Pith. sign in

REVIEW 2 major objections 7 minor 295 references

Subgraph counting estimation for the $\beta$-model in sparse networks

T0 review · 2 major / 7 minor · reviewed 2026-07-07 · glm-5.2

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

Pith's one-line read Triangle counts crack sparse-network parameter estimation

desk verdict 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. read the letter →

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

Qunqiang Feng , Jiashun Jin , Yaru Tian , Ting Yan This is my paper · ORCID
classification stat.ME
keywords beta-modelsparsenetworkssubgraphcountingcycleratioestimatorminimaxestimationdegreeheterogeneityasymptoticnormalitynetworkdensity
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 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.

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.

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

Extended reading notes

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,

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.

Editorial extensions

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.

Reading between the lines

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.
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, 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 · score 0.0 of 10

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.

Assumptions & free parameters 1 free parameters · 4 assumptions · 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.
assumptions (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.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Subgraph counting estimation for the $\beta$-model in sparse networks." pith.science (2026). https://pith.science/paper/PJDPPNQ5

@misc{pith2026260705273,
  author       = {Pith},
  title        = {Pith review of: Subgraph counting estimation for the $\beta$-model in sparse networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJDPPNQ5}},
  note         = {Machine review of arXiv:2607.05273}
}
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 the authors.

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 + a2) + · · ·… view at source ↗
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. 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, 2, . . . … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

295 extracted references · 295 canonical work pages

  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

Show all 295 references
  1. [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=

    2025

  2. [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 =

  3. [11]

    Internet Mathematics , doi=

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

  4. [12]

    2000 , publisher=

    Asymptotic statistics , author=. 2000 , publisher=

    2000

  5. [13]

    The -Model Maximum Likelihood, Cram

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

    2017

  6. [14]

    Journal of Business & Economic Statistics , volume=

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

    2025

  7. [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=

    1981

  8. [16]

    Aids , volume=

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

    2007

  9. [17]

    Social Networks , volume=

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

    2009

  10. [18]

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

  11. [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 =

  12. [20]

    Tsybakov, Alexandre B , year=

  13. [21]

    Concentration Inequalities: A Nonasymptotic Theory of Independence

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

    2013 doi

  14. [22]

    1946 , title =

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

    1946

  15. [23]

    and Liu, Richard C

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

    1991

  16. [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 =

  17. [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 =

  18. [27]

    2022 , eprint=

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

    2022

  19. [28]

    The Annals of Statistics , number =

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

    2007

  20. [29]

    The Annals of Applied Probability , number =

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

    2011

  21. [30]

    The Annals of Statistics , number =

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

    2013

  22. [31]

    and Dahmstrm, K

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

  23. [32]

    de la Pena and S

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

    1995

  24. [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 =

    1986 doi

  25. [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 =

    1985 doi

  26. [35]

    and Petrovi

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

    2011

  27. [36]

    Exponential and Moment Inequalities for U-Statistics , booktitle =

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

    2000

  28. [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=

    2008 doi

  29. [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 =

  30. [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 =

    1981 doi

  31. [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 =

    2010 doi

  32. [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 =

    1990 doi

  33. [42]

    The Annals of Statistics , number =

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

    2013

  34. [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 =

  35. [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 =

    2009 doi

  36. [45]

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

  37. [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 =

    1996

  38. [47]

    Biometrika , volume =

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

    2013 doi

  39. [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 =

  40. [49]

    The Annals of Statistics , number =

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

    2016

  41. [50]

    The Annals of Applied Probability , number =

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

    2016

  42. [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

    2018 doi

  43. [52]

    1985 , publisher=

    Advanced econometrics , author=. 1985 , publisher=

    1985

  44. [53]

    2000 , address=

    Random graphs , author=. 2000 , address=

    2000

  45. [54]

    Functional Analysis in Normed Spaces , author=

  46. [55]

    1991 , journal=

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

    1991

  47. [56]

    1974 , publisher=

    Theoretical Statistics , author=. 1974 , publisher=

    1974

  48. [57]

    2005 , publisher=

    Testing Statistical Hypotheses , author=. 2005 , publisher=

    2005

  49. [58]

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

    2011 doi

  50. [59]

    BMC bioinformatics , volume=

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

    2003

  51. [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 =

    2009 doi

  52. [61]

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

  53. [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

    1988 doi

  54. [63]

    Internet Mathematics , volume =

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

    2011 doi

  55. [64]

    and Savits, Thomas H

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

    1982 doi

  56. [65]

    Biometrika , volume=

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

    1952

  57. [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 =

    1999

  58. [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

    2016

  59. [69]

    Geometric Modeling in Probability and Statistics , pages=

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

    2014

  60. [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

    2013 doi

  61. [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 =

    doi

  62. [72]

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

    1975 doi

  63. [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=

  64. [74]

    2002 , doi =

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

    2002

  65. [75]

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

    1972 doi

  66. [76]

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

    2005

  67. [77]

    Theodore and Grimmett, Geoffrey

    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

    1984 doi

  68. [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

    2016 doi

  69. [79]

    Electronic Journal of Probability , pages=

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

  70. [80]

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

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

  71. [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

    2006 doi

  72. [82]

    The Review of Economics and Statistics , volume =

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

    2019

  73. [83]

    Esary, J. D. and Proschan, F. and Walkup, D. W. Association of random variables, with applications. Ann. Math. Statist. 1967. doi:10.1214/aoms/1177698701

    1967 doi

  74. [84]

    2008 , journal=

    Some finite sample properties of negatively dependent random variables , author=. 2008 , journal=

    2008

  75. [85]

    Fern\'andez-V\'al and M

    I. Fern\'andez-V\'al and M. Weidner. Individual and time effects in nonlinear panel models with large N, T. Journal of Econometrics. 2016. doi:https://doi.org/10.1016/j.jeconom.2015.12.014

    2016 doi

  76. [86]

    Fienberg , title =

    Stephen E. Fienberg , title =. Journal of Computational and Graphical Statistics , volume =. 2012 , publisher =. doi:10.1080/10618600.2012.738106 , URL =

    2012 doi

  77. [87]

    and Zhou, Harrison H

    Gao, Chao and Ma, Zongming and Zhang, Anderson Y. and Zhou, Harrison H. Community detection in degree-corrected block models. Ann. Statist. 2018. doi:10.1214/17-AOS1615

    2018 doi

  78. [88]

    Communications in Statistics - Theory and Methods , volume =

    Malay Ghosh , title =. Communications in Statistics - Theory and Methods , volume =. 1981 , publisher =. doi:10.1080/03610928108828041 , URL =

    1981 doi

  79. [89]

    2010 , volume =

    Foundations and Trends in Machine Learning , title =. 2010 , volume =. doi:10.1561/2200000005 , issn =

    2010 doi

  80. [90]

    W. B. Gragg and R. A. Tapia , title =. SIAM Journal on Numerical Analysis , volume =. 1974 , doi =

    1974

  81. [91]

    , title =

    Graham, Bryan S. , title =. Econometrica , volume =. doi:10.3982/ECTA12679 , pages =

    doi

  82. [92]

    Econometrica , volume =

    Hahn, Jinyong and Newey, Whitney , title =. Econometrica , volume =. 2004 , keywords =. doi:10.1111/j.1468-0262.2004.00533.x , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1468-0262.2004.00533.x , abstract =

    2004 doi

  83. [93]

    Handcock, M. S. , title=. Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers , editor=. 2003 , pages=

    2003

  84. [94]

    Hay and C

    M. Hay and C. Li and G. Miklau and D. Jensen , booktitle=. Accurate estimation of the degree distribution of private networks , year=. doi:10.1109/ICDM.2009.11 , ISSN=

    2009 doi

  85. [95]

    Hay, Michael and Miklau, Gerome and Jensen, David and Towsley, Don and Weis, Philipp , title =. Proc. VLDB Endow. , issue_date =. 2008 , issn =. doi:10.14778/1453856.1453873 , acmid =

    2008 doi

  86. [97]

    Probability inequalities for sums of bounded random variables , volume =

    Wassily Hoeffding , journal =. Probability inequalities for sums of bounded random variables , volume =

  87. [98]

    Journal of the American Statistical Association , volume =

    Peter D Hoff , title =. Journal of the American Statistical Association , volume =. 2005 , publisher =. doi:10.1198/016214504000001015 , URL =

    2005 doi

  88. [99]

    Journal of the American Statistical Association , volume=

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

    1981

  89. [100]

    Ioannides and G.G

    D.A. Ioannides and G.G. Roussas. Exponential inequality for associated random variables. Statistics & Probability Letters. 1999. doi:https://doi.org/10.1016/S0167-7152(98)00240-5

    1999 doi

  90. [101]

    and Pitt, Loren D

    Joag-Dev, Kumar and Perlman, Michael D. and Pitt, Loren D. , doi =. Association of normal random variables and slepian's inequality , url =. The Annals of Probability , month =

  91. [102]

    Negative association of random variables with applications

    Joag-Dev, Kumar and Proschan, Frank. Negative association of random variables with applications. Ann. Statist. 1983. doi:10.1214/aos/1176346079

    1983 doi

  92. [103]

    Journal of Business & Economic Statistics , volume =

    Koen Jochmans , title =. Journal of Business & Economic Statistics , volume =. 2017 , publisher =. doi:10.1080/07350015.2017.1286242 , URL =

    2017 doi

  93. [104]

    The Annals of Statistics , volume=

    Inference using noisy degrees-Differentially private beta model and synthetic graphs , author=. The Annals of Statistics , volume=. 2016 , pages=

    2016

  94. [105]

    Stochastic blockmodels and community structure in networks , author =. Phys. Rev. E , volume =. 2011 , month =. doi:10.1103/PhysRevE.83.016107 , url =

    2011 doi

  95. [106]

    and Xue, Lingzhou and Niu, Xiaoyue

    Kim, Bomin and Lee, Kevin H. and Xue, Lingzhou and Niu, Xiaoyue. A review of dynamic network models with latent variables. Statist. Surv. 2018. doi:10.1214/18-SS121

    2018 doi

  96. [107]

    Kolaczyk , title=

    Eric D. Kolaczyk , title=

  97. [108]

    Science , volume=

    Empirical analysis of an evolving social network , author=. Science , volume=. 2006 , publisher=

    2006

  98. [109]

    2009 , Journal =

    Representing degree distributions, clustering, and homophily in social networks with latent cluster random effects models , Author =. 2009 , Journal =. doi:10.1016/j.socnet.2009.04.001 , Number =

    2009 doi

  99. [110]

    Electronic Journal of Statistics , volume=

    Exponential-family random graph models for valued networks , author=. Electronic Journal of Statistics , volume=. 2012 , publisher=

    2012

  100. [111]

    1993 , publisher=

    Real and Functional Analysis , author=. 1993 , publisher=

    1993

  101. [112]

    Lehmann, E. L. Some concepts of dependence. Ann. Math. Statist. 1966. doi:10.1214/aoms/1177699260

    1966 doi

  102. [113]

    2011 , publisher=

    Probability Inequalities , author=. 2011 , publisher=

    2011

  103. [114]

    Lindeberg, J. W. , journal =. Eine neue herleitung des exponentialgesetzes in der wahrscheinlichkeitsrechnung , url =

  104. [115]

    Probability theory I. 4th ed. , author=. 1977 , publisher=

    1977

  105. [116]

    Econometrica , volume =

    Mele, Angelo , title =. Econometrica , volume =. doi:10.3982/ECTA10400 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.3982/ECTA10400 , year =

    doi

  106. [117]

    Asymptotic independence and limit theorems for positively and negatively dependent random variables

    Newman, Charles M. Asymptotic independence and limit theorems for positively and negatively dependent random variables. Inequalities in Statistics and Probability. 1984. doi:10.1214/lnms/1215465639

    1984 doi

  107. [118]

    Nature , volume=

    Comparative assessment of large-scale data sets of protein-protein interactions , author=. Nature , volume=

  108. [119]

    Generalized Linear Models, Second Edition , publisher=

    McCullagh, Peter and Nelder, John , year=. Generalized Linear Models, Second Edition , publisher=

  109. [120]

    Annual Review of Sociology , volume =

    Miller McPherson and Lynn Smith-Lovin and James M Cook , title =. Annual Review of Sociology , volume =. 2001 , doi =

    2001

  110. [121]

    Detection thresholds for the -model on sparse graphs

    Mukherjee, Rajarshi and Mukherjee, Sumit and Sen, Subhabrata. Detection thresholds for the -model on sparse graphs. Ann. Statist. 2018. doi:10.1214/17-AOS1585

    2018 doi

  111. [122]

    Narayanan and V

    A. Narayanan and V. Shmatikov , booktitle=. De-anonymizing Social Networks , year=. doi:10.1109/SP.2009.22 , ISSN=

    2009 doi

  112. [123]

    Nature methods , volume=

    Detecting overlapping protein complexes in protein-protein interaction networks , author=. Nature methods , volume=. 2012 , publisher=

    2012

  113. [124]

    1948 , number=

    Consistent estimates based on partially consistent observations , author=. 1948 , number=

    1948

  114. [125]

    arXiv preprint arXiv:1211.6537 , year=

    Degree-based network models , author=. arXiv preprint arXiv:1211.6537 , year=

    arXiv

  115. [126]

    and Newman, M

    Park, J. and Newman, M. E. J. , title=. 2004 , journal=

    2004

  116. [127]

    P. O. Perry and P. J. Wolfe , title=. 2012 , journal=

    2012

  117. [128]

    Pitt, Loren D. , doi=. Positively Correlated Normal Variables are Associated , url=. The Annals of Probability , year=

  118. [129]

    Maximum lilkelihood estimation in the -model

    Rinaldo, Alessandro and Petrovi\'c, Sonja and Fienberg, Stephen E. Maximum lilkelihood estimation in the -model. Ann. Statist. 2013. doi:10.1214/12-AOS1078

    2013 doi

  119. [130]

    An introduction to exponential random graph (p*) models for social networks

    Garry Robins and Pip Pattison and Yuval Kalish and Dean Lusher. An introduction to exponential random graph (p*) models for social networks. Social Networks. 2007. doi:https://doi.org/10.1016/j.socnet.2006.08.002

    2007 doi

  120. [131]

    Recent developments in exponential random graph (p*) models for social networks

    Garry Robins and Tom Snijders and Peng Wang and Mark Handcock and Philippa Pattison. Recent developments in exponential random graph (p*) models for social networks. Social Networks. 2007. doi:https://doi.org/10.1016/j.socnet.2006.08.003

    2007 doi

  121. [132]

    Roussas , journal =

    George G. Roussas , journal =. Exponential probability inequalities with some applications , volume =

  122. [133]

    and White, A

    Salter-Townshend, M. and White, A. and Gollini, I. and Murphy, T. B. , year =. Review of statistical network analysis: models, algorithms, and software , journal =. doi:10.1002/sam.11146 , url =

    doi

  123. [134]

    and Pin, Paolo , title =

    Currarini, Sergio and Jackson, Matthew O. and Pin, Paolo , title =. Econometrica , volume =. 2009 , keywords =. doi:10.3982/ECTA7528 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.3982/ECTA7528 , abstract =

    2009 doi

  124. [135]

    Consistency under sampling of exponential random graph models

    Shalizi, Cosma Rohilla and Rinaldo, Alessandro. Consistency under sampling of exponential random graph models. Ann. Statist. 2013. doi:10.1214/12-AOS1044

    2013 doi

  125. [136]

    The Annals of Statistics , volume=

    Asymptotics when the number of parameters tends to infinity in the Bradley-Terry model for paired comparisons , author=. The Annals of Statistics , volume=. 1999 , publisher=

    1999

  126. [137]

    C. M. Stein , booktitle=. A bound for the error in normal approximation to the distribution of a sum of dependent random variables , year=

  127. [138]

    2003 , title=

    Endre S\". 2003 , title=

    2003

  128. [139]

    A note on a network model with degree heterogeneity and homophily

    Liju Su and Xiaodi Qian and Ting Yan. A note on a network model with degree heterogeneity and homophily. Statistics & Probability Letters. 2018. doi:https://doi.org/10.1016/j.spl.2018.02.046

    2018 doi

  129. [140]

    van Duijn, Marijtje A. J. and Snijders, Tom A. B. and Zijlstra, Bonne J. H. , title =. Statistica Neerlandica , volume =. 2004 , keywords =. doi:10.1046/j.0039-0402.2003.00258.x , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1046/j.0039-0402.2003.00258.x , abstract =

    2004 doi

  130. [141]

    Introduction to the non-asymptotic analysis of random matrices , DOI=

    Vershynin, Roman , editor=. Introduction to the non-asymptotic analysis of random matrices , DOI=. Compressed Sensing: Theory and Applications , publisher=. 2012 , pages=

    2012

  131. [142]

    A Berry--Esseen theorem for weakly negatively dependent random variables and its applications

    Wang, Jianfeng and Zhang, Lixin. A Berry--Esseen theorem for weakly negatively dependent random variables and its applications. Acta Mathematica Hungarica. 2006. doi:10.1007/s10474-006-0024-x

    2006 doi

  132. [143]

    Statistics & Probability Letters , volume=

    A note on asymptotic distributions in maximum entropy models for networks , author=. Statistics & Probability Letters , volume=. 2015 , publisher=

    2015

  133. [144]

    arXiv:1803.09971 , year=

    A probit network model with arbitrary dependence , author=. arXiv:1803.09971 , year=

    arXiv

  134. [145]

    Fienberg and Chenlei Leng , title =

    Ting Yan and Binyan Jiang and Stephen E. Fienberg and Chenlei Leng , title =. Journal of the American Statistical Association , volume =. 2019 , publisher =. doi:10.1080/01621459.2018.1448829 , URL =

    2019 doi

  135. [146]

    2016 , journal=

    Asymptotics in directed exponential random graph models with an increasing bi-degree sequence , author=. 2016 , journal=

    2016

  136. [147]

    Asymptotics in undirected random graph models parameterized by the strengths of vertices , volume =

    Ting Yan and Hong Qin and Hansheng Wang , journal =. Asymptotics in undirected random graph models parameterized by the strengths of vertices , volume =

  137. [148]

    Biometrika , volume=

    A central limit theorem in the -model for undirected random graphs with a diverging number of vertices , author=. Biometrika , volume=. 2013 , publisher=

    2013

  138. [149]

    Asymptotic normality in the maximum entropy models on graphs with an increasing number of parameters

    Yan, Ting and Zhao, Yunpeng and Qin, Hong. Asymptotic normality in the maximum entropy models on graphs with an increasing number of parameters. Journal of Multivariate Analysis. 2015. doi:https://doi.org/10.1016/j.jmva.2014.08.013

    2015 doi

  139. [150]

    High-dimensional wilks phenomena in some exponential random graph models , author=

  140. [151]

    and Goldberg, M

    Zhou, Y. and Goldberg, M. and Magdon-Ismail, M. and Wallace, W. A. , booktitle=. Strategies for cleaning organizational emails with an application to Enron email dataset , year=

  141. [152]

    Journal of the American Statistical Association , volume =

    Jingfei Zhang and Yuguo Chen , title =. Journal of the American Statistical Association , volume =. 2013 , publisher =. doi:10.1080/01621459.2012.758587 , URL =

    2013 doi

  142. [153]

    Consistency of community detection in networks under degree-corrected stochastic block models

    Zhao, Yunpeng and Levina, Elizaveta and Zhu, Ji. Consistency of community detection in networks under degree-corrected stochastic block models. Ann. Statist. 2012. doi:10.1214/12-AOS1036

    2012 doi

  143. [154]

    Linear Algebra & Its Applications , volume=

    Approximating the inverse of a symmetric positive definite matrix , author=. Linear Algebra & Its Applications , volume=

  144. [155]

    Uspekhi Mat Nauk , pages=

    Functional analysis and applied mathematics , author=. Uspekhi Mat Nauk , pages=

  145. [156]

    1986 , issue_date =

    Yamamoto, Tetsuro , title =. 1986 , issue_date =. doi:10.1007/BF01389444 , journal =

    1986 doi

  146. [157]

    Ting-Fan Wu and Chih-Jen Lin and Ruby C. Weng. Probability Estimates for Multi-class Classification by Pairwise Coupling. Journal of Machine Learning Research. 2004

    2004

  147. [158]

    Classification by pairwise coupling

    Trevor Hastie and Robert Tibshirani. Classification by pairwise coupling. Annals of Statistics. 1998

    1998

  148. [159]

    2020 , eprint=

    A Multiclass Classification Approach to Label Ranking , author=. 2020 , eprint=

    2020

  149. [160]

    and Wodon, Adam , year=

    Whelan, John T. and Wodon, Adam , year=. Prediction and evaluation in College Hockey using the Bradley-Terry-Zermelo model , volume=. Mathematics for Application , publisher=. doi:10.13164/ma.2019.09 , number=

    2019 doi

  150. [161]

    2019 , eprint=

    Rank Aggregation via Heterogeneous Thurstone Preference Models , author=. 2019 , eprint=

    2019

  151. [162]

    Statistical modelling of citation exchange between statistics journals

    Cristiano Varin and Manuela Cattelan and David Firth. Statistical modelling of citation exchange between statistics journals. Journal of The Royal Statistical Society Series A-statistics in Society. 2016

    2016

  152. [163]

    Stephen M. Stigler. Citation Patterns in the Journals of Statistics and Probability. Statistical Science. 1994

    1994

  153. [164]

    Dynamic Bradley-Terry modelling of sports tournaments

    Manuela Cattelan and Cristiano Varin and David Firth. Dynamic Bradley-Terry modelling of sports tournaments. Journal of The Royal Statistical Society Series C-applied Statistics. 2013

    2013

  154. [165]

    The ranking lasso and its application to sport tournaments

    Guido Masarotto and Cristiano Varin. The ranking lasso and its application to sport tournaments. The Annals of Applied Statistics. 2012

    2012

  155. [166]

    Active exploration for learning rankings from clickthrough data

    Filip Radlinski and Thorsten Joachims. Active exploration for learning rankings from clickthrough data. Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining. 2007

    2007

  156. [167]

    Asymptotic Enumeration of Orientations of a Graph as a Function of the Out-Degree Sequence , volume=

    Isaev, Mikhail and Iyer, Tejas and McKay, Brendan D , year=. Asymptotic Enumeration of Orientations of a Graph as a Function of the Out-Degree Sequence , volume=. The Electronic Journal of Combinatorics , publisher=. doi:10.37236/8929 , number=

    doi

  157. [168]

    A Survey of Real-Time Strategy Game AI Research and Competition in StarCraft

    Santiago Onta\ n \' o n and Gabriel Synnaeve and Alberto Uriarte and Florian Richoux and David Churchill and Mike Preuss. A Survey of Real-Time Strategy Game AI Research and Competition in StarCraft. IEEE Transactions on Computational Intelligence and AI in Games. 2013

    2013

  158. [169]

    Shah and Marc Oliu Simon

    Baiyu Chen and Sergio Escalera and Isabelle Guyon and V\' o ctor Ponce-L\' o pez and Nihar B. Shah and Marc Oliu Simon. Overcoming Calibration Problems in Pattern Labeling with Pairwise Ratings: Application to Personality Traits. European Conference on Computer Vision (ECCV 20...

    2016

  159. [170]

    Minimax-optimal Inference from Partial Rankings

    Bruce Hajek and Sewoong Oh and Jiaming Xu. Minimax-optimal Inference from Partial Rankings. Advances in Neural Information Processing Systems 27. 2014

    2014

  160. [171]

    Shah and Sivaraman Balakrishnan and Joseph Bradley and Abhay Parekh and Kannan Ramchandran and Martin J

    Nihar B. Shah and Sivaraman Balakrishnan and Joseph Bradley and Abhay Parekh and Kannan Ramchandran and Martin J. Wainwright. Estimation from pairwise comparisons: sharp minimax bounds with topology dependence. Journal of Machine Learning Research. 2016

    2016

  161. [172]

    2019 , eprint=

    Stretching the Effectiveness of MLE from Accuracy to Bias for Pairwise Comparisons , author=. 2019 , eprint=

    2019

  162. [173]

    Rui Xia and Vincent Y. F. Tan and Louis Filstroff and C\' e dric F\' e votte. A Ranking Model Motivated by Nonnegative Matrix Factorization with Applications to Tennis Tournaments. arXiv preprint arXiv:1903.06500. 2019

    1903 arXiv

  163. [174]

    The semi-algebraic geometry of optimal designs for the Bradley-Terry model

    Thomas Kahle and Frank R\" o ttger and Rainer Schwabe. The semi-algebraic geometry of optimal designs for the Bradley-Terry model. arXiv preprint arXiv:1901.02375. 2019

    1901 arXiv

  164. [175]

    Convergence Rates of Gradient Descent and MM Algorithms for Generalized Bradley-Terry Models

    Milan Vojnovic and Seyoung Yun and Kaifang Zhou. Convergence Rates of Gradient Descent and MM Algorithms for Generalized Bradley-Terry Models. arXiv preprint arXiv:1901.00150. 2019

    1901 arXiv

  165. [176]

    Phelan and John T

    Gabriel C. Phelan and John T. Whelan. Hierarchical Bayesian Bradley-Terry for applications in Major League Baseball. Mathematics for Application. 2018

    2018

  166. [177]

    Alkeos Tsokos and Santhosh Narayanan and Ioannis Kosmidis and Gianluca Baio and Mihai Cucuringu and Gavin Whitaker and Franz J. Kir??ly. Modeling outcomes of soccer matches. Machine Learning. 2019

    2019

  167. [178]

    John T. Whelan. Prior Distributions for the Bradley-Terry Model of Paired Comparisons. arXiv preprint arXiv:1712.05311. 2017

    2017 arXiv

  168. [180]

    Kenneth Butler and John T. Whelan. The existence of maximum likelihood estimates in the Bradley-Terry model and its extensions. arXiv preprint arXiv:math/0412232. 2004

    2004 arXiv

  169. [181]

    and Redner, S

    Sire, C. and Redner, S. , year=. Understanding baseball team standings and streaks , volume=. The European Physical Journal B , publisher=. doi:10.1140/epjb/e2008-00405-5 , number=

    doi

  170. [182]

    Efficient Bayesian Inference for Generalized Bradley-Terry Models , volume=

    Caron, Fran?ois and Doucet, Arnaud , year=. Efficient Bayesian Inference for Generalized Bradley-Terry Models , volume=. Journal of Computational and Graphical Statistics , publisher=. doi:10.1080/10618600.2012.638220 , number=

    2012 doi

  171. [183]

    A grade of membership model for rank data

    Isobel Claire Gormley and Thomas Brendan Murphy. A grade of membership model for rank data. Bayesian Analysis. 2009

    2009

  172. [184]

    Models for Paired Comparison Data: A Review with Emphasis on Dependent Data , volume=

    Cattelan, Manuela , year=. Models for Paired Comparison Data: A Review with Emphasis on Dependent Data , volume=. Statistical Science , publisher=. doi:10.1214/12-sts396 , number=

    doi

  173. [186]

    The number of potential winners in BradleyCTerry model in random environment , volume=

    Chetrite, Raphael and Diel, Roland and Lerasle, Matthieu , year=. The number of potential winners in BradleyCTerry model in random environment , volume=. The Annals of Applied Probability , publisher=. doi:10.1214/16-aap1231 , number=

    doi

  174. [187]

    David R. Hunter. MM algorithms for generalized Bradley-Terry models. Annals of Statistics. 2003

    2003

  175. [188]

    1929 , journal=

    Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung , author=. 1929 , journal=

    1929

  176. [189]

    Mathematische Zeitschrift , volume=

    Die berechnung der turnier-ergebnisse als ein maximumproblem der wahrscheinlichkeitsrechnung , author=. Mathematische Zeitschrift , volume=. 1929 , publisher=

    1929

  177. [190]

    Ralph Allan Bradley and Milton E. Terry. Rank analysis of incomplete block designs I. The method of paired comparisons. Biometrika. 1952

    1952

  178. [191]

    Representation and uniqueness of the Bradley-Terry-Luce model for pair comparisons

    Hans Colonius. Representation and uniqueness of the Bradley-Terry-Luce model for pair comparisons. British Journal of Mathematical and Statistical Psychology. 1980

    1980

  179. [192]

    1957 , journal=

    Solution of a Ranking Problem from Binary Comparisons , author=. 1957 , journal=

    1957

  180. [193]

    P. V. Rao and L. L. Kupper. Ties in Paired-Comparison Experiments: A Generalization of the Bradley-Terry Model. Journal of the American Statistical Association. 1967

    1967

  181. [194]

    Davidson

    Roger R. Davidson. On Extending the Bradley-Terry Model to Accommodate Ties in Paired Comparison Experiments. Journal of the American Statistical Association. 1970

    1970

  182. [195]

    Luce, R. D. , year=. Individual Choice Behavior , publisher=

  183. [196]

    David, H. A. , year=. The Method of Paired Comparisons, 2nd ed. , publisher=

  184. [197]

    Weng and Chih-Jen Lin

    Tzu-Kuo Huang and Ruby C. Weng and Chih-Jen Lin. Generalized Bradley-Terry Models and Multi-Class Probability Estimates. Journal of Machine Learning Research. 2006

    2006

  185. [198]

    Statistical Science , volume=

    Elo Ratings and the Sports Model: A Neglected Topic in Applied Probability? , author=. Statistical Science , volume=

  186. [199]

    Kir\' a ly

    Alkeos Tsokos and Santhosh Narayanan and Ioannis Kosmidis and Gianluca Baio and Mihai Cucuringu and Gavin Whitaker and Franz J. Kir\' a ly. Modeling outcomes of soccer matches. Machine Learning. 2019

    2019

  187. [200]

    Just Sort It! A Simple and Effective Approach to Active Preference Learning

    Lucas Maystre and Matthias Grossglauser. Just Sort It! A Simple and Effective Approach to Active Preference Learning. Proceedings of the 34th International Conference on Machine Learning - Volume 70. 2017

    2017

  188. [201]

    Accelerated Spectral Ranking

    Arpit Agarwal and Prathamesh Patil and Shivani Agarwal. Accelerated Spectral Ranking. ICML 2018: Thirty-fifth International Conference on Machine Learning. 2018

    2018

  189. [202]

    Parameter Estimation for Thurstone Choice Models , url=

    Vojnovic, Milan and Yun, Seyoung , year=. Parameter Estimation for Thurstone Choice Models , url=. doi:10.2172/1353026 , publisher=

    doi

  190. [203]

    Rank centrality: Ranking from pairwise comparisons

    Sahand Negahban and Sewoong Oh and Devavrat Shah. Rank centrality: Ranking from pairwise comparisons. Operations Research. 2017

    2017

  191. [204]

    Graph Resistance and Learning from Pairwise Comparisons

    Julien Hendrickx and Alexander Olshevsky and Venkatesh Saligrama. Graph Resistance and Learning from Pairwise Comparisons. ICML 2019 : Thirty-sixth International Conference on Machine Learning. 2019

    2019

  192. [205]

    Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons

    Yuxin Chen and Changho Suh. Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons. Proceedings of The 32nd International Conference on Machine Learning. 2015

    2015

  193. [206]

    2012 , publisher=

    Categorical Data Analysis, 3rd Edition , author=. 2012 , publisher=

    2012

  194. [207]

    3rd Edition , author=

    Matrix Computations. 3rd Edition , author=. 1996 , address=

    1996

  195. [208]

    Neyman and Elizabeth L

    J. Neyman and Elizabeth L. Scott. Consistent Estimates Based on Partially Consistent Observations. Econometrica. 1948

    1948

  196. [209]

    Sparse paired comparisons in the Bradley--Terry model

    Ting Yan and Yaning Yang and Jinfeng Xu. Sparse paired comparisons in the Bradley--Terry model. Statistica Sinica. 2012

    2012

  197. [210]

    and Ye, R

    Han, R. and Ye, R. and Tan, C. and Chen, K. , year=. Asymptotic Theory of Sparse Bradley-Terry Model , journal=

  198. [211]

    L. L. Thurstone. A law of comparative judgment. Psychological Review. 1927

    1927

  199. [212]

    Brown, L. D. , year=. Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory (Vol. 9) , address=

  200. [213]

    Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity

    Portnoy, Stephen. Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity. Ann. Statist. 1988. doi:10.1214/aos/1176350710

    1988 doi

  201. [214]

    Asymptotic Behavior of M Estimators of p Regression Parameters when p^2 / n is Large; II

    Portnoy, Stephen. Asymptotic Behavior of M Estimators of p Regression Parameters when p^2 / n is Large; II. Normal Approximation. Ann. Statist. 1985. doi:10.1214/aos/1176349744

    1985 doi

  202. [215]

    GEE analysis of clustered binary data with diverging number of covariates

    Wang, Lan. GEE analysis of clustered binary data with diverging number of covariates. Ann. Statist. 2011. doi:10.1214/10-AOS846

    2011 doi

  203. [216]

    Maximum likelihood estimation in logistic regression models with a diverging number of covariates

    Liang, Hua and Du, Pang. Maximum likelihood estimation in logistic regression models with a diverging number of covariates. Electron. J. Statist. 2012. doi:10.1214/12-EJS731

    2012 doi

  204. [217]

    Maximum Likelihood Estimates in Exponential Response Models

    Haberman, Shelby J. Maximum Likelihood Estimates in Exponential Response Models. Ann. Statist. 1977. doi:10.1214/aos/1176343941

    1977 doi

  205. [218]

    and Rinaldo, Alessandro

    Fienberg, Stephen E. and Rinaldo, Alessandro. Maximum likelihood estimation in log-linear models. Ann. Statist. 2012. doi:10.1214/12-AOS986

    2012 doi

  206. [219]

    and Fienberg, Stephen E

    Erosheva, Elena A. and Fienberg, Stephen E. and Joutard, Cyrille. Describing disability through individual-level mixture models for multivariate binary data. Ann. Appl. Stat. 2007. doi:10.1214/07-AOAS126

    2007 doi

  207. [220]

    Fienberg and Alessandro Rinaldo , title =

    Stephen E. Fienberg and Alessandro Rinaldo , title =. Journal of Statistical Planning and Inference , year =

  208. [221]

    Spectral method and regularized MLE are both optimal for top- K ranking

    Chen, Yuxin and Fan, Jianqing and Ma, Cong and Wang, Kaizheng. Spectral method and regularized MLE are both optimal for top- K ranking. Ann. Statist. 2019. doi:10.1214/18-AOS1745

    2019 doi

  209. [222]

    Asymptotic normality and strong consistency of maximum quasi-likelihood estimates in generalized linear models

    Changming Yin and Lincheng Zhao and Chengdong Wei. Asymptotic normality and strong consistency of maximum quasi-likelihood estimates in generalized linear models. Science China-mathematics. 2006

    2006

  210. [223]

    Communication-efficient distributed estimator for generalized linear models with a diverging number of covariates

    Ping Zhou and Zhen Yu and Jingyi Ma and Maozai Tian and Ye Fan. Communication-efficient distributed estimator for generalized linear models with a diverging number of covariates. Computational Statistics & Data Analysis. 2021. doi:https://doi.org/10.1016/j.csda.2020.107154

    2021 doi

  211. [224]

    Luce, R. D. , year=. Individual choice behavior: A theoretical analysis , address=

  212. [225]

    Robust Regression: Asymptotics, Conjectures and Monte Carlo

    Huber, Peter J. Robust Regression: Asymptotics, Conjectures and Monte Carlo. Ann. Statist. 1973. doi:10.1214/aos/1176342503

    1973 doi

  213. [226]

    Bai and Y

    Z.D. Bai and Y. Wu. Limiting Behavior of M-Estimators of Regression Coefficients in High Dimensional Linear Models I. Scale Dependent Case. Journal of Multivariate Analysis. 1994. doi:https://doi.org/10.1006/jmva.1994.1059

    1994 doi

  214. [227]

    On Parameters of Increasing Dimensions

    Xuming He and Qi-Man Shao. On Parameters of Increasing Dimensions. Journal of Multivariate Analysis. 2000. doi:https://doi.org/10.1006/jmva.1999.1873

    2000 doi

  215. [228]

    Strong consistency of maximum quasi-likelihood estimators in generalized linear models with fixed and adaptive designs

    Chen, Kani and Hu, Inchi and Ying, Zhiliang. Strong consistency of maximum quasi-likelihood estimators in generalized linear models with fixed and adaptive designs. Ann. Statist. 1999. doi:10.1214/aos/1017938919

    1999 doi

  216. [229]

    A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

    Chernoff, Herman. A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations. Ann. Math. Statist. 1952. doi:10.1214/aoms/1177729330

    1952 doi

  217. [230]

    Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume =

    Chen, Mingli and Kato, Kengo and Leng, Chenlei , title =. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume =. doi:https://doi.org/10.1111/rssb.12444 , url =. https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/rssb.12444 , abstract =

    doi

  218. [231]

    S. S. Wilks , title =. The Annals of Mathematical Statistics , number =. 1938 , doi =

    1938

  219. [232]

    The Annals of Statistics , number =

    Jianqing Fan and Chunming Zhang and Jian Zhang , title =. The Annals of Statistics , number =. 2001 , doi =

    2001

  220. [233]

    The -Model Maximum Likelihood, Cram\'

    Wahlstr\". The -Model Maximum Likelihood, Cram\'. IEEE Transactions on Signal Processing , volume=

  221. [234]

    Pinhan Chen and Chao Gao and Anderson Y. Zhang. Partial Recovery for Top- k Ranking: Optimality of MLE and Sub-Optimality of Spectral Method. arXiv preprint arXiv:2006.16485. 2020

    2006 arXiv

  222. [235]

    Matrix estimation by Universal Singular Value Thresholding

    Chatterjee, Sourav. Matrix estimation by Universal Singular Value Thresholding. Ann. Statist. 2015. doi:10.1214/14-AOS1272

    2015 doi

  223. [236]

    and Courtade, Thomas A

    Pananjady, Ashwin and Mao, Cheng and Muthukumar, Vidya and Wainwright, Martin J. and Courtade, Thomas A. Worst-case versus average-case design for estimation from partial pairwise comparisons. Ann. Statist. 2020. doi:10.1214/19-AOS1838

    2020 doi

  224. [237]

    P. C. Sham and D. Curtis. An extended transmission/disequilibrium test (TDT) for multi-allele marker loci. Annals of Human Genetics. 1995

    1995

  225. [238]

    On the Distribution of the Likelihood Ratio

    Herman Chernoff. On the Distribution of the Likelihood Ratio. Annals of Mathematical Statistics. 1954

    1954

  226. [239]

    Cand\` e s

    Pragya Sur and Yuxin Chen and Emmanuel J. Cand\` e s. The likelihood ratio test in high-dimensional logistic regression is asymptotically a rescaled Chi-square. Probability Theory and Related Fields. 2019

    2019

  227. [240]

    2012 , eprint=

    Inverses of symmetric, diagonally dominant positive matrices and applications , author=. 2012 , eprint=

    2012

  228. [241]

    Community detection in dense random networks

    Ery Arias-Castro and Nicolas Verzelen. Community detection in dense random networks. Annals of Statistics. 2014

    2014

  229. [242]

    Sussman and Vince Lyzinski and Youngser Park and Carey E

    Minh Tang and Avanti Athreya and Daniel L. Sussman and Vince Lyzinski and Youngser Park and Carey E. Priebe. A Semiparametric Two-Sample Hypothesis Testing Problem for Random Graphs. Journal of Computational and Graphical Statistics. 2017

    2017

  230. [243]

    Sussman and Vince Lyzinski and Carey E

    Minh Tang and Avanti Athreya and Daniel L. Sussman and Vince Lyzinski and Carey E. Priebe. A nonparametric two-sample hypothesis testing problem for random graphs. Bernoulli. 2017

    2017

  231. [244]

    Testing Network Structure Using Relations Between Small Subgraph Probabilities

    Chao Gao and John Lafferty. Testing Network Structure Using Relations Between Small Subgraph Probabilities. arXiv preprint arXiv:1704.06742. 2017

    2017 arXiv

  232. [245]

    Community detection in sparse random networks

    Nicolas Verzelen and Ery Arias-Castro. Community detection in sparse random networks. Annals of Applied Probability. 2015

    2015

  233. [246]

    The Annals of Statistics , number =

    Jing Lei , title =. The Annals of Statistics , number =. 2016 , doi =

    2016

  234. [247]

    The Annals of Statistics , number =

    Debarghya Ghoshdastidar and Maurilio Gutzeit and Alexandra Carpentier and Ulrike von Luxburg , title =. The Annals of Statistics , number =. 2020 , doi =

    2020

  235. [248]

    The Annals of Statistics , number =

    Jiashun Jin and Zheng Tracy Ke and Shengming Luo , title =. The Annals of Statistics , number =. 2021 , doi =

    2021

  236. [249]

    Journal of the American Statistical Association , volume =

    Jianwei Hu and Jingfei Zhang and Hong Qin and Ting Yan and Ji Zhu , title =. Journal of the American Statistical Association , volume =. 2021 , publisher =. doi:10.1080/01621459.2020.1722676 , URL =

    2021 doi

  237. [250]

    The Annals of Probability , number =

    Magda Peligrad , title =. The Annals of Probability , number =. 1987 , doi =

    1987

  238. [251]

    Probability theory and related fields , volume=

    Central limit theorems for dependent variables, II , author=. Probability theory and related fields , volume=. 1987 , publisher=

    1987

  239. [252]

    Electronic Journal of Probability , volume=

    Weighted dependency graphs , author=. Electronic Journal of Probability , volume=. 2018 , publisher=

    2018

  240. [253]

    The Annals of Probability , volume=

    On normal approximations of distributions in terms of dependency graphs , author=. The Annals of Probability , volume=. 1989 , publisher=

    1989

  241. [254]

    The Annals of Probability , pages=

    Normal convergence by higher semiinvariants with applications to sums of dependent random variables and random graphs , author=. The Annals of Probability , pages=. 1988 , publisher=

    1988

  242. [255]

    Probability Theory and Related Fields , pages=

    A central limit theorem for generalized quadratic forms , author=. Probability Theory and Related Fields , pages=. 1987 , volume=

    1987

  243. [256]

    1984 , issn =

    Central limit theorem for integrated square error of multivariate nonparametric density estimators , journal =. 1984 , issn =. doi:https://doi.org/10.1016/0047-259X(84)90044-7 , url =

    1984 doi

  244. [257]

    B. M. Brown , title =. The Annals of Mathematical Statistics , number =. 1971 , doi =

    1971

  245. [258]

    1980 , publisher=

    Martingale limit theory and its application , author=. 1980 , publisher=

    1980

  246. [259]

    2023 , eprint=

    L_2 Regularized maximum likelihood for -model in large and sparse networks , author=. 2023 , eprint=

    2023

  247. [260]

    Wang and George Y

    Yuchung J. Wang and George Y. Wong , title =. Journal of the American Statistical Association , volume =. 1987 , publisher =. doi:10.1080/01621459.1987.10478385 , URL =

    1987 doi

  248. [261]

    O'Malley, A. J. and Marsden, P. V. , year=. The Analysis of Social Networks , url=. Health services & outcomes research methodology , volume=

  249. [262]

    Modeling Dyadic Interactions and Networks in Marketing , urldate =

    Dawn Iacobucci and Nigel Hopkins , journal =. Modeling Dyadic Interactions and Networks in Marketing , urldate =

  250. [263]

    Journal of Social and Personal Relationships , volume =

    Tasuku Igarashi and Jiro Takai and Toshikazu Yoshida , title =. Journal of Social and Personal Relationships , volume =. 2005 , doi =. https://doi.org/10.1177/0265407505056492 , abstract =

    2005 doi

  251. [264]

    Fienberg and Stanley Wasserman , title =

    Stephen E. Fienberg and Stanley Wasserman , title =. Journal of the American Statistical Association , volume =. 1981 , publisher =. doi:10.1080/01621459.1981.10477600 , URL =

    1981 doi

  252. [265]

    Probability: Theory and Examples , DOI=

    Durrett, Rick , year=. Probability: Theory and Examples , DOI=

  253. [266]

    2010 , isbn =

    Leskovec, Jure and Huttenlocher, Daniel and Kleinberg, Jon , title =. 2010 , isbn =. doi:10.1145/1753326.1753532 , booktitle =

    2010 doi

  254. [267]

    Electronic Journal of Statistics , number =

    Karim Lounici , title =. Electronic Journal of Statistics , number =. 2008 , doi =

    2008

  255. [268]

    Health Services and Outcomes Research Methodology , year=

    The analysis of social networks , author=. Health Services and Outcomes Research Methodology , year=

  256. [269]

    Network linear discriminant analysis , journal =

    Wei Cai and Guoyu Guan and Rui Pan and Xuening Zhu and Hansheng Wang , keywords =. Network linear discriminant analysis , journal =. 2018 , issn =. doi:https://doi.org/10.1016/j.csda.2017.07.007 , url =

    2018 doi

  257. [270]

    Asymptotic statistics (Cambridge series in statistical and probabilistic mathematics) , author=

  258. [271]

    Statistica Sinica , year =

    Yan, Ting , title =. Statistica Sinica , year =. doi:10.5705/ss.202019.0215 , fjournal =

    doi

  259. [272]

    2024 , journal =

    Edge differentially private estimation in the -model via jittering and method of moments , author=. 2024 , journal =

    2024

  260. [273]

    arXiv preprint arXiv:2010.13604 , year=

    A Sparse -Model with Covariates for Networks , author=. arXiv preprint arXiv:2010.13604 , year=

    2010 arXiv

  261. [274]

    2001 , Journal =

    Random graphs with arbitrary degree distributions and their applications , Author =. 2001 , Journal =. doi:10.1103/physreve.64.026118 , Number =

    2001 doi

  262. [275]

    and Diaconis, P

    Blitzstein, J. and Diaconis, P. (2011). A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Mathematics , 6(4):489--522

    2011

  263. [276]

    D., Yao, Q., and Yi, F

    Chang, J., Hu, Q., Kolaczyk, E. D., Yao, Q., and Yi, F. (2024). Edge differentially private estimation in the -model via jittering and method of moments. The Annals of Statistics , 52(2):708--728

    2024

  264. [277]

    Chatterjee, S., Diaconis, P., and Sly, A. (2011). Random graphs with a given degree sequence . The Annals of Applied Probability , 21(4):1400 -- 1435

    2011

  265. [278]

    Chen, M., Kato, K., and Leng, C. (2021). Analysis of networks via the sparse -model. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 83(5):887--910

    2021

  266. [279]

    and Lu, L

    Chung, F. and Lu, L. (2002). The average distances in random graphs with given expected degrees. Proceedings of the National Academy of Sciences , 99(25):15879--15882

    2002

  267. [280]

    Donoho, D. L. and Liu, R. C. (1991). Geometrizing Rates of Convergence, II . Ann. Statist. , 19(2):633--667

    1991

  268. [281]

    Du, Y., Qu, L., Yan, T., and Zhang, Y. (2023). Time-varying -model for dynamic directed networks. Scandinavian Journal of Statistics , 50(4):1687--1715

    2023

  269. [282]

    and R \'e nyi, A

    Erd o s, P. and R \'e nyi, A. (1960). On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci , 5:17--61

    1960

  270. [283]

    Feng, Q., Jin, J., Tian, Y., and Yan, T. (2026). Optimal estimators and tests for reciprocal effects. arXiv preprint arXiv:2601.01325

    2026

  271. [284]

    Fienberg, S. E. and Wasserman, S. (1981). An exponential family of probability distributions for directed graphs: Comment. Journal of the American Statistical Association , 76(373):54--57

    1981

  272. [285]

    Graham, B. S. (2017). An econometric model of network formation with degree heterogeneity. Econometrica , 85(4):1033--1063

    2017

  273. [286]

    and Kohler, H.-P

    Helleringer, S. and Kohler, H.-P. (2007). Sexual network structure and the spread of hiv in africa: evidence from likoma island, malawi. Aids , 21(17):2323--2332

    2007

  274. [287]

    and Wibisono, A

    Hillar, C. and Wibisono, A. (2013). Maximum entropy distributions on graphs. arXiv preprint arXiv:1301.3321

    2013 arXiv

  275. [288]

    Holland, P. W. and Leinhardt, S. (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association , 76(373):33--50

    1981

  276. [289]

    Jiang, B., Leng, C., Yan, T., Yao, Q., and Yu, X. (2025). A two-way heterogeneity model for dynamic networks. The Annals of Statistics , 53(6):2617--2641

    2025

  277. [290]

    T., and Luo, S

    Jin, J., Ke, Z. T., and Luo, S. (2018). Network global testing by counting graphlets. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsm \" a ssan, Stockholm, Sweden, July 10-15, 2018 , pages 2338--2346

    2018

  278. [291]

    T., and Luo, S

    Jin, J., Ke, Z. T., and Luo, S. (2021). Optimal adaptivity of signed-polygon statistics for network testing . The Annals of Statistics , 49(6):3408 -- 3433

    2021

  279. [292]

    and Slavkovi\'c, A

    Karwa, V. and Slavkovi\'c, A. (2016). Inference using noisy degrees-differentially private beta model and synthetic graphs. The Annals of Statistics , 44:87--112

    2016

  280. [293]

    and Leskovec, J

    Mcauley, J. and Leskovec, J. (2012). Learning to discover social circles in ego networks. NIPS , 1:539--547

    2012

  281. [294]

    and Bhattacharya, B

    Nandy, S. and Bhattacharya, B. B. (2024). Degree heterogeneity in higher-order networks: Inference in the hypergraph -model. IEEE Transactions on Information Theory , 70(8):6000--6024

    2024

  282. [295]

    and Newman, M

    Park, J. and Newman, M. E. J. (2004). Statistical mechanics of networks. Physical Review E , 70(6):066117

    2004

  283. [296]

    Perry, P. O. and Wolfe, P. J. (2012). Null models for network data. Available at http://arxiv.org/abs/1201.5871

    2012 arXiv

  284. [297]

    Rinaldo, A., Petrovi\'c, S., and Fienberg, S. E. (2013). Maximum lilkelihood estimation in the -model. Ann. Statist. , 41(3):1085--1110

    2013

  285. [298]

    Robins, G., Pattison, P., and Wang, P. (2009). Closure, connectivity and degree distributions: Exponential random graph (p*) models for directed social networks. Social Networks , 31(2):105--117

    2009

  286. [299]

    Shao, M., Zhang, Y., Wang, Q., Zhang, Y., Luo, J., and Yan, T. (2023). l_2 regularized maximum likelihood for -model in large and sparse networks

    2023

  287. [300]

    Stein, S., Feng, R., and Leng, C. (2025). A sparse beta regression model for network analysis. Journal of the American Statistical Association , 120(550):1281--1293

    2025

Pith tools

Reviewed July 7, 2026 · model on record in the stance chip above.