Statistics of Cycles: How Loopy is your Network?

Details Statistics of Cycles: How Loopy is your Network? Statistics of Cycles: How Loopy is your Network?

2004-03-21

arxiv.org/abs/cond-mat/0403536v2

J. Phys. A 38, 4589-4595 (2005)

Physics

Condensed Matter

Disordered Systems and Neural Networks

Further work presented and conclusions revised, following referee reports

Scientific paper

10.1088/0305-4470/38/21/005

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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