Global spectral clustering in dynamic networks

Significance Statistical theory has mostly focused on static networks observed as a single snapshot in time. In reality, networks are generally dynamic, and it is of substantial interest to discover the clusters within each network to visualize and model their connectivities. We propose the persistent communities by eigenvector smoothing algorithm for detecting time-varying community structure and apply it to a recent dataset in which gene expression is measured during a broad range of developmental periods in rhesus monkey brains. The analysis suggests the existence of change points as well as periods of persistent community structure; these are not well estimated by standard methods due to the small sample size of any one developmental period or region of the brain. Community detection is challenging when the network structure is estimated with uncertainty. Dynamic networks present additional challenges but also add information across time periods. We propose a global community detection method, persistent communities by eigenvector smoothing (PisCES), that combines information across a series of networks, longitudinally, to strengthen the inference for each period. Our method is derived from evolutionary spectral clustering and degree correction methods. Data-driven solutions to the problem of tuning parameter selection are provided. In simulations we find that PisCES performs better than competing methods designed for a low signal-to-noise ratio. Recently obtained gene expression data from rhesus monkey brains provide samples from finely partitioned brain regions over a broad time span including pre- and postnatal periods. Of interest is how gene communities develop over space and time; however, once the data are divided into homogeneous spatial and temporal periods, sample sizes are very small, making inference quite challenging. Applying PisCES to medial prefrontal cortex in monkey rhesus brains from near conception to adulthood reveals dense communities that persist, merge, and diverge over time and others that are loosely organized and short lived, illustrating how dynamic community detection can yield interesting insights into processes such as brain development.

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