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Biogeographical regions (bioregions) reveal how different sets of species are spatially grouped and therefore are important units for conservation, historical biogeography, ecology, and evolution. Several methods have been developed to identify bioregions based on species distribution data rather than expert opinion. One approach successfully applies network theory to simplify and highlight the underlying structure in species distributions. However, this method lacks tools for simple and efficient analysis. Here, we present Infomap Bioregions, an interactive web application that inputs species distribution data and generates bioregion maps. Species distributions may be provided as georeferenced point occurrences or range maps, and can be of local, regional, or global scale. The application uses a novel adaptive resolution method to make best use of often incomplete species distribution data. The results can be downloaded as vector graphics, shapefiles, or in table format. We validate the tool by processing large data sets of publicly available species distribution data of the world's amphibians using species ranges, and mammals using point occurrences. We then calculate the fit between the inferred bioregions and WWF ecoregions. As examples of applications, researchers can reconstruct ancestral ranges in historical biogeography or identify indicator species for targeted conservation.

Comprehending complex systems by simplifying and highlighting important dynamical patterns requires modeling and mapping higher-order network flows. However, complex systems come in many forms and demand a range of representations, including memory and multilayer networks, which in turn call for versatile community-detection algorithms to reveal important modular regularities in the flows. Here we show that various forms of higher-order network flows can be represented in a unified way with networks that distinguish physical nodes for representing a complex system's objects from state nodes for describing flows between the objects. Moreover, these so-called sparse memory networks allow the information-theoretic community detection method known as the map equation to identify overlapping and nested flow modules in data from a range of different higher-order interactions such as multistep, multi-source, and temporal data. We derive the map equation applied to sparse memory networks and describe its search algorithm Infomap, which can exploit the flexibility of sparse memory networks. Together they provide a general solution to reveal overlapping modular patterns in higher-order flows through complex systems.

Unreliable network data can cause community-detection methods to overfit and highlight spurious structures with misleading information about the organization and function of complex systems. Here we show how to detect significant flow-based communities in sparse networks with missing links using the map equation. Since the map equation builds on Shannon entropy estimation, it assumes complete data such that analyzing undersampled networks can lead to overfitting. To overcome this problem, we incorporate a Bayesian approach with assumptions about network uncertainties into the map equation framework. Results in both synthetic and real-world networks show that the Bayesian estimate of the map equation provides a principled approach to revealing significant structures in undersampled networks.

Detecting significant community structure in networks with incomplete observations is challenging because the evidence for specific solutions fades away with missing data. For example, recent research shows that flow-based community detection methods can highlight spurious communities in sparse undirected and unweighted networks with missing links. Current Bayesian approaches developed to overcome this problem do not work for incomplete observations in weighted and directed networks that describe network flows. To overcome this gap, we extend the idea behind the Bayesian estimate of the map equation for unweighted and undirected networks to enable more robust community detection in weighted and directed networks. We derive an empirical Bayes estimate of the transitions rates that can incorporate metadata information and show how an efficient implementation in the community-detection method Infomap provides more reliable communities even with a significant fraction of data missing.