Physical Sciences and Mathematics Commons^™

Kinetic Theory Of Random Graphs: From Paths To Cycles, E. Ben-Naim, P.L. Krapivsky

Eli Ben-Naim

Structural properties of evolving random graphs are investigated. Treating linking as a dynamic aggregation process, rate equations for the distribution of node to node distances (paths) and of cycles are formulated and solved analytically. At the gelation point, the typical length of paths and cycles, l, scales with the component size k as l ~ k^{1/2}. Dynamic and finite-size scaling laws for the behavior at and near the gelation point are obtained. Finite-size scaling laws are verified using numerical simulations.