# Scaling Limits of Random Graphs from Subcritical Classes

@article{Panagiotou2014ScalingLO,
title={Scaling Limits of Random Graphs from Subcritical Classes},
author={Konstantinos Panagiotou and Benedikt Stufler and Kerstin Weller},
journal={arXiv: Probability},
year={2014}
}
• Published 2014
• Mathematics
• arXiv: Probability
We study the uniform random graph $\mathsf{C}_n$ with $n$ vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph $\mathsf{C}_n / \sqrt{n}$ converges to the Brownian Continuum Random Tree $\mathcal{T}_{\mathsf{e}}$ multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter $\text{D}(\mathsf{C}_n)$ and height $\text{H}(\mathsf{C}_n^\bullet)$ of the rooted… Expand
36 Citations

#### Figures and Tables from this paper

Asymptotic Enumeration of Graph Classes with Many Components
• Mathematics, Computer Science
• ANALCO
• 2018
The central idea in the approach is to sample objects of $\cal G$ randomly by so-called Boltzmann generators in order to translate enumerative problems to the analysis of iid random variables. Expand
Limits of random tree-like discrete structures
We study a model of random $\mathcal{R}$-enriched trees that is based on weights on the $\mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. WeExpand
Invariance principles for random walks in random environment on trees.
In arXiv:1609.05666v1 [math.PR] a functional limit theorem was proved. It states that symmetric processes associated with resistance metric measure spaces converge when the underlying spaces convergeExpand
Random cographs: Brownian graphon limit and asymptotic degree distribution
• Mathematics
• 2019
We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then showExpand
Random graphs from a block-stable class
• Mathematics, Computer Science
• Eur. J. Comb.
• 2016
It is shown that, as for trees, for most $n-vertex graphs in such a class, each vertex is in at most$(1+o(1)) \log n / \log\log n$blocks, and each path passes through at most$5 (n 1/2)^{1/2}$blocks. Expand 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 assignedExpand Universal height and width bounds for random trees • Mathematics • 2021 We prove non-asymptotic stretched exponential tail bounds on the height of a randomly sampled node in a random combinatorial tree, which we use to prove bounds on the heights and widths of randomExpand Graph limits of random graphs from a subset of connected k‐trees • Mathematics, Computer Science • Random Struct. Algorithms • 2019 It is proved that Gn,k, scaled by (kHkσΩ)/(2n) where H k is the kth harmonic number and σ Ω > 0, converges to the continuum random tree Te. Expand An Asymptotic Analysis of Labeled and Unlabeled k-Trees • Mathematics, Computer Science • Algorithmica • 2015 The number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. Expand The boundary of random planar maps via looptrees • Mathematics • 2018 We study the scaling limits of the boundary of Boltzmann planar maps conditioned on having a large perimeter. We first deal with the non-generic critical regime, where the degree of a typical faceExpand #### References SHOWING 1-10 OF 49 REFERENCES Asymptotic Study of Subcritical Graph Classes • Mathematics, Computer Science • SIAM J. Discret. Math. • 2011 A unified general method for the asymptotic study of graphs from the so-called subcritical graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs, which works in both the labelled and unlabelled framework. Expand Random Graphs from Planar and Other Addable Classes • Mathematics • 2006 We study various properties of a random graph R n , drawn uniformly at random from the class $$\mathcal{A}_n$$ of all simple graphs on n labelled vertices that satisfy some given property, such asExpand Scaling Limit of Random Planar Quadrangulations with a Boundary We discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence$(\sigma_n)$of integers suchExpand The Scaling Limit of Random Outerplanar Maps A planar map is outerplanar if all its vertices belong to the same face. We show that random uniform outerplanar maps with$n$vertices suitably rescaled by a factor$1/ \sqrt{n}\$ converge in theExpand
Scaling limits of random planar maps with a unique large face
• Mathematics
• 2015
We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional toExpand
Maximal biconnected subgraphs of random planar graphs
• Computer Science, Mathematics
• TALG
• 2010
It is obtained that the class of planar graphs belongs to category (1), in contrast to that, outerplanar and series-parallel graphs belong to categories (2) and (3). Expand
Probabilistic and fractal aspects of Lévy trees
• Mathematics
• 2005
Abstract.We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of theseExpand
Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees
• Mathematics
• 2012
We consider a family of random trees satisfying a Markov branching property. Roughly, this property says that the subtrees above some given height are independent with a law that depends only onExpand
The CRT is the scaling limit of random dissections
• Mathematics, Computer Science
• Random Struct. Algorithms
• 2015
It is shown that these random graphs, rescaled by n-1/2, converge in the Gromov-Hausdorff sense towards a multiple of Aldous' Brownian tree when the weights decrease sufficiently fast. Expand
The Degree Sequence of Random Graphs from Subcritical Classes†
• Computer Science, Mathematics
• Combinatorics, Probability and Computing
• 2009
The expected number of vertices of degree k = k(n) in a graph with n vertices that is drawn uniformly at random from a subcritical graph class is determined. Expand