Loop statistics in complex networks

We study a scaling property of the number M _h(N) of loops of size h in complex networks with respect to a network size N. For networks with a bounded second moment of degree, we find two distinct scaling behaviors: M _h(N) ~ (constant) and M _h(N) ~ lnN as N increases. Uncorrelated random networks specified only with a degree distribution and Markovian networks specified only with a nearest neighbor degree-degree correlation display the former scaling behavior, while growing network models display the latter. The difference is attributed to structural correlation that cannot be captured by a short-range degree-degree correlation.