Community detection tasks have received a lot of attention across statistics, machine learning, and information theory with a large body of work concentrating on theoretical guarantees for the stochastic block model. One line of recent work has focused on modeling the spectral embedding of a network using Gaussian mixture models (GMMs) in scaling regimes where the ability to detect community memberships improves with the size of the network. However, these regimes are not very realistic. This paper provides tractable methodology motivated by new theoretical results for networks with non-vanishing noise. We present a procedure for community detection using GMMs that incorporates certain truncation and shrinkage effects that arise in the non-vanishing noise regime. We provide empirical validation of this new representation using both simulated and real-world data.

The study of network data in the social and health sciences frequently concentrates on two distinct tasks (1) detecting community structures among nodes and (2) associating covariate information to edge formation. In much of this data, it is likely that the effects of covariates on edge formation differ between communities (e.g. age might play a different role in friendship formation in communities across a city). In this work, we introduce a latent space network model where coefficients associated with certain covariates can depend on latent community membership of the nodes. We show that ignoring such structure can lead to either over-or under-estimation of covariate importance to edge formation and propose a Markov Chain Monte Carlo approach for simultaneously learning the latent community structure and the community specific coefficients. We leverage efficient spectral methods to improve the computational tractability of our approach.

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Let G be a finitely generated group, and let Σ be a finite subset that generates G as a monoid. The word problem of G with respect to Σ consists of all words in the free monoid Σ * that are equal to the identity in G. The co-word problem of G with respect to Σ is the complement in Σ * of the word problem. We say that a group G is coCF if its co-word problem with respect to some (equivalently, any) finite generating set Σ is a context-free language.We describe a generalized Thompson group V (G,θ) for each finite group G and homomorphism θ: G → G. Our group is constructed using the cloning systems introduced by Witzel and Zaremsky. We prove that V (G,θ) is coCF for any homomorphism θ and finite group G by constructing a pushdown automaton and showing that the co-word problem of V (G,θ) is the cyclic shift of the language accepted by our automaton.A version of a conjecture due to Lehnert says that a group has context-free co-word problem exactly if it is a finitely generated subgroup of V. The groups V (G,θ) where θ is not the identity homomorphism do not appear to have obvious embeddings into V, and may therefore be considered possible counterexamples to the conjecture.Demonstrative subgroups of V , which were introduced by Bleak and Salazar-Diaz, can be used to construct embeddings of certain wreath products and amalgamated free products into V . We extend the class of known finitely generated demonstrative subgroups of V to include all virtually cyclic groups.2010 Mathematics Subject Classification. 20F10, 20E06.

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