Exchangeable Random Networks

We introduce and study a class of exchangeable random graph ensembles. They can be used as statistical null models for empirical networks, and as a tool for theoretical investigations. We provide general theorems that carachterize the degree distribution of the ensemble graphs, together with some features that are important for applications, such as subgraph distributions and kernel of the adjacency matrix. These results are used to compare to other models of simple and complex networks. A particular case of directed networks with power-law out--degree is studied in more detail, as an example of the flexibility of the model in applications.Comment: to appear on "Internet Mathematics

On Finite Exchangeability and Conditional Independence

We study the independence structure of finitely exchangeable distributions over random vectors and random networks. In particular, we provide necessary and sufficient conditions for an exchangeable vector so that its elements are completely independent or completely dependent. We also provide a sufficient condition for an exchangeable vector so that its elements are marginally independent. We then generalize these results and conditions for exchangeable random networks. In this case, it is demonstrated that the situation is more complex. We show that the independence structure of exchangeable random networks lies in one of six regimes that are two-fold dual to one another, represented by undirected and bidirected independence graphs in graphical model sense with graphs that are complement of each other. In addition, under certain additional assumptions, we provide necessary and sufficient conditions for the exchangeable network distributions to be faithful to each of these graphs.Comment: 25 pages, 2 figure

Network inference and community detection, based on covariance matrices, correlations and test statistics from arbitrary distributions

In this paper we propose methodology for inference of binary-valued adjacency matrices from various measures of the strength of association between pairs of network nodes, or more generally pairs of variables. This strength of association can be quantified by sample covariance and correlation matrices, and more generally by test-statistics and hypothesis test p-values from arbitrary distributions. Community detection methods such as block modelling typically require binary-valued adjacency matrices as a starting point. Hence, a main motivation for the methodology we propose is to obtain binary-valued adjacency matrices from such pairwise measures of strength of association between variables. The proposed methodology is applicable to large high-dimensional data-sets and is based on computationally efficient algorithms. We illustrate its utility in a range of contexts and data-sets

Sampling designs and robustness for the analysis of network data

This manuscript addresses three new practical methodologies for topics on Bayesian analysis regarding sampling designs and robustness on network data: / In the first part of this thesis we propose a general approach for comparing sampling designs. The approach is based on the concept of data compression from information theory. The criterion for comparing sampling designs is formulated so that the results prove to be robust with respect to some of the most widely used loss functions for point estimation and prediction. The rationale behind the proposed approach is to find sampling designs such that preserve the largest amount of information possible from the original data generating mechanism. The approach is inspired by the same principle as the reference prior, with the difference that, for the proposed approach, the argument of the optimization is the sampling design rather than the prior. The information contained in the data generating mechanism can be encoded in a distribution defined either in parameter’s space (posterior distribution) or in the space of observables (predictive distribution). The results obtained in this part enable us to relate statements about a feature of an observed subgraph and a feature of a full graph. It is proven that such statements can not be connected by invoking conditional statements only; it is necessary to specify a joint distribution for the random graph model and the sampling design for all values of fully and partially observed random network features. We use this rationale to formulate statements at the level of the sampling graph that help to make non-trivial statements about the full network. The joint distribution of the underlying network and the sampling mechanism enable the statistician to relate both type of conditional statements. Thus, for random network partially and fully observed features joint distribution is considered and useful statements for practitioners are provided. / The second general theme of this thesis is robustness on networks. A method for robustness on exchangeable random networks is developed. The approach is inspired by the concept of graphon approximation through a stochastic block model. An exchangeable model is assumed to infer a feature of a random networks with the objective to see how the quality of that inference gets degraded if the model is slightly modified. Decision theory methods are considered under model misspecification by quantifying stability of optimal actions to perturbations to the approximating model within a well defined neighborhood of model space. The approach is inspired by all recent developments across the context of robustness in recent research in the robust control, macroeconomics and financial mathematics literature. / In all topics, simulation analysis is complemented with comprehensive experimental studies, which show the benefits of our modeling and estimation methods

A hierarchical Bayesian model for predicting ecological interactions using scaled evolutionary relationships

Identifying undocumented or potential future interactions among species is a challenge facing modern ecologists. Recent link prediction methods rely on trait data, however large species interaction databases are typically sparse and covariates are limited to only a fraction of species. On the other hand, evolutionary relationships, encoded as phylogenetic trees, can act as proxies for underlying traits and historical patterns of parasite sharing among hosts. We show that using a network-based conditional model, phylogenetic information provides strong predictive power in a recently published global database of host-parasite interactions. By scaling the phylogeny using an evolutionary model, our method allows for biological interpretation often missing from latent variable models. To further improve on the phylogeny-only model, we combine a hierarchical Bayesian latent score framework for bipartite graphs that accounts for the number of interactions per species with the host dependence informed by phylogeny. Combining the two information sources yields significant improvement in predictive accuracy over each of the submodels alone. As many interaction networks are constructed from presence-only data, we extend the model by integrating a correction mechanism for missing interactions, which proves valuable in reducing uncertainty in unobserved interactions.Comment: To appear in the Annals of Applied Statistic