Global Calibration with Subgraph Features

• Global calibration using subgraph features is a method that transforms local motif counts into statistically calibrated global test statistics.
• It leverages central limit theorems and covariance derivations to achieve minimax-optimal power in network testing under null and stochastic block model alternatives.
• The approach enables efficient distributed spectral optimization through local subgraph updates that ensure global descent and scalability.

Global calibration using subgraph features refers to statistical and optimization methodologies that aggregate local subgraph statistics or optimization routines to infer or control global properties of networks. In statistical testing, this approach leverages the frequencies of small induced subgraphs—such as edges, V-shapes, and triangles—to design test statistics that are globally calibrated for null models like Erdős–Rényi random graphs, yielding tractable asymptotic laws and minimax-optimal power. In distributed graph optimization, subgraph-based schemes decompose global spectral objectives into locally tractable problems, using subgraph moments and alignment criteria to ensure globally coherent descent steps while respecting constraints.

1. Subgraph-Based Global Testing: Definitions and Motivations

Calibrating global tests from local subgraph frequencies involves transforming counts of small subgraphs into test statistics with analytically tractable distributions under null models. Let $G=(V,E)$ denote an undirected graph with $n=|V|$ nodes and adjacency matrix $A=(A_{ij})$. Frequencies of key 3-node subgraphs are defined as follows (Gao et al., 2017, Gao et al., 2017):

• Edge frequency: $\hat p = \frac{1}{\binom n2}\sum_{i<j}A_{ij}$
• Triangle frequency: $$\hat F_3 = \frac{1}{\binom n3}\sum_{i<j<k}A_{ij}A_{jk}A_{ki}$$
• V-shape (wedge) frequency: $$\hat F_2 = \frac{1}{\binom n3} \sum_{i<j<k} [(1 - A_{ij})A_{jk}A_{ki} + A_{ij}(1 - A_{jk})A_{ki} + A_{ij}A_{jk}(1 - A_{ki})]$$

For each $m\in\{0,1,2,3\}$, the deviation statistic $T_m = m \hat p^m (1-\hat p)^{3-m} - \hat F_m$ quantifies the discrepancy between empirical and null-model-expected frequencies for each 3-node motif.

Key invariants, $T_2$ and $T_3$, are selected due to their asymptotic independence in sparse regimes and their sufficiency for capturing the "global" deviation from random structure.

2. CLT-Based Calibration and Global Goodness-of-Fit

Under the Erdős–Rényi null $ER(n,p)$, theoretical calculations yield closed-form expressions for the expectations and covariances of local subgraph counts. Applying U-statistic and martingale CLT theory, one obtains an explicit joint central limit theorem (Gao et al., 2017, Gao et al., 2017):

$$\sqrt{\binom n3} \begin{pmatrix} T_2 \ T_3 \end{pmatrix} \xrightarrow{\;\;\;\;} N\left(0, \Sigma_p\right)$$

where the explicit covariance matrix $\Sigma_p$ is a function of $p$.

This supports construction of a global $\chi^2_2$-calibrated test statistic: $$T^2 = \binom n3 \left[ \frac{T_2^2}{3\hat p^2(1-\hat p)^2(1-3\hat p)^2 + 9\hat p^3(1-\hat p)^3} + \frac{T_3^2}{\hat p^3 (1-\hat p)^3 + 3 \hat p^4(1-\hat p)^2} \right]$$ which converges in law to $\chi^2_2$ under the null. The procedure admits optimal Type I error control, and the explicit calibration ensures that global decision thresholds can be set with high theoretical guarantee (Gao et al., 2017).

3. Minimax Power and Comparison With Community Detection

The resulting global subgraph-based test achieves minimax detection rates under stochastic block model (SBM) alternatives, without requiring explicit community recovery. For SBM with $k$ blocks, the test has power tending to $1$ provided the SNR condition

$\frac{n(a-b)^2}{k^{4/3}(a+b)} \rightarrow \infty$

is satisfied, compared to $k^2$ (or $k^3$) scaling in many community detection algorithms (Gao et al., 2017). For degree-corrected SBMs, mean shifts under the null and alternative can be calculated in closed form in terms of $a, b, k$, and degree factors, allowing precise global power calculations (Gao et al., 2017). This implies global calibration via subgraph features achieves weaker signal requirements than most algorithms for weak recovery.

4. Subgraph Sampling and Computational Trade-offs

Full enumeration of $\binom n3$ subgraphs can be computationally demanding. To address this, sampling schemes are statistically analyzed (Gao et al., 2017):

• Vertex-centric sampling: Sample $m$ nodes, compute all subgraph counts over triples containing them. Asymptotic validity is retained if $p^3 m n^2 \to \infty$.
• Triple-based sampling: Sample $|\Delta|$ unordered triples directly. Power and asymptotic approximations are preserved if $|\Delta| p^3 \to \infty$.

Sampling modifies the covariance scaling and test statistic normalization, but enables variance-cost trade-offs that maintain global calibration, given appropriate sample size and SNR scaling.

Sampling Scheme	Condition	Asymptotic Law
All triples	$n \to \infty$	$\chi^2_2$
$m$ vertices	$p^3 m n^2 \to \infty$	$\chi^2_2$ (rescaled)
$|\Delta|$ triples	$|\Delta| p^3 \to \infty$	$\chi^2_2$ (rescaled)

The theoretical framework quantifies the minimal sampling levels needed for valid global inference, further justifying subgraph-local calibration.

5. Decentralized Global Calibration in Graph Optimization

In distributed spectral optimization, global calibration is realized through decomposition of the global cost into subgraph-local optimization problems that, when strategically aligned, move the global objective in descent directions (Liu et al., 14 Nov 2025). The scheme is as follows:

• The global objective $J_G(w)$ is recast as a bilinear form $J_G(w)=\frac12 v(w)^{\!T} C v(w)$ in terms of moment vectors $v(w)$ of Laplacian powers.
• For centers $v\in V'$, local subgraph problems are defined on $a$-hop supports $H_v$, optimizing over 1-hop core edge weights subject to local budget and positivity constraints.
• SVD-based alignment on the $ZC$ matrix tests whether subgraph-local gradients approximate global gradients (rank-one dominant regime). Only locally-aligned subgraphs are updated.
• An iterate-and-embed algorithm advances the system by parallel, overlapping local updates, maintaining feasibility globally by disjointness and local constraint satisfaction.

Warm-starts via quadratic degree-regularization based on randomized gossip efficiently push node degrees toward their global average before full spectral optimization, accelerating convergence and achieving $>95\%$ of the centralized performance (Liu et al., 14 Nov 2025).

6. Learning-Based Local Proposers and Practical Considerations

To reduce per-node computational cost, a learning-based proposer uses a deep neural network trained to mimic optimal centralized one-shot updates for maximal 1-hop embeddings. This DNN is applied locally for edge updates and either serves as an initial warm-start or a refinement to the convex subgraph QPs. Empirical evidence indicates that such integration recovers $>95\%$ of centralized optimization gains after only a few passes, while a purely learning-only approach achieves about $30\%$ of centralized gains (Liu et al., 14 Nov 2025).

Practical attributes of this modular global calibration pipeline include:

• Low-degree polynomiality of spectral objectives,
• Strict feasibility preservation under local updates,
• Scalability to geometric graphs of hundreds of thousands of nodes,
• Use only of local (d-hop) information and neighbor-to-neighbor communication,
• Variance/computation trade-offs tunable by subgraph selection and sampling.

These features collectively demonstrate the operational benefits and scalability of globally calibrated subgraph-based frameworks in both statistical testing and distributed optimization.

7. Extensions and Theoretical Significance

Global calibration using subgraph features unifies local motif statistics and optimization algorithms under asymptotic and convex-analytic frameworks. The approach is distinguished by:

• Rigorous asymptotic null distribution derivations and explicit power characterizations against SBMs and degree-corrected models,
• Explicit mapping between local subgraph structure and global graph properties,
• Adaptability to Gaussian and weighted networks by analogous statistics and moment calculations (Gao et al., 2017),
• Full decentralized implementation potential in large-scale networked systems.

A plausible implication is that these subgraph-calibrated approaches provide a robust foundation for global inference and control in network science, especially in settings prohibitive of global enumeration or centralized computation. The methods leverage only localized structural information, yet realize global performance bounds and theoretically optimal or near-optimal guarantees.