Alignment strength and correlation for graphs

Fishkind, Donniell E.; Meng, Lingyao; Sun, Aixin; Priebe, Carey E.; Lyzinski, Vince

doi:10.1016/j.patrec.2019.05.008

Cited by 11 publications

(16 citation statements)

References 23 publications

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“…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; however, we point out here that str is not a balanced statistic, hence, as noted above, the mean squared error in estimating T is reduced by using str instead. We then prove (in Theorems 7, 8, 9) that there do not exist unbiased estimators for several correlation parameters, including T .…”

Section: Overviewmentioning

confidence: 82%

“…It is always the case that 0 ≤ T ≤ 1. In [4] it was empirically demonstrated-for the correlated graphs in broad families within our model-that graph matching complexity as well as graph matchability are each functions of total correlation, hence the importance of total correlation.…”

Section: Important Statistics and Functions Of The Parametersmentioning

confidence: 90%

“…Our second group of contributions of this paper focuses on very recent advances in [4] regarding the correlated Bernoulli random graph model. Specifically, the paper [4] introduced and showed the importance of a novel model parameter called total correlation, which combines inter-and intra-graph contributions to a unified measure of the correlation between the pair of graphs. The authors empirically demonstrated-in broad families within the modelthat graph matching complexity and matchability are each functions of total correlation.…”

Section: Overviewmentioning

confidence: 99%

“…It is not hard to see that that the choice of these parameters uniquely specifies the joint distribution of the two graphs (see Appendix A of [4]). Indeed, we can sample from the distribution in the following manner.…”

Section: Correlated Bernoulli Random Graphsmentioning

confidence: 99%

See 3 more Smart Citations

On a complete and sufficient statistic for the correlated Bernoulli random graph model

Fishkind

Athreya

Meng

et al. 2021

Electron. J. Statist.

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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 balancingthat 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 es-Complete & sufficient statistic for correlated Bernoulli random graph 2337 timators (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.

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Section: Overviewmentioning

confidence: 82%

Section: Important Statistics and Functions Of The Parametersmentioning

confidence: 90%

Section: Overviewmentioning

confidence: 99%

Section: Correlated Bernoulli Random Graphsmentioning

confidence: 99%

See 2 more Smart Citations

On a complete and sufficient statistic for the correlated Bernoulli random graph model

Fishkind

Athreya

Meng

et al. 2021

Electron. J. Statist.

Self Cite

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“…Two recent results for graphs with the same number of vertices do provide some hints. [45] investigates the ratio of A − B 2…”

Section: Discussionmentioning

confidence: 99%

Matched Filters for Noisy Induced Subgraph Detection

Sussman

Park

Priebe

et al. 2019

IEEE Trans. Pattern Anal. Mach. Intell.

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The problem of finding the vertex correspondence between two noisy graphs with different number of vertices where the smaller graph is still large has many applications in social networks, neuroscience, and computer vision. We propose a solution to this problem via a graph matching matched filter: centering and padding the smaller adjacency matrix and applying graph matching methods to align it to the larger network. The centering and padding schemes can be incorporated into any algorithm that matches using adjacency matrices. Under a statistical model for correlated pairs of graphs, which yields a noisy copy of the small graph within the larger graph, the resulting optimization problem can be guaranteed to recover the true vertex correspondence between the networks. However, there are currently no efficient algorithms for solving this problem. To illustrate the possibilities and challenges of such problems, we use an algorithm that can exploit a partially known correspondence and show via varied simulations and applications to Drosophila and human connectomes that this approach can achieve good performance.

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