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Real-world networks have a complex topology comprising many elements often structured into communities. Revealing these communities helps researchers uncover the organizational and functional structure of the system that the network represents. However, detecting community structures in complex networks requires selecting a community detection method among a multitude of alternatives with different network representations, community interpretations, and underlying mechanisms. This tutorial focuses on a popular community detection method called the map equation and its search algorithm Infomap. The map equation framework for community detection describes communities by analyzing dynamic processes on the network. Thanks to its flexibility, the map equation provides extensions that can incorporate various assumptions about network structure and dynamics. To help decide if the map equation is a suitable community detection method for a given complex system and problem at hand - and which variant to choose - we review the map equation's theoretical framework and guide users in applying the map equation to various research problems.

Many real-world systems, from social networks to protein-protein interactions and species distributions, exhibit overlapping flow-based communities that reflect their functional organisation. However, reliably identifying such overlapping flow-based communities requires higher-order relational data, which are often unavailable. To address this challenge, we capitalise on the flow model underpinning the representation-learning algorithm node2vec and model higher-order flows through memory-biased random walks on first-order networks. Instead of simulating these walks, we model their higher-order dynamic constraints with compact models and control model complexity with an information-theoretic approach. Using the map equation framework, we identify overlapping modules in the resulting higher-order networks. Our compact-model approach proves robust across synthetic benchmark networks, reveals interpretable overlapping communities in empirical networks, and scales to large networks.

Life on Earth is a mosaic distributed across biogeographical regions. Their regional species pools have experienced distinct historical and eco-evolutionary pressures, leading to an expected context-dependent organization of biodiversity. Here we identify a general spatial organization within biogeographical regions of terrestrial and marine vertebrates, invertebrates and plants (more than 30,000 species). We detect seven types of areas in these biogeographical regions that reflect unique combinations of four fundamental aspects of biodiversity (species richness, range size, endemicity and biogeographical transitions). These areas form ordered layers from the core to the transition zones of the biogeographical regions, reflecting gradients in the biodiversity aspects, experiencing distinct environmental conditions, and exhibiting taxonomic dissimilarities due to nestedness. These findings suggest this ubiquitous organization is mainly driven by the action of two complementary environmental filters, one acting on species from regional hotspots and the other on species from permeable biogeographical boundaries. The influence of these regional filters extends across spatial scales and shapes global patterns of species richness. Regional biodiversity follows a universal core-to-transition organization governed by general forces operating across the tree of life and space.

Deep learning (DL) models require extensive data to achieve strong performance and generalization. Deep generative models (DGMs) offer a solution by synthesizing data. Yet current approaches for tabular data often fail to preserve feature correlations and distributions during training, struggle with multi-metric hyperparameter selection, and lack comprehensive evaluation protocols. We address this gap with a unified framework that integrates training, hyperparameter tuning, and evaluation. First, we introduce a novel correlation- and distribution-aware loss function that regularizes DGMs, enhancing their ability to generate synthetic tabular data that faithfully represents the underlying data distributions. Theoretical analysis establishes stability and consistency guarantees. To enable principled hyper-parameter search via Bayesian optimization (BO), we also propose a new multi-objective aggregation strategy based on iterative objective refinement Bayesian optimization (IORBO), along with a comprehensive statistical testing framework. We validate the proposed approach using a benchmarking framework with twenty real-world datasets and ten established tabular DGM baselines. The correlation-aware loss function significantly improves the synthetic data fidelity and downstream machine learning (ML) performance, while IORBO consistently outperforms standard Bayesian optimization (SBO) in hyper-parameter selection. The unified framework advances tabular generative modeling beyond isolated method improvements. Code is available at: https://github.com/vuhoangminh/TabGen-Framework.

Predicting future interactions or novel links in networks is an indispensable tool across diverse domains, including genetic research, online social networks, and recommendation systems. Among the numerous techniques developed for link prediction, those leveraging the networks' community structure have proven highly effective. For example, the recently proposed MapSim predicts links based on a similarity measure derived from the code structure of the map equation, a community-detection objective function that operates on network flows. However, the standard map equation assumes complete observations and typically identifies many small modules in networks where the nodes connect through only a few links. This aspect can degrade MapSim's performance on sparse networks. To overcome this limitation, we propose to incorporate a global regularization method based on a Bayesian estimate of the transition rates along with three local regularization methods. The regularized versions of the map equation compensate for incomplete observations and mitigate spurious community fragmentation in sparse networks. The regularized methods outperform standard MapSim and several state-of-the-art embedding methods in highly sparse networks. This performance holds across multiple real-world networks with randomly removed links, simulating incomplete observations. Among the proposed regularization methods, the global approach provides the most reliable community detection and the highest link prediction performance across different network densities. The principled method requires no hyperparameter tuning and runs at least an order of magnitude faster than the embedding methods.

Network methods capture the interplay between structure and dynamics of complex systems across scales by modeling indirect interactions as random walks. However, path data from real-world systems frequently exhibit memory effects that this first-order Markov model fails to capture. Although higher-order Markov models can capture these effects, they grow rapidly in size and require large amounts of data, making them prone to overfitting some parts and underfitting others in systems with uneven coverage. To address this challenge, we construct concise networks from path data by interpolating between first-order and second-order Markov models. We prioritize simplicity and interpretability by creating state nodes that capture prominent second-order effects and by proposing a transparent measure that balances model size and quality. Our concise networks reveal large-scale memory patterns in both synthetic and real-world systems while remaining far simpler than full second-order models.

Alternative splicing (AS) enables cells to produce multiple protein isoforms from single genes, fine-tuning protein function across numerous cellular processes. However, despite its biological importance, researchers lack effective tools to compare the domain composition of AS-derived protein isoforms because such comparisons require both structural data and specialized methods. Recent advances in AI-driven protein structure prediction, particularly AlphaFold2, now make accurate structural determination of splicing isoforms accessible, enabling functional AS analysis at the protein structure level. Here, we present IsoformMapper, a web resource that analyzes AS through network community analysis of protein structures. This approach captures 3D physical interactions between protein regions often missed by traditional domain analysis, enabling structural comparisons of isoforms across any biological system. We illustrate our tool by analyzing validated human Bcl-X protein isoforms, revealing how AS creates distinct community structures with antagonistic functional roles. As a proof of concept, we apply our tool to investigate how GENOMES UNCOUPLED1 (GUN1)-dependent retrograde signaling regulates plant de-etiolation through alternative splicing in Arabidopsis. In response to light, gun1 shows alterations in spliceosome component expression, suggesting that GUN1 contributes to AS regulation of genes essential for photosynthetic establishment. The gun1 mutant displays altered splice variant ratios for PNSL2, CHAOS, and SIG5. Our tool reveals that these isoforms form distinct protein community structures, demonstrating how AS impacts protein function and validating IsoformMapper's practical value.

Pathogenic bacteria encounter various stressors while residing in the host. They respond through intricate mechanisms of gene expression regulation, ensuring their survival and adaptation. Understanding how bacteria adapt to different stress conditions through regulatory processes of specific genes requires exploring complex transcriptional responses using gene co-expression networks. We employed a large transcriptome data set comprising 32 diverse human bacterial pathogens exposed to the same 11 host-mimicking stress conditions. Using the weighted gene co-expression network analysis algorithm, we generated bacterial gene co-expression networks. By associating modular eigengene expression with specific stress conditions, we identified gene co-expression modules and stress-specific stimulons, including genes with unique expression patterns under specific stress conditions. Suggesting a new potential role of the frm operon in responding to bile stress in enteropathogenic bacteria demonstrates the effectiveness of our approach. We also revealed the regulation of streptolysin S genes, involved in the production, processing, and export of streptolysin S, a toxin responsible for the beta-hemolytic phenotype of group A Streptococcus. In a comparative analysis of stress responses in three Escherichia coli strains from the core transcriptome, we revealed shared and unique expression patterns across the strains, offering insights into convergent and divergent stress responses. To help researchers perform similar analyses, we created the user-friendly web application Co-PATHOgenex. This tool aids in deepening our understanding of bacterial adaptation to stress conditions and in deciphering complex transcriptional responses of bacterial pathogens.IMPORTANCEUnveiling gene co-expression networks in bacterial pathogens has the potential for gaining insights into their adaptive strategies within the host environment. Here, we developed Co-PATHOgenex, an interactive and user-friendly web application that enables users to construct networks from gene co-expressions using custom-defined thresholds (https://avicanlab.shinyapps.io/copathogenex/). The incorporated search functions and visualizations within the tool simplify the usage and facilitate the interpretation of the analysis output. Co-PATHOgenex also includes stress stimulons for various bacterial species, which can help identify gene products not previously associated with a particular stress condition. Unveiling gene co-expression networks in bacterial pathogens has the potential for gaining insights into their adaptive strategies within the host environment. Here, we developed Co-PATHOgenex, an interactive and user-friendly web application that enables users to construct networks from gene co-expressions using custom-defined thresholds (https://avicanlab.shinyapps.io/copathogenex/). The incorporated search functions and visualizations within the tool simplify the usage and facilitate the interpretation of the analysis output. Co-PATHOgenex also includes stress stimulons for various bacterial species, which can help identify gene products not previously associated with a particular stress condition.

Inferring relations from correlational data allows researchers across the sciences to uncover complex connections between variables for insights into the underlying mechanisms. The researchers often represent inferred relations using Gaussian graphical models, requiring regularization to sparsify the models. Acknowledging that the modular structure of these inferred networks is often studied, we suggest module-based regularization to balance under- and overfitting. Compared with the graphical lasso, a standard approach using the Gaussian log-likelihood for estimating the regularization strength, this approach better recovers and infers modular structure in noisy synthetic and real data. The module-based regularization technique improves the usefulness of Gaussian graphical models in the many applications where they are employed.

Rapid metabolic responses to pathogens are essential for plant survival and depend on numerous transcription factors. Mediator is the major transcriptional co-regulator for integration and transmission of signals from transcriptional regulators to RNA polymerase II. Using four Arabidopsis Mediator mutants, med16, med18, med25 and cdk8, we studied how differences in regulation of their transcript and metabolite levels correlate to their responses to Pseudomonas syringae infection. We found that med16 and cdk8 were susceptible, while med25 showed increased resistance. Glucosinolate, phytoalexin and carbohydrate levels were reduced already before infection in med16 and cdk8, but increased in med25, which also displayed increased benzenoids levels. Early after infection, wild type plants showed reduced glucosinolate and nucleoside levels, but increases in amino acids, benzenoids, oxylipins and the phytoalexin camalexin. The Mediator mutants showed altered levels of these metabolites and in regulation of genes encoding key enzymes for their metabolism. At later stage, mutants displayed defective levels of specific amino acids, carbohydrates, lipids and jasmonates which correlated to their infection response phenotypes. Our results reveal that MED16, MED25 and CDK8 are required for a proper, coordinated transcriptional response of genes which encode enzymes involved in important metabolic pathways for Arabidopsis responses to Pseudomonas syringae infections.