Topology of Multi-species Localization

Abstract: Spatial relationships in multi-species data can indicate and affect system outcomes and behaviors, ranging from disease progression in cancer to coral reef resilience in ecology; therefore, quantifying these relationships is an important problem across scientific disciplines. Persistent homology (PH), a key mathematical and computational tool in topological data analysis (TDA), provides a multiscale description of the shape of data. While it effectively describes spatial organization of species, such as cellular patterns in pathology, it cannot detect the shape relations between different types of species. Traditionally, PH analyzes single-species data, which limits the spatial analysis of interactions between different species. Leveraging recent developments in TDA and computational geometry, we introduce a scalable approach to quantify higher-order interactions in multi-species data. The framework can distinguish the presence of shape features or patterns in the data that are (i) common to multiple species of points, (ii) present in some species but disappear in the presence of other species, (iii) only visible when multiple species are considered together, and (iv) formed by some species and remain visible in the presence of others. We demonstrate our approach on two example applications. We identify (1) different behavioral regimes in a synthetic tumor micro-environment model, and (2) interspecies spatial interactions that are most significantly altered in colorectal cancer tissue samples during disease progression.