Community detection has attracted increasing attention during the past decade, and many algorithms have been proposed to find the underlying community structure in a given network. Many of these algorithms are based on modularity maximization, and these methods suffer from the resolution limit. In order to detect the underlying cluster structure, we propose a new convex formulation to decompose a partially observed adjacency matrix of a network into low-rank and sparse components. In such decomposition, the low-rank component encodes the cluster structure under certain assumptions. We also devise an alternating direction method of multipliers with increasing penalty sequence to solve this problem; and compare it with Louvain method, which maximizes the modularity, on some synthetic randomly generated networks. Numerical results show that our method outperforms Louvain method on the randomly generated networks when variance among cluster sizes increases. Moreover, empirical results also demonstrate that our formulation is indeed tighter than the robust PCA formulation, and is able to find the true clustering when the robust PCA formulation fails.
The field of landscape genetics enables the study of infectious disease dynamics by connecting the landscape features with evolutionary changes. Quantifying genetic correlation across space is helpful in providing insight into the rate of spread of an infectious disease. We investigate two genetic patterns in spatially referenced singlenucleotide polymorphisms (SNPs): isolation by distance and isolation by resistance.We model the data using a Generalized Linear Mixed effect Model (GLMM) with spatially referenced random effects and provide a novel approach for estimating parameters in spatial GLMMs. In this approach, we use the links between binary probit models and bivariate normal probabilities to directly compute the model-based covariance function for spatial binary data. Parameter estimation is based on minimizing sum of squared distance between the elements of sample covariance and model-based covariance matrices. We analyze data including Brucella Abortus SNPs from spatially referenced hosts in the Greater Yellowstone Ecosystem (GYE).
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