1. Department of Computer Science, and,
2. Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA
3. Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
4. Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA

Correspondence to: Aaron Clauset^1,3 Correspondence and requests for materials should be addressed to A.C. (Email: aaronc@santafe.edu).

Abstract

Networks have in recent years emerged as an invaluable tool for describing and quantifying complex systems in many branches of science^{1, 2, 3}. Recent studies suggest that networks often exhibit hierarchical organization, in which vertices divide into groups that further subdivide into groups of groups, and so forth over multiple scales. In many cases the groups are found to correspond to known functional units, such as ecological niches in food webs, modules in biochemical networks (protein interaction networks, metabolic networks or genetic regulatory networks) or communities in social networks^{4, 5, 6, 7}. Here we present a general technique for inferring hierarchical structure from network data and show that the existence of hierarchy can simultaneously explain and quantitatively reproduce many commonly observed topological properties of networks, such as right-skewed degree distributions, high clustering coefficients and short path lengths. We further show that knowledge of hierarchical structure can be used to predict missing connections in partly known networks with high accuracy, and for more general network structures than competing techniques⁸. Taken together, our results suggest that hierarchy is a central organizing principle of complex networks, capable of offering insight into many network phenomena.

1. Department of Computer Science, and,
2. Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA
3. Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
4. Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA

Correspondence to: Aaron Clauset^1,3 Correspondence and requests for materials should be addressed to A.C. (Email: aaronc@santafe.edu).