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arxiv: 2606.22148 · v1 · pith:S7O45DTQnew · submitted 2026-06-20 · 📊 stat.ME · stat.CO

Ordering Stochastic Block Models via prior transitivity

Lapo Santi , Nial Friel , Pierpaolo De Blasi This is my paper

Pith reviewed 2026-06-26 11:30 UTC · model grok-4.3

classification 📊 stat.ME stat.CO
keywords transitive stochastic block modelordered blocksdirected networksBayesian inferencestochastic block modelsnetwork transitivitypartition recoverydirectional imbalance
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The pith

The Transitive Stochastic Block Model imposes order on directed network blocks by placing transitivity-inducing priors on directional imbalance rather than interaction volume.

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

The paper presents the Transitive Stochastic Block Model, a Bayesian approach for directed weighted networks that jointly infers node groups and an ordering among those groups. It factors each edge into a total volume term and a conditional direction term, then restricts the direction probabilities to obey either weak or strong stochastic transitivity. An age-ordered partition prior replaces exchangeable labels so that the number of blocks and their rank order are learned together via Gibbs sampling with Pólya-Gamma augmentation. Simulation results indicate that the ordering constraint improves both predictive accuracy and recovery of the true partition, with the largest gains appearing in sparse regimes. On six real directed networks the model raises predictive performance in four cases while also flagging situations, such as citation graphs, where the transitivity assumption reduces fit.

Core claim

The central claim is that order-restricted priors applied to the direction component of a stochastic block model yield a coherent probabilistic representation of hierarchical block structure in directed networks; the resulting TSBM improves out-of-sample prediction and partition quality relative to unconstrained block models whenever the observed directional imbalances are consistent with transitivity, while the same construction supplies a direct diagnostic for when that consistency fails.

What carries the argument

The Transitive Stochastic Block Model (TSBM), which factors each directed edge into a volume parameter and a direction probability that is constrained by either weak-stochastic-transitivity or Toeplitz strong-stochastic-transitivity priors, together with an age-ordered partition prior that breaks label exchangeability.

If this is right

  • Order-constrained priors raise predictive accuracy and block recovery, especially when networks are sparse.
  • The age-ordered partition prior allows the number of blocks to be inferred jointly with their rank order.
  • The model distinguishes cases in which transitive block structure improves fit from cases in which it harms fit.
  • Posterior inference remains feasible through a standard Gibbs sampler that uses Pólya-Gamma augmentation.
  • Partitions recovered under the model display visibly clearer ordered structure on empirical directed networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The volume-direction split could be reused in other directed network models that currently treat edge weight and orientation jointly.
  • Comparing the weak and strong transitivity versions on the same data supplies a direct test of whether rank separation alone explains observed imbalances.
  • The framework supplies a natural null model against which to measure the strength of non-transitive patterns such as rock-paper-scissors motifs.
  • Because the ordering is learned from the data rather than imposed externally, the same sampler could be applied to networks whose hierarchy is only partially known.

Load-bearing premise

Directional imbalances between blocks obey a transitive ordering without cycles.

What would settle it

A directed network in which the inferred blocks exhibit cyclic dominance (A beats B, B beats C, C beats A) at rates that the model assigns near-zero probability would falsify the ordering benefit.

Figures

Figures reproduced from arXiv: 2606.22148 by Lapo Santi, Nial Friel, Pierpaolo De Blasi.

Figure 1
Figure 1. Figure 1: Side-by-side comparison of (a) the WST and (b) the Toeplitz SST. In panel (b), lower-triangular [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Geometry of support restrictions in ψ-space for WST, SST, Toeplitz SST, and LST at K = 3, shown as shaded support surfaces with annotated vertices and a common viewing angle. Setting (x, y, z) = (ψ12, ψ23, ψ13), the coordinate z = ψ13 is the long-range comparison and sits on the vertical axis. CWST consists of x, y, z ≥ 0, which fills the cube. CSST has x, y ≥ 0, z ≥ x, z ≥ y, which is a pyramid inside the… view at source ↗
Figure 3
Figure 3. Figure 3: DAG and generative process for the TSBM. The [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: VI distance between the minVI partition estimate and the true partition for WST-generated (left) and SST-generated (right) scenarios. Each panel is a combination of ψ ⋆ ∈ {0.2, 1.3} rows and K⋆ ∈ {3, 8} columns. Increasing κ ⋆ values on the x-axis lead to more edges and thus, better signal. On the y−axis we have the VI distance from the true partition, where lower is better. Boxplots span 5 replicates over… view at source ↗
Figure 5
Figure 5. Figure 5: Partition point estimates for the bighorn sheep network (Hass, [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Empirical forward share for the spotted hyenas network ( [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Empirical forward share matrices for the high school network (Kunegis, [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: VI distance between the minimum-VI partition estimate and the true partition across SST [PITH_FULL_IMAGE:figures/full_fig_p040_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Adjusted Rand Index between the minimum-VI partition estimate and the true partition for [PITH_FULL_IMAGE:figures/full_fig_p041_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Prior sensitivity of the age-ordered prior at [PITH_FULL_IMAGE:figures/full_fig_p046_10.png] view at source ↗
read the original abstract

In directed networks, nodes may form groups with similar interaction patterns, while these groups may themselves follow an ordered structure. Existing methods typically treat these features separately, either clustering nodes without enforcing a coherent block order, or ranking individual nodes without allowing for structurally equivalent groups. We introduce the Transitive Stochastic Block Model (TSBM), a Bayesian model for directed weighted networks that uses transitivity-inducing priors to infer ordered blocks. The model separates the total volume of interaction between two nodes from the direction of interaction conditional on interaction occurring, so that hierarchy is imposed on directional imbalance rather than interaction frequency. We consider two order-restricted specifications: a flexible weak-stochastic-transitivity version, which excludes cyclic dominance patterns while allowing heterogeneous block-pair strengths, and a Toeplitz strong-stochastic-transitivity version, in which directional advantage increases with rank separation. Posterior inference is performed through a Gibbs sampler using P\'olya-Gamma data augmentation. Since ordered block labels are not exchangeable, we introduce an age-ordered partition prior to infer the number of blocks jointly with node allocation. Simulation studies show that order-constrained priors improve prediction and partition recovery, especially in sparse networks. Across six empirical directed networks, the TSBM improves predictive performance in four cases and yields partitions with clearer ordered structure. The results also identify cases, such as nearly deterministic dominance networks or non-transitive citation networks, where imposing ordered blocks can harm prediction. The TSBM therefore provides a probabilistic framework for estimating ordered groups and assessing when a transitive block structure is supported by the data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper introduces the Transitive Stochastic Block Model (TSBM) for directed weighted networks. It separates interaction volume from conditional direction and imposes transitivity-inducing priors (weak stochastic transitivity or Toeplitz strong stochastic transitivity) on the direction component to infer ordered blocks. Posterior inference uses a Gibbs sampler with Pólya-Gamma augmentation together with an age-ordered partition prior that jointly infers the number of blocks. Simulation studies report improved prediction and partition recovery (especially in sparse networks). On six empirical directed networks the TSBM improves predictive performance in four cases and produces partitions with clearer ordered structure, while explicitly identifying degradation on nearly deterministic dominance networks and non-transitive citation networks.

Significance. If the reported gains hold, the TSBM supplies a coherent Bayesian framework that bridges clustering and ranking by allowing the data to indicate when a transitive block ordering is supported. The separation of volume and direction, the two order-restricted specifications, the age-ordered prior, and the explicit counter-examples where the model harms performance are all positive features. The skeptic's concern that the transitivity premise is load-bearing is mitigated by the manuscript's own identification of settings in which the prior conflicts with the likelihood and harms prediction.

major comments (1)
  1. [Abstract] Abstract: the headline claim that order-constrained priors improve prediction rests on the data satisfying the transitivity premise. While the abstract correctly flags degradation on non-transitive citation networks, a concrete diagnostic (e.g., a posterior predictive check or a model-comparison statistic) for deciding when the TSBM should be preferred to an unconstrained SBM would make the practical recommendation load-bearing rather than post-hoc.
minor comments (1)
  1. Notation for the two transitivity specifications (weak vs. Toeplitz strong) and for the age-ordered partition prior should be introduced with explicit equations in the model section so that the Gibbs sampler steps can be verified without ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive suggestion regarding practical model selection. We address the comment below and will incorporate revisions to strengthen the guidance on when the TSBM is appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline claim that order-constrained priors improve prediction rests on the data satisfying the transitivity premise. While the abstract correctly flags degradation on non-transitive citation networks, a concrete diagnostic (e.g., a posterior predictive check or a model-comparison statistic) for deciding when the TSBM should be preferred to an unconstrained SBM would make the practical recommendation load-bearing rather than post-hoc.

    Authors: We agree that an explicit, pre-specified diagnostic would make the recommendation more actionable. The current manuscript already reports out-of-sample predictive performance (log predictive density) as the primary basis for comparing TSBM variants against the unconstrained SBM, and explicitly identifies the two empirical networks where the order constraint degrades performance. In the revision we will add a dedicated subsection on model comparison that formalizes this comparison via (i) posterior predictive checks for transitivity violations (e.g., frequency of observed cycles relative to posterior predictive replicates) and (ii) WAIC computed from the Pólya-Gamma-augmented MCMC output. These quantities will be reported alongside the predictive scores in both the simulation and empirical sections, and the abstract will be updated to reference the availability of these diagnostics. This change directly addresses the concern while remaining within the scope of the existing computational framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity; performance claims rest on external simulation and data validation

full rationale

The TSBM defines new transitivity-inducing priors (weak or Toeplitz strong stochastic transitivity) on directional imbalance after separating volume from conditional direction, then uses an age-ordered partition prior and Pólya-Gamma Gibbs sampler for inference. Simulation and empirical results (improved prediction in four of six networks) are obtained by applying the model to held-out data and comparing against baselines; these are not reductions by the paper's own equations to quantities already fitted inside the model. The abstract explicitly flags degradation on non-transitive networks, confirming the framework does not force its own success by construction. No self-citation load-bearing steps, fitted-input-called-prediction patterns, or ansatz smuggling appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the validity of the transitivity priors and the separation of volume and direction; these are domain assumptions rather than derived quantities. No explicit free parameters are named beyond those inferred by the model; the invented entity is the TSBM itself.

axioms (1)
  • domain assumption Directional imbalance between blocks obeys weak or strong stochastic transitivity.
    This premise is invoked when the model imposes the order-restricted priors on the direction component.
invented entities (1)
  • Transitive Stochastic Block Model (TSBM) no independent evidence
    purpose: To infer ordered blocks in directed weighted networks by combining SBM with transitivity priors.
    New model introduced by the paper; no independent evidence supplied in the abstract.

pith-pipeline@v0.9.1-grok · 5807 in / 1434 out tokens · 27039 ms · 2026-06-26T11:30:47.602777+00:00 · methodology

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Works this paper leans on

298 extracted references · 171 canonical work pages · 9 internal anchors

  1. [1]

    Advances in Applied Probability , year =

    The ages of alleles and a coalescent , author =. Advances in Applied Probability , year =. doi:10.2307/1427237 , number =

    work page doi:10.2307/1427237

  2. [2]

    Advances in Applied Probability , year =

    Consistent ordered sampling distributions: characterization and convergence , author =. Advances in Applied Probability , year =. doi:10.2307/1427746 , number =

    work page doi:10.2307/1427746

  3. [3]

    Regenerative composition structures , url =

    Gnedin, Alexander and Pitman, Jim , doi =. Regenerative composition structures , url =

  4. [4]

    2025 , eprint =

    Bayesian nonparametric community detection in assortative stochastic block models , author =. 2025 , eprint =

    2025

  5. [5]

    Journal of Computational and Graphical Statistics , author =

    A split-merge. Journal of Computational and Graphical Statistics , author =. 2004 , pages =. doi:10.1198/1061860043001 , number =

    work page doi:10.1198/1061860043001 2004

  6. [6]

    Computational Statistics & Data Analysis , author =

    The. Computational Statistics & Data Analysis , author =. 2007 , pages =. doi:10.1016/j.csda.2007.01.024 , language =

    work page doi:10.1016/j.csda.2007.01.024 2007

  7. [7]

    Bayesian Analysis , author =

    Improved criteria for clustering based on the posterior similarity matrix , volume =. Bayesian Analysis , author =. doi:10.1214/09-BA414 , number =

    work page doi:10.1214/09-ba414

  8. [8]

    Journal of Computational and Graphical Statistics , author =

    Dependent. Journal of Computational and Graphical Statistics , author =. 2022 , pages =. doi:10.1080/10618600.2021.1987255 , language =

    work page doi:10.1080/10618600.2021.1987255 2022

  9. [9]

    Bayesian Analysis , author =

    Bayesian. Bayesian Analysis , author =. doi:10.1214/24-BA1415 , number =

    work page doi:10.1214/24-ba1415

  10. [10]

    Journal of Computational and Graphical Statistics , author =

    An. Journal of Computational and Graphical Statistics , author =. 2010 , pages =. doi:10.1198/jcgs.2010.09008 , language =

    work page doi:10.1198/jcgs.2010.09008 2010

  11. [11]

    Proceedings of the 21st National Conference on Artificial Intelligence (AAAI) , volume=

    Learning systems of concepts with an infinite relational model , author=. Proceedings of the 21st National Conference on Artificial Intelligence (AAAI) , volume=

  12. [12]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , author =

    Dealing. Journal of the Royal Statistical Society Series B: Statistical Methodology , author =. 2000 , pages =. doi:10.1111/1467-9868.00265 , abstract =

    work page doi:10.1111/1467-9868.00265 2000

  13. [13]

    Educational and Psychological Measurement , author =

    A. Educational and Psychological Measurement , author =. 2021 , pages =. doi:10.1177/0013164420970614 , abstract =

    work page doi:10.1177/0013164420970614 2021

  14. [14]

    Gaucher, Solenne and Klopp, Olga , year =. Maximum. doi:10.48550/ARXIV.1902.10605 , abstract =

    work page doi:10.48550/arxiv.1902.10605 1902

  15. [15]

    The Annals of Applied Statistics , author =

    Uncovering latent structure in valued graphs:. The Annals of Applied Statistics , author =. doi:10.1214/10-AOAS361 , number =

    work page doi:10.1214/10-aoas361

  16. [16]

    Biometrika , author =

    Conjugate. Biometrika , author =. 2019 , pages =. doi:10.1093/biomet/asz034 , abstract =

    work page doi:10.1093/biomet/asz034 2019

  17. [17]

    Journal of the American Statistical Association , author =

    Bayesian. Journal of the American Statistical Association , author =. 1993 , pages =. doi:10.1080/01621459.1993.10476321 , language =

    work page doi:10.1080/01621459.1993.10476321 1993

  18. [18]

    Bayesian Analysis , author =

    Bayesian. Bayesian Analysis , author =. doi:10.1214/17-BA1073 , number =

    work page doi:10.1214/17-ba1073

  19. [19]

    Statistics and Computing , author =

    Optimal. Statistics and Computing , author =. 2018 , pages =. doi:10.1007/s11222-017-9786-y , number =

    work page doi:10.1007/s11222-017-9786-y 2018

  20. [20]

    Combinatorial

    Pitman, Jim , year =. Combinatorial. doi:10.1007/b11601500 , language =

    work page doi:10.1007/b11601500

  21. [21]

    The Annals of Applied Probability , author =

    Conditional formulae for. The Annals of Applied Probability , author =. doi:10.1214/12-AAP843 , number =

    work page doi:10.1214/12-aap843

  22. [22]

    Methodology and Computing in Applied Probability , author =

    On a. Methodology and Computing in Applied Probability , author =. 2012 , pages =. doi:10.1007/s11009-010-9202-y , language =

    work page doi:10.1007/s11009-010-9202-y 2012

  23. [23]

    PLOS ONE , volume =

    Contact Patterns in a High School: A Comparison between Data Collected Using Wearable Sensors, Contact Diaries and Friendship Surveys , author =. PLOS ONE , volume =

  24. [24]

    Matrilineal rank Inheritance varies with absolute rank in Japanese macaques , author =

  25. [25]

    Faulkenberry and Osvaldo A Martin and Alex Kipnis and Salvador Balkus and lfkdlfdlk and Carsten Allefeld and Heiner Atze and Adam Knapp and Ciarán D

    Joram Soch and The Book of Statistical Proofs and Karahan Sarıtaş and Maja and Pietro Monticone and Thomas J. Faulkenberry and Osvaldo A Martin and Alex Kipnis and Salvador Balkus and lfkdlfdlk and Carsten Allefeld and Heiner Atze and Adam Knapp and Ciarán D. McInerney and Lo4ding00 and Valeriu Ohan and amvosk and maxgrozo , month = jan, year =. doi:10.52...

    work page doi:10.5281/zenodo.4305949

  26. [26]

    A new parametrization of the Gnedin-Fisher species sampling model

    Cerquetti, Annalisa , year =. A new parametrization of the. doi:10.48550/ARXIV.1008.2285 , abstract =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1008.2285

  27. [27]

    The Annals of Applied Probability , author =

    Bayesian nonparametric estimators derived from conditional. The Annals of Applied Probability , author =. doi:10.1214/07-AAP495 , number =

    work page doi:10.1214/07-aap495

  28. [28]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , author =

    Bayesian. Journal of the Royal Statistical Society Series B: Statistical Methodology , author =. 2009 , pages =. doi:10.1111/j.1467-9868.2009.00717.x , abstract =

    work page doi:10.1111/j.1467-9868.2009.00717.x 2009

  29. [29]

    Biometrics , author =

    A. Biometrics , author =. 2012 , pages =. doi:10.1111/j.1541-0420.2012.01793.x , abstract =

    work page doi:10.1111/j.1541-0420.2012.01793.x 2012

  30. [30]

    Biometrika , author =

    Bayesian cluster analysis , volume =. Biometrika , author =. 1978 , pages =. doi:10.1093/biomet/65.1.31 , language =

    work page doi:10.1093/biomet/65.1.31 1978

  31. [31]

    , month = apr, year =

    Charalambides, Charalambos A. , month = apr, year =. Combinatorial. doi:10.1002/0471733180 , language =

    work page doi:10.1002/0471733180

  32. [32]

    Discrete Mathematics , author =

    Generalized. Discrete Mathematics , author =. 1996 , pages =. doi:10.1016/0012-365X(95)00112-A , language =

    work page doi:10.1016/0012-365x(95)00112-a 1996

  33. [33]

    Electronic Communications in Probability , author =

    A. Electronic Communications in Probability , author =. 2010 , pages =. doi:10.1214/ECP.v15-1532 , number =

    work page doi:10.1214/ecp.v15-1532 2010

  34. [34]

    Journal of Mathematical Sciences , author =

    Exchangeable. Journal of Mathematical Sciences , author =. 2006 , pages =. doi:10.1007/s10958-006-0335-z , language =

    work page doi:10.1007/s10958-006-0335-z 2006

  35. [35]

    2025 , eprint =

    Santi, Lapo and Friel, Nial , title =. 2025 , eprint =

    2025

  36. [36]

    2021 IEEE International Symposium on Information Theory (ISIT) , pages =

    Hsiao, Chen-Hao and Wang, I-Hsiang , title =. 2021 IEEE International Symposium on Information Theory (ISIT) , pages =. 2021 , publisher =

    2021

  37. [37]

    and Leinhardt, Samuel , title =

    Holland, Paul W. and Leinhardt, Samuel , title =. Journal of the American Statistical Association , volume =. 1981 , doi =

    1981

  38. [38]

    and Wasserman, Stanley S

    Fienberg, Stephen E. and Wasserman, Stanley S. , title =. Sociological Methodology 1981 , editor =

    1981

  39. [39]

    and Wong, George Y

    Wang, Yuchung J. and Wong, George Y. , title =. Journal of the American Statistical Association , volume =. 1987 , doi =

    1987

  40. [40]

    Tournament Solutions and Majority Voting , series =

    Laslier, Jean-Fran. Tournament Solutions and Majority Voting , series =. 1997 , doi =

    1997

  41. [41]

    Handbook of Computational Social Choice , editor =

    Brandt, Felix and Brill, Markus and Harrenstein, Paul , title =. Handbook of Computational Social Choice , editor =. 2016 , doi =

    2016

  42. [42]

    , title =

    Hass, Christine C. , title =. Journal of Zoology , volume =. 1991 , doi =

    1991

  43. [43]

    and DeCasien, Alex R

    Strauss, Eli D. and DeCasien, Alex R. and Galindo, Gabriela and Hobson, Elizabeth A. and Shizuka, Daizaburo and Curley, James P. , title =. Philosophical Transactions of the Royal Society B: Biological Sciences , volume =. 2022 , doi =

    2022

  44. [44]

    Journal of Machine Learning Research , volume =

    Bengio, Yoshua and Grandvalet, Yves , title =. Journal of Machine Learning Research , volume =

  45. [45]

    Statistics and Computing , volume =

    Vehtari, Aki and Gelman, Andrew and Gabry, Jonah , title =. Statistics and Computing , volume =. 2017 , doi =

    2017

  46. [46]

    , title =

    Elo, Arpad E. , title =. 1978 , isbn =

    1978

  47. [47]

    2025 , eprint =

    Grazian, Clara , title =. 2025 , eprint =

    2025

  48. [48]

    and Titterington, D

    Celeux, Gilles and Forbes, Florence and Robert, Christian P. and Titterington, D. M. , title =. Bayesian Analysis , year =

  49. [49]

    , title =

    MacEachern, Steven N. , title =. ASA Proceedings of the Section on Bayesian Statistical Science , publisher =

  50. [50]

    and Steel, Mark F

    Griffin, Jim E. and Steel, Mark F. J. , title =. Journal of the American Statistical Association , year =

  51. [51]

    Quintana, Fernando A. and M. The Dependent Dirichlet Process and Related Models , journal =. 2022 , volume =

    2022

  52. [52]

    Bayesian Dependent Mixture Models: A Predictive Comparison and Survey , journal =

    Wade, Sara and In. Bayesian Dependent Mixture Models: A Predictive Comparison and Survey , journal =. 2025 , volume =

    2025

  53. [53]

    , title =

    Park, Ju-Hyun and Dunson, David B. , title =. Statistica Sinica , year =

  54. [54]

    A Product Partition Model with Regression on Covariates , journal =

    M. A Product Partition Model with Regression on Covariates , journal =. 2011 , volume =

    2011

  55. [55]

    and Frazier, Peter I

    Blei, David M. and Frazier, Peter I. , title =. Journal of Machine Learning Research , year =

  56. [56]

    and Quintana, Fernando A

    Page, Garritt L. and Quintana, Fernando A. , title =. Bayesian Analysis , year =

  57. [57]

    and Day, Ryan and Tsai, Jerry W

    Dahl, David B. and Day, Ryan and Tsai, Jerry W. , title =. Journal of the American Statistical Association , year =

  58. [58]

    Biometrics , year =

    Argiento, Raffaele and Corradin, Riccardo and Guglielmi, Alessandra and Lanzarone, Ettore , title =. Biometrics , year =

  59. [59]

    and Quintana, Fernando A

    Paganin, Sally and Page, Garritt L. and Quintana, Fernando A. , title =. Electronic Journal of Statistics , year =

  60. [60]

    Assessing a Mixture Model for Clustering with the Integrated Completed Likelihood , journal =

    Biernacki, Christophe and Celeux, Gilles and Govaert, G. Assessing a Mixture Model for Clustering with the Integrated Completed Likelihood , journal =. 2000 , volume =

    2000

  61. [61]

    The Annals of Statistics , year =

    Nobile, Agostino , title =. The Annals of Statistics , year =

  62. [62]

    , title =

    Richardson, Sylvia and Green, Peter J. , title =. Journal of the Royal Statistical Society: Series B (Methodological) , year =

  63. [63]

    , title =

    Jain, Sonia and Neal, Radford M. , title =. Journal of Computational and Graphical Statistics , year =

  64. [64]

    and Harrison, Matthew T

    Miller, Jeffrey W. and Harrison, Matthew T. , title =. Journal of the American Statistical Association , year =

  65. [65]

    Generalized Mixtures of Finite Mixtures and Telescoping Sampling , journal =

    Fr. Generalized Mixtures of Finite Mixtures and Telescoping Sampling , journal =. 2021 , volume =

    2021

  66. [66]

    Probability Theory and Related Fields , year =

    Pitman, Jim , title =. Probability Theory and Related Fields , year =

  67. [67]

    , title =

    Ferguson, Thomas S. , title =. The Annals of Statistics , year =

  68. [68]

    , title =

    Antoniak, Charles E. , title =. The Annals of Statistics , year =

  69. [69]

    Journal of Mathematical Sciences , year =

    Gnedin, Alexander and Pitman, Jim , title =. Journal of Mathematical Sciences , year =

  70. [70]

    , title =

    Hartigan, John A. , title =. Communications in Statistics -- Theory and Methods , year =

  71. [71]

    , title =

    Barry, Daniel and Hartigan, John A. , title =. The Annals of Statistics , year =

  72. [72]

    Cluster Analysis, Model Selection, and Prior Distributions on Models , journal =

    Casella, George and Moreno, El. Cluster Analysis, Model Selection, and Prior Distributions on Models , journal =. 2014 , volume =

    2014

  73. [73]

    2025 , eprint =

    Advances in Bayesian random partition models: A comprehensive review , author =. 2025 , eprint =

    2025

  74. [74]

    Are Gibbs-Type Priors the Most Natural Generalization of the Dirichlet Process? , journal =

    De Blasi, Pierpaolo and Favaro, Stefano and Lijoi, Antonio and Mena, Rams. Are Gibbs-Type Priors the Most Natural Generalization of the Dirichlet Process? , journal =. 2015 , volume =

    2015

  75. [75]

    Page, Lawrence and Brin, Sergey and Motwani, Rajeev and Winograd, Terry , title =

  76. [76]

    and Moore, Cristopher , title =

    De Bacco, Caterina and Larremore, Daniel B. and Moore, Cristopher , title =. Science Advances , year =

  77. [77]

    Ryan, Matthew , title =

  78. [78]

    The American Mathematical Monthly , volume =

    Harary, Frank and Leo Moser , title =. The American Mathematical Monthly , volume =. 1966 , pages =

    1966

  79. [79]

    Communications Physics , volume =

    Morel-Balbi, Sebastian and Kirkley, Alec , title =. Communications Physics , volume =. 2026 , doi =

    2026

  80. [80]

    Journal of the Royal Society Interface , author =

    The network motif architecture of dominance hierarchies , volume =. Journal of the Royal Society Interface , author =. 2015 , pages =. doi:10.1098/rsif.2015.0080 , number =

    work page doi:10.1098/rsif.2015.0080 2015

Showing first 80 references.