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14
ASYMPTOTIC STUDY OF SUBCRITICAL GRAPH CLASSES
"... We present a unified general method for the asymptotic study of graphs from the socalled “subcritical” graph classes, which include the classes of cacti graphs, outerplanar graphs, and seriesparallel graphs. This general method works both in the labelled and unlabelled framework. The main results ..."
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We present a unified general method for the asymptotic study of graphs from the socalled “subcritical” graph classes, which include the classes of cacti graphs, outerplanar graphs, and seriesparallel graphs. This general method works both in the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number gn/n! (resp. gn) of labelled (resp. unlabelled) graphs on n vertices from a subcritical graph class G = ∪nGn satisfies asymptotically the universal behaviour gn = c n −5/2 γ n (1 + o(1)) for computable constants c, γ, e.g. γ ≈ 9.38527 for unlabelled seriesparallel graphs, and that the number of vertices of degree k (k fixed) in a graph chosen uniformly at random from Gn, converges (after rescaling) to a normal law as n → ∞.
RANDOM SAMPLING OF PLANE PARTITIONS
"... abstract. This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is O(n(ln n) ..."
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abstract. This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is O(n(ln n) 3) in approximatesize sampling and O(n4/3) in exactsize sampling (under a realarithmetic computation model). To our knowledge, these are the first polynomialtime samplers for plane partitions according to the size (there exist polynomialtime samplers of another type, which draw plane partitions that fit inside a fixed bounding box). The same principles yield efficient samplers for (a × b)boxed plane partitions (plane partitions with two dimensions bounded), and for skew plane partitions. The random samplers allow us to perform simulations and observe limit shapes and frozen boundaries, which have been analysed recently by Cerf and Kenyon for plane partitions, and by Okounkov and Reshetikhin for skew plane partitions.
The Maximum Degree of Random Planar Graphs
"... Let Pn denote a graph drawn uniformly at random from the class of all simple planar graphs with n vertices. We show that the maximum degree of a vertex in Pn is with probability 1−o(1) asymptotically equal to c log n, where c ≈ 2.529 is determined explicitly. A similar result is also true for random ..."
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Let Pn denote a graph drawn uniformly at random from the class of all simple planar graphs with n vertices. We show that the maximum degree of a vertex in Pn is with probability 1−o(1) asymptotically equal to c log n, where c ≈ 2.529 is determined explicitly. A similar result is also true for random 2connected planar graphs. Our analysis combines two orthogonal methods that complement each other. First, in order to obtain the upper bound, we resort to exact methods, i.e., to generating functions and analytic combinatorics. This allows us to obtain fairly precise asymptotic estimates for the expected number of vertices of any given degree in Pn. On the other hand, for the lower bound we use Boltzmann sampling. In particular, by tracing the execution of an adequate algorithm that generates a random planar graph, we are able to explicitly find vertices of sufficiently high degree in Pn.
Infinite Boltzmann Samplers and Applications to Branching Processes, in "GASCom  8th edition of the conference GASCom on random generation of combinatorial structures  2012
, 2012
"... ABSTRACT. In this short note, we extend the Boltzmann model for combinatorial random sampling [8] to allow for infinite size objects; in particular, this extension now fully includes GaltonWatson processes. We then illustrate our idea with two examples, one of which is the generation of prefixes of ..."
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ABSTRACT. In this short note, we extend the Boltzmann model for combinatorial random sampling [8] to allow for infinite size objects; in particular, this extension now fully includes GaltonWatson processes. We then illustrate our idea with two examples, one of which is the generation of prefixes of infinite Cayley trees. 1.
contraction of an imbalance sequence, ergodic Markov chain, mixing time
"... Abstract. The imbalance of a vertex v in a digraph D is defined as a(v) = d + (v)−d − (v), where d + (v) and d − (v) respectively denote the outdegree and indegree of vertex v. The imbalance sequence of D is formed by listing vertex imbalances in nondecreasing order. We define a minimally cyclic di ..."
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Abstract. The imbalance of a vertex v in a digraph D is defined as a(v) = d + (v)−d − (v), where d + (v) and d − (v) respectively denote the outdegree and indegree of vertex v. The imbalance sequence of D is formed by listing vertex imbalances in nondecreasing order. We define a minimally cyclic digraph as a connected digraph which is either acyclic or has exactly one oriented cycle whose removal disconnects the digraph. In this paper we consider the problem of sampling minimally cyclic digraphs with a given imbalance sequence. We present an algorithm for generating all minimally cyclic graphs realizing an imbalance sequence with each realization produced in linear time. We then construct a Markov chain on the set M of all minimally cyclic digraphs with a given imbalance sequence and prove that this Markov chain is ergodic, reversible and rapidly mixing. Thus we can sample through M in polynomial time.
Random road networks: the quadtree model
, 1008
"... What does a typical road network look like? Existing generative models tend to focus on one aspect to the exclusion of others. We introduce the generalpurpose quadtree model and analyze its shortest paths and maximum flow. 1 ..."
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What does a typical road network look like? Existing generative models tend to focus on one aspect to the exclusion of others. We introduce the generalpurpose quadtree model and analyze its shortest paths and maximum flow. 1
9. Bibliography............................................................................101. Team
"... c t i v it y e p o r t 2008 Table of contents 1. Team.................................................................................... 1 ..."
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c t i v it y e p o r t 2008 Table of contents 1. Team.................................................................................... 1
ANALYTIC COMBINATORICS OF CHORD AND HYPERCHORD DIAGRAMS WITH k CROSSINGS
"... Using methods from Analytic Combinatorics, we study the families of perfect matchings, partitions, chord diagrams, and hyperchord diagrams on a disk with a prescribed number of crossings. For each family, we express the generating function of the configurations with exactly k crossings as a rationa ..."
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Using methods from Analytic Combinatorics, we study the families of perfect matchings, partitions, chord diagrams, and hyperchord diagrams on a disk with a prescribed number of crossings. For each family, we express the generating function of the configurations with exactly k crossings as a rational function of the generating function of crossingfree configurations. Using these expressions, we study the singular behavior of these generating functions and derive asymptotic results on the counting sequences of the configurations with precisely k crossings. Limiting distributions and random generators are also studied.