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Communities are a common and widely studied structure in networks, typically assum- ing that the network is fully and correctly observed. In practice, network data are often collected by querying nodes about their connections. In some settings, all edges of a sam- pled node will be recorded, and in others, a node may be asked to name its connections. These sampling mechanisms introduce noise and bias, which can obscure the community structure and invalidate assumptions underlying standard community detection methods. We propose a general model for a class of network sampling mechanisms based on recording edges via querying nodes, designed to improve community detection for network data col- lected in this fashion. We model edge sampling probabilities as a function of both individual preferences and community parameters, and show community detection can be performed by spectral clustering under this general class of models. We also propose, as a special case of the general framework, a parametric model for directed networks we call the nomination stochastic block model, which allows for meaningful parameter interpretations and can be fitted by the method of moments. In this case, spectral clustering and the method of mo- ments are computationally ecient and come with theoretical guarantees of consistency. We evaluate the proposed model in simulation studies on unweighted and weighted net- works and under misspecified models. The method is applied to a faculty hiring dataset, discovering a meaningful hierarchy of communities among US business schools. 

Functional connections in the brain are frequently represented by weighted networks, with nodes representing locations in the brain and edges representing the strength of connectivity between these locations. One challenge in analyzing such data is that inference at the individual edge level is not particularly biologically meaningful; interpretation is more useful at the level of so-called functional systems or groups of nodes and connections between them; this is often called “graph-aware” inference in the neuroimaging literature. However, pooling over functional regions leads to significant loss of information and lower accuracy. Another challenge is correlation among edge weights within a subject which makes inference based on independence assumptions unreliable. We address both of these challenges with a linear mixed effects model, which accounts for functional systems and for edge dependence, while still modeling individual edge weights to avoid loss of information. The model allows for comparing two populations, such as patients and healthy controls, both at the functional regions level and at individual edge level, leading to biologically meaningful interpretations. We fit this model to resting state fMRI data on schizophrenic patients and healthy controls, obtaining interpretable results consistent with the schizophrenia literature. 

Summary Latent space models are frequently used for modelling single-layer networks and include many popular special cases, such as the stochastic block model and the random dot product graph. However, they are not well developed for more complex network structures, which are becoming increasingly common in practice. In this article we propose a new latent space model for multiplex networks, i.e., multiple heterogeneous networks observed on a shared node set. Multiplex networks can represent a network sample with shared node labels, a network evolving over time, or a network with multiple types of edges. The key feature of the proposed model is that it learns from data how much of the network structure is shared between layers and pools information across layers as appropriate. We establish identifiability, develop a fitting procedure using convex optimization in combination with a nuclear-norm penalty, and prove a guarantee of recovery for the latent positions provided there is sufficient separation between the shared and the individual latent subspaces. We compare the model with competing methods in the literature on simulated networks and on a multiplex network describing the worldwide trade of agricultural products.