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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...
Brownian Excursions, Critical Random Graphs and the Multiplicative Coalescent
, 1996
"... Let (B t (s); 0 s ! 1) be reflecting inhomogeneous Brownian motion with drift t \Gamma s at time s, started with B t (0) = 0. Consider the random graph G(n; n \Gamma1 +tn \Gamma4=3 ), whose largest components have size of order n 2=3 . Normalizing by n \Gamma2=3 , the asymptotic joint d ..."
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Cited by 86 (10 self)
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Let (B t (s); 0 s ! 1) be reflecting inhomogeneous Brownian motion with drift t \Gamma s at time s, started with B t (0) = 0. Consider the random graph G(n; n \Gamma1 +tn \Gamma4=3 ), whose largest components have size of order n 2=3 . Normalizing by n \Gamma2=3 , the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of B t (Corollary 2). The dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process. The states of this process are vectors x of nonnegative real cluster sizes (x i ), and clusters with sizes x i and x j merge at rate x i x j . The multiplicative coalescent is shown to be a Feller process on l 2 . The random graph limit specifies the standard multiplicative coalescent, which starts from infinitesimally small clusters at time \Gamma1: the existence of such a process is not obvious. AMS 1991 subject classifications. 60C05, 60J50, Key words and phras...
The Standard Additive Coalescent
, 1997
"... Regard an element of the set \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; X i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent of Evans and Pitman (1997) is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; x j g mer ..."
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Cited by 63 (22 self)
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Regard an element of the set \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; X i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent of Evans and Pitman (1997) is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; x j g merge into a cluster of mass x i +x j at rate x i +x j . They showed that a version (X 1 (t); \Gamma1 ! t ! 1) of this process arises as a n !1 weak limit of the process started at time \Gamma 1 2 log n with n clusters of mass 1=n. We show this standard additive coalescent may be constructed from the continuum random tree of Aldous (1991,1993) by Poisson splitting along the skeleton of the tree. We describe the distribution of X 1 (t) on \Delta at a fixed time t. We show that the size of the cluster containing a given atom, as a process in t, has a simple representation in terms of the stable subordinator of index 1=2. As t ! \Gamma1, we establish a Gaussian limit for (centered and norm...
Construction Of Markovian Coalescents
 Ann. Inst. Henri Poincar'e
, 1997
"... Partitionvalued and measurevalued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass m into a finite or countably infinite number of masses with sum m, and whose evolution is determined by the following intuitive prescription: each pair of ma ..."
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Cited by 44 (20 self)
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Partitionvalued and measurevalued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass m into a finite or countably infinite number of masses with sum m, and whose evolution is determined by the following intuitive prescription: each pair of masses of magnitudes x and y runs the risk of a binary collision to form a single mass of magnitude x+y at rate (x; y), for some nonnegative, symmetric collision rate kernel (x; y). Such processes with finitely many masses have been used to model polymerization, coagulation, condensation, and the evolution of galactic clusters by gravitational attraction. With a suitable choice of state space, and under appropriate restrictions on and the initial distribution of mass, it is shown that such processes can be constructed as Feller or Fellerlike processes. A number of further results are obtained for the additive coalescent with collision kernel (x; y) = x + y. This process, which arises fro...
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...
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...
Ranked fragmentations
 ESAIM P&S
"... distributions for random partitions generated by a ..."
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Cited by 13 (3 self)
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distributions for random partitions generated by a
Gibbs distributions for random partitions generated by a fragmentation process
, 2006
"... process ..."
Globs in the Primordial Soup The Emergence of Connected Crowds in Mobile Wireless Networks
"... In many practical scenarios, nodes gathering at points of interest yield sizable connected components (clusters), which sometimes comprise the majority of nodes. While recent analysis of mobile networks focused on the process governing node encounters (“contacts”), this model is not particularly sui ..."
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Cited by 2 (0 self)
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In many practical scenarios, nodes gathering at points of interest yield sizable connected components (clusters), which sometimes comprise the majority of nodes. While recent analysis of mobile networks focused on the process governing node encounters (“contacts”), this model is not particularly suitable for gathering behavior. In this paper, we propose a model of stochastic coalescence (merge) and fragmentation (split) of clusters. We implement this process as a Markov chain and derive analytically the exact stationary distribution of cluster size. Further, we prove that, as the number of nodes grows, the clustering behavior converges to a mean field, which is obtained as a closedform expression. This expression translates the empirical merge and split rate of a scenario, a microscopic property, to an important macroscopic property—the cluster size distribution—with surprising accuracy. We validate all results with synthetic as well as realworld mobility traces from conference visitors and taxicabs with several thousand nodes.