# Tangled Paths: A Random Graph Model from Mallows Permutations

We introduce the random graph š«(n,q) which results from taking the union of two paths of length nā„ 1, where the vertices of one of the paths have been relabelled according to a Mallows permutation with real parameter 0<q(n)ā¤ 1. This random graph model, the tangled path, goes through an evolution: if q is close to 0 the graph bears resemblance to a path and as q tends to 1 it becomes an expander. In an effort to understand the evolution of š«(n,q) we determine the treewidth and cutwidth of š«(n,q) up to log factors for all q. We also show that the property of having a separator of size one has a sharp threshold. In addition, we prove bounds on the diameter, and vertex isoperimetric number for specific values of q.

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