In network science, a group of nodes connected with each other at higher probability than with those outside the group is referred to as a community. From the perspective that individual communities are associated with functional modules constituting complex systems described by networks, discovering communities is primarily important for understanding overall functions of these systems. Much effort has been devoted to developing methods to detect communities in networks since the early days of network science. Nevertheless, the method to reveal key characteristics of communities in real-world network remains to be established. Here we formulate decomposition of a random walk spreading over the entire network into local modules as proxy for communities. This formulation will reveal the pervasive structure of communities and their hierarchical organization, which are the hallmarks of real-world networks but are out of reach of most existing methods.
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