Asymptotic Properties of Random Unlabelled Block-Weighted Graphs

We study the asymptotic shape of random unlabelled graphs subject to certain subcriticality conditions. The graphs are sampled with probability proportional to a product of Boltzmann weights assigned to their $2$-connected components. As their number of vertices tends to infinity, we show that they admit the Brownian tree as Gromov–Hausdorff–Prokhorov scaling limit, and converge in a strengthened Benjamini–Schramm sense toward an infinite random graph. We also consider models of random graphs that are allowed to be disconnected. Here a giant connected component emerges and the small fragments converge without any rescaling towards a finite random limit graph. Our main application of these general results treats subcritical classes of unlabelled graphs. We study the special case of unlabelled outerplanar graphs in depth and calculate its scaling constant.

[1]  O. Cohen.  Recurrence of Distributional Limits of Finite Planar Graphs , 2000 .

[2]  Benedikt Stufler.  Random Enriched Trees with Applications to Random Graphs , 2018, Electron. J. Comb..

[3]  Konstantinos Panagiotou,et al.  On properties of random dissections and triangulations , 2008, SODA '08.

[4]  Jim Pitman,et al.  Tree-valued Markov chains derived from Galton-Watson processes , 1998 .

[5]  Grégory Miermont,et al.  Scaling limits of random planar maps with large faces , 2011 .

[6]  Thomas Duquesne.  A limit theorem for the contour process of condidtioned Galton--Watson trees , 2003 .

[7]  Manuel Bodirsky,et al.  Boltzmann Samplers, Pólya Theory, and Cycle Pointing , 2010, SIAM J. Comput..

[8]  R. Otter.  The Number of Trees , 1948 .

[9]  D. Aldous.  Stochastic Analysis: The Continuum random tree II: an overview , 1991 .

[10]  Nicolas Curien,et al.  The CRT is the scaling limit of random dissections , 2013, Random Struct. Algorithms.

[11]  K. Panagiotou,et al.  Scaling limits of random Pólya trees , 2015, 1502.07180.

[12]  Michael Wallner,et al.  A note on the scaling limits of random Pólya trees , 2016, ANALCO.

[13]  Agelos Georgakopoulos,et al.  Subcritical Graph Classes Containing All Planar Graphs , 2017, Combinatorics, Probability and Computing.

[14]  Benedikt Stufler,et al.  Gibbs partitions: The convergent case , 2016, Random Struct. Algorithms.

[15]  Jean-François Marckert,et al.  The CRT is the scaling limit of unordered binary trees , 2009, Random Struct. Algorithms.

[16]  Minmin Wang.  Scaling limits for a family of unrooted trees , 2016, 1604.08287.

[17]  A. Joyal.  Une théorie combinatoire des séries formelles , 1981 .

[18]  Philippe Flajolet,et al.  The distribution of height and diameter in random non‐plane binary trees , 2010, Random Struct. Algorithms.

[19]  Benedikt Stufler.  Limits of random tree-like discrete structures , 2016, Probability Surveys.

[20]  Colin McDiarmid,et al.  Random graphs from a block-stable class , 2014, Eur. J. Comb..

[21]  Benedikt Stufler,et al.  The continuum random tree is the scaling limit of unlabeled unrooted trees , 2014, Random Struct. Algorithms.

[22]  Alessandra Caraceni.  The Scaling Limit of Random Outerplanar Maps , 2014, 1405.1971.

[23]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[24]  Colin McDiarmid,et al.  Random Graphs from a Weighted Minor-Closed Class , 2012, Electron. J. Comb..

[25]  Jakob E. Bjornberg,et al.  Recurrence of bipartite planar maps , 2013, 1311.0178.

[26]  Michael Drmota,et al.  Asymptotic Study of Subcritical Graph Classes , 2010, SIAM J. Discret. Math..

[27]  S. Lane.  Categories for the Working Mathematician , 1971 .

[28]  Gr'egory Miermont,et al.  Tessellations of random maps of arbitrary genus , 2007, 0712.3688.

[29]  Benedikt Stufler,et al.  Geometry of large Boltzmann outerplanar maps , 2017, Random Struct. Algorithms.

[30]  David Aldous,et al.  The Continuum Random Tree III , 1991 .

[31]  Dominic R. Verity,et al.  ∞-Categories for the Working Mathematician , 2018 .

[32]  V. Kraus.  The degree distribution in unlabelled $2$-connected graph families , 2010 .

[33]  Agelos Georgakopoulos,et al.  Limits of subcritical random graphs and random graphs with excluded minors , 2015, 1512.03572.

[34]  Konstantinos Panagiotou,et al.  Scaling Limits of Random Graphs from Subcritical Classes , 2014, 1411.1865.

[35]  Benedikt Stufler,et al.  Local Convergence of Random Planar Graphs , 2019, Trends in Mathematics.

[36]  L. Richier.  Limits of the boundary of random planar maps , 2017, 1704.01950.

[37]  Yann Ponty,et al.  Multi-dimensional Boltzmann Sampling of Languages , 2010, 1002.0046.

[38]  Patrick Billingsley,et al.  Weak convergence of measures - applications in probability , 1971, CBMS-NSF regional conference series in applied mathematics.

[39]  Robin Stephenson,et al.  Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps , 2014, 1412.6911.

[40]  Colin McDiarmid,et al.  Random Graphs from a Minor-Closed Class , 2009, Combinatorics, Probability and Computing.

[41]  Michael Drmota,et al.  Subgraph statistics in subcritical graph classes , 2015, Random Struct. Algorithms.

[42]  Gilbert Labelle,et al.  Combinatorial species and tree-like structures , 1997, Encyclopedia of mathematics and its applications.

[43]  R. Abraham,et al.  A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces , 2012, 1202.5464.

[44]  Donald L. Iglehart,et al.  Functionals of Brownian meander and Brownian excursion , 1977 .

[45]  Marc Noy,et al.  Extremal Parameters in Sub-Critical Graph Classes , 2013, ANALCO.

[46]  Konstantinos Panagiotou,et al.  The Degree Sequence of Random Graphs from Subcritical Classes† , 2009, Combinatorics, Probability and Computing.

[47]  S. Janson,et al.  Sesqui-type branching processes , 2017, Stochastic Processes and their Applications.

[48]  Michael Wallner,et al.  On the shape of random Pólya structures , 2017, Discret. Math..

[49]  Manuel Bodirsky,et al.  Enumeration and Asymptotic Properties of Unlabeled Outerplanar Graphs , 2007, Electron. J. Comb..

[50]  Omer Angel,et al.  Voronoi tessellations in the CRT and continuum random maps of finite excess , 2018, SODA.

[51]  Stanley Burris,et al.  Counting Rooted Trees: The Universal Law t(n)~C ρ-n n-3/2 , 2006, Electron. J. Comb..

[52]  HEIGHT AND DIAMETER OF BROWNIAN TREE , 2015, 1503.05014.

[53]  B'en'edicte Haas,et al.  Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees , 2010, 1003.3632.

[54]  Frank Harary,et al.  Graph Theory , 2016 .

[55]  Svante Janson,et al.  Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation , 2011, 1112.0510.

[56]  Benedikt Stufler,et al.  Unlabelled Gibbs partitions , 2016, Combinatorics, Probability and Computing.

[57]  Sigurdur Orn Stef'ansson,et al.  Scaling limits of random planar maps with a unique large face , 2012, 1212.5072.

[58]  Guillem Perarnau,et al.  Connectivity in bridge-addable graph classes: the mcdiarmid-steger-welsh conjecture , 2016, SODA 2016.

[59]  Colin McDiarmid,et al.  Random Graphs, Geometry and Asymptotic Structure , 2016, London Mathematical Society student texts.

[60]  Benedikt Stufler.  Scaling limits of random outerplanar maps with independent link-weights , 2015 .

[61]  Omer Giménez,et al.  Asymptotic enumeration and limit laws of planar graphs , 2005, math/0501269.

[62]  Local Convergence and Stability of Tight Bridge-addable Classes , 2016, Canadian Journal of Mathematics.