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Abstract Functional connectivity networks of the human brain are commonly studied using tools from complex network theory. Existing methods focus on functional connectivity within a single frequency band. However, it is well-known that higher order brain functions rely on the integration of information across oscillations at different frequencies. Therefore, there is a need to study these cross-frequency interactions. In this paper, we use multilayer networks to model functional connectivity across multiple frequencies, where each layer corresponds to a different frequency band. We then introduce the multilayer modularity metric to develop a multilayer community detection algorithm. The proposed approach is applied to electroencephalogram (EEG) data collected during a study of error monitoring in the human brain. The differences between the community structures within and across different frequency bands for two response types, i.e. error and correct, are studied. The results indicate that following an error response, the brain organizes itself to form communities across frequencies, in particular between theta and gamma bands while a similar cross-frequency community formation is not observed following the correct response. 

Networks offer a compact representation of complex systems such as social, communication, and biological systems. Traditional network models are often inadequate to capture the diverse nature of contemporary networks, which may exhibit temporal variation and multiple types of interactions between entities. Multilayer networks (MLNs) provide a more comprehensive representation by allowing interactions between nodes to be represented by different types of links, each reflecting a distinct type of interaction. Community detection reveals meaningful structure and provides a better understanding of the overall functioning of networks. Current approaches to multilayer community detection are either limited to community detection over the aggregated network or are extensions of single-layer community detection methods with simplifying assumptions such as a common community structure across layers. Moreover, most of the existing methods are limited to multiplex networks with no inter-layer edges. In this paper, we introduce a spectral-clustering-based community detection method for two-layer MLNs. The problem of detecting the community structure is formulated as an optimization problem where the normalized cut for each layer is minimized simultaneously with the normalized cut for the bipartite network along with regularization terms that ensure the consistency of the within- and across-layer community structures. The proposed method is evaluated on both synthetic and real networks and compared to state-of-the-art methods. MLNs. The problem of detecting the community structure is formulated as an optimization problem where the normalized cut for each layer is minimized simultaneously with the normalized cut for the bipartite network along with regularization terms that ensure the consistency of the intra- and inter-layer community structures. The proposed method is evaluated on both synthetic and real networks and compared to state-of-the-art methods. 

Abstract BackgroundCharacterizing the topology of gene regulatory networks (GRNs) is a fundamental problem in systems biology. The advent of single cell technologies has made it possible to construct GRNs at finer resolutions than bulk and microarray datasets. However, cellular heterogeneity and sparsity of the single cell datasets render void the application of regular Gaussian assumptions for constructing GRNs. Additionally, most GRN reconstruction approaches estimate a single network for the entire data. This could cause potential loss of information when single cell datasets are generated from multiple treatment conditions/disease states. ResultsTo better characterize single cell GRNs under different but related conditions, we propose the joint estimation of multiple networks using multiple signed graph learning (scMSGL). The proposed method is based on recently developed graph signal processing (GSP) based graph learning, where GRNs and gene expressions are modeled as signed graphs and graph signals, respectively. scMSGL learns multiple GRNs by optimizing the total variation of gene expressions with respect to GRNs while ensuring that the learned GRNs are similar to each other through regularization with respect to a learned signed consensus graph. We further kernelize scMSGL with the kernel selected to suit the structure of single cell data. ConclusionsscMSGL is shown to have superior performance over existing state of the art methods in GRN recovery on simulated datasets. Furthermore, scMSGL successfully identifies well-established regulators in a mouse embryonic stem cell differentiation study and a cancer clinical study of medulloblastoma.