Tangled Paths: A Random Graph Model from Mallows Permutations

TL;DR

The paper studies a random graph model $\mathcal{P}(n,q)$ formed by uniting two $n$-paths, one relabeled by a Mallows permutation with parameter $q$, interpolating between a path ($q\approx 0$) and a random expander ($q\approx 1$). It introduces the $q$-Mallows process to generate Mallows permutations and develops local/flush concentration tools to analyze structural properties. The main contributions include (i) a sharp separator threshold at $q_c=1-\frac{\pi^2}{6\log n}$, (ii) up to log-factor bounds on treewidth and cutwidth for all $q$, and (iii) diameter bounds showing linear growth for fixed $q<1$ and $O(\log n)$ growth near $q=1$, with expander behavior for $q$ very close to 1. These results illuminate how the Mallows parameter governs global–local graph structure and provide a tractable framework bridging path-like and expander regimes, with implications for algorithmic tractability and understanding layered graph models. All statements hold w.h.p. and all mathematical quantities are formalized in terms of $n$ and $q$.

Abstract

We introduce the random graph $\mathcal{P}(n,q)$ which results from taking the union of two paths of length $n\geq 1$, where the vertices of one of the paths have been relabelled according to a Mallows permutation with parameter $0<q(n)\leq 1$. 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 $\mathcal{P}(n,q)$ we determine the treewidth and cutwidth of $\mathcal{P}(n,q)$ 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 $q$.

Figures (6)

• Figure 1: The diagram above gives a representation of our main results. All results above hold with high probability, and we say that $f(n) = \widetilde{\Theta}(g(n))$ if there exist constants $c,C>0$ and $n_0$ such that $c\cdot g(n)/\log g(n)\leqslant f_n \leqslant C\cdot g(n)\cdot \log g(n)$ for all $n\geqslant n_0$.
• Figure 2: The table on the left gives the sequences of permutations $(r_n)$ generated by the sequence $(v_n)$ for $i=1, \dots, 6$. On the right we have a tangled path generated by $r_6$, where the edges of $r_6(P_{6})$ are dotted.
• Figure 3: A representation of a flush event $\mathcal{F}_5$ (see \ref{['eq:flushingevent']}) holding in the graph $\mathop{\mathrm{\mathsf{layer}}}\nolimits(P_9,\sigma(P_9))$. In this example $\sigma = (7,9,6,8,4,2,5,3,1)$ was generated by the sequence $\mathbf{x}=(1,1,2,1,3,1,1,3,2)$ satisfying $\mathcal{F}_5$. Observe that only one edge crosses vertex $5$.
• Figure 4: A representation of the event $\mathcal{C}_5^\mathcal{F}$ from \ref{['eq:E1and2']} holding in the graph $\mathop{\mathrm{\mathsf{layer}}}\nolimits(P_9,\sigma(P_9))$. In this example $\sigma = (7,9,6,8,5,2,4,1,3)$ was generated by the sequence $\mathbf{x}=(1,1,3,2,1,1,1,3,2)$ satisfying $\mathcal{C}_5^\mathcal{F}$. Observe that vertex $5$ is a cut vertex.
• Figure 5: An example of Lemma \ref{['lem:concpattern']}. Let $\sigma=(3,4,2,1,5)$ and $\pi=(1,3,5,7,4,2,9,6,8)$, thus $\mathsf{st}(5,7,4,2,9) = \sigma$. The graph induced by the crossed and dotted edges is isomorphic to $\mathop{\mathrm{\mathsf{layer}}}\nolimits(\sigma(P_5),P_5)$. The graph induced by the dashed, solid and dotted edges is $\mathop{\mathrm{\mathsf{layer}}}\nolimits(\pi(P_9),P_9)$. The crossed edges can be subdivided to give a path on $\{2,\dots, 9 \}$.

Theorems & Definitions (65)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1: Folklore, see Kloks
• Lemma 2.2: JansonTail
• Lemma 2.3: PandC
• Lemma 3.1: BhatSubSeq
• Theorem 3.2: BhatSubSeq
• Lemma 3.3