Tangled Paths: A Random Graph Model from Mallows Permutations

EUROCOMB’23

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 real 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 $q$.

Pages:
407–415
References

Bogdan Alecu, Vadim Lozin, Daniel A. Quiroz, Roman Rabinovich, Igor Razgon, and Viktor Zamaraev. The treewidth and pathwidth of graph unions, 2022.

Nayantara Bhatnagar and Ron Peled. Lengths of monotone subsequences in a Mallows permutation. Probab. Theory Related Fields, 161(3-4):719-780, 2015.
https://doi.org/10.1007/s00440-014-0559-7

Mark Braverman and Elchanan Mossel. Sorting from noisy information, 2009.

Yijia Chen and Jörg Flum. On the ordered conjecture. In Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science, LICS 2012, pages 225-234. IEEE Computer Society, 2012.
https://doi.org/10.1109/LICS.2012.33

Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
https://doi.org/10.1007/978-3-319-21275-3

Persi Diaconis and Arun Ram. Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques. Michigan Math. J., 48:157-190, 2000.
https://doi.org/10.1307/mmj/1030132713

Jessica Enright and Kitty Meeks. Two dichotomies for model-checking in multi-layer structures. CoRR, abs/1710.08758, 2017.

P. Erdős and A. Rényi. On random graphs. I. Publ. Math. Debrecen, 6:290-297, 1959.
https://doi.org/10.5486/PMD.1959.6.3-4.12

Ehud Friedgut and Gil Kalai. Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc., 124(10):2993-3002, 1996.
https://doi.org/10.1090/S0002-9939-96-03732-X

Alexey Gladkich and Ron Peled. On the cycle structure of Mallows permutations. Ann. Probab., 46(2):1114-1169, 2018.
https://doi.org/10.1214/17-AOP1202

Alexander E. Holroyd, Tom Hutchcroft, and Avi Levy. Mallows permutations and finite dependence. Ann. Probab., 48(1):343-379, 2020.
https://doi.org/10.1214/19-AOP1363

Jeong Han Kim and Nicholas C. Wormald. Random matchings which induce Hamilton cycles and hamiltonian decompositions of random regular graphs. J. Comb. Theory, Ser. B, 81(1):20-44, 2001.
https://doi.org/10.1006/jctb.2000.1991

Mikko Kivelä, Alex Arenas, Marc Barthelemy, James P. Gleeson, Yamir Moreno, and Mason A. Porter. Multilayer networks. J. Complex Networks, 2(3):203-271, 2014.
https://doi.org/10.1093/comnet/cnu016

Ton Kloks. Treewidth, Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer, 1994.
https://doi.org/10.1007/BFb0045375

Choongbum Lee, Joonkyung Lee, and Sang-il Oum. Rank-width of random graphs. J. Graph Theory, 70(3):339-347, 2012.
https://doi.org/10.1002/jgt.20620

Richard J. Lipton and Robert Endre Tarjan. A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 36(2):177-189, 1979.
https://doi.org/10.1137/0136016

Colin L. Mallows. Non-null ranking models. I. Biometrika, 44:114-130, 1957. Jaroslav Nesetril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012.

Shai Pilosof, Mason A Porter, Mercedes Pascual, and Sonia Kéfi. The multilayer nature of ecological networks. Nature Ecology & Evolution, 1(4):1-9, 2017.
https://doi.org/10.1038/s41559-017-0101

Ross G. Pinsky. Permutations avoiding a pattern of length three under mallows distributions. Random Struct. Algorithms, 58(4):676-690, 2021.
https://doi.org/10.1002/rsa.20988

Ross G. Pinsky. Clustering of consecutive numbers in permutations under Mallows distributions and super-clustering under general p-shifted distributions. Electronic Journal of Probability, 27:1 - 20, 2022.
https://doi.org/10.1214/22-EJP812

Daniel A Quiroz. Chromatic and structural properties of sparse graph classes. PhD thesis, The London School of Economics and Political Science (LSE), 2017.

Metrics

0

Views

0

PDF views