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42
Evolution of networks
 Adv. Phys
, 2002
"... We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence rece ..."
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Cited by 269 (2 self)
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We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number of them are scalefree and show striking resilience against random breakdowns. In spite of large sizes of these networks, the distances between most their vertices are short — a feature known as the “smallworld” effect. We discuss how growing networks selforganize into scalefree structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems
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 84 (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...
Structure Learning in Conditional Probability Models via an Entropic Prior and Parameter Extinction
, 1998
"... We introduce an entropic prior for multinomial parameter estimation problems and solve for its maximum... ..."
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Cited by 66 (0 self)
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We introduce an entropic prior for multinomial parameter estimation problems and solve for its maximum...
The Wiener Index Of Simply Generated Random Trees
 Random Struct. Alg
, 2003
"... Asymptotics are obtained for the mean, variance and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the in ..."
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Cited by 35 (14 self)
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Asymptotics are obtained for the mean, variance and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the internal path length, as well as asymptotics for the covariance and other mixed moments. The limit laws are described using functionals of a Brownian excursion. The methods include both Aldous' theory of the continuum random tree and analysis of generating functions. 1.
The Analysis of a ListColoring Algorithm on a Random Graph (Extended Abstract)
, 1997
"... We introduce a natural kcoloring algorithm and analyze its performance on random graphs with constant expected degree c (Gn,p=c/n). For k = 3 our results imply that almost all graphs with n vertices and 1.923 n edges are 3colorable. This improves the lower bound on the threshold for random 3col ..."
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Cited by 32 (5 self)
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We introduce a natural kcoloring algorithm and analyze its performance on random graphs with constant expected degree c (Gn,p=c/n). For k = 3 our results imply that almost all graphs with n vertices and 1.923 n edges are 3colorable. This improves the lower bound on the threshold for random 3colorability significantly and settles the last case of Q longstanding open question of Bollobas [5]. We also provide a tight asymptotic analysis of the algorithm. We show that for all k 2 3, if c 5 klnk 3/2k then the algorithm almost surely succeeds, while for any E> 0, and k sufficiently large, if c 2 (1 + E)k In k then the algorithm almost surely fails. The analysis is based on the use of differential equations to approximate the mean path of certain Markov chains.
Analytic Variations On The Airy Distribution
, 2001
"... . The Airy distribution (of the \area" type) occurs as limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curio ..."
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Cited by 23 (4 self)
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. The Airy distribution (of the \area" type) occurs as limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curious explicit evaluations for certain moments of the Airy distribution, including moments of orders 1; 3; 5; &c, as well as + 1 3 ; 5 3 ; 11 3 ; &c. and 7 3 ; 13 3 ; 19 3 ; &c . Our proofs are based on integral transforms of the Laplace and Mellin type and they rely essentially on \nonprobabilistic" arguments like analytic continuation. A byproduct of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function Ai(z) at +1, and power symmetric functions of the zeros k of Ai(z). For probabilists, the Airy distribution considered here is nothing but the distribution of the area under the Brownian excursion. The ...
Extremal properties of threedimensional sensor networks with applications
 IEEE Transactions on Mobile Computing
"... In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in threedimensional sensor networks. As in other largescale complex systems, many global parameters of sensor networks undergo phase transitions: For a given property of the network, there is a cri ..."
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Cited by 20 (1 self)
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In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in threedimensional sensor networks. As in other largescale complex systems, many global parameters of sensor networks undergo phase transitions: For a given property of the network, there is a critical threshold, corresponding to the minimum amount of the communication effort or power expenditure by individual nodes, above (resp. below) which the property exists with high (resp. a low) probability. For sensor networks, properties of interest include simple and multiple degrees of connectivity/coverage. First, we investigate the network topology according to the region of deployment, the number of deployed sensors and their transmitting/sensing ranges. More specifically, we consider the following problems: Assume that n nodes, each capable of sensing events within a radius of r, are randomly and uniformly distributed in a 3dimensional region R of volume V, how large must the sensing range rSense be to ensure a given degree of coverage of the region to monitor? For a given transmission range rTrans, what is the minimum (resp. maximum) degree of the network? What is then the typical hopdiameter of the underlying network? Next, we show how these results affect algorithmic aspects of the network by designing specific distributed protocols for sensor networks. Keywords Sensor networks, ad hoc networks; coverage, connectivity; hopdiameter; minimum/maximum degrees; transmitting/sensing ranges; analytical methods; energy consumption; topology control. I.