Maps of random walks on complex networks reveal community structure

Abstract

To comprehend the multipartite organization of large-scale biological and social systems, we introduce an information theoretic approach that reveals community structure in weighted and directed networks. We use the probability flow of random walks on a network as a proxy for information flows in the real system and decompose the network into modules by compressing a description of the probability flow. The result is a map that both simplifies and highlights the regularities in the structure and their relationships. We illustrate the method by making a map of scientific communication as captured in the citation patterns of >6,000 journals. We discover a multicentric organization with fields that vary dramatically in size and degree of integration into the network of science. Along the backbone of the network-including physics, chemistry, molecular biology, and medicine-information flows bidirectionally, but the map reveals a directional pattern of citation from the applied fields to the basic sciences.clustering ͉ compression ͉ information theory ͉ map of science ͉ bibiometrics B iological and social systems are differentiated, multipartite, integrated, and dynamic. Data about these systems, now available on unprecedented scales, often are schematized as networks. Such abstractions are powerful (1, 2), but even as abstractions they remain highly complex. It therefore is helpful to decompose the myriad nodes and links into modules that represent the network (3-5). A cogent representation will retain the important information about the network and reflect the fact that interactions between the elements in complex systems are weighted, directional, interdependent, and conductive. Good representations both simplify and highlight the underlying structures and the relationships that they depict; they are maps (6, 7).To create a good map, the cartographer must attain a fine balance between omitting important structures by oversimplification and obscuring significant relationships in a barrage of superfluous detail. The best maps convey a great deal of information but require minimal bandwidth: the best maps are also good compressions. By adopting an information-theoretic approach, we can measure how efficiently a map represents the underlying geography, and we can measure how much detail is lost in the process of simplification, which allows us to quantify and resolve the cartographer's tradeoff. Network Maps and Coding TheoryIn this article, we use maps to describe the dynamics across the links and nodes in directed, weighted networks that represent the local interactions among the subunits of a system. These local interactions induce a system-wide flow of information that characterizes the behavior of the full system (8-12). Consequently, if we want to understand how network structure relates to system behavior, we need to understand the flow of information on the network. We therefore identify the modules that compose the network by finding an efficiently coarse-grained description of how information flows on the ne...

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