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A classic problem in computational biology is the identi cation of altered subnetworks: subnetworks of an interaction network that contain genes/proteins that are di erentially expressed, highly mutated, or otherwise aberrant compared to other genes/proteins. Numerous methods have been developed to solve this problem under various assumptions, but the statistical properties of these methods are o en unknown. For example, some widely-used methods are reported to output very large subnetworks that are di cult to interpret biologically. In this work, we formulate the identi cation of altered subnetworks as the problem of estimating the parameters of a class of probability distributions which we call the Altered Subset Distribution (ASD). We derive a connection between a popular method, jActiveModules, and the maximum likelihood estimator (MLE) of the ASD. We show that the MLE is statistically biased, explaining the large subnetworks output by jActiveModules. We introduce NetMix, an algorithm that uses Gaussian mixture models to obtain less biased estimates of the parameters of the ASD. We demonstrate that NetMix outperforms existing methods in identifying altered subnetworks on both simulated and real data, including the identi cation of di erentially expressed genes from both microarray and RNA-seq experiments and the identi cation of cancer driver genes in somatic mutation data. Availability: NetMix is available online at h ps://github.com/raphael-group/netmix.

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Recent studies suggest that social media usage -while linked to an increased diversity of information and perspectives for users -has exacerbated user polarization on many issues. A popular theory for this phenomenon centers on the concept of " lter bubbles": by automatically recommending content that a user is likely to agree with, social network algorithms create echo chambers of similarly-minded users that would not have arisen otherwise [54]. However, while echo chambers have been observed in real-world networks, the evidence for lter bubbles is largely post-hoc.In this work, we develop a mathematical framework to study the lter bubble theory. We modify the classic Friedkin-Johnsen opinion dynamics model by introducing another actor, the network administrator, who lters content for users by making small changes to the edge weights of a social network (for example, adjusting a news feed algorithm to change the level of interaction between users).On real-world networks from Reddit and Twitter, we show that when the network administrator is incentivized to reduce disagreement among users, even relatively small edge changes can result in the formation of echo chambers in the network and increase user polarization. We theoretically support this observed sensitivity of social networks to outside intervention by analyzing synthetic graphs generated from the stochastic block model. Finally, we show that a slight modi cation to the incentives of the network administrator can mitigate the lter bubble e ect while minimally a ecting the administrator's target objective, user disagreement.

Hypergraphs are used in machine learning to model higher-order relationships in data. While spectral methods for graphs are well-established, spectral theory for hypergraphs remains an active area of research. In this paper, we use random walks to develop a spectral theory for hypergraphs with edge-dependent vertex weights: hypergraphs where every vertex has a weight γ e ( ) for each incident hyperedge e that describes the contribution of to the hyperedge e. We derive a random walk-based hypergraph Laplacian, and bound the mixing time of random walks on such hypergraphs. Moreover, we give conditions under which random walks on such hypergraphs are equivalent to random walks on graphs. As a corollary, we show that current machine learning methods that rely on Laplacians derived from random walks on hypergraphs with edge-independent vertex weights do not utilize higher-order relationships in the data. Finally, we demonstrate the advantages of hypergraphs with edge-dependent vertex weights on ranking applications using real-world datasets.

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Spatially resolved transcriptomics (SRT) technologies simultaneously measure gene expression and spatial location of cells in a 2D tissue slice, enabling the study of spatial patterns of gene expression. Current approaches to model spatial variation in gene expression assume either that gene expression is determined by discrete cell types or that gene expression varies continuously across a tissue slice. However, neither of these modeling assumptions adequately represent spatial variation in gene expression: the first assumption ignores continuous variation within a spatial region containing cells of the same type, while the second assumption does not allow for discontinuous changes in expression, e.g., due to a sharp change in cell type composition. We propose a model of spatial patterns in gene expression that incorporates both discontinuous and continuous spatial variation in gene expression. Specifically, we model the expression of a gene in a layered tissue slice as a piecewise linear function of a single spatial coordinate with potential discontinuities at tissue layer boundaries. We formulate the problem of inferring tissue layer boundaries and gene expression functions for all genes simultaneously. We derive a dynamic programming algorithm to find the optimal boundaries of tissue layers when these boundaries are lines parallel to a coordinate axis. We generalize this algorithm to arbitrarily curved tissue layers by transforming the tissue geometry to be axis-aligned using the theory of conformal maps from complex analysis. We implement these algorithms in a method called Belayer. Applying Belayer to simulated data and to spatial transcriptomics data from the human dorsolateral prefrontal cortex, we demonstrate that Belayer achieves both higher accuracy in clustering tissue layers compared to state-of-the-art SRT clustering methods, and higher accuracy in identifying cortical layer marker genes compared to commonly-used methods for identification of spatially varying genes.