paper

Offdiagonal Complexity: A computationally quick complexity measure for graphs and networks

arXiv:q-bio/0410024 · doi:10.1016/j.physa.2006.08.067

Abstract

A vast variety of biological, social, and economical networks shows topologies drastically differing from random graphs; yet the quantitative characterization remains unsatisfactory from a conceptual point of view. Motivated from the discussion of small scale-free networks, a biased link distribution entropy is defined, which takes an extremum for a power law distribution. This approach is extended to the node-node link cross-distribution, whose nondiagonal elements characterize the graph structure beyond link distribution, cluster coefficient and average path length. From here a simple (and computationally cheap) complexity measure can be defined. This Offdiagonal Complexity (OdC) is proposed as a novel measure to characterize the complexity of an undirected graph, or network. While both for regular lattices and fully connected networks OdC is zero, it takes a moderately low value for a random graph and shows high values for apparently complex structures as scale-free networks and hierarchical trees. The Offdiagonal Complexity apporach is applied to the Helicobacter pylori protein interaction network and randomly rewired surrogates.

12 pages, revised version, to appear in Physica A

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