The Incipient Giant Component in Bond Percolation on General Finite Weighted Graphs

TL;DR

This paper studies the emergence of a giant component in a finite weighted graph during bond percolation, showing that the critical time for its appearance is tightly concentrated around its expected value.

Contribution

It introduces minimal assumptions under which the onset of the giant component in weighted graphs is shown to be weakly concentrated around its mean.

Findings

Giant component emergence time is tightly concentrated.

Results apply to graphs with arbitrary edge weights.

Provides probabilistic bounds on the critical percolation time.

Abstract

On a large finite connected graph let edges $e$ become "open" at independent random Exponential times of arbitrary rates $w_e$. Under minimal assumptions, the time at which a giant component starts to emerge is weakly concentrated around its mean.