Pith. sign in

REVIEW 3 major objections 4 minor 45 references

A Bayesian approach to out-of-sample network reconstruction

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A Bayesian update using a single fitness parameter and a rolling empirical prior can reconstruct next-week network structure from past snapshots alone, matching an in-sample model that observes the true link count.

desk verdict The one-step-ahead Bayesian reconstruction is clean and worth publishing, but the 'self-sustained' headline claim is a heuristic plug-in (Eq. 39/F.1) rather than a derived posterior update, so the paper needs revision before the self-sustained story can be taken at face value. read the letter →

arxiv 2602.21869 v3 pith:QD3A4PWK submitted 2026-02-25 physics.soc-ph physics.app-phphysics.data-anq-fin.ST

Mattia Marzi , Tiziano Squartini This is my paper · ORCID
classification physics.soc-phphysics.app-phphysics.data-anq-fin.ST
keywords networkreconstructionlinkpredictionBayesianinferenceposteriorpredictivedistributionexponentialrandomgraphsfitnessmodelinterbanknetworksout-of-sample
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 tries to establish that out-of-sample network reconstruction can be carried out by carrying Bayesian priors across time rather than refitting a model to each snapshot. The proposed posterior predictive scheme, instantiated with a single-parameter fitness model, converts the observed network at time t into a posterior over the model parameter and then averages the link probabilities at t+1 over that posterior. Tested on eleven years of weekly eMID interbank snapshots, the method recovers the total number of links with relative error around 0.1 and captures per-node degree heterogeneity far better than a homogeneous Bernoulli version. Its stronger claim is that the procedure is self-sustaining: once seeded with one adjacency matrix, each predicted snapshot serves as the prior for the next, and the resulting forecasts match an in-sample model that is given the true link count. This matters for any setting where only aggregate node-level data are reliably available but future network structure is needed for risk or contagion analysis.

What carries the argument

The load-bearing object is the posterior predictive distribution, instantiated through the density-corrected Gravity Model's one-parameter link probability p_ij(z) = z s_i s_j / (1 + z s_i s_j), with node strengths s_i as exogenous fitnesses. An empirical prior π(z), fit as a Gamma distribution to point estimates of z from a rolling three-year calibration window, is updated with each new snapshot; Gauss-Hermite quadrature (with slice sampling as a check) evaluates the integral. For the self-sustained regime, the estimated probability matrix Q_t replaces the observed adjacency matrix A_t inside the likelihood, a plug-in device that lets the algorithm propagate without new topological data.

What would settle it

Take a network with a known structural break—e.g., a regulatory shock or a crisis—and run the self-sustained routine with no post-break adjacency data. If the relative error on the next snapshot's link count or degree sequence is no better than simply re-estimating link density from the last observed snapshot, the claim that predicted snapshots are reliable priors is falsified.

Watch

Extended reading notes

Core claim

The central claim is that a posterior predictive distribution P(A_{t+1}|A_t) = ∫ P(A_{t+1}|z) P(A_t|z) π(z) dz / P(A_t), with a single fitness parameter z and link probability p_ij = z s_i s_j / (1 + z s_i s_j), carries structural information forward in time. Fitting a Gamma distribution to past point estimates of z provides an empirical prior; the posterior is updated on a rolling three-year window; and the resulting q_ij are expected link probabilities for the next week. On the eMID data, the BFM recovers per-node degrees at the next weekly snapshot with TPR ≈ 0.40 and JI ≈ 0.20, and in self-sustained mode it ties the in-sample density-corrected Gravity Model that observes the true link co

Load-bearing premise

The self-sustained step treats the matrix of estimated link probabilities Q_t as though it were the observed binary adjacency matrix inside the likelihood; if this plug-in approximation is not a valid posterior update, the claim that the procedure can keep predicting from its own estimates lacks a rigorous grounding.

Editorial extensions

If this is right

  • With only node strengths and past snapshots, the BFM produces next-week link probabilities, expected total link counts, and per-node degrees; on eMID, it recovers the degree sequence far more accurately than the homogeneous BERM.
  • A rolling empirical prior removes the need to recalibrate the model from scratch at each time step; the prior is simply enriched with each new weekly estimate.
  • After initialization with one observed adjacency matrix, the self-sustained mode continues predicting snapshots from its own estimates alone, matching the in-sample dcGM that is handed the true link count.
  • The framework yields uncertainty measures through the posterior predictive distribution—e.g., a beta-binomial predictive for the total number of links and expected confusion-matrix entries.
  • Ranking diagnostics (TPR about 0.40, JI about 0.20, AUROC above chance) show that the BFM induces a meaningful ordering of candidate links, something a homogeneous model cannot do.

Reading between the lines

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

  • The paper only tests the binary, undirected, weekly aggregation of one interbank market; an immediate untested extension is whether the same posterior-predictive recipe transfers to directed networks or to coarser monthly snapshots.
  • In the sparse regime the paper shows the link ranking reduces to products of node strengths (preferential attachment), independent of the prior; that suggests the ranking part of the method may transfer to other networks where only strengths are known, even when the prior is misspecified.
  • The paper's reported error bump around 2008 suggests a structural break stresses the prior; one could stress-test the self-sustained loop by deliberately withholding post-break data and measuring how quickly predictive accuracy degrades.
  • If the plug-in replacement of A_t by Q_t were replaced by a proper latent-variable treatment, predictive uncertainty would likely widen; the reported point estimates in self-sustained mode may understate the true uncertainty.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 framework for out-of-sample reconstruction of evolving binary networks. It instantiates the framework with two models: the Bayesian Erdős–Rényi Model (BERM), with a Beta prior, and the Bayesian Fitness Model (BFM), a single-parameter fitness model with an empirically calibrated Gamma prior. The authors derive posterior predictive link probabilities for time t+1 conditional on the snapshot at time t, and test the procedure on weekly eMID interbank snapshots over 1999–2012. In addition to one-step-ahead forecasts, they propose a "self-sustained" recursion in which the estimated probability matrix Q_t replaces the observed adjacency matrix A_t in the likelihood (Eq. 39), and they claim that each predicted snapshot then serves as a reliable prior for the next. The BFM recovers per-node degrees with TPR ≈ 0.40 and JI ≈ 0.20, and the self-sustained version is reported to closely match an in-sample dcGM that observes the true link count.

Significance. If the self-sustained recursion were given a principled Bayesian derivation, the paper would offer a novel and practically relevant temporal extension of entropy-based network reconstruction. The one-step-ahead posterior predictive construction is clean, the rolling-window empirical prior is carefully designed, the comparison against an in-sample dcGM is a sensible control, and the open-source OR4CLE implementation is a concrete asset. However, the headline claim about self-sustained reconstruction currently rests on a plug-in substitution in Eq. (39) that is not derived from a probability model for Q_t, so the central methodological advance is not yet rigorously established.

major comments (3)
  1. [Self-sustained inference, Eq. (39); Appendix F, Eq. (E1)] The self-sustained update replaces the observed binary adjacency matrix A_t with the estimated probability matrix Q_t inside the Bernoulli likelihood, writing P(Q_t|z)=\prod (z s_i s_j)^{q_ij}/(1+ z s_i s_j). This expression is a valid likelihood only when the exponents are 0/1; for q_ij in [0,1] it is not a probability mass or density over the continuous matrix Q_t, and no sampling model for Q_t is specified. The resulting object P(z|Q_t) ∝ P(Q_t|z)π(z) is therefore a pseudo-posterior, and the recursion "Q_t is a reliable prior for Q_{t+1}" has no formal justification. The empirical stability in Figs. 3 and F.1 is suggestive but does not supply the missing derivation. This is load-bearing because the abstract's central claim about self-sustained reconstruction rests on Eq. (39). Please either derive the update from a proper probabilistic model (e.g., via expected sufficient statistics,
  2. [Self-sustained inference, Eq. (39); Appendix E] The recursion is not self-contained in the sense of requiring only the initial adjacency matrix: Eq. (39) uses the future strengths s_i^{t+1}, s_j^{t+1}, which in the eMID application are computed from the future weekly weighted matrix (Appendix E). Thus each step receives new node-level information beyond the initial topology. This can be a legitimate reconstruction scenario if future aggregate strengths are assumed available, but the paper should state this boundary condition explicitly. As written, the phrases "self-sustained" and "minimal amount of additional data" overstate the degree to which the algorithm propagates with no new input.
  3. [Section "Out-of-sample versus in-sample reconstruction", Fig. 5] The text says the Bayesian predictor "frequently matches, and sometimes exceeds" the in-sample dcGM, which is an appropriate characterization. However, the abstract's phrase "outperforming probabilistic benchmarks" is considerably stronger: the reported differences in TPR, PPV, TNR, and ACC are at the fourth decimal (0.2498 vs 0.2494, etc.), and the degree-level errors ARE_k and MRE_k are slightly in favor of the dcGM. In addition, the only external link-prediction benchmark cited is the DBCM result in [39]. The wording should be tempered or supplemented with error bars / significance tests and additional standard baselines.
minor comments (4)
  1. [Eqs. (40)–(41)] The Kullback-Leibler divergences D_KL(A||Q) and D_KL(A||R) are defined with terms a_ij ln(a_ij/q_ij). This is fine when a_ij=0, but if any q_ij or r_ij is 0 while a_ij=1 the expression is infinite. It would be useful to state the convention used for such cases, even if none occur in practice.
  2. [Appendix C, Fig. C.3 caption] Typographical errors: "toal" should be "total" and "constituing" should be "constituting". Similar typos appear in Fig. B.1 captions.
  3. [Methods, Prior calibration] The text says the prior is updated by adding each new weekly value, but also says the prior rolls over a three-year window. Please clarify whether the window strictly drops observations older than three years or whether the prior distribution is refit each week on the trailing window. The current wording is ambiguous.
  4. [Appendix D, Slice sampling] The slice-sampling implementation uses M=3000 post-burn-in draws with no reported convergence diagnostics (e.g., effective sample size or multiple chains). A brief note on Monte Carlo error would strengthen the numerical results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the out-of-sample predictive chain and the in-sample dcGM control are genuine.

full rationale

The paper's core derivation is a standard posterior-predictive construction (eqs. 7-9): P(A_{t+1}|A_t) = ∫ P(A_{t+1}|z) P(A_t|z) π(z) dz / P(A_t). The prior π(z) is calibrated exclusively on the 1999-2001 window and, later, on rolling past data; no parameter is fitted to the future adjacency matrices used in the evaluation. The comparison with the in-sample dcGM is a real control: the dcGM is given the true link count L*, whereas the Bayesian predictor is not, and the reported scores (TPR 0.2498 vs 0.2494) are close rather than identical, showing the comparison is not vacuous. The only potentially worrying step is the 'self-sustained' plug-in in eq. (39)/Appendix F, where the continuous probability matrix Q_t replaces the binary adjacency matrix A_t in the likelihood. This is indeed a heuristic substitution rather than a derived probabilistic update, and it is a legitimate concern about statistical validity. However, it is not circular: R_{t+1} is a non-trivial integral involving Q_t, and its similarity to Q_{t+1} (Figs. 3 and F.1) is an empirical finding, not an identity forced by construction. Self-citations ([35], [39]) appear as data sources or benchmarks, not as load-bearing justifications for the derivation. No step in the paper reduces, by its own equations, to its inputs.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The method's core is a standard Bayesian update on an existing fitness model. The genuinely paper-specific elements are the empirical prior construction (free hyperparameters α, β, κ, θ; rolling window) and the self-sustained plug-in of Q_t into the likelihood. These are not derived from first principles and are the main sources of assumption burden. No new physical entities are proposed.

free parameters (5)
  • α, β (Beta prior hyperparameters for BERM) = α≈58.261, β≈541.543
    Fitted to p* point estimates from the 1999-2001 calibration window (jack-knife augmented); govern the BERM predictive distribution (eq. 22).
  • κ, θ (Gamma prior hyperparameters for BFM) = κ≈27.538, θ≈0.007
    Fitted to z* point estimates from the 1999-2001 calibration window; selected as the only two-parameter family not rejected by KS test; used in the BFM posterior predictive (eq. 25).
  • Rolling window length = 3 years
    Empirical prior is updated over a three-year rolling window (Methods); no sensitivity analysis is provided.
  • Initial calibration period = 1999-2001 (3 years)
    Defines the first prior; data from the test period are excluded, so it is not a fit to the target, but the choice of the period length is ad hoc.
  • Jack-knife augmentation = 7 sub-weeks per week
    Prior calibration augments the sample by removing one trading day at a time (Methods); this is a roughness penalty choice affecting prior shape.
assumptions (6)
  • domain assumption A_{t+1} is conditionally independent of A_t given z
    Eq. 7, third equality; the predictive factorization P(A_{t+1}|A_t)=∫P(A_{t+1}|z)P(z|A_t)dz assumes all temporal dependence is mediated by z. If link persistence or other autoregressive structure exists beyond the shared density, the posterior predictive is misspecified.
  • domain assumption Link probability follows the density-corrected gravity model p_ij = z s_i s_j / (1 + z s_i s_j)
    Eq. 23; the single-parameter fitness ansatz with strengths as exogenous fitnesses; UBCM reduced to one global parameter. This is the generative model the paper chooses, not derived within the paper.
  • ad hoc to paper Empirical prior: π(z) is a Gamma density fitted to point estimates of z from past snapshots
    Section 'Towards a Bayesian approach' and Methods; the functional form of the prior is selected by KS test on the calibration window, not derived from a generative process. This is an empirical-Bayes step specific to this paper.
  • ad hoc to paper Self-sustained substitution: P(Q_t|z) with Q_t probabilistic
    Eq. 39 and Appendix F; the estimated probability matrix Q_t is inserted into the Bernoulli likelihood as if its entries were binary (q_ij used as exponents). No derivation justifies this as a posterior update.
  • domain assumption Target-snapshot strengths s_i^{t+1} are observable
    Eq. 25; the predictive probabilities require the strengths of nodes at time t+1; the method reconstructs links from aggregate node-level data, not from a pure time-series extrapolation of the topology.
  • domain assumption Log-posterior is approximately quadratic for Gauss-Hermite quadrature
    Appendix D, eq. D7; the default integration scheme replaces g(u) by a Gaussian centered at the posterior mode; if the posterior is multi-modal or skewed, the quadrature approximation degrades (though slice sampling is a fallback).

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Bayesian approach to out-of-sample network reconstruction." pith.science (2026). https://pith.science/paper/QD3A4PWK

@misc{pith2026260221869,
  author       = {Pith},
  title        = {Pith review of: A Bayesian approach to out-of-sample network reconstruction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QD3A4PWK}},
  note         = {Machine review of arXiv:2602.21869}
}
read the original abstract

Networks underpin systems that range from finance to biology, yet their structure is often only partially observed. Current reconstruction methods typically fit the parameters of a model anew to each snapshot, thus offering no guidance to predict future configurations. Here, we develop a Bayesian approach that uses the information about past network snapshots to inform a prior and predict the subsequent ones, while quantifying uncertainty. Instantiated with a single-parameter fitness model, our method infers link probabilities from node strengths and carries information forward in time. When applied to the Electronic Market for Interbank Deposit across the years 1999-2012, our method accurately recovers the number of connections per bank at subsequent times, outperforming probabilistic benchmarks designed for analogous, link prediction tasks. Notably, each predicted snapshot serves as a reliable prior for the next one, thus enabling self-sustained, out-of-sample reconstruction of evolving networks with a minimal amount of additional data.

Figures

Figures reproduced from arXiv: 2602.21869 by the authors.

Figure 1
Figure 1. FIG. 1: Top panels: empirical values of the total number of links (red) and node degrees (blue) scattered versus the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Left panel evolution of the TPR, the JI and AUROC across the weeks constituting our dataset. Right panel: [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Left panel: values of the Kullback-Leibler divergence between [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Left panel: empirical adjacency matrix [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Metric-specific distribution of the improvement of the ‘self-sustained’ Bayesian predictor with respect to [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 1 canonical work pages

  1. [39]

    Parisi, G

    F. Parisi, G. Caldarelli, and T. Squartini, Entropy-based approach to missing-links prediction, Appl. Netw. Sci.3, 17 (2018)

    2018

  2. [1]

    Colizza, A

    V. Colizza, A. Barrat, M. Barth´ elemy, and A. Vespig- nani, The role of the airline transportation network in the prediction and predictability of global epidemics, Proc. Natl. Acad. Sci. U.S.A.103, 2015 (2006)

    2015

  3. [2]

    Barrat, M

    A. Barrat, M. Barth´ elemy, and A. Vespignani,Dynami- cal Processes on Complex Networks(Cambridge Univer- sity Press, Cambridge, 2008)

    2008

  4. [3]

    M. E. J. Newman,Networks: An Introduction(Oxford University Press, Oxford, 2010)

    2010

  5. [4]

    Pastor-Satorras, C

    R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys.87, 925 (2015). 11

    2015

  6. [5]

    Squartini, I

    T. Squartini, I. van Lelyveld, and D. Garlaschelli, Early- warning signals of topological collapse in interbank net- works, Sci. Rep.3, 3357 (2013)

    2013

  7. [6]

    Battiston, J

    S. Battiston, J. D. Farmer, A. Flache, D. Garlaschelli, A. G. Haldane, H. Heesterbeek, C. Hommes, C. Jaeger, R. May, and M. Scheffer, Complexity theory and finan- cial regulation, Science351, 818 (2016)

    2016

  8. [7]

    Bardoscia, S

    M. Bardoscia, S. Battiston, F. Caccioli, and G. Cal- darelli, Pathways towards instability in financial net- works, Nat. Commun.8, 14416 (2017)

    2017

Show all 45 references
  1. [8]

    Macchiati, E

    V. Macchiati, E. Marchese, P. Mazzarisi, D. Gar- laschelli, and T. Squartini, Spectral signatures of struc- tural change in financial networks, Chaos, Solit. Fractals 193, 116065 (2025)

    2025

  2. [9]

    Park and M

    J. Park and M. E. J. Newman, Statistical mechanics of networks, Phys. Rev. E70, 066117 (2004)

    2004

  3. [10]

    Bianconi, Entropy of network ensembles, Phys

    G. Bianconi, Entropy of network ensembles, Phys. Rev. E79, 036114 (2009)

    2009

  4. [11]

    Squartini and D

    T. Squartini and D. Garlaschelli, Analytical maximum- likelihood method to detect patterns in real networks, New J. Phys.13, 083001 (2011)

    2011

  5. [12]

    Fronczak and P

    A. Fronczak and P. Fronczak, Statistical mechanics of the international trade network: Structural correlations and modeling, Phys. Rev. E85, 056113 (2012)

    2012

  6. [13]

    Squartini, R

    T. Squartini, R. Mastrandrea, and D. Garlaschelli, Un- biased sampling of network ensembles, New J. Phys.17, 023052 (2015)

    2015

  7. [14]

    Saracco, R

    F. Saracco, R. Di Clemente, A. Gabrielli, and T. Squar- tini, Randomizing bipartite networks: The case of the world trade web, Sci. Rep.5, 10595 (2015)

    2015

  8. [15]

    Cimini, T

    G. Cimini, T. Squartini, F. Saracco, D. Garlaschelli, A. Gabrielli, and G. Caldarelli, The statistical physics of real-world networks, Nat. Rev. Phys.1, 58 (2019)

    2019

  9. [16]

    E. T. Jaynes, Information theory and statistical mechan- ics, Phys. Rev.106, 620 (1957)

    1957

  10. [17]

    Squartini and D

    T. Squartini and D. Garlaschelli,Maximum-Entropy Networks, SpringerBriefs in Complexity (Springer In- ternational Publishing, Cham, 2017)

    2017

  11. [18]

    Garlaschelli and M

    D. Garlaschelli and M. I. Loffredo, Maximum likeli- hood: Extracting unbiased information from complex networks, Phys. Rev. E78, 015101 (2008)

    2008

  12. [19]

    Vallarano, M

    N. Vallarano, M. Bruno, E. Marchese, G. Trapani, F. Saracco, G. Cimini, M. Zanon, and T. Squartini, Fast and scalable likelihood maximization for exponen- tial random graph models with local constraints, Sci. Rep.11, 15227 (2021)

    2021

  13. [20]

    Di Vece, D

    M. Di Vece, D. Garlaschelli, and T. Squartini, Deter- ministic, quenched, and annealed parameter estimation for heterogeneous network models, Phys. Rev. E108, 054301 (2023)

    2023

  14. [21]

    Squartini, G

    T. Squartini, G. Caldarelli, G. Cimini, A. Gabrielli, and D. Garlaschelli, Reconstruction methods for networks: The case of economic and financial systems, Phys. Rep. 757, 1 (2018)

    2018

  15. [22]

    Zhou, Progresses and challenges in link prediction, iScience24, 103217 (2021)

    T. Zhou, Progresses and challenges in link prediction, iScience24, 103217 (2021)

    2021

  16. [23]

    L¨ u and T

    L. L¨ u and T. Zhou, Link prediction in complex networks: A survey, Physica A390, 1150 (2011)

    2011

  17. [24]

    Santucci, G

    F. Santucci, G. Cimini, and T. Squartini, Missing links prediction: comparing machine learning with physics- rooted approaches, arXiv 10.48550/arXiv.2601.23061 (2026)

    2026 doi

  18. [25]

    Bardoscia, P

    M. Bardoscia, P. Barucca, S. Battiston, F. Caccioli, G. Cimini, D. Garlaschelli, F. Saracco, T. Squartini, and G. Caldarelli, The physics of financial networks, Nature Reviews Physics3, 490 (2021)

    2021

  19. [26]

    L. N. Ialongo, C. de Valk, E. Marchese, F. Jansen, H. Zmarrou, T. Squartini, and D. Garlaschelli, Re- constructing firm-level interactions in the dutch input– output network from production constraints, Scientific Reports12, 11847 (2022)

    2022

  20. [27]

    Mungo, F

    L. Mungo, F. Lafond, P. Astudillo-Est´ evez, and J. D. Farmer, Reconstructing production networks using ma- chine learning, Journal of Economic Dynamics and Con- trol148, 104607 (2023)

    2023

  21. [28]

    Gelman, J

    A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin,Bayesian Data Analysis, 3rd ed. (Chapman and Hall/CRC, New York, 2013)

    2013

  22. [29]

    Cimini, T

    G. Cimini, T. Squartini, D. Garlaschelli, and A. Gabrielli, Systemic risk analysis on reconstructed eco- nomic and financial networks, Sci. Rep.5, 15758 (2015)

    2015

  23. [30]

    Mazzarisi and F

    P. Mazzarisi and F. Lillo, Methods for reconstructing in- terbank networks from limited information: A compari- son, inEconophysics and Sociophysics: Recent Progress and Future Directions, New Economic Windows, edited by F. Abergel, B. K. Chakrabarti, A. Ghosh, M. Mi- tra, M. ...

  24. [31]

    Anand, I

    K. Anand, I. van Lelyveld, ´A. Banai, S. Friedrich, R. Garratt, G. Ha laj, J. Fique, I. Hansen, S. M. Jaramillo, H. Lee, J. L. Molina-Borboa, S. Nobili, S. Ra- jan, D. Salakhova, T. C. Silva, L. Silvestri, and S. R. S. de Souza, The missing links: A global study on uncov- erin...

    2018

  25. [32]

    Lebacher, S

    M. Lebacher, S. Cook, N. Klein, and G. Kauermann, In search of lost edges: A case study on reconstructing financial networks, J. Network Theory in Finance5, 29 (2019)

    2019

  26. [33]

    Ramadiah, F

    A. Ramadiah, F. Caccioli, and D. Fricke, Reconstructing and stress testing credit networks, J. Econ. Dyn. Control 111, 103817 (2020)

    2020

  27. [34]

    Cimini, R

    G. Cimini, R. Mastrandrea, and T. Squartini,Recon- structing Networks, Elements in the Structure and Dy- namics of Complex Networks (Cambridge Univ. Press, 2021)

    2021

  28. [35]

    Marzi, F

    M. Marzi, F. Giuffrida, D. Garlaschelli, and T. Squar- tini, Reproducing the first and second moments of em- pirical degree distributions, Phys. Rev. Res.8, 013047 (2026)

    2026

  29. [36]

    G. Iori, G. D. Masi, O. Precup, G. Gabbi, and G. Cal- darelli, A network analysis of the italian overnight money market, J. Econ. Dyn. Control32, 259 (2008)

    2008

  30. [37]

    Finger, D

    K. Finger, D. Fricke, and T. Lux,Network analysis of the e-MID overnight money market: The informational value of different aggregation levels for intrinsic dynamic processes, Kiel Working Paper 1782 (Kiel Institute for the World Economy (IfW Kiel), 2012)

    2012

  31. [38]

    F. J. Massey, The kolmogorov-smirnov test for goodness of fit, J. Am. Stat. Assoc.46, 68 (1951)

    1951

  32. [40]

    T. P. Peixoto, Reconstructing networks with unknown and heterogeneous errors, Phys. Rev. X8, 041011 (2018). 12

    2018

  33. [41]

    T. P. Peixoto, Network reconstruction via the minimum description length principle, Phys. Rev. X15, 011065 (2025)

    2025

  34. [42]

    Young, G

    J.-G. Young, G. T. Cantwell, and M. E. J. Newman, Bayesian inference of network structure from unreliable data, J. Complex Netw.8, cnaa046 (2020)

    2020

  35. [43]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, eds.,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, Vol. 55 (National Bureau of Standards, Washington, D.C., 1964)

    1964

  36. [44]

    QN i=1 Q j(>i) pt+1 ij 1−pt+1 ij at+1 ij 1−p t+1 ij # QN i=1 Q j(>i) pt ij 1−pt ij at ij 1−p t ij π(z)dz R +∞ 0 QN i=1 Q j(>i) pt ij 1−pt ij at ij 1−p t ij π(z)dz = R +∞ 0

    R. M. Neal, Slice sampling, Ann. Statist.31, 705 (2003). ACKNOWLEDGMENTS MM and TS acknowledge support from the project ‘SoBigData.it - Strengthening the Italian RI for So- cial Mining and Big Data Analytics’ - IR0000013 - CUP B53C22001760006, financed by European Union - Next...

    2003

  37. [2002]

    Bottom right: evolution of the relative error on the total number of links (red) and the average relative error on the node degrees (blue), across the weeks constituting the year 2002. that can be further simplified into P(A t+1|At) = Nt+1Y i=1 Y j(>i) st+1 i st+1 j at+1 ij R ...

    2002

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.