Abstract
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.
Highlights
To connect structure and dynamics in complex systems, researchers model, for example, people navigating the web [1], rumors wandering around among citizens [2], and passengers traveling through airports [3], as flows on networks with random walkers
This dynamical process corresponds to a first-order Markov model of network flows: a passenger arriving in an airport will randomly continue to an airport proportional to the air traffic volume to that airport
We show that describing higher-order network flows with so-called sparse memory networks and identifying modules with long flow persistence times, such as groups of airports that contain frequently traveled routes, provides a general solution to reveal modular patterns in higher-order flows through complex systems
Summary
To connect structure and dynamics in complex systems, researchers model, for example, people navigating the web [1], rumors wandering around among citizens [2], and passengers traveling through airports [3], as flows on networks with random walkers. The links show where flows coming in to the center node are constrained to go depending on where they come from; (f) The system represented as a multilayer network with physical nodes for the objects and state nodes in layers corresponding to different data sources. State nodes are not bound to represent, for example, previous steps in memory networks or layers in multilayer networks, but are free to represent abstract states such as lumped states [18] or states in multilayer memory networks, which we demonstrate with multistep and multiquarter air traffic data In this way, a sparse memory network is a MultiAspect Graph with two aspects: the physical object and the flow state such as memory or layer [17]. We provide a detailed derivation of the information-theoretic map equation for identifying hierarchically nested modules with long flow persistence times in sparse memory networks, and introduce a new version of the map equation’s search algorithm Infomap that exploits the flexibility of sparse memory networks
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