Kinetic Theory of Random Graphs: from Paths to Cycles

TL;DR

This paper develops a kinetic theory for evolving random graphs, deriving analytical rate equations for paths and cycles, and reveals scaling laws at the gelation point, validated by simulations.

Contribution

It introduces a kinetic framework that analytically describes the distribution of paths and cycles in evolving random graphs, including scaling laws at gelation.

Findings

Path and cycle lengths scale as k^{1/2} at gelation

Finite-size scaling laws are confirmed by simulations

Analytical solutions for distributions are derived

Abstract

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.