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.
Highlights
Take a finite connected graph (V, E) with edge-weights w =, where we > 0 ∀e ∈ E
In the language of percolation theory, say that edge e becomes open at time ξe
In the special cases of the complete graph and the 2-dimensional discrete torus we are essentially dealing with component sizes in the classical Erdos-Renyi and the bond percolation on Z2 processes, for which much stronger results are known about the “scaling window” of time over which the giant component emerges [10, 11]
Summary
In the special cases of the complete graph and the 2-dimensional discrete torus (with constant edge-weights) we are essentially dealing with component sizes in the classical Erdos-Renyi and the bond percolation on Z2 processes, for which much stronger results are known about the “scaling window” of time over which the giant component emerges [10, 11]. Such stronger results have been generalized (again with constant edge-weights) in several directions, for instance to random subgraphs of certain transitive finite graphs [12, 13] or to random subgraphs of graphs under assumptions that force the critical subgraphs to be “tree-like” as in the Erdos-Renyi case [14]. “Big picture” discussions of various random processes over finite edge-weighted graphs can be found in [1] and [3]
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