Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments | Paper Citation Record

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Unresolved cited work

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This paper cites Fine asymptotics of the magnetization of the annealed dilute Curie-Weiss model.arXiv:2603.09672, 2026.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Fine asymptotics of the magnetization of the annealed dilute Curie-Weiss model.arXiv:2603.09672, 2026

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This paper cites Carlin, and Alan E.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Carlin, and Alan E

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This paper cites Universality of the mean-field for the Potts model.Probability Theory and Related Fields, 168(3):557–600, 2017.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Universality of the mean-field for the Potts model.Probability Theory and Related Fields, 168(3):557–600, 2017

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This paper cites Exact recovery in the Ising blockmodel.The Annals of Statistics, 47(4):1805–1834, August 2019.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Exact recovery in the Ising blockmodel.The Annals of Statistics, 47(4):1805–1834, August 2019

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This paper cites Spatial interaction and the statistical analysis of lattice systems.Journal of the Royal Statistical Society.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Spatial interaction and the statistical analysis of lattice systems.Journal of the Royal Statistical Society

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This paper cites Statistical analysis of non-lattice data.The Statistician, 24(3):179–195, 1975.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Statistical analysis of non-lattice data.The Statistician, 24(3):179–195, 1975

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This paper cites Universal limit theorems in graph coloring problems with connections to extremal combinatorics.The Annals of Applied Probability, 27(1):337–394, 2017.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Universal limit theorems in graph coloring problems with connections to extremal combinatorics.The Annals of Applied Probability, 27(1):337–394, 2017

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This paper cites Inference in Ising models.Bernoulli, 24(1):493–525, 2018.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Inference in Ising models.Bernoulli, 24(1):493–525, 2018

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This paper cites The Ising Model on a Two-Community Stochastic Block Model.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments The Ising Model on a Two-Community Stochastic Block Model

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This paper cites A nonparametric view of network models and Newman–Girvan and other modularities.Proceedings of the National Academy of Sciences, 106(50):21068–21073, 2009.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments A nonparametric view of network models and Newman–Girvan and other modularities.Proceedings of the National Academy of Sciences, 106(50):21068–21073, 2009

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This paper cites Bickel, Aiyou Chen, and Elizaveta Levina.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Bickel, Aiyou Chen, and Elizaveta Levina

Reference 12

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This paper cites The phase transition in inhomogeneous random graphs.Random Structures & Algorithms, 31(1):3–122, 2007.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments The phase transition in inhomogeneous random graphs.Random Structures & Algorithms, 31(1):3–122, 2007

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Unresolved cited work

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This paper cites Chayes, L´ aszl´ o Lov´ asz, Vera T.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Chayes, L´ aszl´ o Lov´ asz, Vera T

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This paper cites Chayes, L´ aszl´ o Lov´ asz, Vera T.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Chayes, L´ aszl´ o Lov´ asz, Vera T

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Observation 3722dd41-e291-4f3a-ba0b-9523b9b97556 · outbound

This paper cites The thermodynamics of the Curie–Weiss model with random couplings.Journal of Statistical Physics, 72(3-4):643–664, 1993.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments The thermodynamics of the Curie–Weiss model with random couplings.Journal of Statistical Physics, 72(3-4):643–664, 1993

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This paper cites Efficiently learning Ising models on arbitrary graphs.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Efficiently learning Ising models on arbitrary graphs

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This paper cites High temperature structure detection in ferromagnets.Infor- mation and Inference: A Journal of the IMA, 11(1):55–102, 2022.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments High temperature structure detection in ferromagnets.Infor- mation and Inference: A Journal of the IMA, 11(1):55–102, 2022

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This paper cites Estimation in spin glasses: A first step.The Annals of Statistics, 35(5):1931–1946, 2007.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Estimation in spin glasses: A first step.The Annals of Statistics, 35(5):1931–1946, 2007

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This paper cites Random graphs with a given degree sequence.The Annals of Applied Probability, 21(4):1400–1435, 2011.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Random graphs with a given degree sequence.The Annals of Applied Probability, 21(4):1400–1435, 2011

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This paper cites Joint parameter estimations for spin glasses.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Joint parameter estimations for spin glasses

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This paper cites Connected components in random graphs with given expected degree sequences.Annals of Combinatorics, 6:125–145, 2002.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Connected components in random graphs with given expected degree sequences.Annals of Combinatorics, 6:125–145, 2002

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Fluctuations of the

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This paper cites On consistency of a class of estimators for exponential families of Markov random fields on the lattice.The Annals of Statistics, 20(1):455–468, 1992.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments On consistency of a class of estimators for exponential families of Markov random fields on the lattice.The Annals of Statistics, 20(1):455–468, 1992

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This paper cites Asymptotics of Maximum Likelihood Estimators for the Curie-Weiss Model.The Annals of Statistics, 19(2):557–578, 1991.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Asymptotics of Maximum Likelihood Estimators for the Curie-Weiss Model.The Annals of Statistics, 19(2):557–578, 1991

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This paper cites CRC Press, 2018.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments CRC Press, 2018

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Learning Ising models from one or multiple samples

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This paper cites Testing Ising models.IEEE Trans- actions on Information Theory, 65(11):6829–6852, 2019.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Testing Ising models.IEEE Trans- actions on Information Theory, 65(11):6829–6852, 2019

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Regression from dependent observa- tions

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Logistic regression with peer-group effects via inference in higher-order Ising models

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This paper cites Pivotal CLTs for pseudolikelihood via conditional centering in dependent random fields.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Pivotal CLTs for pseudolikelihood via conditional centering in dependent random fields

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This paper cites Detecting structured signals in Ising models.The Annals of Applied Probability, 34(1A):1–45, 2024.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Detecting structured signals in Ising models.The Annals of Applied Probability, 34(1A):1–45, 2024

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Fluctuations in mean-field Ising models.The Annals of Applied Probability, 33(3):1961–2003, 2023

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This paper cites Ising model of financial markets with many assets.Physica A: Statistical Mechanics and its Applications, 462:250–254, 2016.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Ising model of financial markets with many assets.Physica A: Statistical Mechanics and its Applications, 462:250–254, 2016

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This paper cites Ellis.Entropy, Large Deviations, and Statistical Mechanics.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Ellis.Entropy, Large Deviations, and Statistical Mechanics

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Ellis, Charles M

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Ellis and Kongming Wang

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Markov random field image models and their applications to computer vision

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This paper cites Joint estimation of parameters in Ising model.The Annals of Statistics, 48(2):785–810, 2020.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Joint estimation of parameters in Ising model.The Annals of Statistics, 48(2):785–810, 2020

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Consistency of maximum likelihood and pseudolikelihood estimators for Gibbs distri- butions

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Green and Sylvia Richardson

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Springer, 2006

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This paper cites Information theoretic properties of Markov Random Fields, and their algorithmic applications.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Information theoretic properties of Markov Random Fields, and their algorithmic applications

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Latent space approaches to social network analysis.Journal of The American Statistical Association, 97(460):1090–1098, 2002

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Stochastic blockmodels: First steps.Social Networks, 5(2):109–137, 1983

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Hopfield

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This paper cites Beitrag zur theorie des ferromagnetismus.Zeitschrift f¨ ur Physik, 31:253–258, 1925.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Beitrag zur theorie des ferromagnetismus.Zeitschrift f¨ ur Physik, 31:253–258, 1925

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Fluctuations of the magnetization for Ising models on dense Erd˝ os–R´ enyi random graphs.Journal of Statistical Physics, 177:78–94, 2019

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Unresolved cited work

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Observation 3a0a2e78-2be5-44a1-9489-3471b43dd8e9 · outbound

This paper cites Fluctuations of the magnetization for Ising models on Erd˝ os–R´ enyi random graphs – the regimes of low temperature and external magnetic field.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Fluctuations of the magnetization for Ising models on Erd˝ os–R´ enyi random graphs – the regimes of low temperature and external magnetic field

Reference 52

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Observation c8f8feb2-7d07-4119-815c-9221125acc25 · outbound

This paper cites Statistically valid variational Bayes al- gorithm for ising model parameter estimation.Journal of Computational and Graphical Statistics, 33(1):75–84, 2024.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Statistically valid variational Bayes al- gorithm for ising model parameter estimation.Journal of Computational and Graphical Statistics, 33(1):75–84, 2024

Reference 53

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Observation 2603ae4b-d8dc-4e43-9c2c-d3d474296d8c · outbound

This paper cites Learning graphical models using multiplicative weights.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Learning graphical models using multiplicative weights

Reference 54

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Observation 38eec58a-3970-4084-9e43-74a7ff96b79d · outbound

This paper cites Limits of experiments.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Limits of experiments

Reference 55

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Observation 217aeab8-0bb1-43e5-8827-213645fbe62e · outbound

This paper cites CLT in high-dimensional Bayesian linear regression with low SNR.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments CLT in high-dimensional Bayesian linear regression with low SNR

Reference 56

Resolution: verified exact; arxiv_id, observed 2026-07-02T20:17:22.197805Z

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Observation 72496f2e-2afd-4b18-8581-9abb2b2b021e · outbound

This paper cites Fluctuations in random field Ising models.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Fluctuations in random field Ising models

Reference 57

Resolution: verified exact; local_arxiv, observed 2026-07-02T20:17:22.192897Z

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Observation bcc2dcdf-2fd9-4e39-97d6-28d13dbe4eb2 · outbound

This paper cites American Mathematical Soc., 2012.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments American Mathematical Soc., 2012

Reference 58

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Observation fd328dcc-e2e1-4f3a-89fc-f1300bbcf71d · outbound

This paper cites Inference in high-dimensional logistic regression under tensor network dependence.arXiv:2603.20082, 2026.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Inference in high-dimensional logistic regression under tensor network dependence.arXiv:2603.20082, 2026

Reference 59

Resolution: verified exact; arxiv_id, observed 2026-07-02T20:17:22.187952Z

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Observation 31a064f9-3087-4136-81b9-5f278151d5b7 · outbound

This paper cites The spread of innovations in social networks.Proceedings of the National Academy of Sciences, 107(47):20196–20201, 2010.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments The spread of innovations in social networks.Proceedings of the National Academy of Sciences, 107(47):20196–20201, 2010

Reference 60

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Observation 6fad9cab-c33d-4547-ac5d-a8d43a3d0ffe · outbound

This paper cites Global testing against sparse alternatives under Ising models.The Annals of Statistics, 46(5):2062–2093, 2018.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Global testing against sparse alternatives under Ising models.The Annals of Statistics, 46(5):2062–2093, 2018

Reference 61

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Observation 98026030-2bd2-47a1-875a-879526824306 · outbound

This paper cites On testing for parameters in Ising models.Annales de l’Institut Henri Poincar´ e – Probabilit´ es et Statistiques, 58(1):164–187, 2022.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments On testing for parameters in Ising models.Annales de l’Institut Henri Poincar´ e – Probabilit´ es et Statistiques, 58(1):164–187, 2022

Reference 62

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Observation 3b852b9e-629e-4f42-af79-bc6550f71edf · outbound

This paper cites Bhattacharya, and George Michailidis.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Bhattacharya, and George Michailidis

Reference 63

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Observation 4912e709-4d59-4803-89c9-756e302194c5 · outbound

This paper cites Property testing in high-dimensional Ising models.The Annals of Statistics, 47(5):2472–2503, 2019.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Property testing in high-dimensional Ising models.The Annals of Statistics, 47(5):2472–2503, 2019

Reference 64

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Observation d0d25a04-8bfd-4eb5-b993-4be5db7a1969 · outbound

This paper cites Asymptotic distributions of weightedU-statistics of degree 2.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Asymptotic distributions of weightedU-statistics of degree 2

Reference 65

Resolution: unresolved; no resolver link, observed 2026-06-27T20:40:02.905546Z

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Observation 3b5dd612-c0e6-41f4-9bb4-50474418e1d7 · outbound

This paper cites an unresolved cited work.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Unresolved cited work

Reference 66

Resolution: unresolved; no resolver link, observed 2026-06-27T20:40:02.905546Z

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Observation 08e7548a-cd18-4206-9653-697c5883283c · outbound

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Unresolved cited work

Reference 67

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Observation f595df91-f2ee-41c8-b88c-1e3b3d8bd447 · outbound

This paper cites an unresolved cited work.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Unresolved cited work

Reference 68

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Observation 6171035d-4a0c-4e5e-babf-e8ba38122769 · outbound

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Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Unresolved cited work

Reference 69

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Observation 2ebec0c7-f87f-4506-82aa-1a6685431ff5 · outbound

This paper cites arXiv preprint arXiv:2603.28702 , year=.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments arXiv preprint arXiv:2603.28702 , year=

Reference 70

Resolution: metadata mismatch; arxiv_id, observed 2026-07-02T20:17:22.190382Z

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Observation 5da1b02a-8d06-4e00-8997-1b662b92511f · outbound

This paper cites A polynomial-time algorithm for image segmentation using Ising models.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments A polynomial-time algorithm for image segmentation using Ising models

Reference 71

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Observation ce51659f-2095-44e5-9864-877503b1fdef · outbound

This paper cites Wainwright, and John D.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Wainwright, and John D

Reference 72

Resolution: unresolved; no resolver link, observed 2026-06-27T20:40:02.905546Z

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Observation 098ed46b-9758-462b-ba21-32a48b8d1a35 · outbound

This paper cites Hanson-wright inequality and sub-gaussian concentration.Elec- tronic Communications in Probability, 18:1–9, 2013.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Hanson-wright inequality and sub-gaussian concentration.Elec- tronic Communications in Probability, 18:1–9, 2013

Reference 73

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Observation 03439604-9131-4a92-b874-a27a21820be9 · outbound

This paper cites Santhanam and Martin J.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Santhanam and Martin J

Reference 74

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Observation 96d0fc06-a6f2-4f63-aae2-23a06e9c3a03 · outbound

This paper cites Stoˇ si´ c and Ivon P.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Stoˇ si´ c and Ivon P

Reference 75

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Observation 20402dd2-0376-4895-82c8-60c8d999f178 · outbound

This paper cites Cambridge university press, 2000.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Cambridge university press, 2000

Reference 76

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Observation d6df6637-aabb-4715-8a02-70f342f872de · outbound

This paper cites van der Vaart and Jon A.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments van der Vaart and Jon A

Reference 77

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Observation 45469419-5101-492c-9e90-bc6b9fd53db1 · outbound

This paper cites Cambridge university press, 2018.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Cambridge university press, 2018

Reference 78

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Observation b423a38f-b042-46e5-afe9-0ce93ee41175 · outbound

This paper cites Interaction Screening: Efficient and Sample-Optimal Learning of Ising Models.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments Interaction Screening: Efficient and Sample-Optimal Learning of Ising Models

Reference 79

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Observation 44408ca6-0364-4abf-835c-041ece31ab96 · outbound

This paper cites 1. Then for anypě1, eγϑN `γ 2ηN “1`γ ϑN `γ 2ηN `o Lp ˆ 1?N θN ˙ .(A.7) Furthermore, ifN θ 2 N.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments 1. Then for anypě1, eγϑN `γ 2ηN “1`γ ϑN `γ 2ηN `o Lp ˆ 1?N θN ˙ .(A.7) Furthermore, ifN θ 2 N

Reference 80

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Observation ef4fcd4e-903f-48eb-a66b-81e79e8e6df4 · outbound

This paper cites ˜XN ` ¯XN, Lemma A.8 implies thatE µN |UN |1`δ ≲1, for someδą0. Therefore, by H¨ older’s inequality, EµN |VN | ď ´ EµN |UN |1`δ ¯ 1 1`δ ˜ EµN.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments ˜XN ` ¯XN, Lemma A.8 implies thatE µN |UN |1`δ ≲1, for someδą0. Therefore, by H¨ older’s inequality, EµN |VN | ď ´ EµN |UN |1`δ ¯ 1 1`δ ˜ EµN

Reference 81

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Observation afd7db0f-cc1d-4330-8160-bf566fcc0f99 · outbound

This paper cites P ˜ N ˜ 0, 2´ r0,1s2 Wpx, yqdxdy ¸ ďt ¸ . This completes the proof of Theorem 2.1 (1).2 C.1.2.Proof of Theorem 2.1p2q.Recall from the proof of Theorem 5.1, that for everytPR, Eβ.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments P ˜ N ˜ 0, 2´ r0,1s2 Wpx, yqdxdy ¸ ďt ¸ . This completes the proof of Theorem 2.1 (1).2 C.1.2.Proof of Theorem 2.1p2q.Recall from the proof of Theorem 5.1, that for everytPR, Eβ

Reference 82

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Observation 5914e451-ab60-47d0-acd8-6a6cf3bf4abe · outbound

This paper cites e řN i“1 f p1q i ` ř 1ďiăjďN f p2q ij pσiσjq.(F.5) Expanding the exponential and the logarithm inf 1 i and using the factN θ N.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments e řN i“1 f p1q i ` ř 1ďiăjďN f p2q ij pσiσjq.(F.5) Expanding the exponential and the logarithm inf 1 i and using the factN θ N

Reference 83

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Observation 5dd07101-4f20-4c79-90ac-b43a447d3535 · outbound

This paper cites 1, such that E.

Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments 1, such that E

Reference 84

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