Pith. sign in

REVIEW 2 major objections 7 minor 295 references

Triangle counts crack sparse-network parameter estimation

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · glm-5.2

2026-07-07 20:09 UTC pith:PJDPPNQ5

load-bearing objection Cycle-counting estimator for β-model parameters in sparse networks: novel construction, real contribution, but the minimax claim is overstated relative to what is proven. the 2 major comments →

arxiv 2607.05273 v1 pith:PJDPPNQ5 submitted 2026-07-06 stat.ME

Subgraph counting estimation for the β-model in sparse networks

Qunqiang Feng , Jiashun Jin , Yaru Tian , Ting Yan This is my paper
classification stat.ME
keywords beta-modelsparse networkssubgraph countingcycle counting ratio estimatorminimax estimationdegree heterogeneityasymptotic normalitynetwork density
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proposes the Cycle Counting Ratio (CCR) estimator for the beta-model of undirected networks. The beta-model assigns each node a degree-heterogeneity parameter that controls its propensity to form edges, and the link probability between two nodes is a logistic function of the sum of their parameters. The CCR estimator works by counting two specific types of 3-node subgraphs (triangles) involving the target node -- one where the node has two edges and one non-edge to its neighbors, and one where it has two non-edges and one edge -- and taking half the log of their ratio. This ratio isolates the target node's parameter because the contributions from all other nodes cancel exactly. The paper proves that under the conditions that the maximum link parameter goes to zero (ensuring sparsity) and the product of a node's parameter with the sum of all parameters diverges, the CCR estimator is consistent and achieves the minimax mean-squared-error rate of 1/(theta_t times the L1 norm of the parameter vector). This rate persists even when overall network density approaches the Erdos-Renyi connectivity threshold of log n / n. The authors also establish asymptotic normality with variance matching that of the maximum likelihood estimator, and uniform consistency across all nodes under a slightly stronger condition.

Core claim

The central mechanism is that the ratio of expected counts of two complementary triangle types around a node equals exp(2*beta_t), so beta_t can be recovered by a simple log-ratio of observable subgraph counts. This converts a high-dimensional likelihood problem into n independent counting problems, each solvable in time proportional to the maximum degree. The minimax rate 1/(theta_t ||theta||_1) is established by matching an upper bound on the CCR estimator's MSE against an information-theoretic lower bound obtained via a two-point testing argument, where the chi-squared distance between distributions differing only in one node's parameter goes to zero precisely when the perturbation is on,

What carries the argument

The estimator is half the log of the ratio of T_{n,t}(a) = sum of A_{ti} B_{ij} A_{jt} to T_{n,t}(b) = sum of B_{ti} A_{ij} B_{jt}, where A is the adjacency matrix and B records non-edges. By Lemma 1, these sums equal diagonal entries of ABA and BAB respectively, enabling matrix-based computation. A threshold at log n prevents infinite values when counts are zero. The signal-to-noise ratio is shown to be sqrt(theta_t ||theta||_1), making the squared SNR the inverse of the minimax rate.

Load-bearing premise

The upper bound on the estimator's error requires a condition that is log-squared-n times more restrictive than the condition needed for the matching lower bound, so the minimax optimality is established over a parameter space that requires both conditions to hold simultaneously. Whether the minimax rate holds under the weaker condition alone remains open.

What would settle it

If networks with density near log n / n and heterogeneous degree parameters were generated and the CCR estimator systematically failed to concentrate (e.g., MSE not scaling as 1/(theta_t ||theta||_1)), the main theoretical claims would be refuted. Additionally, if the threshold choice at log n were shown to introduce non-vanishing bias for specific parameter configurations, the consistency results would not hold as stated.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Networks near the Erdos-Renyi connectivity threshold log n / n -- where the MLE frequently fails to exist -- can still yield per-node parameter estimates with provable optimality guarantees.
  • The explicit closed-form estimator avoids iterative optimization, making it scalable to networks with millions of nodes provided the maximum degree is not too large.
  • The asymptotic variance matching the MLE suggests no statistical efficiency is lost by abandoning likelihood-based methods in sparse regimes.
  • The testing framework based on pairwise parameter differences enables hypothesis tests about degree heterogeneity structure even in very sparse networks.

Where Pith is reading between the lines

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

  • The cycle-counting ratio approach could extend to directed networks or models with reciprocity parameters, since the cancellation mechanism relies only on the additive structure of parameters in the link probability.
  • The gap between the condition needed for the upper bound (theta_t ||theta||_1 / (log n)^2 -> infinity) and the lower bound (theta_t ||theta||_1 -> infinity) suggests that a sharper concentration argument for the subgraph counts could close this log-squared gap.
  • If the asymptotic independence of estimators extends to growing dimensions, simultaneous confidence bands for the full parameter vector could be constructed, enabling global goodness-of-fit testing for the beta-model.
  • The choice of 3-cycles over longer cycles trades statistical efficiency (negligible improvement per simulations) for computational simplicity, but in regimes where 3-cycle counts are frequently zero, longer cycles might provide a fallback with weaker but non-degenerate estimates.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 7 minor

Summary. This paper proposes a Cycle Counting Ratio (CCR) estimator for node-specific parameters in the β-model of undirected networks, targeting moderate to extremely sparse regimes. The estimator is based on the log-ratio of two 3-cycle counting statistics and has a closed-form expression, avoiding the iterative algorithms and non-existence issues of the MLE in sparse settings. The authors establish a signal-to-noise ratio (SNR) of θ_t‖θ‖_1, derive an MSE upper bound (Theorem 2), a matching minimax lower bound via the two-point χ² method (Theorems 3–4), and a CLT (Theorem 5). The central claim is that the CCR estimator achieves the minimax MSE rate 1/(θ_t‖θ‖_1) under the conditions θ_max → 0 and θ_t‖θ‖_1 → ∞, which are weaker than those in prior work.

Significance. The paper addresses an important open problem: estimation in the β-model under weak conditions in sparse networks, without requiring the structured parameter assumptions of Chen et al. (2021) or Shao et al. (2023). The closed-form, computationally scalable estimator is a practical strength. The minimax lower bound via the two-point method and the matching upper bound constitute a genuine theoretical contribution. The CLT and variance estimator (Theorems 5–6) enable inference. Simulations and a real-data application illustrate the method's viability. The gap between the upper and lower bound conditions (see major comments) does not negate the contribution but does affect how the central claim should be stated.

major comments (2)
  1. The abstract states: 'Under the very weak conditions that max_t θ_t → 0 and θ_t‖θ‖_1 → ∞, we show that the CCR estimator is consistent and achieves the minimax rate.' However, the MSE upper bound (Theorem 2, §3.2) explicitly requires the stronger condition (15): θ_t‖θ‖_1/(log n)² → ∞. The minimax optimality (Theorem 4) is established over Θ(ε_n) ⊂ Θ_0, where Θ_0 includes the requirement θ_t‖θ‖_1/(log n)² ≥ (1/2)log(log n). Thus the parameter space over which minimax optimality is proven already embeds the stronger condition. The lower bound (Theorem 3) only needs condition (7). This creates a genuine gap: the lower bound holds under weaker conditions than the upper bound. The CLT (Theorem 5) does hold under condition (7) alone, but convergence in distribution does not directly imply the MSE bound without a uniform integrability argument, which is not provided. The claim that the minimax率
  2. The relationship between Theorem 5 (CLT under condition (7)) and Theorem 2 (MSE bound under condition (15)) should be clarified. If the authors can provide a uniform integrability argument to bridge the CLT to the MSE bound under condition (7) alone, the gap would be closed. If not, the abstract and Theorem 4 statement should be revised to accurately reflect that the MSE rate is proven under condition (15), not condition (7). The current phrasing 'Under a slight stronger condition' in the abstract for asymptotic normality suggests the authors are aware of the distinction, but the minimax claim does not make the same distinction clear. The parameter space Θ_0 in §3.2 should be discussed more transparently in relation to condition (7), not only in Section 3.2.
minor comments (7)
  1. In the definition of Θ_0 (§3.2), the term log(log n) is said to be chosen 'only for convenience and can be replaced by other diverging sequences.' A brief remark on how sensitive the results are to this choice would help the reader.
  2. The threshold estimator (3) truncates at log n. The sensitivity of the theoretical results to this specific threshold is not rigorously bounded. A brief discussion of whether other thresholds (e.g., C log n for some constant C) would yield the same asymptotic properties would improve clarity.
  3. Table 1 caption mentions 'based on 100 generated networks in each simulation,' but the text in §4.1 states 'Each simulation is repeated 500 times.' Please reconcile.
  4. In Section 4.2, the phrase 'the leading estimator is meaningfulness' appears to be a grammatical error; consider revising to 'the leading estimator is not meaningful' or similar.
  5. The reference to 'Feng et al. (2026)' in the introduction and reference list appears to be a future-dated preprint (arXiv:2601.01325). Confirming the correct year and providing complete bibliographic details would be helpful.
  6. Remark 2 states that the asymptotic variance of the CCR estimator matches that of the MLE. This is an interesting point; a brief discussion of whether this holds only under θ_max → 0 or under broader conditions would strengthen the remark.
  7. The condition θ_t‖θ‖_1 → ∞ in (7) is described as 'necessary' in §3.1. A one-sentence justification (e.g., SNR → 0 otherwise) is given, but making this more explicit would help readers less familiar with the signal-to-noise framework.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies a gap between the conditions under which our lower and upper bounds hold, and between the CLT and the MSE bound. We address both major comments below and commit to revising the manuscript accordingly.

read point-by-point responses

  1. Referee: The abstract states the minimax rate is achieved under conditions (7) alone, but Theorem 2 (upper bound) requires the stronger condition (15), and the parameter space Θ_0 for Theorem 4 embeds condition (15). The lower bound (Theorem 3) only needs condition (7). The CLT (Theorem 5) holds under (7) but does not directly imply the MSE bound without a uniform integrability argument. The minimax claim should be revised.

    Authors: The referee is correct on all counts. There is a genuine gap between the conditions for the lower bound (condition (7)) and the upper bound (condition (15)). We have carefully re-examined whether a uniform integrability argument can bridge the CLT (Theorem 5, under condition (7)) to the MSE bound under condition (7) alone. While the CLT gives convergence in distribution at the correct rate, we have not been able to establish the uniform integrability of the squared studentized statistic under condition (7) alone; the technical difficulty is that the tail behavior of the log-ratio statistic depends on the probability that the cycle counts T_{n,t}(a) or T_{n,t}(b) are very small, and controlling this requires the stronger condition (15). We will therefore revise the abstract, the introduction, and the statement of Theorem 4 to accurately reflect that the MSE rate 1/(θ_t‖θ‖_1) is proven under condition (15), not condition (7). The lower bound (Theorem 3) holds under condition (7), and the CLT (Theorem 5) holds under condition (7), but the matching upper bound requires the stronger condition. We will state this distinction clearly throughout the paper. revision: yes

  2. Referee: The relationship between Theorem 5 (CLT under condition (7)) and Theorem 2 (MSE bound under condition (15)) should be clarified. The parameter space Θ_0 should be discussed more transparently in relation to condition (7).

    Authors: We agree that the relationship between these results and the role of Θ_0 need to be stated more transparently. In the revision, we will: (1) add an explicit remark explaining that Theorem 5 (CLT) holds under condition (7) alone, while Theorem 2 (MSE upper bound) requires the stronger condition (15), and that the gap reflects the absence of a uniform integrability argument; (2) clarify in Section 3.2 that Θ_0 embeds condition (15) via the requirement θ_t‖θ‖_1/(log n)^2 ≥ (1/2)log(log n), and that this is stronger than condition (7); (3) add a sentence noting that if one neglects the log n factors, Θ_0 essentially corresponds to condition (7), but that the log n factors are essential for the upper bound proof. We will also adjust the abstract's phrasing to make the distinction between the consistency/minimax result (under condition (15)) and the CLT (under condition (7)) unambiguous. revision: yes

Circularity Check

0 steps flagged

No circularity found: the CCR estimator is derived from a structural model identity (Proposition 1), and the minimax lower bound uses an external method (Donoho and Liu, 1991).

full rationale

The paper's derivation chain is self-contained and does not exhibit circularity. The CCR estimator (Eq. 4) is motivated by Proposition 1, which states that the ratio of expected cycle counts g_{t,m}(a)/g_{t,m}(b) equals exp(c*beta_t) purely as a consequence of the beta-model's probability structure (Eq. 1). This is a structural identity, not a fitted input renamed as a prediction. The minimax lower bound (Theorem 3) is derived using the standard two-point chi-squared distance method attributed to Donoho and Liu (1991), an external reference. The upper bound (Theorem 2) is derived from variance calculations (Theorem 1) on the cycle counting statistics. The parameter space Theta(epsilon_n) over which minimax optimality (Theorem 4) is established embeds the stronger condition (15), which is a restriction on the scope of the optimality claim rather than a circular definition. While the skeptic correctly identifies a gap between the abstract's advertised conditions and the theorem's actual requirements, this is a correctness/overstatement concern, not circularity. No step in the derivation reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

No new entities, particles, forces, or dimensions are introduced. The estimator is a function of observable network data (adjacency matrix entries).

free parameters (1)
  • Truncation threshold log(n) = log(n)
    The threshold estimator (3) truncates β̂_t at ±log(n). This is chosen heuristically: 'Since such graphs obey far away from real-world networks, we use the threshold |β_t| < log(n).' Not fitted to data but chosen by argument about realistic network density.
axioms (4)
  • domain assumption Edges {A_ij, 1≤i<j≤n} are independent Bernoulli random variables with P(A_ij=1) = e^{β_i+β_j}/(1+e^{β_i+β_j})
    Equation (1), §2. This is the β-model specification. All results depend on this independence assumption, which is standard in the β-model literature but strong for real-world networks.
  • domain assumption θ_max → 0 (network sparsity)
    Condition (7), §3. Ensures the network is sparse, consistent with observed real-world networks.
  • domain assumption θ_t‖θ‖₁ → ∞ (sufficient signal-to-noise ratio)
    Condition (7), §3. Necessary for consistency; the paper shows SNR² = (1+o(1))θ_t‖θ‖₁, so this condition is equivalent to SNR → ∞.
  • standard math Two-point testing method for minimax lower bounds (Donoho and Liu, 1991)
    §3.2, Theorem 3. Standard technique: if χ²-distance between null and alternative goes to zero, the minimax MSE is bounded below by Cδ².

pith-pipeline@v1.1.0-glm · 19069 in / 3540 out tokens · 73510 ms · 2026-07-07T20:09:28.425748+00:00 · methodology

0 comments
read the original abstract

The $\beta$-model is popular for characterizing the commonly observed degree heterogeneity phenomenon in real-world networks. In this study, we develop a cycle counting approach to estimate $n$ node-specific parameters in the $\beta$-model for moderate or extremely sparse networks. Our proposed estimators, called \emph{Cycle Counting Ratio (CCR) Estimator}, are based on the log-ratios of two network cycle counting statistics with explicit expressions and therefore easy to compute. We focus on conditions to guarantee statistical properties of the single estimator for each node. Under the very weak conditions that $\max_t \theta_t \to 0$ and $\theta_t \|\theta\|_1 \to \infty$, we show that the CCR estimator is consistent and achieves the minimax rate in terms of the mean squared error, which is the squared signal-to-noise ratio for $\hat{\beta}_t$ up to a constant factor. Here, $\hat{\beta}_t$ is the CCR estimator of the node-specific parameter $\beta_t$, $\theta_t = \exp(\beta_t)$ and $\theta=(\theta_1, \ldots, \theta_n)$. Even if the whole network density is close to the Erd\H{o}s-R\'{e}nyi lower bound $\log n/n$, the CCR estimator for the single parameter $\beta_t$ is still consistent as long as $\theta_t \|\theta\|_1 \to \infty$. To the best of our knowledge, this is the first time to derive the minimax rate and consistency result under such weak conditions. Under a slight stronger condition, we further establish its uniform consistency and asymptotic normality, whose asymptotic variance is $\theta_t \|\theta\|_1$. Numerical studies and an application to a sparse network data set demonstrate our theoretical findings.

Figures

Figures reproduced from arXiv: 2607.05273 by Jiashun Jin, Qunqiang Feng, Ting Yan, Yaru Tian.

Figure 1
Figure 1. Figure 1: All non-isomorphic cycles with 3 nodes. where Ft,m = {(i1, i2, . . . , im) : i1, i2, . . . , im are distinct and ij ∈ [n]\{t} for j = 1, . . . , m}. Let gt,m(a) = E[Gt,m(a)]. Because At,i1 , Ai1,i2 , · · · , Aim,t are independent when (i1, i2, . . . , im) ∈ Ft,m, we have gt,m(a) = X (i1,i2,...,im)∈Fm 1 (1 + e βt+βi1 )(1 + e βi1+βi2 )· · ·(1 + e βim−1+βim )(1 + e βim+βt ) · exp  βt(a1 + am+1) + βi1 (a1 + a… view at source ↗
Figure 2
Figure 2. Figure 2: QQ-plots of (βˆ i − βˆ j − (βi − βj ))/σˆij (n = 1000). where ˆσ 2 i is defined in (17). We show the QQ plots in case of n = 1000 in [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Histogram of the βˆ, with the red color denoting the density estimator. Next, we test the equality of several pairs of parameters. We use the test statistic in (18) to test the null H0 : βi = βj against the alternative H1 : βi ̸= βj . At level α = 0.05, the hypothesis can be rejected if | ˆξij | > z1−α/2, where zτ denotes the τ -quantile of the standard normality. We random select 10 nodes from the set {1,… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

295 extracted references · 295 canonical work pages · 12 internal anchors

  1. [1]

    IEEE Transactions on Information Theory , volume=

    Degree heterogeneity in higher-order networks: Inference in the hypergraph -model , author=. IEEE Transactions on Information Theory , volume=. 2024 , publisher=

    work page 2024

  2. [2]

    Journal of the American Statistical Association , volume=

    Likelihood ratio tests in random graph models with increasing dimensions , author=. Journal of the American Statistical Association , volume=. 2025 , publisher=

    work page 2025

  3. [3]

    Scandinavian Journal of Statistics , volume =

    Du, Yuqing and Qu, Lianqiang and Yan, Ting and Zhang, Yuan , title =. Scandinavian Journal of Statistics , volume =. doi:https://doi.org/10.1111/sjos.12650 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1111/sjos.12650 , abstract =

    work page doi:10.1111/sjos.12650

  4. [4]

    The Annals of Statistics , volume=

    A two-way heterogeneity model for dynamic networks , author=. The Annals of Statistics , volume=. 2025 , publisher=

    work page 2025

  5. [5]

    Electronic Journal of Statistics , number =

    Jiayi Deng and Danyang Huang and Xiangyu Chang and Bo Zhang , title =. Electronic Journal of Statistics , number =. 2024 , doi =

    work page 2024

  6. [6]

    On the evolution of random graphs , author=. Publ. Math. Inst. Hungar. Acad. Sci , volume=. 1960 , publisher=

    work page 1960

  7. [7]

    LH-moment-based regional flood frequency analysis framework to determine design floods in Krishna River basin , journal =

    Amit Kumar Singh and Sagar Rohidas Chavan , keywords =. LH-moment-based regional flood frequency analysis framework to determine design floods in Krishna River basin , journal =. 2025 , issn =. doi:https://doi.org/10.1016/j.ejrh.2025.102282 , url =

    work page doi:10.1016/j.ejrh.2025.102282 2025

  8. [8]

    Estimation of multiple networks with common structures in heterogeneous subgroups , journal =

    Xing Qin and Jianhua Hu and Shuangge Ma and Mengyun Wu , keywords =. Estimation of multiple networks with common structures in heterogeneous subgroups , journal =. 2024 , issn =. doi:https://doi.org/10.1016/j.jmva.2024.105298 , url =

    work page doi:10.1016/j.jmva.2024.105298 2024

  9. [9]

    Journal of the American Statistical Association , volume=

    A sparse beta regression model for network analysis , author=. Journal of the American Statistical Association , volume=. 2025 , publisher=

    work page 2025

  10. [10]

    Learning to discover social circles in ego networks , volume =

    Mcauley, Jim and Leskovec, Jure , year =. Learning to discover social circles in ego networks , volume =

    work page

  11. [11]

    Internet Mathematics , doi=

    Concentration Inequalities and Martingale Inequalities: A Survey , volume=3, author=. Internet Mathematics , doi=

    work page

  12. [12]

    2000 , publisher=

    Asymptotic statistics , author=. 2000 , publisher=

    work page 2000

  13. [13]

    The -Model Maximum Likelihood, Cram

    Wahlstr. The -Model Maximum Likelihood, Cram. IEEE Transactions on Signal Processing , volume=. 2017 , publisher=

    work page 2017

  14. [14]

    Journal of Business & Economic Statistics , volume=

    Group Sparse -Model for Network , author=. Journal of Business & Economic Statistics , volume=. 2025 , publisher=

    work page 2025

  15. [15]

    Journal of the American Statistical Association , volume=

    An exponential family of probability distributions for directed graphs: Comment , author=. Journal of the American Statistical Association , volume=. 1981 , publisher=

    work page 1981

  16. [16]

    Aids , volume=

    Sexual network structure and the spread of HIV in Africa: evidence from Likoma Island, Malawi , author=. Aids , volume=. 2007 , publisher=

    work page 2007

  17. [17]

    Social Networks , volume=

    Closure, connectivity and degree distributions: Exponential random graph (p*) models for directed social networks , author=. Social Networks , volume=. 2009 , publisher=

    work page 2009

  18. [18]

    Vaart, A. W. van der , year=. Asymptotic Statistics , publisher=

    work page

  19. [19]

    Item Response Theory-A Statistical Framework for Educational and Psychological Measurement , journal =

    Chen, Yunxiao and Li, Xiaoou and Liu, Jingchen Liu and Ying, zhiliang , year=. Item Response Theory-A Statistical Framework for Educational and Psychological Measurement , journal =

    work page

  20. [20]

    Tsybakov, Alexandre B , year=

    work page

  21. [21]

    The Black Book of Quantum Chromodynam- ics: a Primer for the LHC Era,

    Boucheron, St\'. Concentration Inequalities: A Nonasymptotic Theory of Independence. 2013 , month =. doi:10.1093/acprof:oso/9780199535255.001.0001 , url =

    work page doi:10.1093/acprof:oso/9780199535255.001.0001 2013

  22. [22]

    1946 , title =

    Bernstein, S N , publisher =. 1946 , title =

    work page 1946

  23. [23]

    and Liu, Richard C

    Donoho, David L. and Liu, Richard C. , title =. Ann. Statist. , number =. 1991 , doi =

    work page 1991

  24. [24]

    Network global testing by counting graphlets , year =

    Jiashun Jin and Zheng Tracy Ke and Shengming Luo , booktitle =. Network global testing by counting graphlets , year =

    work page

  25. [26]

    and Petrovic, Sonja and Fienberg, S

    Rinaldo, A. and Petrovic, Sonja and Fienberg, S. , year =. On the existence of the MLE for a directed random graph network model with reciprocation , eprint =

    work page

  26. [27]

    2022 , eprint=

    Existence of maximum likelihood estimates in exponential random graph models , author=. 2022 , eprint=

    work page 2022

  27. [28]

    The Annals of Statistics , number =

    Sourav Chatterjee , title =. The Annals of Statistics , number =. 2007 , doi =

    work page 2007

  28. [29]

    The Annals of Applied Probability , number =

    Sourav Chatterjee and Persi Diaconis and Allan Sly , title =. The Annals of Applied Probability , number =. 2011 , doi =

    work page 2011

  29. [30]

    The Annals of Statistics , number =

    Sourav Chatterjee and Persi Diaconis , title =. The Annals of Statistics , number =. 2013 , doi =

    work page 2013

  30. [31]

    and Dahmstrm, K

    Corander, J. and Dahmstrm, K. and Dahmstrm, P. , year =. Maximum likelihood estimation for exponential random graph models. , journal =

    work page

  31. [32]

    de la Pena and S

    Victor H. de la Pena and S. J. Montgomery-Smith , title =. The Annals of Probability , number =. 1995 , doi =

    work page 1995

  32. [33]

    Journal of the American Statistical Association , volume =

    Ove Frank and David Strauss , title =. Journal of the American Statistical Association , volume =. 1986 , publisher =. doi:10.1080/01621459.1986.10478342 , URL =

    work page doi:10.1080/01621459.1986.10478342 1986

  33. [34]

    Fienberg and Michael M

    Stephen E. Fienberg and Michael M. Meyer and Stanley S. Wasserman , title =. Journal of the American Statistical Association , volume =. 1985 , publisher =. doi:10.1080/01621459.1985.10477129 , URL =

    work page doi:10.1080/01621459.1985.10477129 1985

  34. [35]

    and Petrovi

    Fienberg, Stephen E. and Petrovi. Algebraic Statistics for p_1 Random Graph Models: Markov Bases and Their Uses , booktitle =. 2011 , publisher =

    work page 2011

  35. [36]

    Exponential and Moment Inequalities for U-Statistics , booktitle =

    Gin. Exponential and Moment Inequalities for U-Statistics , booktitle =. 2000 , publisher =

    work page 2000

  36. [37]

    Journal of Statistical Software , author=

    statnet: Software Tools for the Representation, Visualization, Analysis and Simulation of Network Data , volume=. Journal of Statistical Software , author=. 2008 , pages=. doi:10.18637/jss.v024.i01 , abstract=

    work page doi:10.18637/jss.v024.i01 2008

  37. [38]

    Estimation of exponential random graph models for large social networks via graph limits

    He, Ran and Zheng, Tian , year =. Estimation of exponential random graph models for large social networks via graph limits. , journal =

    work page

  38. [39]

    Holland and Samuel Leinhardt , title =

    Paul W. Holland and Samuel Leinhardt , title =. Journal of the American Statistical Association , volume =. 1981 , publisher =. doi:10.1080/01621459.1981.10477598 , URL =

    work page doi:10.1080/01621459.1981.10477598 1981

  39. [40]

    Algebraic statistics for a directed random graph model with reciprocation , BOOKTITLE =

    Petrovi\'. Algebraic statistics for a directed random graph model with reciprocation , BOOKTITLE =. 2010 , MRCLASS =. doi:10.1090/conm/516/10180 , URL =

    work page doi:10.1090/conm/516/10180 2010

  40. [41]

    Journal of the American Statistical Association , volume =

    David Strauss and Michael Ikeda , title =. Journal of the American Statistical Association , volume =. 1990 , publisher =. doi:10.1080/01621459.1990.10475327 , URL =

    work page doi:10.1080/01621459.1990.10475327 1990

  41. [42]

    The Annals of Statistics , number =

    Alessandro Rinaldo and Sonja Petrovi. The Annals of Statistics , number =. 2013 , doi =

    work page 2013

  42. [43]

    Markov chain monte carlo estimation of exponential random graph models

    Snijders, Tom , year =. Markov chain monte carlo estimation of exponential random graph models. , volume =

    work page

  43. [44]

    A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models , journal =

    Marijtje A.J. A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models , journal =. 2009 , issn =. doi:https://doi.org/10.1016/j.socnet.2008.10.003 , url =

    work page doi:10.1016/j.socnet.2008.10.003 2009

  44. [45]

    Wasserman, Stan and Stanley, and Faust, and Katherine, and Iacobucci, Dawn and Dawn, , year =

    work page

  45. [46]

    An introduction to Markov graphs and p^* , author =

    Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and p^* , author =. 1996 , journal =

    work page 1996

  46. [47]

    Biometrika , volume =

    Yan, Ting and Xu, Jinfeng , title = ". Biometrika , volume =. 2013 , month =. doi:10.1093/biomet/ass084 , url =

    work page doi:10.1093/biomet/ass084 2013

  47. [48]

    A simulation study of the p_1 model for directed random graphs

    Yan, Ting and Leng, Chenlei , year =. A simulation study of the p_1 model for directed random graphs. , volume =. Statistics and Its Interface , doi =

    work page

  48. [49]

    The Annals of Statistics , number =

    Ting Yan and Chenlei Leng and Ji Zhu , title =. The Annals of Statistics , number =. 2016 , doi =

    work page 2016

  49. [50]

    The Annals of Applied Probability , number =

    Mei Yin and Alessandro Rinaldo and Sukhada Fadnavis , title =. The Annals of Applied Probability , number =. 2016 , doi =

    work page 2016

  50. [51]

    Methods to identify linear network models: a review

    Advani, Arun and Malde, Bansi. Methods to identify linear network models: a review. Swiss Journal of Economics and Statistics. 2018. doi:10.1186/s41937-017-0011-x

    work page doi:10.1186/s41937-017-0011-x 2018

  51. [52]

    1985 , publisher=

    Advanced econometrics , author=. 1985 , publisher=

    work page 1985

  52. [53]

    2000 , address=

    Random graphs , author=. 2000 , address=

    work page 2000

  53. [54]

    Functional Analysis in Normed Spaces , author=

    work page

  54. [55]

    1991 , journal=

    On a theorem of Janson , author=. 1991 , journal=

    work page 1991

  55. [56]

    1974 , publisher=

    Theoretical Statistics , author=. 1974 , publisher=

    work page 1974

  56. [57]

    2005 , publisher=

    Testing Statistical Hypotheses , author=. 2005 , publisher=

    work page 2005

  57. [58]

    Backstrom, Lars and Dwork, Cynthia and Kleinberg, Jon , title =. Commun. ACM , volume =. 2011 , issn =. doi:10.1145/2043174.2043199 , acmid =

    work page doi:10.1145/2043174.2043199 2011

  58. [59]

    BMC bioinformatics , volume=

    An automated method for finding molecular complexes in large protein interaction networks , author=. BMC bioinformatics , volume=. 2003 , publisher=

    work page 2003

  59. [60]

    Bhagat, Smriti and Cormode, Graham and Krishnamurthy, Balachander and Srivastava, Divesh , title =. Proc. VLDB Endow. , issue_date =. 2009 , issn =. doi:10.14778/1687627.1687714 , acmid =

    work page doi:10.14778/1687627.1687714 2009

  60. [61]

    Billingsley, P. , year=. Probability and measure. 3rd edition. , publisher=

    work page

  61. [62]

    On the convergence rate in the central limit theorem for associated processes

    Birkel, Thomas. On the convergence rate in the central limit theorem for associated processes. Ann. Probab. 1988. doi:10.1214/aop/1176991591

    work page doi:10.1214/aop/1176991591 1988

  62. [63]

    Internet Mathematics , volume =

    Joseph Blitzstein and Persi Diaconis , title =. Internet Mathematics , volume =. 2011 , publisher =. doi:10.1080/15427951.2010.557277 , URL =

    work page doi:10.1080/15427951.2010.557277 2011

  63. [64]

    de Acosta,Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm, Ann

    Block, Henry W. and Savits, Thomas H. and Shaked, Moshe. Some concepts of negative dependence. Ann. Probab. 1982. doi:10.1214/aop/1176993784

    work page doi:10.1214/aop/1176993784 1982

  64. [65]

    Biometrika , volume=

    Rank analysis of incomplete block designs the method of paired comparisons , author=. Biometrika , volume=. 1952 , publisher=

    work page 1952

  65. [66]

    Emergence of scaling in random networks , volume =

    Barab. Emergence of scaling in random networks , volume =. 1999 , doi =. http://science.sciencemag.org/content/286/5439/509.full.pdf , journal =

    work page 1999

  66. [67]

    Spectral Clustering and Block Models: A Review and a New Algorithm

    Bhattacharyya, Sharmodeep and Bickel, Peter J. Spectral Clustering and Block Models: A Review and a New Algorithm. Statistical Analysis for High-Dimensional Data. 2016

    work page 2016

  67. [69]

    Geometric Modeling in Probability and Statistics , pages=

    Maximum entropy distributions , author=. Geometric Modeling in Probability and Statistics , pages=. 2014 , publisher=

    work page 2014

  68. [70]

    Estimating and understanding exponential random graph models

    Chatterjee, Sourav and Diaconis, Persi. Estimating and understanding exponential random graph models. Ann. Statist. 2013. doi:10.1214/13-AOS1155

    work page doi:10.1214/13-aos1155 2013

  69. [71]

    , title =

    Charbonneau, Karyne B. , title =. The Econometrics Journal , volume =. doi:10.1111/ectj.12093 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1111/ectj.12093 , abstract =

    work page doi:10.1111/ectj.12093

  70. [72]

    Chen, Louis H. Y. Poisson Approximation for Dependent Trials. Ann. Probab. 1975. doi:10.1214/aop/1176996359

    work page doi:10.1214/aop/1176996359 1975

  71. [73]

    Myers and Jure Leskovec , year=

    Eunjoon Cho and Seth A. Myers and Jure Leskovec , year=. In Proc. Int. Conf. on Knowledge Discovery and Data Mining , title=

    work page

  72. [74]

    2002 , doi =

    Chung, Fan and Lu, Linyuan , title =. 2002 , doi =. http://www.pnas.org/content/99/25/15879.full.pdf , journal =

    work page 2002

  73. [75]

    Cocke, W. J. Central limit theorems for sums of dependent vector variables. Ann. Math. Statist. 1972. doi:10.1214/aoms/1177692559

    work page doi:10.1214/aoms/1177692559 1972

  74. [76]

    Cohen, William W. , year=. Enron Email Dataset (Retrieved March 12, 2005) , url=

    work page 2005

  75. [77]

    de Acosta,Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm, Ann

    Cox, J. Theodore and Grimmett, Geoffrey. Central limit theorems for associated random variables and the percolation model. Ann. Probab. 1984. doi:10.1214/aop/1176993303

    work page doi:10.1214/aop/1176993303 1984

  76. [78]

    A simple model of homophily in social networks

    Sergio Currarini and Jesse Matheson and Fernando Vega-Redondo. A simple model of homophily in social networks. European Economic Review. 2016. doi:https://doi.org/10.1016/j.euroecorev.2016.03.011

    work page doi:10.1016/j.euroecorev.2016.03.011 2016

  77. [79]

    Electronic Journal of Probability , pages=

    Bernard Delyon , title=. Electronic Journal of Probability , pages=

    work page

  78. [80]

    33rd International Colloquium on Automata, Languages and Programming , title=

    Cynthia Dwork , year=. 33rd International Colloquium on Automata, Languages and Programming , title=

    work page

  79. [81]

    Calibrating noise to sensitivity in private data analysis

    Dwork, Cynthia and McSherry, Frank and Nissim, Kobbi and Smith, Adam. Calibrating noise to sensitivity in private data analysis. Theory of Cryptography: Third Theory of Cryptography Conference, TCC 2006, New York, NY, USA, March 4-7, 2006. Proceedings. 2006. doi:10.1007/11681878_14

    work page doi:10.1007/11681878_14 2006

  80. [82]

    The Review of Economics and Statistics , volume =

    Dzemski, Andreas , title =. The Review of Economics and Statistics , volume =. 2019 , doi =

    work page 2019

Showing first 80 references.