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

Abstract: In this paper, we develop an inferential framework with sharp asymptotic optimality guarantees for Ising models on inhomogeneous random graphs in the subcritical parameter regime. We begin by characterizing the asymptotic distribution of the maximum likelihood (ML) estimate of the natural parameter, based on a single sample from the underlying model, covering both sparse and dense network regimes. Next, to overcome the computational intractability of the ML method, we propose a simple closed-form estimate obtained from a one-step approximation to the likelihood equation. We show that this estimate attains the same asymptotic distribution and variance as the ML estimate, thereby yielding a computationally efficient and asymptotically valid confidence interval for the natural parameter. We complement these inferential results by establishing a Hájek--Le Cam-type local asymptotic minimax theorem, showing that the proposed estimate achieves the smallest possible asymptotic maximum risk, both in rate and in leading constant, over shrinking neighborhoods of the true parameter. We also derive the corresponding limit of experiments. To the best of our knowledge, these are among the first sharp asymptotic optimality results for network-dependent data. Finally, we study goodness-of-fit testing for the natural parameter, deriving the local power of the likelihood ratio test and minimax detection rates. Our analysis relies on new fluctuation results for the sufficient statistic (Hamiltonian) and for the random partition function of Ising models on inhomogeneous random graphs, which are of independent interest.

• The paper establishes asymptotically sharp inferential results for the Ising model parameter, revealing distinct statistical behaviors in sparse versus dense network regimes.
• It derives exact asymptotic distributions for the maximum likelihood estimator, showing Gaussian consistency in sparse graphs and non-Gaussian limits in dense graphs.
• The study introduces a computationally efficient one-step estimator and proves local asymptotic minimax optimality, providing actionable insights for inference on complex network data.

Ising Models on Inhomogeneous Random Graphs: Inference and Asymptotic Optimality

Introduction and Model Framework

This paper develops a comprehensive inferential theory for Ising models defined on inhomogeneous random graphs, focusing particularly on the subcritical (high-temperature) regime and networks generated via general graphons. The Ising model, parameterized by $\beta$, is defined with an interaction matrix $J_N$ corresponding to the normalized adjacency matrix of the observed random graph, and the sufficient statistic is a quadratic form (Hamiltonian) that aggregates pairwise node interactions.

Networks are sampled via the inhomogeneous random graph (graphon) model, $G(N, \theta_N, W)$, allowing latent node-level heterogeneity through a symmetric measurable function $W:[0,1]^2 \rightarrow [0,1]$ and a sparsity parameter $\theta_N$. This broad framework encompasses Erdős-Rényi graphs, stochastic block models, and rank-one models as special cases.

The authors focus on sharp inferential characterizations for $\beta$ under a single observation of the global configuration, highlighting the pronounced differences in statistical behavior between the sparse ($\theta_N \ll 1$) and dense ($\theta_N \sim 1$) graph regimes.

Asymptotic Behavior of the Maximum Likelihood Estimate

An exact asymptotic distribution for the maximum likelihood (ML) estimate $\hat{\beta}_N$ is derived, conditional on the network realization:

• Sparse Regime ($\theta_N \ll 1$): $J_N$0 is $J_N$1-consistent, with asymptotically normal fluctuations:

$J_N$2

where $J_N$3 is the mean edge probability under $J_N$4.

• Dense Regime ($J_N$5): $J_N$6 is not consistent; its limit is non-Gaussian, involving a random mixture with probability mass at $J_N$7. The distribution is expressed in terms of the spectral decomposition of the graphon $J_N$8 and random quadratic forms of Gaussian variables, making explicit the dependence on the global network structure.

Notably, in the dense regime, the paper demonstrates the impossibility of consistent estimation for $J_N$9 in the subcritical phase, aligning with previous minimax lower bounds. This sharply contradicts prior intuition from the i.i.d. exponential family, marking a clear phase transition in inferential feasibility tied to network sparsity.

Computationally Efficient Optimal Estimation

Given the computational intractability of ML estimation (owing to the partition function), the authors introduce a closed-form estimator based on a one-step approximation to the likelihood equation. For the sparse regime, this estimator achieves the identical asymptotic variance as the ML estimator:

$G(N, \theta_N, W)$0

This result enables the construction of explicit, computationally efficient confidence intervals for $G(N, \theta_N, W)$1 with asymptotically valid coverage.

Local Asymptotic Minimaxity and Limit of Experiments

A major technical contribution is the proof of a local asymptotic minimax theorem (in the spirit of Hájek–Le Cam) for the estimation of $G(N, \theta_N, W)$2 in the sparse regime. The introduced estimator is shown to achieve the minimax risk, both in rate and leading constant, for shrinking neighborhoods of size $G(N, \theta_N, W)$3 around the true parameter.

The limit of statistical experiments is derived in full detail:

• Sparse Graphs: The limiting experiment is Gaussian (i.e., LAN holds).
• Dense Graphs: The limiting experiment is non-Gaussian and depends intricately on the graphon spectrum, involving a sum of independent non-central $G(N, \theta_N, W)$4 random variables. This is, to the best knowledge of the authors, the first precise characterization of a non-Gaussian limiting experiment for network-dependent data of this type.

The paper provides a table of these regimes, but here we clarify:

Regime	Fluctuations of MLE	Limiting Experiment Type	Minimax Rate
Sparse ($G(N, \theta_N, W)$5)	Gaussian, consistent	Gaussian/LAN	$G(N, \theta_N, W)$6
Dense ($G(N, \theta_N, W)$7)	Non-Gaussian, inconsistent	Non-Gaussian (spectral)	Impossible

Minimax Hypothesis Testing and Goodness-of-Fit

For hypothesis testing of $G(N, \theta_N, W)$8, likelihood ratio tests are shown to be minimax rate-optimal. The detection boundary is $G(N, \theta_N, W)$9 in the sparse regime; it is proven that no test (including those with adaptive thresholds) can separate alternatives differing from the null at a smaller rate.

A notable feature is that the developed test statistics and thresholds can be computed without knowledge of the underlying graphon $W:[0,1]^2 \rightarrow [0,1]$0: in the sparse regime by empirical plug-in estimators; in the dense regime via calibrated multiplier-bootstrap using graphon spectral estimates computed from the observed adjacency matrix.

The derivation of local power stretches to scaling alternatives at the detection rate, with explicit limiting values, and a full minimax lower bound demonstrates the sharpness of these results.

Technical Contributions: Fluctuations of Sufficient Statistic and Partition Function

The paper provides detailed fluctuation results for the Ising model sufficient statistic and (random) partition function on general inhomogeneous random graphs:

• Fluctuation regimes are precisely identified in terms of network sparsity, connecting to classical and recent random graph results.
• These fluctuation results not only resolve open questions for general graphons but also specialize to the homogeneous (Erdős–Rényi) and stochastic block models, recovering and generalizing earlier results (see, e.g., [kabluchko2021fluctuations]).

Theoretical and Practical Implications

The findings fundamentally alter the prospect for statistical inference on dependencies in networked data:

• In the sparse regime, classical inference remains possible and optimality is attainable.
• In the dense regime, consistent estimation and testing are infeasible—network dependence fundamentally limits statistical resolution, independent of sample size.

Practical implications include:

• Guidance for statistical modeling on real-world network data, especially regarding the feasibility of parameter inference in high-dependence regimes.
• Generalizability to a broad class of graphical models and consideration of nuisance parameters (e.g., in the graphon) in statistically optimal testing.

From a theoretical perspective:

• This work sharpens the boundary between high- and low-temperature statistical physics models, highlighting the quantifiable and provable effects of network heterogeneity on inference.
• The non-Gaussian limiting experiment in the dense regime opens new directions for both probability theory and statistics in high-dependence data.

Conclusion

This paper establishes a self-contained, sharp inferential theory for Ising models on general inhomogeneous random graphs. Through rigorous asymptotic analysis and novel fluctuation results, it precisely quantifies where inference is possible, where it is fundamentally impossible, and how to achieve minimax optimality when possible. It lays the technical groundwork for further study of inference in dependent-data models, especially as motivated by complex network applications. The advanced spectral analysis and bootstrapping methods developed may find further use in broader classes of graphical and spatial models.