The evolution of random reversal graph

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 $λ_n=1/\binom{n+1}{2}$. For $λ_n=(1-ε)/\binom{n+1}{2}$, the random graph consists of components of size at most $O(n\ln(n))$ a.s. and for $(1+ε)/\binom{n+1}{2}$, there emerges a unique largest component of size $\sim \wp(ε) \cdot 2^n\cdot n$!$ a.s.. This "giant" component is furthermore dense in the reversal graph.

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