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

TL;DR

This paper analytically investigates the prevalence and size distribution of loops and Hamilton cycles in random scale-free networks, revealing their typical sizes, frequency, and conditions for their existence.

Contribution

It introduces an analytical method to evaluate the average number of loops of any size in scale-free networks and examines the rarity of Hamilton cycles under certain degree distribution conditions.

Findings

Most frequent loop size scales with network size N.

Small loops are more common when the degree distribution's second moment diverges.

Hamiltonian cycles are rare and may not exist near a degree exponent of 2.

Abstract

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 like 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 2 even for minimal connectivity grater than 3.