  We present a Markov Chain Monte Carlo method for sampling cycle length in
large graphs. Cycles are treated as microstates of a system with many degrees
of freedom. Cycle length corresponds to energy such that the length histogram
is obtained as the density of states from Metropolis sampling. In many growing
networks, mean cycle length increases algebraically with system size. The cycle
exponent $\alpha$ is characteristic of the local growth rules and not
determined by the degree exponent $\gamma$. For example, $\alpha=0.76(4)$ for
the Internet at the Autonomous Systems level.
