To comprehend the hierarchical organization of large integrated
systems, we introduce the hierarchical map equation that reveals
multilevel structures in networks. In this information-theoretic
approach, we exploit the duality between compression and pattern
detection; by compressing a description of a random walker as a
proxy for real flow on a network, we find regularities in the
network that induce this system-wide flow. Finding the shortest
multilevel description of the random walker therefore gives us the
best hierarchical clustering of the network — the optimal
number of levels and modular partition at each level — with
respect to the dynamics on the network. With a novel search
algorithm, we extract and illustrate the rich multilevel
organization of several large social and biological networks. For
example, from the global air traffic network we uncover countries
and continents, and from the pattern of scientific communication
we reveal more than 100 scientific fields organized in four major
disciplines: life sciences, physical sciences, ecology and earth
sciences, and social sciences. In general, we find shallow
hierarchical structures in globally interconnected systems, such
as neural networks, and rich multilevel organizations in systems
with highly separated regions, such as road networks.