Counting connected graphs inside-out
Journal of Combinatorial Theory Series B
Diameters in supercritical random graphs via first passage percolation
Combinatorics, Probability and Computing
The diameter of sparse random graphs
Combinatorics, Probability and Computing
Component behavior near the critical point of the random graph process
Random Structures & Algorithms
Cycles in a random graph near the critical point
Random Structures & Algorithms
Anatomy of a young giant component in the random graph
Random Structures & Algorithms
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In a recent work of the authors and Kim, we derived a complete description of the largest component of the Erdos-Renyi random graph G(n,p) as it emerges from the critical window, i.e. for p=(1+@e)/n where @e^3n-~ and @e=o(1), in terms of a tractable contiguous model. Here we provide the analogous description for the supercritical giant component, i.e. the largest component of G(n,p) for p=@l/n where @l1 is fixed. The contiguous model is roughly as follows. Take a random degree sequence and sample a random multigraph with these degrees to arrive at the kernel; replace the edges by paths whose lengths are i.i.d. geometric variables to arrive at the 2-core; attach i.i.d. Poisson-Galton-Watson trees to the vertices for the final giant component. As in the case of the emerging giant, we obtain this result via a sequence of contiguity arguments at the heart of which are Kim's Poisson-cloning method and the Pittel-Wormald local limit theorems.