The Annals of Applied Probability

Degree sequence of random permutation graphs

Bhaswar B. Bhattacharya and Sumit Mukherjee

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In this paper, we study the asymptotics of the degree sequence of permutation graphs associated with a sequence of random permutations. The limiting finite-dimensional distributions of the degree proportions are established using results from graph and permutation limit theories. In particular, we show that for a uniform random permutation, the joint distribution of the degree proportions of the vertices labeled $\lceil nr_{1}\rceil,\lceil nr_{2}\rceil,\ldots,\lceil nr_{s}\rceil$ in the associated permutation graph converges to independent random variables $D(r_{1}),D(r_{2}),\ldots,D(r_{s})$, where $D(r_{i})\sim\operatorname{Unif}(r_{i},1-r_{i})$, for $r_{i}\in[0,1]$ and $i\in\{1,2,\ldots,s\}$. Moreover, the degree proportion of the mid-vertex (the vertex labeled $n/2$) has a central limit theorem, and the minimum degree converges to a Rayleigh distribution after an appropriate scaling. Finally, the asymptotic finite-dimensional distributions of the permutation graph associated with a Mallows random permutation is determined, and interesting phase transitions are observed. Our results extend to other nonuniform measures on permutations as well.

Article information

Ann. Appl. Probab., Volume 27, Number 1 (2017), 439-484.

Received: April 2015
Revised: March 2016
First available in Project Euclid: 6 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05A05: Permutations, words, matrices 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability

Combinatorial probability graph limit limit theorems Mallow’s model permutation limit


Bhattacharya, Bhaswar B.; Mukherjee, Sumit. Degree sequence of random permutation graphs. Ann. Appl. Probab. 27 (2017), no. 1, 439--484. doi:10.1214/16-AAP1207.

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  • [1] Acan, H. and Pittel, B. (2013). On the connected components of a random permutation graph with a given number of edges. J. Combin. Theory Ser. A 120 1947–1975.
  • [2] Awasthi, P., Blum, A., Sheffet, O. and Vijayaraghavan, A. (2014). Learning mixtures of ranking models. Available at arXiv:1410.8750.
  • [3] Bafna, V. and Pevzner, P. A. (1996). Genome rearrangements and sorting by reversals. SIAM J. Comput. 25 272–289.
  • [4] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178.
  • [5] Beŕard, S., Bergeron, A., Chauve, C. and Paul, C. (2007). Perfect sorting by reversals is not always difficult. IEEE/ACM Transactions on Computational Biology and Bioinformatics 4 4–16.
  • [6] Bhatnagar, N. and Peled, R. (2015). Lengths of monotone subsequences in a Mallows permutation. Probab. Theory Related Fields 161 719–780.
  • [7] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [8] Bóna, M. (2004). Combinatorics of Permutations. Chapman & Hall/CRC, Boca Raton, FL.
  • [9] Bóna, M. (2007). The copies of any permutation pattern are asymptotically normal. Available at arXiv:0712.2792.
  • [10] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008). Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math. 219 1801–1851.
  • [11] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2012). Convergent sequences of dense graphs II. Multiway cuts and statistical physics. Ann. of Math. (2) 176 151–219.
  • [12] Braverman, M. and Mossel, E. (2009). Sorting from noisy information. Available at arXiv:0910.1191.
  • [13] Chen, H., Branavan, S. R. K., Barzilay, R. and Karger, D. R. (2009). Content modeling using latent permutations. J. Artificial Intelligence Res. 36 129–163.
  • [14] Diaconis, P. (1988). Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series 11. IMS, Hayward, CA.
  • [15] Diaconis, P., Graham, R. and Holmes, S. P. (2001). Statistical problems involving permutations with restricted positions. In State of the Art in Probability and Statistics (Leiden, 1999). Institute of Mathematical Statistics Lecture Notes—Monograph Series 36 195–222. IMS, Beachwood, OH.
  • [16] Diaconis, P., Holmes, S. and Janson, S. (2008). Threshold graph limits and random threshold graphs. Internet Math. 5 267–320 (2009).
  • [17] Diaconis, P., Holmes, S. and Janson, S. (2013). Interval graph limits. Ann. Comb. 17 27–52.
  • [18] Diaconis, P. and Ram, A. (2000). Analysis of systematic scan Metropolis algorithms using Iwahori–Hecke algebra techniques. Michigan Math. J. 48 157–190.
  • [19] Even, S., Pnueli, A. and Lempel, A. (1972). Permutation graphs and transitive graphs. J. Assoc. Comput. Mach. 19 400–410.
  • [20] Feigin, P. and Cohen, A. (1978). On a model for concordance between judges. J. Roy. Statist. Soc. Ser. B 40 203–213.
  • [21] Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge Univ. Press, Cambridge.
  • [22] Fulman, J. (2004). Stein’s method and non-reversible Markov chains. In Stein’s Method: Expository Lectures and Applications. Institute of Mathematical Statistics Lecture Notes—Monograph Series 46 69–77. IMS, Beachwood, OH.
  • [23] Glebov, R., Grzesik, A., Klimošová, T. and Král’, D. (2015). Finitely forcible graphons and permutons. J. Combin. Theory Ser. B 110 112–135.
  • [24] Glebov, R., Hoppen, C., Klimošová, T., Kohayakawa, Y., Král’, D. and Liu, H. (2014). Large permutations and parameter testing. Available at arXiv:1412.5622.
  • [25] Golumbic, M. C. (1980). Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York.
  • [26] Hoppen, C., Kohayakawa, Y., Moreira, C. G., Ráth, B. and Menezes Sampaio, R. (2013). Limits of permutation sequences. J. Combin. Theory Ser. B 103 93–113.
  • [27] Hoppen, C., Kohayakawa, Y., Moreira, C. G. and Sampaio, R. M. (2011). Testing permutation properties through subpermutations. Theoret. Comput. Sci. 412 3555–3567.
  • [28] Huang, J., Guestrin, C. and Guibas, L. (2009). Fourier theoretic probabilistic inference over permutations. J. Mach. Learn. Res. 10 997–1070.
  • [29] Janson, S., Nakamura, B. and Zeilberger, D. (2015). On the asymptotic statistics of the number of occurrences of multiple permutation patterns. J. Comb. 6 117–143.
  • [30] Knuth, D. E. (1998). The Art of Computer Programming. Sorting and Searching. Vol. 3. Addison-Wesley, Reading, MA. 2nd ed. [of MR0445948].
  • [31] Knuth, D. E. (2005). The Art of Computer Programming: Generating All Tuples and Permutations. Vol. 4, Fasc. 2. Addison-Wesley, Upper Saddle River, NJ.
  • [32] Kondor, R., Howard, A. and Jebara, T. (2007). Multi-object tracking with representations of the symmetric group. In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics (AISTATS) 211–218.
  • [33] Král’, D. and Pikhurko, O. (2013). Quasirandom permutations are characterized by 4-point densities. Geom. Funct. Anal. 23 570–579.
  • [34] Lahiri, S. N. and Chatterjee, A. (2007). A Berry–Esseen theorem for hypergeometric probabilities under minimal conditions. Proc. Amer. Math. Soc. 135 1535–1545.
  • [35] Lebanon, G. and Lafferty, J. (2002). Cranking: Combining rankings using conditional probability models on permutations. In Proceedings of the 19th International Conference on Machine Learning 363–370.
  • [36] Lebanon, G. and Mao, Y. (2008). Non-parametric modeling of partially ranked data. J. Mach. Learn. Res. 9 2401–2429.
  • [37] Lovász, L. (2012). Large Networks and Graph Limits. American Mathematical Society Colloquium Publications 60. Amer. Math. Soc., Providence, RI.
  • [38] Mallows, C. L. (1957). Non-null ranking models. I. Biometrika 44 114–130.
  • [39] Meila, M. and Bao, L. (2008). Estimation and clustering with infinite rankings. In Proceedings of the 24th Conference in Uncertainty in Artificial Intelligence 393–402.
  • [40] Meilă, M. and Bao, L. (2010). An exponential model for infinite rankings. J. Mach. Learn. Res. 11 3481–3518.
  • [41] Meila, M., Phadnis, K., Patterson, A. and Blimes, J. (2007). Consensus ranking under the exponential model. Statistics Technical Report 515, Univ. Washington, Seattle, WA.
  • [42] Mueller, C. and Starr, S. (2013). The length of the longest increasing subsequence of a random Mallows permutation. J. Theoret. Probab. 26 514–540.
  • [43] Mukherjee, S. (2015). Estimation of parameters in non-uniform models on permutations. Ann. Statist. To appear. Available at arXiv:1307.0978.
  • [44] Pnueli, A., Lempel, A., Even, S. and Pnueli, A. (1971). Transitive orientation of graphs and identification of permutation graphs. Canad. J. Math. 23 160–175.
  • [45] Sherwani, N. A. (1999). Algorithms for VLSI Physical Design Automation, 3rd ed. Kluwer Academic, Boston, MA.
  • [46] Skala, M. (2013). Hypergeometric tail inequalities: Ending the insanity. Available at arXiv:1311.5939.
  • [47] Starr, S. (2009). Thermodynamic limit for the Mallows model on $S_{n}$. J. Math. Phys. 50 095208, 15.
  • [48] Starr, S. and Walters, M. (2015). Phase uniqueness for the Mallows measure on permutations. Available at arXiv:1502.03727.
  • [49] Sweeting, T. J. (1989). On conditional weak convergence. J. Theoret. Probab. 2 461–474.