Cyclization in finite random graphs

This paper reports results of a study of cyclization kinetics in finite random graphs. The relationship between the generating function for the number of edges in the graph and the generating function of linked components applies for deriving the exact expression for the spectrum of linked components in finite random graphs with a given number of edges. The spectrum of cycled linked components of the random graph is shown to be a product of two terms one of which is independent of the component structures and sensitive to the total size of the graph while the second one depends on the fine structure of the graph and does not include any dependence on the global structure of the graph. The latter term is expressed via quite analyzable integrals the asymptotics of which at large orders g of the components defines the critical properties of the spectra of k-cycled linked components. Cyclization dominates after emergence of the giant component, with the k-cycled components being algebraically distributed as U-k(g) proportional to g(()(3k-5)()/2) at the critical point. The spectra are shown to be modulated by a factor independent of the complexity of the graph.