Loops of any size and Hamilton cycles in random scale-free networks

Loops are subgraphs responsible for the multiplicity of paths going from one to another generic node in a given network. In this paper we present an analytic approach for the evaluation of the average number of loops in random scale-free networks valid at fixed number of nodes N and for any length L of the loops. We bring evidence that the most frequent loop size in a scale-free network of N nodes is of the order of N as in random regular graphs, while small loops are more frequent when the second moment of the degree distribution diverges. In particular, we find that finite loops of sizes larger than a critical one almost surely pass from any node, thus casting some doubts on the validity of the random tree approximation for the solution of lattice models on these graphs. Moreover, we show that Hamiltonian cycles are rare in random scale-free networks and may fail to appear if the power-law exponent of the degree distribution is close to two even for minimal connectivity k(min) >= 3.