A Random Graph Growth Model
Abstract
A growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start of the process the pool contains $N$ virtual vertices and no edges. Each time a vertex is sampled and occupied, the edges linking the vertex to previously occupied vertices are added to the pool of virtual elements. We focus on the edge-counting at times when the graph has $n\leq N$ occupied vertices. Two different Poisson limits are identified for $n\asymp N^{1/3}$ and $N-n\asymp 1$. For the bulk of the process, when $n\asymp N$, the scaled number of edges is shown to fluctuate about a deterministic curve, with fluctuations being of the order of $N^{3/2}$ and approximable by a Gaussian bridge.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2023
- DOI:
- 10.48550/arXiv.2301.07809
- arXiv:
- arXiv:2301.07809
- Bibcode:
- 2023arXiv230107809F
- Keywords:
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- Mathematics - Probability;
- Mathematics - Combinatorics;
- 05C80;
- 60B20
- E-Print:
- 21 pages, 1 figure