Bayesian Spiked Laplacian Graphs
Authors: Leo L Duan, George Michailidis, Mingzhou Ding; Volume 24(3):1−35, 2023.
Abstract
In the field of network analysis, it is common to encounter a collection of graphs that exhibit heterogeneity. For instance, there is an increasing availability of neuroimaging data from patient cohorts. An essential analytical task is to identify communities within these graphs, and graph Laplacian-based methods are frequently employed for this purpose. However, these methods currently have limitations: they can only handle a single network and do not provide measures of uncertainty for community assignments. In this study, we propose a probabilistic network model called the “Spiked Laplacian Graph,” which views an observed network as a transformation of the Laplacian and degree matrices of the network generating process. The Laplacian eigenvalues are modeled using a modified spiked structure, reducing the number of parameters in the eigenvectors. The sign patterns of these eigenvectors enable efficient estimation of the underlying community structure. Moreover, the posterior distribution of the eigenvectors provides uncertainty quantification for the community estimates. Additionally, we introduce a Bayesian non-parametric approach to address heterogeneity in a collection of graphs. Theoretical results establish the posterior consistency of the procedure and offer insights into the trade-off between model resolution and accuracy. We demonstrate the performance of this methodology using synthetic datasets and a neuroscience study related to brain activity in working memory.
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