Hierarchy and Anti-Hierarchy in Real and Scale Free networks

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  • 18 August 2003
Abstract
Using node degree as a proxy for importance, we quantify the hierarchical fraction of a complex network and discuss structures of the hardwired Internet, protein interaction networks and artificially constructed scale free networks. While Internet is found to be hierarchical, and protein interaction network anti-hierarchical, the random scale free networks exhibit two "hierarchy" transitions as function of the slope of the degree distribution. In particular at slope -2 a random network is naturally self hierarchical, whereas networks with slope steeper than -3 are naturally anti-hierarchical and as a consequence modular. At any degree distribution we explicitly construct networks with maximal hierarchical, respectively anti-hierarchical organization.

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