Consistent estimation of the number of communities via regularized network embedding

The network analysis plays an important role in numerous application domains including biomedicine. Estimation of the number of communities is a fundamental and critical issue in network analysis. Most existing studies assume that the number of communities is known a priori, or lack of rigorous theoretical guarantee on the estimation consistency. In this paper, we propose a regularized network embedding model to simultaneously estimate the community structure and the number of communities in a unified formulation. The proposed model equips network embedding with a novel composite regularization term, which pushes the embedding vector toward its center and pushes similar community centers collapsed with each other. A rigorous theoretical analysis is conducted, establishing asymptotic consistency in terms of community detection and estimation of the number of communities. Extensive numerical experiments have also been conducted on both synthetic networks and brain functional connectivity network, which demonstrate the superior performance of the proposed method compared with existing alternatives.

Bayesian Fusion Model Enhanced Codfish Classification Using Near Infrared and Raman Spectrum

In this study, a Bayesian-based decision fusion technique was developed for the first time to quickly and non-destructively identify codfish using near infrared (NIRS) and Raman spectroscopy (RS). NIRS and RS spectra from 320 codfish samples were collected, and separate partial least squares discriminant analysis (PLS-DA) models were developed to establish the relationship between the raw data and cod identity for each spectral technique. Three decision fusion methods: decision fusion, data layer or feature layer, were tested and compared. The decision fusion model based on the Bayesian algorithm (NIRS-RS-B) was developed on the optimal discrimination features of NIRS and RS data (NIRS-RS) extracted by the PLS-DA method whereas the other fusion models followed conventional, non-Bayesian approaches. The Bayesian model showed enhanced classification metrics (92% sensitivity, 98% specificity, 98% accuracy) that were significantly superior to those demonstrated by any of other two spectroscopic methods (NIRS, RS) and the two data fusion methods (data layer fused, NIRS-RS-D, or feature layer fused, NIRS-RS-F). This novel proposed approach can provide an alternative classification for codfish and potentially other food speciation cases.

A Useful Criterion on Studying Consistent Estimation in Community Detection

In network analysis, developing a unified theoretical framework that can compare methods under different models is an interesting problem. This paper proposes a partial solution to this problem. We summarize the idea of using a separation condition for a standard network and sharp threshold of the Erdös-Rényi random graph to study consistent estimation, and compare theoretical error rates and requirements on the network sparsity of spectral methods under models that can degenerate to a stochastic block model as a four-step criterion SCSTC. Using SCSTC, we find some inconsistent phenomena on separation condition and sharp threshold in community detection. In particular, we find that the original theoretical results of the SPACL algorithm introduced to estimate network memberships under the mixed membership stochastic blockmodel are sub-optimal. To find the formation mechanism of inconsistencies, we re-establish the theoretical convergence rate of this algorithm by applying recent techniques on row-wise eigenvector deviation. The results are further extended to the degree-corrected mixed membership model. By comparison, our results enjoy smaller error rates, lesser dependence on the number of communities, weaker requirements on network sparsity, and so forth. The separation condition and sharp threshold obtained from our theoretical results match the classical results, so the usefulness of this criterion on studying consistent estimation is guaranteed. Numerical results for computer-generated networks support our finding that spectral methods considered in this paper achieve the threshold of separation condition.

Two provably consistent divide-and-conquer clustering algorithms for large networks

In this article, we advance divide-and-conquer strategies for solving the community detection problem in networks. We propose two algorithms that perform clustering on several small subgraphs and finally patch the results into a single clustering. The main advantage of these algorithms is that they significantly bring down the computational cost of traditional algorithms, including spectral clustering, semidefinite programs, modularity-based methods, likelihood-based methods, etc., without losing accuracy, and even improving accuracy at times. These algorithms are also, by nature, parallelizable. Since most traditional algorithms are accurate, and the corresponding optimization problems are much simpler in small problems, our divide-and-conquer methods provide an omnibus recipe for scaling traditional algorithms up to large networks. We prove the consistency of these algorithms under various subgraph selection procedures and perform extensive simulations and real-data analysis to understand the advantages of the divide-and-conquer approach in various settings.

Doubly Stochastic Normalization of the Gaussian Kernel Is Robust to Heteroskedastic Noise

A fundamental step in many data-analysis techniques is the construction of an affinity matrix describing similarities between data points. When the data points reside in Euclidean space, a widespread approach is to from an affinity matrix by the Gaussian kernel with pairwise distances, and to follow with a certain normalization (e.g. the row-stochastic normalization or its symmetric variant). We demonstrate that the doubly-stochastic normalization of the Gaussian kernel with zero main diagonal (i.e., no self loops) is robust to heteroskedastic noise. That is, the doubly-stochastic normalization is advantageous in that it automatically accounts for observations with different noise variances. Specifically, we prove that in a suitable high-dimensional setting where heteroskedastic noise does not concentrate too much in any particular direction in space, the resulting (doubly-stochastic) noisy affinity matrix converges to its clean counterpart with rate m ^-1/2, where m is the ambient dimension. We demonstrate this result numerically, and show that in contrast, the popular row-stochastic and symmetric normalizations behave unfavorably under heteroskedastic noise. Furthermore, we provide examples of simulated and experimental single-cell RNA sequence data with intrinsic heteroskedasticity, where the advantage of the doubly-stochastic normalization for exploratory analysis is evident.

Network cross-validation by edge sampling

While many statistical models and methods are now available for network analysis, resampling of network data remains a challenging problem. Cross-validation is a useful general tool for model selection and parameter tuning, but it is not directly applicable to networks since splitting network nodes into groups requires deleting edges and destroys some of the network structure. In this paper we propose a new network resampling strategy, based on splitting node pairs rather than nodes, that is applicable to cross-validation for a wide range of network model selection tasks. We provide theoretical justification for our method in a general setting and examples of how the method can be used in specific network model selection and parameter tuning tasks. Numerical results on simulated networks and on a statisticians’ citation network show that the proposed cross-validation approach works well for model selection.

Global spectral clustering in dynamic networks

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.