[Paper Review] On the Theoretical Properties of the Network Jackknife

Qiaohui Lin, Robert Lunde+1 more|arXiv (Cornell University)|Apr 19, 2020

Limits and Structures in Graph Theory38 references4 citations

TL;DR

This paper establishes theoretical properties of the network jackknife—a leave-node-out resampling method—for estimating variance in network statistics under the sparse graphon model. It proves an Efron-Stein-type inequality showing conservative variance estimation in expectation and establishes consistency for a general class of count functionals, demonstrating competitive finite-sample performance compared to subsampling and other resampling methods in simulations and real data.

ABSTRACT

We study the properties of a leave-node-out jackknife procedure for network data. Under the sparse graphon model, we prove an Efron-Stein-type inequality, showing that the network jackknife leads to conservative estimates of the variance (in expectation) for any network functional that is invariant to node permutation. For a general class of count functionals, we also establish consistency of the network jackknife. We complement our theoretical analysis with a range of simulated and real-data examples and show that the network jackknife offers competitive performance in cases where other resampling methods are known to be valid. In fact, for several network statistics, we see that the jackknife provides more accurate inferences compared to related methods such as subsampling.

Motivation & Objective

• To develop a theoretically grounded resampling method for network statistics that is robust to model misspecification and does not require full graphon estimation.
• To establish conservative variance estimation for permutation-invariant network functionals under the sparse graphon model.
• To prove consistency of the network jackknife for a broad class of count functionals, such as subgraph counts and transitivity measures.
• To compare the finite-sample performance of the network jackknife with existing methods like subsampling and bootstrap in simulated and real network data.
• To demonstrate computational efficiency and practical viability of the jackknife in large-scale network inference tasks.

Proposed method

• Proposes a leave-node-out jackknife procedure where the variance of a network functional is estimated by averaging squared deviations from leave-one-node-out estimates.
• Applies an Efron-Stein-type inequality to show that the expected jackknife variance estimate is an upper bound on the true variance for permutation-invariant functionals.
• Uses a filtration argument and uniform integrability to establish asymptotic normality and convergence of the jackknife variance estimator.
• Derives a conservative variance inequality (Proposition 4) for a general functional $ Z_n $, showing $ \mathrm{Var}(Z_n) \leq E(\widehat{\mathrm{Var}}_{\mathrm{JACK}} Z_n) $ without requiring permutation invariance.
• Employs subsampling as a benchmark for comparison, using block sizes $ b = 0.05n, 0.1n, 0.2n $, and compares confidence intervals and computation times.
• Validates the method on real Facebook college networks and simulated networks under the sparse graphon model, focusing on triangle density, two-star density, and normalized transitivity.

Experimental results

Research questions

• RQ1 Does the network jackknife produce conservative variance estimates in expectation for permutation-invariant network functionals under the sparse graphon model?
• RQ2 Is the network jackknife variance estimator consistent for general classes of count functionals such as subgraph counts?
• RQ3 How does the finite-sample performance of the network jackknife compare to subsampling and bootstrap methods in terms of coverage and accuracy?
• RQ4 Can the network jackknife be implemented efficiently in large-scale network settings without requiring latent space estimation or large resampling overhead?
• RQ5 Under what conditions does the jackknife outperform subsampling in terms of variance estimation accuracy and computation time?

Key findings

• The network jackknife produces conservative variance estimates in expectation, satisfying $ \mathrm{Var}(Z_n) \leq E(\widehat{\mathrm{Var}}_{\mathrm{JACK}} Z_n) $, which justifies its use as a robust, safety-first estimator.
• For a general class of count functionals, the jackknife variance estimator is consistent under the sparse graphon model, with convergence established via uniform integrability and Slutsky’s theorem.
• In simulations and real data from six college Facebook networks, jackknife-based confidence intervals for triangle density, two-star density, and normalized transitivity were in close agreement with those from subsampling across different block sizes.
• Computationally, the jackknife was often faster than subsampling with larger block sizes ($ b = 0.2n $), especially in large networks, due to the nested structure of subgraph counts enabling efficient computation.
• The jackknife outperformed the nonparametric bootstrap and latent position bootstrap in terms of computational efficiency and did not require accurate estimation of the underlying graphon.
• The alternative jackknife estimator $ \widehat{\mathrm{Var}}_{\mathrm{JACK}} Z_n = \sum_{i=1}^n (Z_n - \widetilde{Z}_{n,i})^2 $, which does not require permutation invariance, still satisfies the conservative inequality, though it may be less precise.