Subgraph counting estimation for the $β$-model in sparse networks

Abstract: The $β$-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 $β$-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 $\max_t θ_t \to 0$ and $θ_t |θ|_1 \to \infty$, 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 $\hatβ_t$ up to a constant factor. Here, $\hatβ_t$ is the CCR estimator of the node-specific parameter $β_t$, $θ_t = \exp(β_t)$ and $θ=(θ_1, \ldots, θ_n)$. Even if the whole network density is close to the Erdős-Rényi lower bound $\log n/n$, the CCR estimator for the single parameter $β_t$ is still consistent as long as $θ_t |θ|_1 \to \infty$. 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 $θ_t |θ|_1$. Numerical studies and an application to a sparse network data set demonstrate our theoretical findings.

• The paper introduces the Cycle Counting Ratio (CCR) estimator to recover node-specific β parameters from 3-node subgraph counts, effectively addressing sparsity challenges.
• It leverages explicit log-ratios of non-isomorphic triangles to bypass iterative MLE, ensuring computational scalability for large sparse networks.
• Theoretical and empirical analyses confirm the estimator's minimax optimality, asymptotic normality, and uniform consistency under weak sparsity conditions.

Subgraph Counting Estimation for the $\beta$-Model in Sparse Networks

Introduction and Motivation

The $\beta$-model serves as a canonical random graph model with node-specific parameters controlling degree heterogeneity. Estimating these parameters in sparse regimes is central to statistical network analysis but presents significant difficulties due to non-standard inference, non-existence of the maximum likelihood estimator (MLE), and computational and statistical instability. The work under discussion provides a comprehensive approach to parameter estimation for the $\beta$-model in sparse networks, introducing the Cycle Counting Ratio (CCR) estimator based on log-ratios of subgraph counting statistics—specifically, 3-node cycles.

Methodological Innovations

The primary contribution is the proposal and analysis of the CCR estimator, grounded in the idea that the ratio of the probabilities of two non-isomorphic cycles yields direct access to the node-specific parameter, $\beta_t$. By focusing on 3-node cycles ("triangles" and their non-isomorphic variants), the approach avoids the combinatorial explosion and analytical complications associated with longer cycles. The use of explicit algebraic operations, rather than iterative likelihood maximization, provides desirable computational scalability for large sparse graphs.

Figure 1: All non-isomorphic cycles with $3$ nodes.

The justification for this focus arises from the underlying combinatorial structure: among the possible 3-node non-isomorphic cycles (depicted in Figure 1), only certain pairs yield valid identifiability via log-ratio conditions required by the theory. The estimator for any node $t$ is given by

$$\hat{\beta}_t = \frac{1}{2}\log\frac{T_{n,t}(a)}{T_{n,t}(b)},$$

where $T_{n,t}(a)$ and $T_{n,t}(b)$ denote appropriate counts of 3-node subgraphs involving $t$ corresponding to different cycle types, and the exponent $\beta$0 and selection of $\beta$1 and $\beta$2 follow from the structure in Proposition 1 (in the text, conditions (critical)).

The computational complexity is linear in the maximum degree, $\beta$3, of the network—making it feasible for massive sparse graphs.

Theoretical Properties

Consistency and Minimax Optimality

Under the sparsity regime $\beta$4 and the effective signal condition $\beta$5—where $\beta$6 and $\beta$7—the CCR estimator is proven to be consistent for each $\beta$8. Notably, these assumptions are strictly weaker than previously required for penalized or regularized MLE approaches, which either constrain parameter configurations or impose higher minimum density.

The mean-squared error (MSE) of the CCR estimator matches the minimax lower bound in the sense that

$\beta$9

and no estimator can uniformly outperform this rate over the natural parameter space. This establishes asymptotic minimaxity for CCR in the sparse $\beta$0-model.

Asymptotic Normality

For each $\beta$1, provided the above conditions and slightly reinforced versions, the CCR estimator is asymptotically normal with variance matching that of the MLE (when the latter exists). Specifically, for a finite collection of nodes,

$\beta$2

with $\beta$3.

Uniform Consistency

Uniform consistency over all node parameters can be achieved under a more stringent but still weak condition on the minimum node strength and network size, specifically when $\beta$4. The maximal error over nodes vanishes with $\beta$5 at a rate determined by network sparsity.

Practical Implications and Empirical Evaluation

Beyond theory, the estimator's robustness and computational tractability are corroborated through extensive simulations and application to real-world data. Key findings include:

• For networks with densities down to the Erdős-Rényi lower bound ($\beta$6), CCR achieves low error and closely matches (or outperforms, when the MLE is undefined) the classical MLE and penalized approaches.
• No substantial empirical gain is realized by using longer cycle statistics (e.g., 5 or 7 nodes), reinforcing 3-node cycle sufficiency both for accuracy and computational speed.
• In observed networks (e.g., Facebook ego-networks), CCR provides interpretable degree parameter estimates and enables reliable hypothesis testing regarding node homogeneity.

Significance in the Landscape of Network Models

This work addresses a fundamental gap in the estimation of random graph models for genuinely sparse regimes, where classical likelihood-based methods and their regularized variants break down either theoretically (due to existence issues, ill-posedness) or computationally. The CCR estimator's reliance on explicit cycle counts and log-ratio representation circumvents these barriers and scales to large $\beta$7.

Methodologically, the subgraph counting approach connects with recent advances in combinatorial and spectral statistics for network inference (e.g., graphlet analysis, signed-polygon statistics), reinforcing the utility of small motif statistics in parameter recovery and optimality characterization.

The weak conditions for identification and consistency may inspire analogous methodologies in broader classes of random graph models, including directed/weighted/hypergraphs, or in settings with covariates or dynamic structure. The approach may also inform differentially private estimation and robust inference under network data perturbations.

Future Prospects

Outstanding theoretical directions include the development of high-dimensional (diverging $\beta$8) joint central limit theorems for correlated parameter estimators, as well as extension of the cycle counting paradigm to more general exponential random graph models (ERGMs), or models with additional structure (e.g., community, attribute, or time dependence).

Practically, the CCR estimator's explicitness and scalability position it as a promising tool for large-network empirical studies, enabling hypothesis testing and parameter estimation in settings inaccessible to conventional techniques.

Conclusion

This paper delivers a substantial advance in sparse network parameter estimation, establishing the CCR estimator as a consistent, computationally efficient, and minimax optimal alternative for the $\beta$9-model under weak assumptions. It demonstrates that motif-based subgraph counts, when judiciously selected, provide both statistical identifiability and scalability—even at near-critical sparsity. The implications extend to both applied network analysis and the theoretical underpinnings of random graph inference, setting a new standard for estimation methodology in the sparse regime (2607.05273).