The map equation

• Published in: The European physical journal. ST, Special topics Special topics, 2009-11, Vol.178 (1), p.13-23
• Main Authors: Rosvall, M., Axelsson, D., Bergstrom, C. T.
• Format: Article
• Language: English
• Subjects: Atomic; Classical and Continuum Physics; Condensed Matter Physics; Materials Science; Measurement Science and Instrumentation; Molecular; Optical and Plasma Physics; Physics; Physics and Astronomy
• ISSN: 1951-6355; 1951-6401; 1951-6401
• DOI: 10.1140/epjst/e2010-01179-1

Many real-world networks are so large that we must simplify their structure before we can extract useful information about the systems they represent. As the tools for doing these simplifications proliferate within the network literature, researchers would benefit from some guidelines about which of the so-called community detection algorithms are most appropriate for the structures they are studying and the questions they are asking. Here we show that different methods highlight different aspects of a network's structure and that the the sort of information that we seek to extract about the system must guide us in our decision. For example, many community detection algorithms, including the popular modularity maximization approach, infer module assignments from an underlying model of the network formation process. However, we are not always as interested in how a system's network structure was formed, as we are in how a network's extant structure influences the system's behavior. To see how structure influences current behavior, we will recognize that links in a network induce movement across the network and result in system-wide interdependence. In doing so, we explicitly acknowledge that most networks carry flow. To highlight and simplify the network structure with respect to this flow, we use the map equation. We present an intuitive derivation of this flow-based and information-theoretic method and provide an interactive on-line application that anyone can use to explore the mechanics of the map equation. The differences between the map equation and the modularity maximization approach are not merely conceptual. Because the map equation attends to patterns of flow on the network and the modularity maximization approach does not, the two methods can yield dramatically different results for some network structures. To illustrate this and build our understanding of each method, we partition several sample networks. We also describe an algorithm and provide source code to efficiently decompose large weighted and directed networks based on the map equation.