Tangled Paths: A Random Graph Model from Mallows Permutations

We introduce the random graph P(n, q) which results from taking the union of two paths of length n > 1, where the vertices of one of the paths have been relabelled according to a Mallows permutation with real parameter 0 < q(n) 6 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 P(n, q) we determine the treewidth and cutwidth of 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.

[1]  Nicholas C. Wormald,et al.  Random Matchings Which Induce Hamilton Cycles and Hamiltonian Decompositions of Random Regular Graphs , 2001, J. Comb. Theory, Ser. B.

[2]  S. Mukherjee,et al.  DEGREE SEQUENCE OF RANDOM PERMUTATION GRAPHS , 2015, 1503.03582.

[3]  P. Diaconis,et al.  Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques , 2004, math/0401318.

[4]  T. Lindvall.  Lectures on the Coupling Method , 1992 .

[5]  Eli Upfal,et al.  Probability and Computing: Randomized Algorithms and Probabilistic Analysis , 2005 .

[6]  A. Holroyd,et al.  Mallows permutations as stable matchings , 2018, Canadian Journal of Mathematics.

[7]  ROSS G. PINSKY.  Permutations avoiding a pattern of length three under Mallows distributions , 2021, Random Struct. Algorithms.

[8]  A. Frieze,et al.  Introduction to Random Graphs , 2016 .

[9]  Navin Goyal,et al.  Expanders via random spanning trees , 2008, SODA.

[10]  Leonard Lewin,et al.  Polylogarithms and Associated Functions , 1981 .

[11]  R. Peled,et al.  On the Cycle Structure of Mallows Permutations , 2016, 1601.06991.

[12]  W. T. Gowers,et al.  RANDOM GRAPHS (Wiley Interscience Series in Discrete Mathematics and Optimization) , 2001 .

[13]  Nicholas C. Wormald,et al.  Lower Bounds for the Isoperimetric Numbers of Random Regular Graphs , 2014, SIAM J. Discret. Math..

[14]  S. Janson.  Tail bounds for sums of geometric and exponential variables , 2017, 1709.08157.

[15]  S. Starr.  THERMODYNAMIC LIMIT FOR THE MALLOWS MODEL ON Sn , 2009, 0904.0696.

[16]  Hans L. Bodlaender,et al.  Some Classes of Graphs with Bounded Treewidth , 1988, Bull. EATCS.

[17]  Mark Braverman,et al.  Sorting from Noisy Information , 2009, ArXiv.

[18]  Harry Crane,et al.  The probability of avoiding consecutive patterns in the Mallows distribution , 2016, Random Struct. Algorithms.

[19]  A. Holroyd,et al.  Mallows permutations and finite dependence , 2017, The Annals of Probability.

[20]  C. L. Mallows.  NON-NULL RANKING MODELS. I , 1957 .

[21]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[22]  G. Kalai,et al.  Every monotone graph property has a sharp threshold , 1996 .

[23]  S. Starr,et al.  The Length of the Longest Increasing Subsequence of a Random Mallows Permutation , 2011, Journal of Theoretical Probability.

[24]  Nayantara Bhatnagar,et al.  Lengths of monotone subsequences in a Mallows permutation , 2013, 1306.3674.

[25]  Julia Böttcher,et al.  Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs , 2009, Eur. J. Comb..

[26]  Jessica Enright,et al.  Two dichotomies for model-checking in multi-layer structures. , 2017 .