common intersection. The argument rides on the equivalence that a k-set of latent points becomes a hyperedge exactly when its smallest enclosing ball has radius at most r_k, which turns hyperedge checking into miniball computations and makes the likelihood evaluable by counting hyperedge matches and mismatches rather than enumerating all possible hyperedges.

What would settle it

Fit the model to a hypergraph containing a high-degree 3-star—one central node in many order-3 hyperedges whose other two nodes share no pairwise hyperedge—and compare the observed star count with the posterior predictive count under the fitted geometry; if the observed count lies far outside the predictive range, the thresholded-ball premise is not generating the data.

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Extended reading notes

Core claim

The central claim is that non-simplicial hypergraph data can be modelled by a non-simplicial random geometric hypergraph built from Cech complexes: for each order k, a separate radius r_k is used, and only hyperedges of exactly order k are selected from the k-skeleton. Taking the union over k removes the simplicial constraint, and independent Bernoulli modifications give the model support on all hypergraphs and a Hamming-distance likelihood that is a product of Bernoulli factors. Conditional on latent coordinates, the geometry is deterministic, so the latent positions carry the structure and the modification probabilities absorb discrepancies. With normal priors on coordinates, the model implies exchangeability of nodes and a higher-order transitivity in which subsets of a latent geometric hyperedge are likely to be present even when they are not logically required.

Load-bearing premise

The load-bearing premise is that a hyperedge of order k forms exactly when the k latent points all lie inside some ball of radius r_k, so that real-world higher-order interactions are well approximated by a purely geometric overlap threshold; if the true process is non-geometric or non-exchangeable, the latent representation is a misspecified projection.

Editorial extensions

If this is right

  • Hypergraphs gain the same visualisation as latent space graphs: posterior mean coordinates place centrally connected actors at the centre and peripheral nodes at the edge, as demonstrated on the Star Wars and coauthorship datasets.
  • Predictive inference for new nodes follows naturally: simulate coordinates from the fitted normal, read off the induced hyperedges, and estimate degree or motif distributions for a next wave of nodes.
  • The degree distribution of order-2 hyperedges is exactly Binomial and order-3 hyperedges are approximately Poisson (Lemma 4.1 and Theorem 4.1), connecting model parameters directly to observable degree summaries.
  • Per-order radii plus directed modification probabilities let the same framework express very different regimes, such as dense pairwise edges with sparse triangles, and broad comparison shows the model covers a wider range of motif counts than two existing hypergraph baselines.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct extension, not pursued in the paper, would replace the Cech skeleton with a Vietoris-Rips construction so that an order-k hyperedge is present when all pairwise balls intersect; the authors note this would scale better in K, and a natural test is whether posterior predictive motif counts change materially across the two complexes.
  • Because the likelihood treats the geometry as a deterministic incidence pattern with independent flips, the same framework could accommodate weighted or repeated hyperedges by letting the modification probabilities depend on multiplicity, which the paper leaves open.
  • The geometric core can only express star-like structures up to a limit set by the latent dimension, so for datasets with very high-degree centres the fitted noise parameters, not the latent positions, will carry the explanatory burden; checking the posterior of the modification probabilities would reveal when the geometric interpretation is strained.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a Bayesian latent space model for non-simplicial hypergraph data. Hyperedges are generated as the union of order-k skeletons of Čech complexes with per-order radii r_k, subject to r_k > r_{k-1}, and then each hyperedge indicator is independently flipped with a small probability. This construction yields likelihoods (10) and (13) that are inexpensive to evaluate, and the authors use Bookstein coordinates to remove rotation, translation, and scaling non-identifiability. Posterior sampling is carried out by a Metropolis-within-Gibbs scheme, Section 4 derives degree-distribution results, and simulation and real-data analyses (Star Wars co-occurrence and a statistician coauthorship subset) illustrate the framework. The central claim is that this is the first computationally tractable latent-position model for non-simplicial hypergraphs.

Significance. The modeling idea is natural and original, and it is developed in a mostly coherent way. The likelihood is derived from the generative mechanism rather than fitted to an external benchmark, which is appropriate for model-based inference; the connection to Čech-complex skeletons is a useful bridge from computational topology to statistical network analysis. The paper also includes predictive degree-distribution studies and two real-data applications, and it gives detailed appendices on initialization, implementation, and misspecification. If the technical problems below are fixed, the framework would be a valuable contribution to the hypergraph and latent-space network literature. The current version, however, contains errors in the theoretical degree-distribution results and a mismatch between the model's support and the MCMC implementation.

major comments (3)
  1. [Section 5.1, Algorithm 3, Eq. (22)] The MCMC scheme does not enforce the ordering constraint r_k > r_{k-1} required by Definition 3.1. In Algorithm 3 each r_k is proposed independently as r_k + epsilon_r, and the acceptance probability in Eq. (22) uses only the independent exponential prior p(r|lambda). The chain can therefore visit parameter values for which the induced hypergraph is simplicial, meaning it samples from a larger model than the one defined in Section 3.2. This is not purely cosmetic: the non-simplicial claims in Sections 7.1 and 7.2 are made for the ordered-radius model. A reparameterization to increments delta_k = r_k - r_{k-1} > 0, or an indicator restricting the prior/support to the ordered set, would fix the problem.
  2. [Theorem 4.1(2), Eq. (20)] The displayed Poisson rate in Eq. (20) is wrong. There are C(N-1,2) hyperedges e3 incident to node i, and Eq. (20) multiplies each of those summands by C(N-1,2). The resulting rate is C(N-1,2)^2 times the per-hyperedge probability rather than C(N-1,2) times it. This algebraic error directly affects the theoretical claim and the comparison made in Figure 11. The binomial coefficient should be removed from the summand, or the entire sum should be replaced by C(N-1,2) times the common per-hyperedge success probability.
  3. [Theorem 4.1(1)-(2), Appendix F.3] The theorem states a marginal degree distribution after integrating over the latent positions, but the indicators of hyperedges incident to node i are not independent conditional only on (phi, Sigma, r). Conditional on the latent position u_i, the indicators are independent, but u_i is itself random; the marginal degree of node i is therefore a mixture of Binomials (and, for k=3, a mixture of Poisson-binomials), not a single Binomial or Poisson law. The proof's phrase 'given conditioning on i' does not remove this dependence; conditioning on the full latent configuration U would, but then the statement should be explicitly conditional on U. This is a load-bearing issue because Section 4 presents these distributions as exact results and Section 7.2 relies on them for the real-data comparison.
minor comments (4)
  1. [Abstract and Section 5] The abstract promises a 'delayed-acceptance MCMC scheme,' but Section 5 describes only a Metropolis-Hastings-within-Gibbs sampler with no delayed-acceptance step; please align the abstract with the actual algorithm.
  2. [Section 3.4] The title 'Can we improved model flexibility?' is a typo, and the phrase 'for k = 2, 4,..., K - 1' should presumably be 'for k = 2, 3, ..., K - 1'.
  3. [Equation (11)] Equation (11) appears to contain a typesetting error: 'Sigma mu ~ W^-1(Phi, nu)' should presumably be 'Sigma ~ W^-1(Phi, nu)'.
  4. [Section 6.2] The text 'we being by simulating a hypergraph' should read 'we begin by simulating a hypergraph'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the nsRGH likelihood is derived from the stated generative mechanism, self-citations are not load-bearing, and held-out predictive checks are genuine.

full rationale

The paper's central derivation chain is self-contained. Definition 3.1 defines the non-simplicial random geometric hypergraph as the union of k-skeletons of Cech complexes with ordered radii; Algorithm 1/2 and equations (10)/(13) construct the likelihood directly as a product of Bernoulli modification probabilities, so there is no fitted input renamed as a prediction. The only self-citations are contextual: Lunagómez et al. (2017) is cited as related work, and the remark about likelihood monotonicity cites Lunagómez et al. (2019, proof of Proposition 3.1), but the monotonicity claim is evident from (10) itself when 0 < phi_k < 1/2, so the citation is not load-bearing. Bookstein coordinates are an identifiability convention, not an imported uniqueness theorem. The Section 7.2 comparison of theoretical and observed degree distributions uses fitted parameters and is an in-sample diagnostic, not a claimed prediction; the actual predictive check targets the next sampling wave, which is not used in fitting. The MCMC issue raised by the skeptic (unconstrained r proposals) is a potential implementation/correctness gap relative to Definition 3.1, but it is not circularity: the model and sampler are not defined in terms of the paper's conclusions. No step in the derivation reduces to its own inputs by construction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 2 invented entities

The model's central machinery rests on a small number of stated generative assumptions: Gaussian latent positions, per-order radii, and independent flips. Standard computational topology facts, such as the Cech complex characterization and the miniball algorithm, supply the geometric computations. No hidden physical entities are introduced; the nsRGH construction is a model device.

free parameters (5)
  • Per-order radii r_k (k=2,...,K) = e.g., rhat=(0.13,0.16) in Section 6.2; estimated per dataset
    Control the geometric scale at which order-k hyperedges can form; inferred from data and subject to the constraint r_k > r_{k-1}.
  • Flip probabilities phi_k or psi_k^(0), psi_k^(1) = e.g., psihat=(0.0058,0.0014),(0.0057,0.0035) in Section 6.2
    Probability of modifying hyperedge states; in Section 7 constrained post hoc by observed hyperedge densities.
  • Latent mean mu = e.g., muhat=(-0.13,0.44) in Section 6.2
    Center of latent positions; inferred with a Normal prior.
  • Latent covariance Sigma = e.g., Sigmahat=[[0.14,-0.0039],[-0.0039,0.078]] in Section 6.2
    Dispersion of latent positions; inferred with an inverse-Wishart prior.
  • Latent dimension d = 2 in most examples; 3 in one simulation case
    Chosen by the user, not inferred; affects the geometric flexibility and computational cost.
assumptions (5)
  • domain assumption Latent coordinates u_i are iid N(mu, Sigma).
    Generative assumption stated in Section 3.3 and Algorithm 1; not derived from data.
  • domain assumption Hyperedges of each order occur independently conditional on U and r.
    Property P2 in Section 4; follows from the Cech complex construction plus independent Bernoulli flips.
  • standard math An order-k hyperedge is present iff the k latent points lie in a ball of radius r_k.
    Cech complex characterization used in Algorithm 1 and Appendix E.1; standard computational topology fact.
  • standard math Poisson approximation for sums of dependent Bernoulli trials is valid when p3 is small.
    Used in Theorem 4.1(2) for the k=3 degree distribution; the paper acknowledges the restriction and cites Teerapabolarn (2014).
  • standard math Bookstein coordinates remove translation, rotation, and scaling non-identifiability.
    Used in Section 3.5 and Appendix A; standard shape analysis result from Bookstein (1986).
invented entities (2)
  • Non-simplicial random geometric hypergraph (nsRGH), union of k-skeletons D^(k)_{r_k}
    purpose: Express non-simplicial hypergraphs via geometric intersection patterns while keeping the likelihood tractable.
    Defined in Definition 3.1; no external empirical handle independently of the paper.
  • Hyperedge flip noise (phi_k or psi_k^(0), psi_k^(1))
    purpose: Extend support of the model to all hypergraphs and make the likelihood continuous.
    Model device introduced to smooth the deterministic geometric construction; not an independently observed quantity.

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Pith. "Pith review of Latent Space Modelling of Hypergraph Data." pith.science (2026). https://pith.science/paper/7Q2B6YBH

@misc{pith2026190900472,
  author       = {Pith},
  title        = {Pith review of: Latent Space Modelling of Hypergraph Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Q2B6YBH}},
  note         = {Machine review of arXiv:1909.00472}
}
read the original abstract

The increasing prevalence of relational data describing interactions among a target population has motivated a wide literature on statistical network analysis. In many applications, interactions may involve more than two members of the population and this data is more appropriately represented by a hypergraph. In this paper, we present a model for hypergraph data which extends the well established latent space approach for graphs and, by drawing a connection to constructs from computational topology, we develop a model whose likelihood is inexpensive to compute. A delayed-acceptance MCMC scheme is proposed to obtain posterior samples and we rely on Bookstein coordinates to remove the identifiability issues associated with the latent representation. We theoretically examine the degree distribution of hypergraphs generated under our framework and, through simulation, we investigate the flexibility of our model and consider estimation of predictive distributions. Finally, we explore the application of our model to two real-world datasets.

Figures

Figures reproduced from arXiv: 1909.00472 by the authors.

Figure 1
Figure 1. Figures 1a and 1b depict two possible hypergraphs, where a node belongs to a hyperedge [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Example of a Cech complex. Left: ˇ Br(ui) for {ui = (ui1, ui2)} 7 i=1 in R 2 . Middle: the graph obtained by taking pairwise intersections. Right: the hypergraph obtained by taking intersections of arbitrary order. The shaded region between nodes 3, 5 and 6 indicates a hyperedge of order 3. For this framework to be computationally appealing, it is important that the sets Ai are easy to parameterise and their interse… view at source ↗
Figure 3
Figure 3. Example of a Cech complex. Left: ˇ Br2 (u) for each of 6 points in R 2 . Middle: Br3 (u) for each of 6 points in R 2 . Right: ∪ 3 k=2D (k) rk . too restrictive and this motivates us to develop a model that is appropriate for non-simplicial hypergraphs. In this section we will build upon the graph and hypergraph models introduced in Section 2, where our model for a hypergraph on N nodes has a maximal hyperedge order … view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Estimate of probability of a hyperedge occurring for [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Comparison of theoretical (black, dashed) and simulated (red, solid) degree distribution [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Depiction of motifs considered in Sections 6 and 7. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Summary of hypergraphs simulated from each of the models considered in Section 6.1. The cases considered are summarised [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Comparison of prior and posterior predictive degree distributions for [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Posterior mean of latent coordinates for the Star Wars dataset with upper limits [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Predicted degree distributions conditional on the fitted model. Given [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: (Left) posterior mean of latent coordinates after 20000 iterations. Hyperedges of order [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: N∗ = 19 predictive subgraph counts. Left to right: motifs depicted in Figure 6a3, 6a4, 6a5, 6b1, 6b2 and 6b3. The red dots correspond to the observed motif counts for the newly sampled nodes. imposes properties on the hypergraphs generated from our model, including a …
Figure 13
Figure 13. Figure 13: Bookstein transformation in R 2 . Left: original coordinates. Right: transformed Bookstein coordinates. The points highlighted in red are mapped to (−1/2, 0) and (1/2, 0). The Bookstein coordinate u B i for i = 4, 5, . . . , N is then given by u B i = R1R2R3(wi1, wi2,…
Figure 14
Figure 14. Figure 14: The blue shaded regions correspond to Br(ui), for i = 1, 2, 3, and the purple shaded region is the smallest enclosing ball of the points. The statements r ∗ < r and Br(u1) ∩ Br(u2) ∩ Br(u3) 6= ∅ are equivalent. Hence, in order to determine whether yek = 1, we must fin…
Figure 15
Figure 15. Figure 15: Comparison of prior and posterior predictive degree distributions for hyperedges [PITH_FULL_IMAGE:figures/full_fig_p039_15.png]
Figure 16
Figure 16. Figure 16: Comparison of prior and posterior predictive degree distributions for hyperedges oc [PITH_FULL_IMAGE:figures/full_fig_p039_16.png]
Figure 17
Figure 17. Figure 17: Summary of misspecification simulation study. Left to right: average degree distri [PITH_FULL_IMAGE:figures/full_fig_p041_17.png]
Figure 18
Figure 18. Figure 18: Posterior mean of the latent coordinates after 25000 post burn-in iterations. Figure [PITH_FULL_IMAGE:figures/full_fig_p043_18.png]
Figure 19
Figure 19. Figure 19: Predictive distributions for motif counts for [PITH_FULL_IMAGE:figures/full_fig_p044_19.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Modeling Hypergraphs with Diversity and Heterogeneous Popularity

    stat.ME 2025-01 conditional novelty 7.0 of 10

    New generative hypergraph model based on determinantal point processes, with consistency and asymptotic normality guarantees for maximum likelihood estimates.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [1]

    Sample i1 with probability p(i1) = N−i1 ∑N−1 i=1 (N−i) , for i1 = 1,..., (N− 1)

    work page

  2. [2]

    A similar procedure can be used for arbitrary k

    Sample i2|i1 with probability p(i2|i1) = 1 N−i1 , for i2 = (i1 + 1),...,N . A similar procedure can be used for arbitrary k. Note that this procedure ignores the dependence between samples since, once a hyperedge is sampled, the remaining hyperedges are sampled from a subset of hyperedges of size (N k ) − 1. However, we expect this effect to be negligible ...

    work page 2012

  3. [3]

    Apply MDS to D to obtain coordinates U0∈ RN×d

    work page

  4. [4]

    The radii r depend on the scale of U, and so they are initialised in terms of U0

    Specify the index of the anchor points, and transform U0 onto Bookstein coordinates (see Appendix A). The radii r depend on the scale of U, and so they are initialised in terms of U0. Given the initial latent coordinates, r0 is chosen to be the minimum radius which induces all edges that are present in hN,K. The noise parameters ψ(0) andψ(1) are initialis...

    work page

  5. [5]

    Let A∈ RN×N denote a weighted adjacency matrix. Fori,j ∈{ 1, 2,...,N }, if{i,j} are connected by a hyperedge - let A(i,j) = 1 if{i,j} are only connected by a hyperedge of order k = 2, - let A(i,j) =λ if{i,j} are connected by a hyperedge of order k> 2

    work page

  6. [6]

    For i =j, let D(i,j) = 0

    Find the distance matrix D∈ RN×N, where D(i,j) is the shortest path between nodes{i,j} in the weighted graph determined by A. For i =j, let D(i,j) = 0

    work page

  7. [9]

    Let n = 0

    Calculate T (hN,K), where T (·) is a vector of hypergraph summary statistics. Let n = 0

    work page

  8. [10]

    -Sample u∗ i∼N (µ∗, Σ∗) for i = 1, 2,...,N

    While n<N smp -Sample µ∗∼N (mµ, Σµ) and Σ∗∼W−1(Φ,ν ). -Sample u∗ i∼N (µ∗, Σ∗) for i = 1, 2,...,N . LetU∗ be the N×d matrix whose ith row is u∗ i . -Given initialr0, determine the hypergraph gN,K(U∗,r0). -Let g∗ N,K by the hypergraph obtained by modifying gN,K(U∗,r0) with noise ψ(0) 0 andψ(1) 0 -Calculate T ( g∗ N,K ) . -If|T (hN,K)−T (g∗ N,K)|<ϵ Accept sa...

    work page

Show all 30 references
  1. [11]

    To initialise the parameters µ and Σ we use an ABC scheme (see Marin et al

    Let µ0 and Σ0 be the average of Nsmp samples. To initialise the parameters µ and Σ we use an ABC scheme (see Marin et al. (2012) for an overview). In this scheme we first sampleµ and Σ from their priors. Conditional on these samples, we then sample a hypergraph. By comparing su...

    2012

  2. [12]

    and uB 2 = (uB 21,uB

  3. [13]

    Determine U0 by applying Algorithm 4

  4. [14]

    Let initial radii r0 be the smallest radii which induce all hyperedges observed in hN,K, conditional on U0

  5. [15]

    Sample ψ(0) 0 andψ(1) 0 from their prior distributions

  6. [16]

    E Practicalities To implement the MCMC scheme given in Algorithm 3 there are a number of practical con- siderations we must address

    Sample µ0 and Σ0 by applying Algorithm 5. E Practicalities To implement the MCMC scheme given in Algorithm 3 there are a number of practical con- siderations we must address. In this section we comment on these where, in E.1 we discuss an approach for determining the presence ...

    2010

  7. [17]

    Determine the smallest enclosing ball B for the coordinates{ui1,ui2,...,u ik}

  8. [18]

    If the radius of B is less than rk, the hyperedge ek ={i1,i 2,...,i k} is present in the hypergraph. To compute the smallest enclosing ball we can rely on the the miniball algorithm (see Section 3.2 of Edelsbrunner and Harer (2010)), which may be also be referred to as the min...

    2010

  9. [19]

    Set σ1 =ek and σ2 =∅

  10. [20]

    (2008), and for efficient implementation of the ˇCech complex we rely on the GUDHI C++ library (The GUDHI Project (2015))

    if σ1 =∅, compute the miniball B of σ2 else choose u∈σ1 -Calculate the miniball B which contains the points σ1\u in its interior and the points σ2 on its boundary -if u /∈B, then set B to be the miniball B which contains the points σ1\u in its interior and the points σ2∪u on i...

    2008

  11. [21]

    All nodes equally likely to form connections βi =−1.4 for i = 1, 2,...,N

  12. [22]

    LetUB denote the Bookstein coordinates andU denote the untransformed coordinates

    to be ( −1/2, 0) and (1/2, 0), respectively. LetUB denote the Bookstein coordinates andU denote the untransformed coordinates. Then UB is given by UB =cR(U−b) = 1√ (uB 21−uB 11)2 + (uB 22−uB 12)2 [ cos(a) sin( a) − sin(a) cos( a) ]( U− 1 2 [uB 11 +uB 21 uB 12 +uB 22 ]) , (23) ...

    1998

  13. [23]

    Some nodes more likely to form connections β = (−0.5,−0.53,..., −1.97,−2) Ng and Murphy (2018)

    2018

  14. [24]

    Hyperedges in a single cluster G =K = 1, a = 1,φi1 = 0.075, π =b1,τ = 1

  15. [25]

    Distinct topic clusters only G = 3,K = 1, a = 1,φi1 = 0.25 for i∈A , φi2 = 0.25 for i∈B , φi3 = 0.25 for i∈C , π = (1/3, 1/3, 1/3),τ = 1

  16. [26]

    Distinct size clusters only G = 1,K = 3, a = (0.2, 0.5, 1), φi1 = 0.15,π = 1,τ = (1/3, 1/3, 1/3)

  17. [27]

    Fuzzy topic clusters G = 2,K = 3, a = (0.4, 1),φi1 = 0.3 for i∈A , φi2 = 0.3 for i∈B , φi1 =φi2 = 0.2 for i∈C , π = (1/2, 1/2),τ = (1/3, 1/3, 1/3) LSH

  18. [28]

    Strongly correlated Σ r = (0.18, 0.3, 0.35), µ = (0, 0), Σ = 0.25 ( 1 0 .9 0.9 1 ), ψ0 = (0.01, 0.01, 0.01), ψ1 = (0.01, 0.01, 0.01)

  19. [29]

    No correlation in Σ r = (0.18, 0.3, 0.35),µ = (0, 0), Σ = 0.25 (1 0 0 1 ), ψ0 = (0.01, 0.01, 0.01), ψ1 = (0.01, 0.01, 0.01)

  20. [30]

    Dense in e2, sparse in e3,e 4 r = (0.2, 0.3, 0.35),µ = (0, 0), Σ = 0.25 (1 0 0 1 ), ψ0 = (0.01, 0.01, 0.01), ψ1 = (0.01, 0.5, 0.01)

  21. [31]

    Sparse e2,e 4, dense in e3 r = (0.1, 0.35, 0.4),µ = (0, 0), Σ = 0.25 (1 0 0 1 ), ψ0 = (0.01, 0.01, 0.01), ψ1 = (0.01, 0.01, 0.01)

  22. [32]

    The case numbers correspond to the labels in Figures 7a, 7b and 7c

    Increase latent dimension from d = 2 to d = 3 r = (0.18, 0.3, 0.35),µ = (0, 0), Σ = 0.25 ( 1 0 0 0 1 0 0 0 1 ) , ψ0 = (0.01, 0.01, 0.01), ψ1 = (0.01, 0.01, 0.01) Table 2: Cases for each hypergraph model considered in the model depth comparison study. The case numbers correspon...

    2019

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