Unicyclic Components in Random Graphs

TL;DR

This paper analytically characterizes the distribution and growth of unicyclic components in random graphs, revealing a self-similar form near gelation and algebraic decay at the transition point.

Contribution

It provides an analytical description of unicyclic component distribution in random graphs, especially near the gelation transition, highlighting their growth and decay behaviors.

Findings

Distribution approaches a self-similar form near gelation

At gelation, distribution decays as 1/(4k) for large k

Number of unicyclic components grows logarithmically with system size

Abstract

The distribution of unicyclic components in a random graph is obtained analytically. The number of unicyclic components of a given size approaches a self-similar form in the vicinity of the gelation transition. At the gelation point, this distribution decays algebraically, U_k ~ 1/(4k) for k>>1. As a result, the total number of unicyclic components grows logarithmically with the system size.