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The handle http://hdl.handle.net/1887/49012 holds various files of this Leiden University dissertation.
Author: Gao, F.
Title: Bayes and networks
Issue Date: 2017-05-23
S U M M A R Y
The dissertation is a collection of five papers in two general themes—
Bayes and Networks and three different subjects.
Part I of the dissertation is on an inverse problem in nonparamet- ric Bayesian statistics. Nonparametric Bayes is the statistical method- ology of recovering nonparametric objects—e.g., densities, distribu- tions and those whom it is not possible to describe with a finite num- ber of parameters—with proper prior knowledge about the object. In- verse problems arise when the object of interest cannot be directly observed and we must account for the noisy observation. In Chapter 1, we study the inference of the mixing distribution from observations corrupted with Laplace errors by putting a Dirichlet process prior on the mixing distribution. We derive a contraction rate for the corre- sponding posterior distribution, both for the mixing distribution rel- ative to the Wasserstein metric and for the mixed density relative to the Hellinger and𝐿𝑞metrics. The result is quite surprising as it contra- dicts the usual interpretation that the Laplace mixtures are1-Hölder smooth, which would suggest a slower rate than that obtained in the chapter.
In Part II of the dissertation, we study statistical inference in pref- erential attachment network model. Preferential attachment model is a popular dynamic network model where the network evolves by con- secutively adding new nodes and the nodes connect preferentially to the existing nodes of high degrees. The model offers a possible expla- nation for the ubiquitousness of scale-free networks. The preferential attachment function, which maps the degrees to the preference, de- termines the behavior of the model and we study the inverse problem of recovering the function from the observed network. For example, suppose we observe a scale-free social network and want to find out the mechanism responsible for the evolution of the network.
Chapter 2 serves as the introduction to Part II. First we sketch the importance of the relatively new discipline of network science and describes how network science has emerged into such a flourishing field. Then we briefly explain the history of the preferential attach- ment model and its relevance in dynamic network modeling and rig- orously formulate the mathematical model of the preferential attach- ment networks.
In Chapter 3, we propose the empirical estimator of the general sublinear preferential attachment function. We design a supercritical CMJ branching process, whose evolution is in a certain sense equiv- alent to preferential attachment model. Under the branching process
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Summary
framework, we proceed to prove almost sure consistency of the pro- posed estimator by applying classical results on branching processes.
Empirical properties of the estimators are studied through numerical illustrations.
In Chapter 4, we consider estimation of the affine parameter and power-law exponent in the preferential attachment model with ran- dom initial degrees. The model combines the “rich-get-richer” and
“rich-by-birth” effects and is sufficiently flexible in modeling some real-world networks. We derive the likelihood, and show that the max- imum likelihood estimator is asymptotically normal and efficient. We also propose a quasi-maximum-likelihood estimator to overcome the dependence on the history of the initial degrees. To demonstrate the power of our idea, we present numerical simulations.
In Chapter 5, we go back to the general sublinear preferential at- tachment model and assume that the preferential attachment func- tion has a parametric form—the function is completely determined by a finite number of parameters. With the supercritical CMJ branch- ing process framework identical to the one in Chapter 3 and the rel- evant classical results, we study the deterministic asymptotic limit of the likelihood function, and show the maximum likelihood estimator is asymptotic normal. We also discuss a slight history-free remedy to the likelihood function so that it no longer relies on the evolution his- tory of the network but its maximizer still tends to the true parameter.
The final Part III of the dissertation concerns modeling the movie- actor network with the preferential attachment network model. With the dataset freely provided by the Internet Movie Database (IMDb), we devise a novel model with two layers—one layer of actors and the other of movies, and connect the nodes in two layers if an actor plays in a movie and if two actors have a common movie neighbor. Each movie arrives with a random number of actors and has to pick a ran- dom proportion of actors from the existing actor network preferen- tially to the actors having played in many movies. We discover that the degree distribution of the network looks like a power law and fit our model to the IMDb dataset. We build simulations on the fitted model to reconstruct the historical evolution of the movie-actor net- work and determine that the proposed model gives a rather realistic fit to the IMDb dataset.
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