Mapping higher-order network flows in memory and multilayer networks with Infomap

Comprehending complex systems by simplifying and highlighting important dynamical patterns often require modeling and mapping of 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. We show that various higher-order network flow representations can be represented with sparse memory networks, in which the multilevel map equation can identify overlapping and nested modules that capture network flows. 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.