Abstract
Inference on vertex-aligned graphs is of wide theoretical and practical importance. There are, however, few flexible and tractable statistical models for correlated graphs, and even fewer comprehensive approaches to parametric inference on data arising from such graphs. In this paper, we consider the correlated Bernoulli random graph model (allowing different Bernoulli coefficients and edge correlations for different pairs of vertices), and we introduce a new variance-reducing technique—called balancing—that can refine estimators for model parameters. Specifically, we construct a disagreement statistic and show that it is complete and sufficient; balancing can be interpreted as Rao-Blackwellization with this disagreement statistic. We show that for unbiased estimators of functions of model parameters, balancing generates uniformly minimum variance unbiased estimators (UMVUEs). However, even when unbiased estimators for model parameters do not exist—which, as we prove, is the case with both the heterogeneity correlation and the total correlation parameters—balancing is still useful, and lowers mean squared error. In particular, we demonstrate how balancing can improve the efficiency of the alignment strength estimator for the total correlation, a parameter that plays a critical role in graph matchability and graph matching runtime complexity.
Highlights
Paired random graphs with a natural alignment between their vertex sets arise in a wide variety of application domains; for example, the interaction dynamics of the same set of users across two social media platforms, or a pair of connectomes as imaged from two different subjects of the same species
Under a nondegeneracy condition, we prove in Theorem 4 that if S is an unbiased estimator of g(θ) S is the UMVU estimator of g(θ); this is because S is a Rao-Blackwellization of S via the disagreement statistic H, and we prove in our main result Theorem 2 that H is complete and sufficient, under the nondegeneracy condition
Our second group of contributions: In the context of a correlated Bernoulli random graph model, the alignment strength statistic str was shown in [4] to be a strongly consistent estimator of total correlation T between the pair of graphs; we point out here that str is not a balanced statistic, as noted above, the mean squared error in estimating T is reduced by using str instead
Summary
Paired random graphs with a natural alignment between their vertex sets arise in a wide variety of application domains; for example, the interaction dynamics of the same set of users across two social media platforms, or a pair of connectomes (brain graphs) as imaged from two different subjects of the same species. The authors empirically demonstrated—in broad families within the model— that graph matching complexity and matchability are each functions of total correlation They proved that the statistic called alignment strength is a strongly consistent estimator of total correlation. Our second group of contributions: In the context of a correlated Bernoulli random graph model, the alignment strength statistic str was shown in [4] to be a strongly consistent estimator of total correlation T between the pair of graphs; we point out here that str is not a balanced statistic, as noted above, the mean squared error in estimating T is reduced by using str instead.
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