Results 1  10
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95
Latent Space Approaches to Social Network Analysis
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2001
"... Network models are widely used to represent relational information among interacting units. In studies of social networks, recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation be ..."
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Cited by 154 (15 self)
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Network models are widely used to represent relational information among interacting units. In studies of social networks, recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation between actors. We develop a class of models where the probability of a relation between actors depends on the positions of individuals in an unobserved "social space." Inference for the social space is developed within a maximum likelihood and Bayesian framework, and Markov chain Monte Carlo procedures are proposed for making inference on latent positions and the effects of observed covariates. We present analyses of three standard datasets from the social networks literature, and compare the method to an alternative stochastic blockmodeling approach. In addition to improving upon model fit, our method provides a visual and interpretable modelbased spatial representation of social relationships, and improves upon existing methods by allowing the statistical uncertainty in the social space to be quantified and graphically represented.
Characterisation results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes
, 1996
"... The areainteraction process and the continuum randomcluster model are characterised in terms of certain functional forms of their respective conditional intensities. In certain cases, these two point process models can be derived from a bivariate point process model which in many respects is simpl ..."
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Cited by 71 (6 self)
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The areainteraction process and the continuum randomcluster model are characterised in terms of certain functional forms of their respective conditional intensities. In certain cases, these two point process models can be derived from a bivariate point process model which in many respects is simpler to analyse and simulate. Using this correspondence we devise a twocomponent Gibbs sampler, which can be used for fast and exact simulation by extending the recent ideas of Propp and Wilson. We further introduce a SwendsenWang type algorithm. The relevance of the results within spatial statistics as well as statistical physics is discussed.
Assessing Degeneracy in Statistical Models of Social Networks
 Journal of the American Statistical Association
, 2003
"... discussions. This paper presents recent advances in the statistical modeling of random graphs that have an impact on the empirical study of social networks. Statistical exponential family models (Wasserman and Pattison 1996) are a generalization of the Markov random graph models introduced by Frank ..."
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Cited by 55 (14 self)
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discussions. This paper presents recent advances in the statistical modeling of random graphs that have an impact on the empirical study of social networks. Statistical exponential family models (Wasserman and Pattison 1996) are a generalization of the Markov random graph models introduced by Frank and Strauss (1986), which in turn are derived from developments in spatial statistics (Besag 1974). These models recognize the complex dependencies within relational data structures. A major barrier to the application of random graph models to social networks has been the lack of a sound statistical theory to evaluate model fit. This problem has at least three aspects: the specification of realistic models, the algorithmic difficulties of the inferential methods, and the assessment of the degree to which the graph structure produced by the models matches that of the data. We discuss these and related issues of the model degeneracy and inferential degeneracy for commonly used estimators.
On some weighted Boolean models.
 Advances in Theory and Applications of Random Sets
, 1997
"... An overview is given of some recent work (joint with Adrian Baddeley of Perth and Colette van Lieshout of Warwick) on a new class of random point and set processes, obtained using a rather natural weighting procedure employing quermass integrals. The concept of exact (or perfect) simulation of point ..."
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Cited by 30 (10 self)
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An overview is given of some recent work (joint with Adrian Baddeley of Perth and Colette van Lieshout of Warwick) on a new class of random point and set processes, obtained using a rather natural weighting procedure employing quermass integrals. The concept of exact (or perfect) simulation of point processes is then introduced, and a discussion is given of possibilities for perfect simulation of quermass weighted processes. 1 Introduction. This short paper is a progress report on some recent work of mine (partly in collaboration with Adrian Baddeley of Perth and Colette van Lieshout of Warwick); concerning new models for point processes and random sets, and the application to them of a new technique for "exact" or "perfect" simulation. 2 A brief overview of quermassinteraction processes In this section we review the idea of weighting point processes and random sets using quermass integrals. Recall Baddeley and Van Lieshout's definition [BvL95] of an areainteraction point process: ...
Hydrodynamical limit for a Hamiltonian system with weak noise
 Comm. Math. Phys
, 1993
"... Abstract. Starting from a general Hamiltonian system with superstable pairwise potential, we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles. We prove that, in the scaling limit, the time conserved quantities, energy, momenta and density, satisf ..."
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Cited by 25 (11 self)
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Abstract. Starting from a general Hamiltonian system with superstable pairwise potential, we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles. We prove that, in the scaling limit, the time conserved quantities, energy, momenta and density, satisfy the Euler equation of conservation laws up to a fixed time t provided that the Euler equation has a smooth solution with a given initial data up to time t. The strength of the noise term is chosen to be very small (but nonvanishing) so that it disappears in the scaling limit. 1.
The Classical Limit of Quantum Partition Functions
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1980
"... We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approach S² classical spins as L> ∞] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight ..."
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Cited by 22 (0 self)
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We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approach S² classical spins as L> ∞] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight vector in an irreducible representation and we prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has "diagonal form").
Markov Properties of Cluster Processes
, 1996
"... We show that a Poisson cluster point process is a nearestneighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finiterange Markov point process in the sense of Ripley and Kelly [11]. Furthermore, when the parent Poisson process is replaced by a M ..."
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Cited by 20 (8 self)
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We show that a Poisson cluster point process is a nearestneighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finiterange Markov point process in the sense of Ripley and Kelly [11]. Furthermore, when the parent Poisson process is replaced by a Markov or nearestneighbour Markov point process, the resulting cluster process is also nearestneighbour Markov, provided all clusters are nonempty. In particular, the nearestneighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned. 1. Introduction Markov or Gibbs point processes [2, 8, 11, 12] form a large, flexible, and understandable class of point process models with many practical advantages (see e.g. [4, 9, 10] for surveys). In this paper we consider the relationship of these models to the basic point process operation of clustering. We ask whether cluster processes are Markov, and whether the Mark...
Symmetry Decomposition of Chaotic Dynamics
, 1993
"... Discrete symmetries of dynamical flows give rise to relations between periodic orbits, reduce the dynamics to a fundamental domain, and lead to factorizations of zeta functions. These factorizations in turn reduce the labor and improve the convergence of cycle expansions for classical and quantum sp ..."
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Cited by 16 (5 self)
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Discrete symmetries of dynamical flows give rise to relations between periodic orbits, reduce the dynamics to a fundamental domain, and lead to factorizations of zeta functions. These factorizations in turn reduce the labor and improve the convergence of cycle expansions for classical and quantum spectra associated with the flow. In this paper the general formalism is developed, with the Ndisk pinball model used as a concrete example and a series of physically interesting cases worked out in detail. 1 permanent address 1 Introduction The periodic orbit theory of classical chaotic dynamical systems has a long and distinguished history; initiated by Poincar'e[1], and developed as a mathematical theory of hyperbolic dynamical systems by Smale, Sinai, Bowen, Ruelle and others[2, 3, 4, 5], it has in recent years been applied to many systems of physical interest[6, 7, 8, 9]. The periodic orbit theory of quantum mechanical systems largely parallels this development; originating in the wor...
Glauber dynamics of continuous particle systems
"... This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ cor ..."
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Cited by 16 (7 self)
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This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ corresponding to a general pair potential φ and activity z> 0. We consider a Dirichlet form E on L2 (Γ,µ) which corresponds to the generator H of the Glauber dynamics. We prove the existence of a Markov process M on Γ that is properly associated with E. In the case of a positive potential φ which satisfies δ: = ∫ Rd(1 − e−φ(x))z dx < 1, we also prove that the generator H has a spectral gap ≥ 1−δ. Furthermore, for any pure Gibbs state µ, we derive a Poincaré inequality. The results about the spectral gap and the Poincaré inequality are a generalization and a refinement of a recent result from [6].
Remarks on decay of correlations and Witten Laplacians  BrascampLieb inequalities and semiclassical limit
, 1997
"... As it appears in recent articles by Helffer or Sjostrand and NaddafSpencer, the analysis, in the context of the statistical mechanics, of measures of the type exp \Gamma\Phi(x) dx is connected with the analysis of suitable Witten Laplacians on 1forms. For illustrating this point of view, we present ..."
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Cited by 14 (2 self)
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As it appears in recent articles by Helffer or Sjostrand and NaddafSpencer, the analysis, in the context of the statistical mechanics, of measures of the type exp \Gamma\Phi(x) dx is connected with the analysis of suitable Witten Laplacians on 1forms. For illustrating this point of view, we present here remarks about the BrascampLieb inequalities and its extensions and prove the decay of the correlation in some cases when \Phi is weakly non convex. 1 Introduction Our aim 1 is to analyze Laplace integrals associated to a measure whose density with respect to the Lebesgue measure takes the form exp \Gamma 1 h \Phi, up to a multiplicative normalization constant, in the case when the potential \Phi is weakly convex or weakly non convex. We analyze as a starting point the 1 A first version of these remarks was diffused in June 1996. BrascampLieb inequality or the Poincar'e inequality in connection with the lowest eigenvalue of a suitable Witten Laplacian on oneforms. The role ...