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29
Deterministic and Stochastic Models for Coalescence (Aggregation, Coagulation): a Review of the MeanField Theory for Probabilists
 Bernoulli
, 1997
"... Consider N particles, which merge into clusters according to the rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x; y)=N , where K is a specified rate kernel. This MarcusLushnikov model of stochastic coalescence, and the underlying deterministic approximation given by ..."
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Cited by 142 (13 self)
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Consider N particles, which merge into clusters according to the rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x; y)=N , where K is a specified rate kernel. This MarcusLushnikov model of stochastic coalescence, and the underlying deterministic approximation given by the Smoluchowski coagulation equations, have an extensive scientific literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x; y) = 1 and K(x; y) = xy. We attempt a wideranging survey. General kernels are only now starting to be studied rigorously, so many interesting open problems appear. Keywords. branching process, coalescence, continuum tree, densitydependent Markov process, gelation, random graph, random tree, Smoluchowski coagulation equation Research supported by N.S.F. Grant DMS9622859 1 Introduction Models, implicitly or explicitly stochastic, of coalescence (= coagulati...
Sudden Emergence Of A Giant kCore In A Random Graph.
 J. Combinatorial Theory, Series B
, 1996
"... The k core of a graph is the largest subgraph with minimum degree at least k . For the ErdosR'enyi random graph G(n; m) on n vertices, with m edges, it is known that a giant 2core grows simultaneously with a giant component, that is when m is close to n=2 . We show that for k 3 , with high proba ..."
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Cited by 105 (8 self)
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The k core of a graph is the largest subgraph with minimum degree at least k . For the ErdosR'enyi random graph G(n; m) on n vertices, with m edges, it is known that a giant 2core grows simultaneously with a giant component, that is when m is close to n=2 . We show that for k 3 , with high probability, a giant k core appears suddenly when m reaches c k n=2 ; here c k = min ?0 = k () and k () = PfPoisson() k \Gamma 1g . In particular, c 3 3:35 . We also demonstrate that, unlike the 2core, when a k core appears for the first time it is very likely to be giant, of size p k ( k )n . Here k is the minimum point of = k () and p k ( k ) = PfPoisson( k ) kg . For k = 3 , for instance, the newborn 3core contains about 0:27n vertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always finds a k core if the graph has one. 1991 Mathematics Subject Classification. Primary 05C80, 05C85, 60C05; Secondary 60F10, 60G42, 60J10.
Coalescent Random Forests
 J. COMBINATORIAL THEORY A
, 1998
"... Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ..."
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Cited by 38 (18 self)
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Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...
Maximum matchings in sparse random graphs: KarpSipser revisited
, 1997
"... We study the average performance of a simple greedy algorithm for finding a matching in a sparse random graph G n;c=n , where c ? 0 is constant. The algorithm was first proposed by Karp and Sipser [12]. We give significantly improved estimates of the errors made by the algorithm. For the subcritica ..."
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Cited by 35 (10 self)
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We study the average performance of a simple greedy algorithm for finding a matching in a sparse random graph G n;c=n , where c ? 0 is constant. The algorithm was first proposed by Karp and Sipser [12]. We give significantly improved estimates of the errors made by the algorithm. For the subcritical case where c ! e we show that the algorithm finds a maximum matching with high probability. If c ? e then with high probability the algorithm produces a matching which is within n 1=5+o(1) of maximum size. 1 Introduction A matching in a graph G = (V; E) is a set of edges in E which are vertex disjoint. A standard problem in algorithmic graph theory is to find the largest possible matching in a graph. The first polynomial time algorithm to solve this problem was devised by Edmonds in 1965 and runs in time O(jV j 4 ) [10]. Over the years, many improvements have been made. Currently the fastest such algorithm is that of Micali and Vazirani which dates back to 1980. Its running time is O(...
The Minimal Spanning Tree In A Complete Graph And A Functional Limit Theorem For Trees In A Random Graph.
, 1997
"... . The minimal weight of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges, is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tr ..."
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Cited by 22 (3 self)
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. The minimal weight of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges, is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tree components of given sizes in a random graph. 1. Introduction and results Assign random weights T ij , 1 i ! j n, to the edges of the complete graph K n with vertex set f1; : : : ; ng, and let W n be the minimum weight of a spanning tree of K n . We assume that the weights are independent and identically distributed, with a uniform distribution on [0; 1]. It was proved by Frieze [5] that W n ! i(3) = 1 X k=1 k \Gamma3 = 1:202 : : : in probability as n ! 1, see also Bollobs [3]. The main purpose of the present paper is to show that W n has an asymptotic normal distribution. Theorem 1. Let W n be the weight of the minimal spanning tree. Then n 1=2 \Gamma W n \Gamma i(3) \Delta ...
Emergence of the Giant Component in Special MarcusLushnikov Processes
 Random Structures and Algorithms
, 1997
"... Component sizes in the usual random graph process are a special case of the MarcusLushnikov process discussed in the scientific literature, so it is natural to ask how theory surrounding emergence of the giant component generalizes to the MarcusLushnikov process. Essentially no rigorous results ar ..."
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Cited by 13 (4 self)
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Component sizes in the usual random graph process are a special case of the MarcusLushnikov process discussed in the scientific literature, so it is natural to ask how theory surrounding emergence of the giant component generalizes to the MarcusLushnikov process. Essentially no rigorous results are known; we make a start by proving a weak result, but our main purpose is to draw this topic to the attention of random graph theorists. 1 Introduction 1.1 Background At time zero there are n separate "atoms"; as time increases, these atoms coalesce into clusters according to the rule for each pair of clusters, of sizes fx; yg say, they coalesce into a single cluster of size x + y at rate K(x; y)=n where K(x; y) = K(y; x) 0 is some specified rate kernel. This rule specifies a continuoustime finitestate Markov process which we shall call the Research supported by N.S.F. Grant DMS9622859 MarcusLushnikov process. The model was introduced by Marcus [16], and further studied by Lush...
The Average Performance Of The Greedy Matching Algorithm
, 1997
"... this paper we discuss the expected performance of the simplest of matching algorithms i.e. the GREEDY (or myopic) algorithm. Given a graph G the ..."
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Cited by 11 (1 self)
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this paper we discuss the expected performance of the simplest of matching algorithms i.e. the GREEDY (or myopic) algorithm. Given a graph G the
Asymptotic normality of the kcore in random graphs
, 2006
"... We study the kcore of a random (multi)graph on n vertices with a given degree sequence. In our previous paper [Random Structures Algorithms 30 (2007) 50–62] we used properties of empirical distributions of independent random variables to give a simple proof of the fact that the size of the giant k ..."
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Cited by 8 (6 self)
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We study the kcore of a random (multi)graph on n vertices with a given degree sequence. In our previous paper [Random Structures Algorithms 30 (2007) 50–62] we used properties of empirical distributions of independent random variables to give a simple proof of the fact that the size of the giant kcore obeys a law of large numbers as n →∞. Here we develop the method further and show that the fluctuations around the deterministic limit converge to a Gaussian law above and near the threshold, and to a nonnormal law at the threshold. Further, we determine precisely the location of the phase transition window for the emergence of a giant kcore. Hence, we deduce corresponding results for the kcore in G(n, p) and G(n, m).
A new approach to the giant component problem
 Random Struct. Alg
, 2008
"... Abstract. We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the ..."
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Cited by 7 (3 self)
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Abstract. We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed [24; 25] on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n, p) with np = 1 + ω(n)n −1/3, where ω(n) → ∞ arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs. 1.