  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.
