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The structure and function of complex networks
 SIAM REVIEW
, 2003
"... Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, ..."
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Cited by 1407 (9 self)
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Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the smallworld effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
The Posterior Probability of Bayes Nets with Strong Dependences
 Soft Computing
, 1999
"... Stochastic independence is an idealized relationship located at one end of a continuum of values measuring degrees of dependence. Modeling real world systems, we are often not interested in the distinction between exact independence and any degree of dependence, but between weak ignorable and strong ..."
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Cited by 14 (1 self)
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Stochastic independence is an idealized relationship located at one end of a continuum of values measuring degrees of dependence. Modeling real world systems, we are often not interested in the distinction between exact independence and any degree of dependence, but between weak ignorable and strong substantial dependence. Good models map significant deviance from independence and neglect approximate independence or dependence weaker than a noise threshold. This intuition is applied to learning the structure of Bayes nets from data. We determine the conditional posterior probabilities of structures given that the degree of dependence at each of their nodes exceeds a critical noise level. Deviance from independence is measured by mutual information. Arc probabilities are determined by the amount of mutual information the neighbors contribute to a node, is greater than a critical minimum deviance from independence. A Ø 2 approximation for the probability density function of mutual info...
Stochastic Evolution via Graph Grammars
 INRIA Research Report #3380
, 1998
"... : This is the second part in the series of papers where we are looking for new connections between computer science, mathematics and physics. These connections go through the central notions of computer science  grammar and graph grammar. In section 2 main definitions concerning random graph gramma ..."
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Cited by 3 (1 self)
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: This is the second part in the series of papers where we are looking for new connections between computer science, mathematics and physics. These connections go through the central notions of computer science  grammar and graph grammar. In section 2 main definitions concerning random graph grammars are given. Section 3 and 4 are devoted to the simplest models of graph dynamics: we study the large time behaviour of local and global characteristics of growing onedimensional complexes. Main emphasis is on looking for correct problems and models, discussing their qualitative behaviour. We consider asymptotic growth of the number of connected components, the degree of local compactness, topological chaos, phases of different topology etc. In section 4 we construct infinite cluster dynamics. We discuss some aspects which distinguish thermodynamic limit for graph grammars from that for Gibbs fields on a lattice. One of the central emerging notions is the statistically homogeneous infinite...
AverageCase Analysis of GraphSearching Algorithms
, 1990
"... Sarantos Kapidakis Advisor  Professor Robert Sedgewick We estimate the expected value of various search quantities for a variety of graphsearching methods, for example depthfirst search and breadthfirst search. Our analysis applies to both directed and undirected random graphs, and it covers th ..."
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Cited by 2 (0 self)
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Sarantos Kapidakis Advisor  Professor Robert Sedgewick We estimate the expected value of various search quantities for a variety of graphsearching methods, for example depthfirst search and breadthfirst search. Our analysis applies to both directed and undirected random graphs, and it covers the range of interesting graph densities, including densities at which a random graph is disconnected with a giant connected component. We estimate the number of edges examined during the search, since this number is proportional to the running time of the algorithm. We find that for hardly connected graphs, all of the edges might be examined, but for denser graphs many fewer edges are generally required. We prove that any searching algorithm examines \Theta(n log n) edges, if present, on all random graphs with n nodes but not necessarily on the complete graphs. One property of some searching algorithms is the maximum depth of the search. In depthfirst search, this depth can be used to estima...
Disintegration, Yule Process and Random Graphs
"... In this paper we demonstrate that the disintegration process of radioactive atoms in the discrete time model has a natural interpretation in the theory of random graphs. Moreover, we reduce some problems in this area to disintegration processes with anomalies. For instance, we derive exact closed fo ..."
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In this paper we demonstrate that the disintegration process of radioactive atoms in the discrete time model has a natural interpretation in the theory of random graphs. Moreover, we reduce some problems in this area to disintegration processes with anomalies. For instance, we derive exact closed formulae for the probabilities and the moments of the number of isolated vertices in Bernoulli graphs and we investigate the connectedness of a graph. With a similar approach we study a different problem, namely the Yule process, and we observe some relationships to the theory of fractals. 1 Introduction In probability theory, one of the dominating processes for analyzing successions of non independent events, is the Markov chain. It is defined as a stochastic process (or more precisely, a family of random variables fX t j t = 0; 1; : : :g), in which the future development depends only on the present state, but not on its past history or the manner in which the present state was reached. Furt...
Weak Thresholds
, 1996
"... . We generalize the notion of a "threshold function" from random graph theory: given a poset with an upwards closed subposet, we ask when a random process going up the poset reaches the subposet. We present a measuretheoretic generalization of a splitting argument to prove that in many such process ..."
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. We generalize the notion of a "threshold function" from random graph theory: given a poset with an upwards closed subposet, we ask when a random process going up the poset reaches the subposet. We present a measuretheoretic generalization of a splitting argument to prove that in many such processes, all `upwards closed' properties have `weak thresholds'. Contents 0. Introductory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1. Processionals : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2. Thresholds : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 3. Marginalia : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 4. Splitting : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 22 5. Powers of Processionals: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29 6. Sections : : : : : : : : : : : : : : : : : :...
Random intersection graphs when m = w(n): an equivalence theorem relating the evolution of the G(n, m, p) and G(n, p) models
, 1998
"... When the random intersection graph G(n, m, p) proposed by Karonski, Scheinerman, and SingerCohen in [8] is compared with the independentedge G(n, p), the evolutions are di#erent under some values of m and equivalent under others. In particular, when m = n # and # > 6, the total variation distanc ..."
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When the random intersection graph G(n, m, p) proposed by Karonski, Scheinerman, and SingerCohen in [8] is compared with the independentedge G(n, p), the evolutions are di#erent under some values of m and equivalent under others. In particular, when m = n # and # > 6, the total variation distance between the graph random variables has limit 0. Key Words: Random graphs, intersection graphs, total variation distance, threshold AMS Subject Classifications: 05C80 Random graphs # Department of Mathematical Sciences, The Johns Hopkins University, email: jimfill@jhu.edu + Department of Mathematical Sciences, The Johns Hopkins University, email: ers@jhu.edu # Department of Mathematics, Wellesley College, email: kcohen@wellesley.edu 1 1
Estimating the Posterior Probability of Bayesian Network
"... Random graphs are used to model the qualitative structure of Bayesian networks. The arc ..."
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Random graphs are used to model the qualitative structure of Bayesian networks. The arc