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REVIEW 3 major objections 6 minor 82 references

On the reconstruction limits of complex networks

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Network reconstruction has an algorithm-independent ceiling set by the mutual information between the hidden graph and the data.

desk verdict A sound information-theoretic core with an honest but unproven empirical extension; the reconstruction index is a useful heuristic, not yet a validated reliability measure. read the letter →

arxiv 2501.01437 v3 pith:JA4ZIS4S submitted 2024-12-23 stat.AP cs.ITmath.ITphysics.data-an

Charles Murphy , Simon Lizotte , François Thibault , Vincent Thibeault , Patrick Desrosiers , Antoine Allard This is my paper · ORCID
classification stat.APcs.ITmath.ITphysics.data-an MSC 62B1094A1505C80
keywords networkreconstructionreconstructabilitymutualinformationdataprocessinginequalityBayesianmodelselectiongainindexneuroscience
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

The paper tries to establish that network reconstruction is governed by an algorithm-independent limit: no method, however clever, can recover more information about a hidden graph than the mutual information between that graph and the observed data. It defines the reconstructability $\Psi^* = I(G^*; X^*)/H(G^*)$ as this limit and proves, through the data processing inequality, that the mutual information between the true graph and any reconstructed graph cannot exceed it. Because the true data-generating process is unknown in practice, the paper introduces the reconstruction index $\psi_M$, a normalized information gain computed from a single dataset, and argues that when coupled with evidence-based model selection it approximates the reconstructability and predicts reconstruction error without the true network. The case studies include synthetic dynamics, two empirical graphs, and spontaneous mouse-brain spiking activity, where the best-evidence model yields a reconstruction index near 67%.

What carries the argument

The machinery is the joint process $(G^*, X^*, \hat{G})$ with distribution $P(G^*)P(X^*|G^*)P(\hat{G}|X^*)$, together with the data processing inequality applied to that chain. The inequality yields the reconstruction limit $I(G^*; \hat{G}) \leq I(G^*, X^*)$, so the reconstructability $\Psi^* = I(G^*; X^*)/H(G^*)$ is the ceiling; Fano's inequality converts the same quantity into a lower bound on the probability of misidentifying the graph. On the empirical side, the engine is the information gain $I_M(x) = D_{\mathrm{KL}}(P(G|X=x) \,\|\, P(G))$, normalized by the cross-entropy upper bound $\Lambda_M(x) = -\mathbb{E}_{G|X=x}[\log P(G)]$, giving the reconstruction index. Evidence-based model selection via Eq. (33) picks which candidate model's index should be trusted as an estimate of the true limit.

What would settle it

On synthetic data with a known true generative process, compute the exact $\Psi^*$ by enumerating all small graphs, then run the full procedure with a candidate set that excludes the true model; the paper's own Fig. 5 shows regimes where $\psi$ increases while posterior loss increases, so finding a dataset where the selected model's $\psi$ is high but the exact $\Psi^*$ is low, or where $\psi$ and reconstruction error decorrelate, would falsify the claim that the index is a trustworthy proxy without model correctness.

Watch

Extended reading notes

Core claim

The central discovery is Eq. (16): for any reconstruction algorithm, $I(G^*; \hat{G}) \leq I(G^*; X^*)$. Since the observed data $X^*$ is a lossy encoding of the true graph $G^*$ by the unknown data-generating process, the mutual information $I(G^*; X^*)$, normalized by the entropy $H(G^*)$ to give the reconstructability $\Psi^*$, is the maximum fraction of structural information any method can recover. Even the ideal reconstruction model, the true generative process itself, cannot exceed this bound. The paper then makes this theoretical ceiling operational: the reconstruction index $\psi_M = I_M(x)/\Lambda_M(x)$, the Kullback-Leibler divergence between a model's posterior and prior normalized by its upper bound, is shown to track posterior loss when the model matches the data process, and the paper proposes selecting a model by evidence and reading the best-evidence model's index as a proxy for the true reconstructability.

Load-bearing premise

The method's guarantee that the reconstruction index tracks true reconstructability depends on the candidate model set containing a model whose evidence distribution matches the true data-generating process, and for real data that match can never be certified since the evidence estimates are themselves biased approximations.

Editorial extensions

If this is right

  • No amount of algorithmic sophistication can beat $\Psi^* H(G^*)$; two reconstruction methods can differ only in how much of the available mutual information they extract.
  • Perfect reconstruction is achievable only when $I(G^*; X^*) = H(G^*)$, and when the data and graph are independent every method is equivalent to random guessing.
  • A correctly calibrated reconstruction index gives a ground-truth-free error estimate, so practitioners can compare reconstruction models and flag unreliable inferred networks without knowing the true network.
  • The bound extends to hierarchical Bayesian models: marginalizing or inferring hyperparameters does not change the limit, since $I(\theta^*, G^*; X^*) = I(G^*; X^*)$.
  • The framework transfers from ensembles to single observed graphs, which is what makes the empirical applications to spiking data and individual networks possible.

Reading between the lines

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

  • A practical consequence the authors leave implicit: the reconstruction index should be reported together with the evidence gap to the next-best model, because a near-tie in evidence means the selected index is not a stable estimate of the true reconstructability.
  • The index is prior-dependent by construction because $\Lambda_M$ shrinks as the prior concentrates on the true graph, so comparing $\psi$ values across models with different priors is not apples-to-apples; the paper's own Fig. 7 shows exactly this slope variation.
  • The misspecification failure shown in Fig. 5 suggests a calibration test for real applications: split the data and check whether $\psi$ and predictive error move together, since a breakdown of that co-movement diagnoses a wrong candidate model set.
  • Because a low $\psi$ means the posterior barely moves from the prior, the method could serve as a screening tool for datasets too uninformative to support any network claim.
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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 / 6 minor

Summary. The paper studies the information-theoretic limits of network reconstruction from indirect observations. It defines the reconstructability of a true data-generating (TDG) process as the normalized mutual information Ψ* = I(G*; X*)/H(G*), and proves, via the data processing inequality, that any reconstructed graph Ĝ satisfies I(G*; Ĝ) ≤ I(G*; X*) (Eq. 16), so that Ψ* is an algorithm-independent upper bound on reconstruction performance. Motivated by this bound, the authors introduce a data-driven reconstruction index ψ_M(x), defined as the normalized information gain of a candidate model M, and propose a procedure (Sec. V) that selects a model by evidence maximization (Eq. 33) and uses the corresponding ψ_M as a proxy for the true reconstructability and as a predictor of reconstruction error without knowing the true graph. The method is tested on synthetic data, and applied to two empirical settings: reconstruction of a mouse brain network from spiking data (Sec. V A) and reconstruction of two empirical social networks from synthetic dynamics (Sec. V B). An analytic single-edge example (Sec. II B) and a delta-prior limit (Appendix H) support the theoretical claims.

Significance. The central information-theoretic statement, I(G*; Ĝ) ≤ I(G*; X*), is correct and cleanly presented; the single-edge example and the ϵ→0 delta-prior limit are instructive and provide analytic confirmation of the framework. If the empirical reconstruction index could be validated as a reliable proxy for reconstructability under realistic model uncertainty, the paper would offer a practical tool for assessing reconstruction quality in settings where ground truth is unavailable, which is a genuine need in network science. The authors are transparent about the main limitation—that the index is trustworthy only when the candidate model set contains a model resembling the TDG process—and they include posterior predictive checks. However, the load-bearing empirical claim that ψ_M predicts reconstruction error on real data is not quantitatively established by the presented evidence, because the paper does not characterize how the index degrades with model misspecification and does not propagate known sources of estimation bias and Monte Carlo variability into the reported values.

major comments (3)
  1. [Sec. IV C and Sec. V, Eq. (33)] The central empirical claim—that the evidence-selected reconstruction index ψ_hatM approximates the true reconstructability Ψ* and predicts posterior loss—is conditional on the candidate set containing a model that 'resembles the TDG process' (Sec. V), but the paper provides no quantitative test of this proximity. The paper itself concedes (Sec. IV C) that minimizing evidence cross-entropy is necessary but not sufficient for model correctness, and Fig. 5 demonstrates that when the coupling constant J is misspecified, ψ_M can increase while the posterior loss worsens. Since this is the key bridge from the exact information-theoretic bound to the practical method, the manuscript needs either a quantitative characterization of how ψ_M deviates from Ψ* as a function of measurable model-misspecification diagnostics, or a more circumscribed statement that limits the practical claim to settings where the candidate set is known to contain the generating model.
  2. [Appendix F, Sec. IV C] The evidence and mutual information estimates used for model selection are biased and are presented without uncertainties: Eq. (F6) is explicitly a variational lower bound, the evidence estimator in Sec. F4 is also a lower bound, and the KDE-based differential entropy estimates for continuous parameters (Sec. F4) have no reported variance. Because Eq. (33) selects the model with maximal estimated evidence, biased evidence estimates can change the selected model and therefore the reported ψ_M. The paper should either quantify the bias in controlled experiments where the true model is known, or report Monte Carlo standard errors for log-evidence and ψ_M and show that model selection is stable with respect to these uncertainties.
  3. [Appendix I, Table III] The semi-greedy edge-count optimization used for all models in Sec. V A is not standard posterior inference, and Table III shows substantial chain-to-chain variability (e.g., SIS-SBM: 1722 ± 90 edges across 8 chains). This variability is not propagated into the reported reconstruction index ψ_M ≈ 0.67 or the evidence values, which are computed conditional on a single frozen edge count. The paper should assess the sensitivity of ψ_M and of the evidence ranking to the edge count, for example by repeating the posterior sampling for several edge counts around the converged value or by jointly sampling the number of edges within the MCMC procedure.
minor comments (6)
  1. [Eq. (26)] The chain rule is stated with the wrong sign: I(θ*, G*; X*) = I(G*; X*) + I(θ*; X* | G*), not minus. Since the conditional term is zero, the final conclusion is unaffected, but the equation as written is incorrect.
  2. [Sec. II B] The sentence 'we avoid this non-identifiability issue by not inferring p and r' appears to contain a typo: the symmetry is between q and r, so it should read 'by not inferring q and r.'
  3. [Fig. 2 caption] The phrase '90% confident interval' should be '90% confidence interval.'
  4. [Appendix F, Eq. after (F13)] The displayed equation for the mean-field evidence estimator appears truncated; it should explicitly show the full binary-entropy sum over pairs (i,j), including the closing brackets and the final expression.
  5. [Sec. V A] When citing the inferred edge count, the text gives 'approximately 1722' without the standard deviation reported in Table III (89.98). Including the variability would avoid overprecision and is relevant to the preceding major comment.
  6. [Sec. IV C] The statement that 'the empirical average of information gain becomes equal to the mutual information for model M' is imprecise: it converges to I(G; X) under the model M, and only coincides with I(G*; X*) if M and M* are equal in distribution. The text should clarify this distinction to avoid conflating the model's reconstructability with the true one.

Circularity Check

2 steps flagged · score 4.0 of 10

Central information-theoretic bound is standard and non-circular, but the reconstruction index's claimed prediction of posterior loss is partly an algebraic identity, and the numerical pipeline leans on a self-cited approximation.

  1. self definitional [Sec. III C Eq. (22); Sec. IV A Eq. (29); Appendix D Eq. (D5)]
    "The reconstruction index is shown to predict the reconstruction error without requiring knowledge of the true underlying network. ... ψM = IM/ΛM. ... when M and M∗ are equal in distribution, EX∗,G∗[L(y, p∗)] ≈ H(G∗|X∗). ... EX∗,G∗[L(y, p∗)] ≈ H(G∗)[1 − Ψ∗]."

    The index is defined as a normalized information gain, i.e., ψ = 1 − H(G|x)/ΛM. Under the paper's own alignment assumption (M equals the TDG model in distribution), E[ΛM] = H(G) and the expected posterior loss equals H(G∗|X∗). Therefore E[ψ] ≈ 1 − E[L]/H(G∗) = Ψ∗ by Eq. (22). The claimed "prediction of reconstruction error" for the posterior loss is thus an algebraic identity relating ψ to the expected log-loss; it is not an independent empirical discovery. Fig. 5 confirms the conditionality by showing the correlation inverts under misspecification, which the paper concedes.

  2. self citation load bearing [Sec. IV C; Appendix F 2, Eq. (F6)]
    "In Ref. [36], we showed that variational mean-field methods provide efficient approximations for both the mutual information and the evidence. The same techniques are used here (see Appendix F). ... Although Eq. (F6) is a biased estimator of the mutual information, it was shown in Ref. [36] that the bias is generally small, especially for large networks."

    All reported values of ψM and evidence used for model selection, including the mouse-brain reconstruction index of about 67%, are computed with these biased variational estimators. The only cited justification that the bias is acceptably small is the authors' own previous paper [36], which is not an independent or machine-checked verification. This makes the empirical reliability of the reconstruction-index pipeline depend on a self-citation, although the central theorem Eq. (16) does not.

full rationale

The main theoretical result, Eq. (16) with the Markov chain G∗ → X∗ → Ĝ, is a standard application of the data-processing inequality, and the definition Ψ∗ = I(G∗; X∗)/H(G∗) is self-contained. The single-edge example and the delta-prior analysis in Appendix H are independent analytical checks with no circularity. The circularity concerns attach to the empirical wrapper rather than the theorem. First, the reconstruction index is defined as a normalized posterior concentration measure; under the paper's stated model-alignment assumption, its relation to the posterior loss is forced algebraically, so presenting that relation as a validated "prediction" is partly self-referential. The paper is transparent about the failure under misspecification (Fig. 5), which limits the severity. Second, the numerical feasibility of the method relies on variational mean-field and KDE estimates whose acceptable-bias claim is supported only by the authors' earlier work, so the empirical pipeline has a self-citation load-bearing component. The paper's own concessions—that evidence cross-entropy minimization is necessary but not sufficient (Sec. IV C) and that determining whether a candidate model resembles the TDG process is hardly feasible experimentally (Sec. V)—are limitations and correctness risks rather than additional circularities, but they reinforce the conditionality of the ψ-based claims. Overall, Eq. (16) and the analytical reconstructability results have independent content; the partial circularity is confined to the data-driven index and its numerical implementation.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The central theoretical bound uses only standard information-theoretic inequalities with no fitted parameters. The free parameters listed are illustrative or tuning choices for examples and applications. The domain assumptions about the true generative process and the candidate model set are load-bearing for the empirical reconstruction index, and the mean-field and edge-factorization approximations are stated but not proven to be accurate in the regimes tested.

free parameters (4)
  • edge prior p in single-edge example = p = 1/2 in Fig. 3
    Chosen to illustrate the analytic calculation; not part of the general framework.
  • noise probabilities q and r = r = 0.2; q in {0.4, 0.2, 0.1} in Fig. 3
    Illustrative values for a single-edge observation model; the paper notes it avoids inferring p and r.
  • artificial spike duration mean = 0.012 s (about 10 time steps)
    Introduced in Appendix I to convert spike timestamps to binary time series; affects all downstream brain-network inference.
  • MCMC proposal width sigma = 0.1
    Chosen in Appendix G for continuous parameter updates; a tuning choice that could affect mixing and posterior estimates.
assumptions (6)
  • standard math Data processing inequality and Fano's inequality
    Used to derive the reconstruction limit and the error lower bound in Secs. III B and III C.
  • domain assumption Existence of a unique true data-generating model M*
    Sec. II A assumes a unique generative process P(G*)P(X*|G*) produced the observations; the whole framework is defined relative to this object.
  • domain assumption Candidate model set contains a model close to M*
    Secs. IV C and V state that the index approximates Psi* only when the selected model minimizes evidence cross-entropy; this is necessary for the empirical method.
  • domain assumption Graphical Markov-chain likelihood factorization
    Eqs. (B1)-(B3) restrict dynamics to node-wise transition probabilities depending only on neighbor states, used in all synthetic and empirical experiments.
  • ad hoc to paper Variational mean-field posterior approximation bias is small
    Appendix F estimates mutual information and evidence via mean-field lower bounds; the bias is asserted to be small with a citation to the authors' prior work [36], not proven here.
  • ad hoc to paper Posterior edge-factorization for the loss relation
    Eq. (22) and Appendix D require the posterior to factor over edges (Eq. D3); the text says equality is achieved when this holds, making the linear loss relation approximate otherwise.
invented entities (1)
  • Reconstruction index psi_M
    purpose: Data-driven proxy for the reconstructability Psi*, intended to predict reconstruction error when the true graph is unknown.
    Defined in Eq. (29) as I_M/Lambda_M; its predictive validity is established in the paper's own simulations under correct-model assumptions, and it can fail under misspecification (Fig. 5).

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Cite this review

Pith. "Pith review of On the reconstruction limits of complex networks." pith.science (2026). https://pith.science/paper/JA4ZIS4S

@misc{pith2026250101437,
  author       = {Pith},
  title        = {Pith review of: On the reconstruction limits of complex networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JA4ZIS4S}},
  note         = {Machine review of arXiv:2501.01437}
}
read the original abstract

Network reconstruction consists in retrieving the hidden interaction structure of a system from observations. Many reconstruction algorithms have been proposed, although less research has been devoted to describe their theoretical limitations. In this work, we take a first-principles approach and build on our earlier definition of reconstructability-the fraction of structural information recoverable from data. We relate this quantity to the true data-generating (TDG) process and delineate an information-theoretic reconstruction limit, i.e., the upper bound of the mutual information between the true underlying graph and any graph reconstructed from observations. These concepts lead us to a principled numerical method to assess the validity of empirically reconstructed networks, based on model selection and a quantity we introduce: the reconstruction index. This index approximates the reconstructability from data, quantifies the variability of the reconstructed network ensemble, and is shown to predict reconstruction error without requiring knowledge of the true underlying network. We characterize this method and test it on empirical time series and networks.

Figures

Figures reproduced from arXiv: 2501.01437 by the authors.

Figure 1
Figure 1. Illustration of the network reconstruction context [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Performance comparison between the TDG model [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Posterior probability of a reconstructed edge: (a) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Comparison between reconstructability and different [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Effect of varying the coupling constant on the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Reconstruction from spontaneous neuronal activity in the mouse brain [ [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: Reconstruction indices of empirical graphs with different graph prior models: (top) SIS dynamics on the Zachary’s [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 9
Figure 9. Figure 9: Posterior predictive checks of the maximum evidence [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 8
Figure 8. Figure 8: Posterior of the maximum evidence model (SIS model [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

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Reference graph

Works this paper leans on

82 extracted references · 79 canonical work pages

  1. [1]

    Select a set of d reconstruction model candidates M = {M1, M2, ..., Md}

  2. [2]

    For each candidate M , sample a reconstructed graph ensemble ˆGM using the posterior of M

  3. [3]

    F for detail)

    Calculate for each candidate M the evidence ζM (x) and the reconstruction index ψM (x) using ˆGM (see Sec. F for detail)

    work page

  4. [4]

    (33) The index ψ ˆM (x) is the output of the procedure

    Choose the reconstruction index ψ ˆM (x) of the model ˆM with the highest evidence: ˆM = arg max M ∈M ζM (x) . (33) The index ψ ˆM (x) is the output of the procedure. Some remarks about this method are in order. Indeed, as shown in Sec. IV C, the validity of the reconstruc- tion index heavily relies on the model’s aptitude to rep- resent the data. Hence, ...

    work page

  5. [5]

    Generate a sample {ˆx1, ...,ˆxK} of synthetic data with ˆM , assuming its parameters (θ, ϕ) are sampled from the model’s posterior given x

    work page

  6. [6]

    Calculate some test quantities τ (ˆxk), i.e., statistics used for comparison, for each sample ˆxk, generating a set of samples T = {τ (ˆx1), ..., τ(ˆxK)}

    work page

  7. [7]

    If τ (x) is typical in T , then we can be sure ˆM is statis- tically similar to M ∗

    Compare τ (x) with T . If τ (x) is typical in T , then we can be sure ˆM is statis- tically similar to M ∗. 11 0 2 4 6 8 10 12 Time (s) 0 183 366 548 731 914 1096 1279 1462Neurons Probe 0 Probe 1 Probe 2 Probe 3 Probe 4 Probe 5 Probe 6 Probe 7 (a) 101 102 103 Number of edges ER 0 1 2 3 4 5 6 7 SIS 0 1 2 3 4 5 6 7 Cowan 0 1 2 3 4 5 6 7 Glauber UCM0 1 2 3 4...

    work page 2000

  8. [8]

    Erd˝ os-R´ enyi model The Erd˝ os-R´ enyi (ER) model corresponds to the maxi- mum entropy random graph model, i.e., the uniform dis- tribution over all simple graphs with N nodes and E edges, such that P (G|E = e) = N (N −1) 2 e −1 , (A1) where we recall that n k is the binomial coefficient. The ER model is also generalizable to loopy multigraphs, where P...

    work page

Show all 82 references
  1. [9]

    Configuration model The configuration model (CM) describes an ensem- ble of loopy multigraphs where the degree sequence is given [53]. From a network reconstruction perspective, the CM can also be used as a prior, assuming that its probability factors as follows P (G, k, E) = ...

  2. [10]

    However, unlike the ER model, each node i is associated with a random block bi ∈ {1, 2,

    Stochastic block model The stochastic block model (SBM), in its microcanon- ical version [54], closely resembles the ER model, where edges are picked uniformly at random. However, unlike the ER model, each node i is associated with a random block bi ∈ {1, 2, . . . , B} and ins...

  3. [11]

    Graph enumeration approach For sufficiently small random graphs ( N ≈ 5), the evidence probability can be computed by enumerating all graphs of G and by adding explicitly each term of Eq. (F1). Using the law of large numbers, we can esti- mate the mutual information I(G; X) ≃ ...

  4. [12]

    [36] which uses a varia- tional mean-field approximation to estimate the posterior probability instead of the evidence probability

    V ariational mean-field approximation This approach is based on Ref. [36] which uses a varia- tional mean-field approximation to estimate the posterior probability instead of the evidence probability. The vari- ational mean-field (MF) approximation assumes the con- ditional in...

  5. [13]

    Graph evidence estimation for the stochastic block model Using the stochastic block model (SBM) as the prior for our reconstruction model and for estimating the mu- tual information is challenging. Indeed, computing the graph entropy H(G) requires that we marginalize the parti...

  6. [14]

    Using the same approach as for Eq

    Evidence estimation for model selection The estimation of the evidence log probability relies on the previously discussed techniques for evaluating the posterior probability. Using the same approach as for Eq. (F8), we obtain log ζ(x) = EG|X=x[log P (X, G)] − H(G|X = x) , (F12...

  7. [15]

    Data preprocessing This dataset is composed of the spontaneous activity of the brains of three mice (Krebs, Robbins and Waks- man) monitored via eight neuropixel probes each. These probes record the time stamps of each spike of individual neurons in different regions of the br...

    2000

  8. [16]

    We consider a model parametrized by X, G and their parameters ϕ and θ, respectively

    Inference procedure The inference procedure is very similar to that pre- sented in Appendix G. We consider a model parametrized by X, G and their parameters ϕ and θ, respectively. How- ever, we have an additional limitation: We do not know the number of edges in the graph. We ...

  9. [17]

    Posterior inspection The graph and parameter marginal posteriors of the maximum evidence model are illustrated in Fig. 8. We also include a validation of the posterior on the inference data. Figure 9 shows the posterior predictive check vali- dation, which includes a predictio...

  10. [18]

    Network neuroscience,

    D. S. Bassett and O. Sporns, “Network neuroscience,” Nat. Neurosci 20, 353 (2017)

    2017

  11. [19]

    On the nature and use of models in network neuroscience,

    D. S Bassett, P. Zurn, and J. I. Gold, “On the nature and use of models in network neuroscience,” Nat. Rev. Neurosci. 19, 566 (2018)

    2018

  12. [20]

    The physics of brain network structure, function and control,

    C. W. Lynn and D. S Bassett, “The physics of brain network structure, function and control,” Nat. Rev. Phys. 1, 318 (2019)

    2019

  13. [21]

    Structure and function of complex brain net- works,

    O. Sporns, “Structure and function of complex brain net- works,” Dialogues Clin. Neurosci. 15, 247–262 (2013)

    2013

  14. [22]

    Network analysis of intrinsic functional brain connectivity in alzheimer’s disease,

    K. Supekar, V. Menon, D. Rubin, M. Musen, and M. D. Greicius, “Network analysis of intrinsic functional brain connectivity in alzheimer’s disease,” PLOS Com- put. Biol. 4, e1000100 (2008)

    2008

  15. [23]

    Using computational models to relate structural and functional brain connectivity,

    J. Hlinka and S. Coombes, “Using computational models to relate structural and functional brain connectivity,” Eur. J. Neurosci. 36, 2137 (2012)

    2012

  16. [24]

    The role of node dynam- ics in shaping emergent functional connectivity patterns in the brain,

    M. Forrester, J. J. Crofts, S. N. Sotiropoulos, S. Coombes, and R. D. O’Dea, “The role of node dynam- ics in shaping emergent functional connectivity patterns in the brain,” Netw. Neurosci. 4, 467 (2020)

    2020

  17. [25]

    Inferring gene regulatory networks from multiple mi- croarray datasets,

    Y. Wang, T. Joshi, X.-S. Zhang, D. Xu, and L. Chen, “Inferring gene regulatory networks from multiple mi- croarray datasets,” Bioinformatics 22, 2413–2420 (2006)

    2006

  18. [26]

    Network reconstruction and community detection from dynamics,

    T. P. Peixoto, “Network reconstruction and community detection from dynamics,” Phys. Rev. Lett. 123, 128301 (2019)

    2019

  19. [27]

    Network-inference-based prediction of the COVID-19 epidemic outbreak in the Chinese province Hubei,

    B. Prasse, M. A. Achterberg, L. Ma, and P. Van Mieghem, “Network-inference-based prediction of the COVID-19 epidemic outbreak in the Chinese province Hubei,” Appl. Netw. Sci. 5, 35 (2020)

    2020

  20. [28]

    Bootstrapping topological properties and systemic risk of complex networks using the fitness model,

    N. Musmeci, S. Battiston, G. Caldarelli, M. Puliga, and A. Gabrielli, “Bootstrapping topological properties and systemic risk of complex networks using the fitness model,” J. Stat. Phys. 151, 720–734 (2013)

    2013

  21. [29]

    Network structure from rich but noisy data,

    M. E. J. Newman, “Network structure from rich but noisy data,” Nat. Phys. 14, 542–545 (2018)

    2018

  22. [30]

    Reconstructing networks with unknown and heterogeneous errors,

    T. P. Peixoto, “Reconstructing networks with unknown and heterogeneous errors,” Phys. Rev. X 8, 041011 (2018)

    2018

  23. [31]

    Bayesian inference of network structure from unreliable data,

    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

  24. [32]

    Statistical inference links data and theory in network science,

    L. Peel, T. P. Peixoto, and M. De Domenico, “Statistical inference links data and theory in network science,” Nat. Commun. 13, 1–15 (2022)

    2022

  25. [33]

    Net- work structure inference, a survey: Motivations, meth- ods, and applications,

    I. Brugere, B. Gallagher, and T. Y. Berger-Wolf, “Net- work structure inference, a survey: Motivations, meth- ods, and applications,” ACM Comput. Surv. 51, 1–39 (2018)

    2018

  26. [34]

    Eichler, Causal Inference in Time Series Analysis (Wiley Online Library, 2012)

    M. Eichler, Causal Inference in Time Series Analysis (Wiley Online Library, 2012)

    2012

  27. [35]

    Network inference with confidence from multi- variate time series,

    M. A. Kramer, U. T. Eden, S. S. Cash, and E. D. Ko- laczyk, “Network inference with confidence from multi- variate time series,” Phys. Rev. E 79, 061916 (2009)

    2009

  28. [36]

    Causal connectivity of evolved neural net- works during behavior,

    A. K. Seth, “Causal connectivity of evolved neural net- works during behavior,” Netw. Comput. Neural Syst. 16, 35–54 (2005)

    2005

  29. [37]

    Measuring Information Transfer,

    T. Schreiber, “Measuring Information Transfer,” Phys. Rev. Lett. 85, 461–464 (2000)

    2000

  30. [38]

    Learning factor graphs in polynomial time and sample complexity,

    P. Abbeel, D. Koller, and A. Y. Ng, “Learning factor graphs in polynomial time and sample complexity,” J. Mach. Learn. Res. 7, 1743–1788 (2006)

    2006

  31. [39]

    Which graphical models are difficult to learn?

    J. Bento and A. Montanari, “Which graphical models are difficult to learn?” in Advances in neural information processing systems (2009) pp. 1303–1311

    2009

  32. [40]

    Reconstruction of markov random fields from samples: some observations and algorithms,

    G. Bresler, E. Mossel, and A. Sly, “Reconstruction of markov random fields from samples: some observations and algorithms,” SIAM J. Comput. 42, 563–578 (2013)

    2013

  33. [41]

    Quantum Boltzmann Machine,

    M. H. Amin, E. Andriyash, J. Rolfe, B. Kulchytskyy, and R. Melko, “Quantum Boltzmann Machine,” Phys. Rev. 24 X 8, 021050 (2018)

    2018

  34. [42]

    Neural relational inference for interacting systems,

    T. Kipf, E. Fetaya, K.-C. Wang, M. Welling, and R. Zemel, “Neural relational inference for interacting systems,” in Proceedings of the 35th International Con- ference on Machine Learning , Proceedings of Machine Learning Research, Vol. 80 (PMLR, 2018) pp. 2688–2697

    2018

  35. [43]

    Hier- archical structure and the prediction of missing links in networks,

    A. Clauset, C. Moore, and M. E. J. Newman, “Hier- archical structure and the prediction of missing links in networks,” Nature 453, 98–101 (2008)

    2008

  36. [44]

    Missing and spuri- ous interactions and the reconstruction of complex net- works,

    R. Guimer` a and M. Sales-Pardo, “Missing and spuri- ous interactions and the reconstruction of complex net- works,” Proc. Natl. Acad. Sci. U.S.A. 106, 22073–22078 (2009)

    2009

  37. [45]

    Link prediction in complex net- works: A survey,

    L. L¨ u and T. Zhou, “Link prediction in complex net- works: A survey,” Physica A Stat. 390, 1150–1170 (2011)

    2011

  38. [46]

    Assessing experimen- tally derived interactions in a small world,

    D. S. Goldberg and F. P´ e Roth, “Assessing experimen- tally derived interactions in a small world,” Proc. Natl. Acad. Sci. U.S.A. 100, 4372–4376 (2003)

    2003

  39. [47]

    Reconstruction of plant–pollinator networks from obser- vational data,

    J.-G. Young, F. S. Valdovinos, and M. E. J. Newman, “Reconstruction of plant–pollinator networks from obser- vational data,” Nat. Commun. 12, 3911 (2021)

    2021

  40. [48]

    Hypergraph re- construction from uncertain pairwise observations,

    S. Lizotte, J.-G. Young, and A. Allard, “Hypergraph re- construction from uncertain pairwise observations,” Sci. Rep. 13, 21364 (2023)

    2023

  41. [49]

    Predicting network dynamics without requiring the knowledge of the in- teraction graph,

    B. Prasse and P. Van Mieghem, “Predicting network dynamics without requiring the knowledge of the in- teraction graph,” Proc. Natl. Acad. Sci. U.S.A. 119, e2205517119 (2022)

    2022

  42. [50]

    Fundamental limitations of network re- construction from temporal data,

    M. T. Angulo, J. A. Moreno, G. Lippner, A.-L. Barab´ asi, and Y.-Y. Liu, “Fundamental limitations of network re- construction from temporal data,” J. R. Soc. Interface 14, 20160966 (2017)

    2017

  43. [51]

    Similar net- work activity from disparate circuit parameters,

    A. A. Prinz, D. Bucher, and E. Marder, “Similar net- work activity from disparate circuit parameters,” Nat. Neurosci. 7, 1345–1352 (2004)

    2004

  44. [52]

    Conse- quences of degeneracy in network function,

    E. C. Cropper, A. M. Dacks, and K. R. Weiss, “Conse- quences of degeneracy in network function,” Curr. Opin. Neurobiol. 41, 62–67 (2016)

    2016

  45. [53]

    Duality between predictability and reconstructability in complex systems,

    C. Murphy, V. Thibeault, A. Allard, and P. Desrosiers, “Duality between predictability and reconstructability in complex systems,” Nat. Commun. 15, 4478 (2024)

    2024

  46. [54]

    Network reconstruction via the minimum description length principle,

    T. P. Peixoto, “Network reconstruction via the minimum description length principle,” (2024), arXiv:2405.01015

    2024 arXiv

  47. [55]

    Im- proved mutual information measure for clustering, clas- sification, and community detection,

    M. E. J. Newman, G. T. Cantwell, and J.-G. Young, “Im- proved mutual information measure for clustering, clas- sification, and community detection,” Phys. Rev. E 101, 042304 (2020)

    2020

  48. [56]

    Mu- tual information and the encoding of contingency tables,

    M. Jerdee, A. Kirkley, and M. E. J. Newman, “Mu- tual information and the encoding of contingency tables,” Phys. Rev. E 110, 064306 (2024)

    2024

  49. [57]

    T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. (Wiley-Interscience, 2006)

    2006

  50. [58]

    M., Machine Learning (McGraw-Hill, 1997)

    Mitchell, T. M., Machine Learning (McGraw-Hill, 1997)

    1997

  51. [59]

    Bayes factors,

    R. E. Kass and A. E. Raftery, “Bayes factors,” J. Am. Stat. Assoc. 90, 773–795 (1995)

    1995

  52. [60]

    Eight-probe Neuropixels recordings dur- ing spontaneous behaviors,

    N. Steinmetz, M. Pachitariu, C. Stringer, M. Carandini, and K. Harris, “Eight-probe Neuropixels recordings dur- ing spontaneous behaviors,” (2019)

    2019

  53. [61]

    Spontaneous behaviors drive multidimensional, brainwide activity,

    C. Stringer, M. Pachitariu, N. Steinmetz, C. Bai Reddy, M. Carandini, and K. D. Harris, “Spontaneous behaviors drive multidimensional, brainwide activity,” Science364, eaav7893 (2019)

    2019

  54. [62]

    Reconstruction of neocortex: Organelles, com- partments, cells, circuits, and activity,

    Nicholas L Turner, Thomas Macrina, J Alexander Bae, Runzhe Yang, Alyssa M Wilson, Casey Schneider-Mizell, Kisuk Lee, Ran Lu, Jingpeng Wu, Agnes L Bodor, et al. , “Reconstruction of neocortex: Organelles, com- partments, cells, circuits, and activity,” Cell 185, 1082– 1100 (2022)

    2022

  55. [63]

    An information flow model for conflict and fission in small groups,

    W. W. Zachary, “An information flow model for conflict and fission in small groups,” J. Anthropol. Res. 33, 452– 473 (1977)

    1977

  56. [64]

    The political books network,

    V. Krebs, “The political books network,” (2004), online; accessed October 28 2024

    2004

  57. [65]

    Inference and phase transitions in the detection of mod- ules in sparse networks,

    A. Decelle, F. Krzakala, C. Moore, and L. Zdeborov´ a, “Inference and phase transitions in the detection of mod- ules in sparse networks,” Phys. Rev. Lett. 107, 065701 (2011)

    2011

  58. [66]

    Detectability thresholds and optimal algorithms for community structure in dynamic networks,

    A. Ghasemian, P. Zhang, A. Clauset, C. Moore, and L. Peel, “Detectability thresholds and optimal algorithms for community structure in dynamic networks,” Phys. Rev. X 6, 031005 (2016)

    2016

  59. [67]

    Finite-size analysis of the de- tectability limit of the stochastic block model,

    J.-G. Young, P. Desrosiers, L. H´ ebert-Dufresne, E. Lau- rence, and L. J. Dub´ e, “Finite-size analysis of the de- tectability limit of the stochastic block model,” Phys. Rev. E 95, 062304 (2017)

    2017

  60. [68]

    Development of structure–function coupling in human brain networks during youth,

    G. L. Baum, D. R. Cui, Z.and Roalf, R. Ciric, R.F . Betzel, B. Larsen, M. Cieslak, P. A. Cook, C. H Xia, T. M Moore, et al. , “Development of structure–function coupling in human brain networks during youth,” Proc. Natl. Acad. Sci. U. S. A. 117, 771–778 (2020)

    2020

  61. [69]

    Structural and genetic determinants of zebrafish functional brain net- works,

    A. Legare, M. Lemieux, V. Boily, S. Poulin, A. Legare, P. Desrosiers, and P. De Koninck, “Structural and genetic determinants of zebrafish functional brain net- works,” bioRxiv , 2024–12 (2024)

    2024

  62. [70]

    Configuring random graph models with fixed degree sequences,

    B. K. Fosdick, D. B. Larremore, J. Nishimura, and J. Ugander, “Configuring random graph models with fixed degree sequences,” SIAM Rev. 60, 315–355 (2018)

    2018

  63. [71]

    Nonparametric bayesian inference of the microcanonical stochastic block model,

    T. P. Peixoto, “Nonparametric bayesian inference of the microcanonical stochastic block model,” Phys. Rev. E95, 012317 (2017)

    2017

  64. [72]

    Edwards, Introduction to Graphical Modelling (Springer Science & Business Media, 2012)

    D. Edwards, Introduction to Graphical Modelling (Springer Science & Business Media, 2012)

    2012

  65. [73]

    Time-Dependent Statistics of the Ising Model,

    R. J. Glauber, “Time-Dependent Statistics of the Ising Model,” J. Math. Phys. 4, 294–307 (1963)

    1963

  66. [74]

    Epidemic processes in complex net- works,

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

    2015

  67. [75]

    A model for spatial conflict,

    P. Clifford and A. Sudbury, “A model for spatial conflict,” Biometrika 60, 581–588 (1973), https://academic.oup.com/biomet/article- pdf/60/3/581/576759/60-3-581.pdf

    1973

  68. [76]

    Stochastic neurodynamics,

    J. D. Cowan, “Stochastic neurodynamics,” in Advances in Neural Information Processing Systems , Vol. 3 (1990) p. 62

    1990

  69. [77]

    R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and control (Oxford university press, 1992)

    1992

  70. [78]

    Beyond Wilson-Cowan dynamics: oscillations and chaos without inhibition,

    V. Painchaud, N. Doyon, and P. Desrosiers, “Beyond Wilson-Cowan dynamics: oscillations and chaos without inhibition,” Biol. Cybern. 116, in press (2022)

    2022

  71. [79]

    Network inference from functional experimental data (Conference Presen- tation),

    P. Desrosiers, S. Labrecque, M. Tremblay, M. B´ elanger, B. De Dorlodot, and D. C. Cˆ ot´ e, “Network inference from functional experimental data (Conference Presen- tation),” in Clinical and Translational Neurophotonics; Neural Imaging and Sensing; and Optogenetics and Op- ti...

    2016

  72. [80]

    netrd: A library for network re- construction and graph distances,

    S. McCabe, L. Torres, T. LaRock, S. Haque, C.-H. Yang, H. Hartle, and B. Klein, “netrd: A library for network re- construction and graph distances,” J. Open Source Softw. 6, 2990 (2021)

    2021

  73. [81]

    Revealing consensus and dissensus be- tween network partitions,

    T. P. Peixoto, “Revealing consensus and dissensus be- tween network partitions,” Phys. Rev. X 11, 021003 (2021)

    2021

  74. [82]

    Markov Chain Monte Carlo sampling of graphs,

    A.C.C. Coolen, A. Annibale, and E.S. Roberts, “Markov Chain Monte Carlo sampling of graphs,” in Generating Random Networks and Graphs (Oxford University Press, 2017)

    2017

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.