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Enumerations Of Trees And Forests Related To Branching Processes And Random Walks
 Microsurveys in Discrete Probability, number 41 in DIMACS Ser. Discrete Math. Theoret. Comp. Sci
, 1997
"... In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p ..."
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Cited by 38 (15 self)
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In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p j+1 for j \Gamma1. The formula for this distribution is a probabilistic expression of the Lagrange inversion formula for the coefficients in the power series expansion of f(z) k in terms of those of g(z) for f(z) defined implicitly by f(z) = zg(f(z)). The Lagrange inversion formula is the analytic counterpart of various enumerations of trees and forests which generalize Cayley's formula kn n\Gammak\Gamma1 for the number of rooted forests labeled by a set of size n whose set of roots is a particular subset of size k. These known results are derived by elementary combinatorial methods without appeal to the Lagrange formula, which is then obtained as a byproduct. This approach unifies an...
TreeValued Markov Chains Derived From GaltonWatson Processes.
 Ann. Inst. Henri Poincar'e
, 1997
"... Let G be a GaltonWatson tree, and for 0 u 1 let G u be the subtree of G obtained by retaining each edge with probability u. We study the treevalued Markov process (G u ; 0 u 1) and an analogous process (G u ; 0 u 1) in which G 1 is a critical or subcritical GaltonWatson tree conditio ..."
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Cited by 34 (9 self)
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Let G be a GaltonWatson tree, and for 0 u 1 let G u be the subtree of G obtained by retaining each edge with probability u. We study the treevalued Markov process (G u ; 0 u 1) and an analogous process (G u ; 0 u 1) in which G 1 is a critical or subcritical GaltonWatson tree conditioned to be infinite. Results simplify and are further developed in the special case of Poisson() offspring distribution. Running head. Treevalued Markov chains. Key words. Borel distribution, branching process, conditioning, GaltonWatson process, generalized Poisson distribution, htransform, pruning, random tree, sizebiasing, spinal decomposition, thinning. AMS Subject classifications 05C80, 60C05, 60J27, 60J80 Research supported in part by N.S.F. Grants DMS9404345 and 9622859 1 Contents 1 Introduction 2 1.1 Related topics : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2 Background and technical setup 5 2.1 Notation and terminology for trees : : : : : : : : : : : : : : :...
A LoadingDependent Model Of Probabilistic Cascading Failure
 Probability in the Engineering and Informational Sciences
, 2004
"... We propose an analytically tractable model of loadingdependent cascading failure that captures some of the salient features of large blackouts of electric power transmission systems. This leads to a new application and derivation of the quasibinomial distribution and its generalization to a saturat ..."
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Cited by 22 (8 self)
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We propose an analytically tractable model of loadingdependent cascading failure that captures some of the salient features of large blackouts of electric power transmission systems. This leads to a new application and derivation of the quasibinomial distribution and its generalization to a saturating form with an extended parameter range. The saturating quasibinomial distribution of the number of failed components has a power law region at a critical loading and a significant probability of total failure at higher loadings.
Parking Functions, Empirical Processes, and the Width of Rooted Labeled Trees
"... This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking f ..."
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Cited by 20 (5 self)
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This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking functions and the empirical processes of mathematical statistics. Our result turns out to be a consequence of the strong convergence of empirical processes to the Brownian bridge (Komlos, Major and Tusnady, 1975).
Universal Scaling Behavior of NonEquilibrium Phase Transitions
 Int. J. Mod. Phys. B
, 2004
"... Nonequilibrium critical phenomena have attracted a lot of research interest in the recent decades. Similar to equilibrium critical phenomena, the concept of universality remains the major tool to order the great variety of nonequilibrium phase transitions systematically. All systems belonging to a ..."
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Cited by 7 (0 self)
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Nonequilibrium critical phenomena have attracted a lot of research interest in the recent decades. Similar to equilibrium critical phenomena, the concept of universality remains the major tool to order the great variety of nonequilibrium phase transitions systematically. All systems belonging to a given universality class share the same set of critical exponents, and certain scaling functions become identical near the critical point. It is known that the scaling functions vary more widely between different universality classes than the exponents. Thus, universal scaling functions offer a sensitive and accurate test for a system’s universality class. On the other hand, universal scaling functions demonstrate the robustness of a given universality class impressively. Unfortunately, most studies focus on the determination of the critical exponents, neglecting the universal scaling functions. In this work a particular class of nonequilibrium critical phenomena is considered, the socalled absorbing phase transitions. Absorbing phase transitions are expected to occur in physical, chemical as well as biological systems, and a detailed introduction is presented. The universal scaling behavior of two different universality classes is
SIMPLY GENERATED TREES, CONDITIONED GALTON–WATSON TREES, RANDOM ALLOCATIONS AND CONDENSATION (EXTENDED ABSTRACT)
, 2012
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A point process describing the component sizes in the critical window of the random graph evolution
 Combinatorics, Probability and Computing
"... Abstract. We study a point process describing the asymptotic behavior of sizes of the largest components of the random graph G(n, p) in the critical window, that is, for p = n −1 + λn −4/3, where λ is a fixed real number. In particular, we show that this point process has a surprising rigidity. Fluc ..."
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Cited by 6 (2 self)
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Abstract. We study a point process describing the asymptotic behavior of sizes of the largest components of the random graph G(n, p) in the critical window, that is, for p = n −1 + λn −4/3, where λ is a fixed real number. In particular, we show that this point process has a surprising rigidity. Fluctuations in the large values will be balanced by opposite fluctuations in the small values such that the sum of the values larger than a small ε (a scaled version of the number of vertices in components of size greater than εn 2/3) is almost constant. 1.
Spatial Random Trees and the CenterSurround Algorithm
 Purdue University, School of Electrical and Computer Engineering
, 2003
"... A new class of multiscale stochastic processes called spatial random trees (SRTs) is introduced and studied. As with previous multiscale stochastic processes, SRTs model multidimensional signals using random processes on trees. Our key innovation, however, is that the tree structure itself is rand ..."
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Cited by 6 (5 self)
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A new class of multiscale stochastic processes called spatial random trees (SRTs) is introduced and studied. As with previous multiscale stochastic processes, SRTs model multidimensional signals using random processes on trees. Our key innovation, however, is that the tree structure itself is random and is generated by a probabilistic contextfree grammar (PCFG) [26]. While PCFGs have been used to model 1D signals, the generalization to multiple dimensions is not direct because the leaves of a tree generated by a PCFG cannot be naturally mapped to a multidimensional lattice. We solve this problem by defining a new class of PCFGs which can produce trees whose leaves are naturally arranged in a multidimensional lattice. We call such trees admissible and show that each of them generates a unique multidimensional signal. Based on this framework, procedures are developed for likelihood calculation, MAP estimation of the processes, and parameter estimation. The new framework is illustrated through simple detection problems.
The asymptotic behavior of the Hurwitz binomial distribution
 Combinatorics, Probability and Computing
, 1998
"... Hurwitz's extension of Abel's binomial theorem defines a probability distribution on the set of integers from 0 to n. This is the distribution of the number of nonroot vertices of a fringe subtree of a suitably defined random tree with n + 2 vertices. The asymptotic behaviour of this distribution i ..."
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Cited by 5 (5 self)
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Hurwitz's extension of Abel's binomial theorem defines a probability distribution on the set of integers from 0 to n. This is the distribution of the number of nonroot vertices of a fringe subtree of a suitably defined random tree with n + 2 vertices. The asymptotic behaviour of this distribution is described in a limiting regime where the distribution of the delabeled fringe subtree approaches that of a GaltonWatson tree with a mixed Poisson offspring distribution. 1 Introduction and statement of results Hurwitz [10] discovered the following identity of polynomials in n + 2 variables x; y and z s ; s 2 [n] := f1; : : : ; ng, which reduces to the binomial expansion of (x + y) n when Research supported in part by N.S.F. Grant DMS9703961 z s j 0: X A`[n] x(x + z A ) jAj\Gamma1 (y + z ¯ A ) j ¯ Aj = (x + y + z [n] ) n (1) where the sum is over all 2 n subsets A of [n], with the notations z A := P s2A z s , and jAj for the number of elements of A, and ¯ A := [n] \...
Susceptibility in subcritical random graphs
 125207. OF RANDOM GRAPHS WITH GIVEN VERTEX DEGREES 25
"... Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its ..."
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Cited by 5 (4 self)
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Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal. 1.