🤖 AI Summary
This paper investigates the structural evolution of the random graph model $mathcal{P}(n,q)$, formed by superimposing two length-$n$ paths—one with vertices relabeled according to a Mallows permutation (parameter $q in (0,1]$), thereby modeling “entangled” or “twisted” path topologies. This constitutes the first application of Mallows permutations in random graph modeling, uncovering deep connections between permutation partial orders and graph connectivity and phase transitions. Using combinatorial probability, random permutation theory, and asymptotic analysis, we precisely characterize the connectivity threshold, the asymptotic size of the largest connected component, and the diameter—each exhibiting nontrivial $q$-dependent scaling. We identify a continuous phase transition: as $q$ increases from $o(1)$ to $1-Theta(1)$, the graph evolves from a path-like structure (low connectivity, large diameter) to a highly expanding one. Our work introduces the “entangled path” paradigm, significantly broadening the scope of random graph phase transition theory.
📝 Abstract
We introduce the random graph $mathcal{P}(n,q)$ which results from taking the union of two paths of length $ngeq 1$, where the vertices of one of the paths have been relabelled according to a Mallows permutation with real parameter $0
