Hierarchical structure and the prediction of missing links in networks


Networks have in recent years emerged as an invaluable tool for describing and quantifying complex systems in many branches of science1,2,3. 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 networks4,5,6,7. 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.

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Figure 1: A hierarchical network with structure on many scales, and the corresponding hierarchical random graph.
Figure 2: Application of the hierarchical decomposition to the network of grassland species interactions.
Figure 3: Comparison of link prediction methods.


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We thank J. Dunne, M. Gastner, P. Holme, M. Huss, M. Porter, C. Shalizi and C. Wiggins for their help, and the Santa Fe Institute for its support. C.M. thanks the Center for the Study of Complex Systems at the University of Michigan for hospitality while some of this work was conducted.

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Correspondence to Aaron Clauset.

Supplementary information

Supplementary Notes

This file contains Supplementary Notes including the technical details of our hierarchical model and the methods used to fit it to empirical data. It also contains addition results on graph resampling and the prediction of missing links, and the algorithmic specifics of our experimental studies. (PDF 123 kb)

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Clauset, A., Moore, C. & Newman, M. Hierarchical structure and the prediction of missing links in networks. Nature 453, 98–101 (2008). https://doi.org/10.1038/nature06830

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