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Statistics of Cycles: How Loopy is your Network?

Hernan D. Rozenfeld·Joseph E. Kirk·Erik M. Bollt·Daniel ben-Avraham
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Abstract

We study the distribution of cycles of length h in large networks (of size N>>1) and find it to be an excellent ergodic estimator, even in the extreme inhomogeneous case of scale-free networks. The distribution is sharply peaked around a characteristic cycle length, h* ~ N^a. Our results suggest that h* and the exponent a might usefully characterize broad families of networks. In addition to an exact counting of cycles in hierarchical nets, we present a Monte-Carlo sampling algorithm for approximately locating h* and reliably determining a. Our empirical results indicate that for small random scale-free nets of degree exponent g, a=1/(g-1), and a grows as the nets become larger.

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Hernan D. Rozenfeld, Joseph E. Kirk, Erik M. Bollt, Daniel ben-Avraham. 2004-11-10. Statistics of Cycles: How Loopy is your Network?. https://doi.org/10.1088/0305-4470%2F38%2F21%2F005

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