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

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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Structure Learning in Coupled Dynamical Systems and Dynamic Causal Modelling

Identifying a coupled dynamical system out of many plausible candidates, each of which could serve as the underlying generator of some observed measurements, is a profoundly ill posed problem that commonly arises when modelling real world phenomena. In this review, we detail a set of statistical procedures for inferring the structure of nonlinear coupled dynamical systems (structure learning), which has proved useful in neuroscience research. A key focus here is the comparison of competing models of (ie, hypotheses about) network architectures and implicit coupling functions in terms of their Bayesian model evidence. These methods are collectively referred to as dynamical casual modelling (DCM). We focus on a relatively new approach that is proving remarkably useful; namely, Bayesian model reduction (BMR), which enables rapid evaluation and comparison of models that differ in their network architecture. We illustrate the usefulness of these techniques through modelling neurovascular coupling (cellular pathways linking neuronal and vascular systems), whose function is an active focus of research in neurobiology and the imaging of coupled neuronal systems

Model Selection for Stochastic Block Models

As a flexible representation for complex systems, networks (graphs) model entities and their interactions as nodes and edges. In many real-world networks, nodes divide naturally into functional communities, where nodes in the same group connect to the rest of the network in similar ways. Discovering such communities is an important part of modeling networks, as community structure offers clues to the processes which generated the graph. The stochastic block model is a popular network model based on community structures. It splits nodes into blocks, within which all nodes are stochastically equivalent in terms of how they connect to the rest of the network. As a generative model, it has a well-defined likelihood function with consistent parameter estimates. It is also highly flexible, capable of modeling a wide variety of community structures, including degree specific and overlapping communities. Performance of different block models vary under different scenarios. Picking the right model is crucial for successful network modeling. A good model choice should balance the trade-off between complexity and fit. The task of model selection is to automatically choose such a model given the data and the inference task. As a problem of wide interest, numerous statistical model selection techniques have been developed for classic independent data. Unfortunately, it has been a common mistake to use these techniques in block models without rigorous examinations of their derivations, ignoring the fact that some of the fundamental assumptions has been violated by moving into the domain of relational data sets such as networks. In this dissertation, I thoroughly exam the literature of statistical model selection techniques, including both Frequentist and Bayesian approaches. My goal is to develop principled statistical model selection criteria for block models by adapting classic methods for network data. I do this by running bootstrapping simulations with an efficient algorithm, and correcting classic model selection theories for block models based on the simulation data. The new model selection methods are verified by both synthetic and real world data sets

Limits of Model Selection, Link Prediction, and Community Detection

Relational data has become increasingly ubiquitous nowadays. Networks are very rich tools in graph theory, which represent real world interactions through a simple abstract graph, including nodes and edges. Network analysis and modeling has gained extremely wide attentions from the researchers in various disciplines, such as computer science, social science, biology, economics, electrical engineering, and physics. Network analysis is the study of the network topology to answer a variety of application-based questions regarding the original real world problem. For example in social network analysis the questions are related to how people interact with each other in online social networks, or in collaboration networks, how diseases propagate or how information flows through a network, or how to control a disease or food outbreak. In electric networks like power grids or in internet networks, the questions can be related to vulnerability assessment of the networks to be prepared for power outage or internet blackout. In biological network analysis, the questions are related to how different diseases are related to each other, which can be useful in discovering new symptoms of diseases and producing and developing new medicines. It appears clearly that the reason of the importance of this interdisciplinary area of science, is due to its widespread applications which involves scientists and researchers with a variety of background and interests. Although networks are much simpler compared to the original complex systems, the interactions among the nodes in the real-world network may seem random, and capturing patterns on these entities is not trivial. There are tremendous questions about inference on networks, which makes this topic very attractive for researchers in the field. In this dissertation we answer some of the questions regarding this topic in two lines of study: one focused on experimental analyses and one focused on theoretical limitations. In Chapter 2 we look at community detection, a common graph mining task in network inference, which seeks an unsupervised decomposition of a network into groups based on statistical regularities in network connectivity. Although many such algorithms exist, community detection’s No Free Lunch theorem implies that no algorithm can be optimal across all inputs. However, little is known in practice about how different algorithms over or underfit to real networks, or how to reliably assess such behavior across algorithms. We present a broad investigation of over and underfitting across 16 state-of-the-art community detection algorithms applied to a novel benchmark corpus of 572 structurally diverse real-world networks. We find that (i) algorithms vary widely in the number and composition of communities they find, given the same input; (ii) algorithms can be clustered into distinct high-level groups based on similarities of their outputs on real-world networks; (iii) algorithmic differences induce wide variation in accuracy on link-based learning tasks; and, (iv) no algorithm is always the best at such tasks across all inputs. Finally, we quantify each algorithm’s overall tendency to over or underfit to network data using a theoretically principled diagnostic, and discuss the implications for future advances in community detection. In Chapter 3 we investigate link prediction problem, another important inference task in complex networks with a wide variety of applications. As we observed in Chapter 2, the community detection algorithmic differences induce wide variation in accuracy on link prediction tasks. On the other hand, many link prediction techniques exist in literature and still there is lack of methodology to analyze and compare these techniques. In Chapter 3, we provide a methodological overview of link prediction techniques and present new results on optimal link prediction and on transfer learning for link prediction. In the former, we investiga

Link Prediction in Complex Networks: A Survey

Link prediction in complex networks has attracted increasing attention from both physical and computer science communities. The algorithms can be used to extract missing information, identify spurious interactions, evaluate network evolving mechanisms, and so on. This article summaries recent progress about link prediction algorithms, emphasizing on the contributions from physical perspectives and approaches, such as the random-walk-based methods and the maximum likelihood methods. We also introduce three typical applications: reconstruction of networks, evaluation of network evolving mechanism and classification of partially labelled networks. Finally, we introduce some applications and outline future challenges of link prediction algorithms.Comment: 44 pages, 5 figure