Statistics of cycles in large networks

The occurrence of self-avoiding closed paths (cycles) in networks is studied under varying rules of wiring. As a main result, we find that the dependence between network size N and typical cycle length is algebraic, h t Nα, with distinct values of α for different wiring rules. The Barabasi-Albert model has α=1. Different preferential and nonpreferential attachment rules and the growing Internet graph yield α<1. Computation of the statistics of cycles at arbitrary length is made possible by the introduction of an efficient sampling algorithm.