Michael Thomas Schaub

Extending community detection from pairwise networks to hypergraphs introduces fundamental theoretical challenges. Hypergraphs exhibit structural heterogeneity with no direct graph analog: Hyperedges of varying orders can connect nodes across communities in diverse configurations. Crucially, this means that any hypergraph community detection algorithm must make a choice regarding which order or balance of hyperedges it prefers to maintain or split during partitioning. We address these challenges by developing a unified framework for community detection in nonuniform hypergraphs under the hypergraph stochastic block model. We introduce a general signal-to-noise ratio that enables a quantitative analysis of trade-offs unique to higher-order networks, such as which hypergedges we choose to split across communities and how we choose to split them. Building on this framework, we derive a Bethe Hessian operator for nonuniform hypergraphs that provides efficient spectral clustering with principled model selection. We characterize the resulting spectral detectability threshold and compare it to belief propagation limits, showing the methods coincide for uniform hypergraphs but diverge in nonuniform settings. Synthetic experiments confirm our analytical predictions and reveal systematic biases toward preserving higher-order and balanced shape hyperedges. Application to empirical data demonstrates the practical relevance of these higher-order detectability trade-offs in real-world systems.

Higher-order interactions fundamentally shape collective dynamics in oscillator networks. The topological Kuramoto model captures these effects by extending synchronization models to include interactions between cells of arbitrary dimension within simplicial and cell complexes. We introduce the topological nonlinear Kirchhoff conditions to characterize all phase-locked states of the topological Kuramoto model. These states are organized by winding numbers associated with generalized independent cycles, which quantify how phases wind around these cycles. Using rings, Platonic solids, and regular simplices as illustrative examples, we uncover a universal rule: boundaries must have at least five elements for multistability to arise. We further find that independent winding numbers associated with lower- and higher-dimensional boundaries generate cascades of multistability across dimensions. These results show how the topology and boundary structure of cell complexes influence phase locking and multistability, and provide a general framework for collective dynamics on cell complexes.

The prevalence of multimorbid patients (i.e., one person with two or more diseases) is critical for patients, medical doctors, and society alike since it complicates treatment and results in high (treatment) costs. However, contemporary medicine mainly specializes in single-organ approaches, frequently overlooking the interactions between different organs causing multimorbidities. Recent research on bilateral organ crosstalk (OCT) reveals the complex interplay between organs and the importance of studying OCT, especially involving more than two organs. Tackling the challenges associated with multi-organ crosstalk requires fundamental methodological advances in computer science, specifically related to handling multi-modal data, the combinatorial explosion of complexity, and data sensitivity. Thus, research should generally be directed toward (i) methods for identifying affected organs, (ii) concepts for discovering disease progression, and (iii) models for validation in clinical studies. When addressing these challenges interdisciplinarily, computer science and medicine will jointly improve the understanding of OCT and multimorbidity, resulting in novel treatment options for multimorbid patients, reduced healthcare costs, and broad benefits for society.

The adult mammalian heart has a limited regenerative capacity. Following injury, cardiomyocytes undergo a hypertrophic response accompanied by polyploidization, which has been described as a barrier to proliferation and regeneration of the heart ^1,2 . However, the unique molecular programs of polyploidy, or genome multiplied cardiomyocytes, and their influence on the disease-related myocardial remodelling process remains unclear. Here, we integrate single-nuclei and high-resolution spatial multi-omics across human, rat, and mouse hearts to define novel cardiac cell states and their tissue niches in ischemic and non-ischemic heart disease. Computational analysis across scales allowed us to generate detailed networks of the cardiac tissue remodelling process as well as tissue and sub-cellular environments uniquely enriched in polyploid cardiomyocytes or their diploid origins. We identify a conserved, dichotomous transcriptional program distinguishing diploid from polyploid cardiomyocytes. Polyploid cardiomyocytes demonstrated rewired metabolic and chromatin-remodeling transcriptional programs and recapitulate the gene signature of immature human fetal cardiomyocytes. Notably, we observe that polyploid cardiomyocytes—rather than the general myocyte population—are the primary sites of enrichment for major heart-failure drug targets, including the mineralocorticoid, β1-adrenergic, and glucagon-like peptide-1 receptors. Based on our cross-species dataset we further identified TNIK, a Wnt-pathway regulator expressed in polyploid cardiomyocytes across species, as a potential therapeutic target and demonstrate that pharmacological TNIK inhibition improves cardiac function after myocardial infarction in rats. Together, this species-spanning, disease-resolved study redefines cardiomyocyte heterogeneity in heart disease and suggests a therapeutic path to heart failure treatment by targeting polyploid cardiomyocytes.

Higher-order interactions are increasingly being recognised as fundamental to our understanding of complex systems, networks, and the development of the next generation of AI algorithms. However, modelling higher-order interactions requires us to go beyond graphs and networks, which can only encode pairwise interactions, and so demands a new theory. Hypergraphs and simplicial complexes (also called higher-order networks), which arise as natural mathematical representations of higher-order complex systems, are therefore attracting increasing attention. The mathematics of higher-order networks is already providing important insights, yet many fundamental mathematical questions remain unsolved; for instance, in spectral graph theory, discrete topology, and higher-order network dynamics. This roadmap summarizes the scientific discussions that took place on these topics between pure mathematicians, theoretical physicists, computer and network scientists at the Newton Institute Satellite meeting on `Hypergraphs: Theory and Applications'. We survey the current state-of-the-art in higher-order network research, and propose some trajectories for future research, including in areas such as extremal and spectral hypergraph theory, discrete topology, higher-order dynamics, higher-order machine learning, and applications in the brain and social sciences.

Computational trajectory analysis is a key computational task for inferring differentiation trees from this single-cell data. An open challenge is the prediction of complex and multi-branching trees from multimodal data. To address these challenges, we present PHLOWER (decomposition of the Hodge Laplacian for inferring trajectories from flows of cell differentiation), which leverages the harmonic component of the Hodge decomposition on simplicial complexes to infer trajectory embeddings from single-cell multimodal data. These natural representations of cell differentiation facilitate the estimation of their underlying differentiation trees. We evaluate PHLOWER through benchmarking with multi-branching differentiation trees and using kidney organoid multimodal and spatial single-cell data. These demonstrate the power of PHLOWER in both the inference of complex trees and the identification of transcription factors regulating off-target cells in kidney organoids. Thus, PHLOWER enables inference of complex branching trajectories and prediction of transcriptional regulators by leveraging multimodal data.

We consider the following inference problem: Given a set of edge-flow signals observed on a graph, lift the graph to a cell complex, such that the observed edge-flow signals can be represented as a sparse combination of gradient and curl flows on the cell complex. Specifically, we aim to augment the observed graph by a set of 2 -cells (polygons encircled by closed, non-intersecting paths), such that the eigenvectors of the Hodge Laplacian of the associated cell complex provide a sparse, interpretable representation of the observed edge flows on the graph. As it has been shown that the general problem is NP-hard in prior work, we here develop a novel matrix-factorization-based heuristic to solve the problem. Using computational experiments, we demonstrate that our new approach is significantly less computationally expensive than prior heuristics, while achieving only marginally worse performance in most settings. In fact, we find that for specifically noisy settings, our new approach outperforms the previous state of the art in both solution quality and computational speed.