The evolution of random reversal graph

Details The evolution of random reversal graph The evolution of random reversal graph

2010-03-03

arxiv.org/abs/1003.0739v1

Mathematics

Combinatorics

5 pages with supplementary materials 13 pages

Scientific paper

The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph changes dramatically at $\lambda_n=1/\binom{n+1}{2}$. For $\lambda_n=(1-\epsilon)/\binom{n+1}{2}$, the random graph consists of components of size at most $O(n\ln(n))$ a.s. and for $(1+\epsilon)/\binom{n+1}{2}$, there emerges a unique largest component of size $\sim \wp(\epsilon) \cdot 2^n\cdot n$!$ a.s.. This "giant" component is furthermore dense in the reversal graph.

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