Mixed membership stochastic blockmodels

Observations consisting of measurements on relationships for pairs of objects arise in many settings, such as protein interaction and gene regulatory networks, collections of author-recipient email, and social networks. Analyzing such data with probabilisic models can be delicate because the simple exchangeability assumptions underlying many boilerplate models no longer hold. In this paper, we describe a latent variable model of such data called the mixed membership stochastic blockmodel. This model extends blockmodels for relational data to ones which capture mixed membership latent relational structure, thus providing an object-specific low-dimensional representation. We develop a general variational inference algorithm for fast approximate posterior inference. We explore applications to social and protein interaction networks.Comment: 46 pages, 14 figures, 3 table

A survey of statistical network models

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference

Stochastic blockmodels with growing number of classes

We present asymptotic and finite-sample results on the use of stochastic blockmodels for the analysis of network data. We show that the fraction of misclassified network nodes converges in probability to zero under maximum likelihood fitting when the number of classes is allowed to grow as the root of the network size and the average network degree grows at least poly-logarithmically in this size. We also establish finite-sample confidence bounds on maximum-likelihood blockmodel parameter estimates from data comprising independent Bernoulli random variates; these results hold uniformly over class assignment. We provide simulations verifying the conditions sufficient for our results, and conclude by fitting a logit parameterization of a stochastic blockmodel with covariates to a network data example comprising a collection of Facebook profiles, resulting in block estimates that reveal residual structure.Comment: 12 pages, 3 figures; revised versio

Bayesian stochastic blockmodeling

This chapter provides a self-contained introduction to the use of Bayesian inference to extract large-scale modular structures from network data, based on the stochastic blockmodel (SBM), as well as its degree-corrected and overlapping generalizations. We focus on nonparametric formulations that allow their inference in a manner that prevents overfitting, and enables model selection. We discuss aspects of the choice of priors, in particular how to avoid underfitting via increased Bayesian hierarchies, and we contrast the task of sampling network partitions from the posterior distribution with finding the single point estimate that maximizes it, while describing efficient algorithms to perform either one. We also show how inferring the SBM can be used to predict missing and spurious links, and shed light on the fundamental limitations of the detectability of modular structures in networks.Comment: 44 pages, 16 figures. Code is freely available as part of graph-tool at https://graph-tool.skewed.de . See also the HOWTO at https://graph-tool.skewed.de/static/doc/demos/inference/inference.htm

Stochastic blockmodels and community structure in networks

Stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks. Most blockmodels, however, ignore variation in vertex degree, making them unsuitable for applications to real-world networks, which typically display broad degree distributions that can significantly distort the results. Here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks. We also propose a heuristic algorithm for community detection using this objective function or its non-degree-corrected counterpart and show that the degree-corrected version dramatically outperforms the uncorrected one in both real-world and synthetic networks.Comment: 11 pages, 3 figure

Involuntary psychiatric admissions: A retrospective study of 460 cases

Introduction: We collected the data relating to involuntary hospital treatment (IHT) in the University Psychiatric Ward at Novara Hospital between 1991 and 2002, and compared them with those relating to Piedmont and the whole of Italy. Methods: The data were collected from the ward medical records. Results: IHT was much more frequent among young male schizophrenics living with their families of origin. Most of the subjects were not working at the time of admission. There was a statistically significant correlation between male gender and the risk of being admitted for a period of less than 12 days. The risk of being admitted for more than 12 days significantly correlated with the province of birth and residence, as well as with a diagnosis of schizophrenic psychosis. Conclusions: Schizophrenia is the diagnosis that is most frequently associated with IHT

Contrasting non-dynamic and dynamic models of the water-energy nexus in small, off-grid Mediterranean islands

Water and energy supply in small Mediterranean islands are strictly interrelated and face a large number of challenging issues, mainly caused by the distance from the mainland, the lack of accessible and safe potable water sources, and the high seasonal variability of the water and energy demands driven by touristic fluxes. The energy system generally relies on carbon intensive, expensive stand-alone diesel generators, while potable water supply is provided by tank vessels. Although this combination provides essential services for local communities, it is often economically and environmentally unsustainable due to high operational costs and greenhouse gas (GHG) emissions. A traditional approach to improve the sustainability and the efficiency of the water and energy systems is to couple renewable energy sources (RES) with water supply technologies (e.g., desalination), in order to obtain efficient planning solutions (i.e. RES capacity, desalination plant capacity) in a least-cost fashion. However, this approach is generally non-dynamic and optimizes the power allocation using fixed electricity loads as a surrogate of the actual water demand supplied by the desalination plant through the water distribution network. Although this load reflects the actual water demand on the long-term (i.e. monthly or annual time scale), it could strongly deviate from the real water demand if we consider shorter time scales (i.e. daily or hourly), over which the water distribution network is able to store and move water in space and time. In this work, we comparatively analyse this traditional non-dynamic model of the water-energy nexus with a novel dynamic modelling approach, where the operation of both the nexus components (i.e. power allocation and operations of the water distribution network) is conjunctively optimized with respect to multiple economic and sustainability indicators (e.g., net present costs, GHG emissions, water supply deficit, RES penetration). This comparative analysis is performed over the real case study of the Italian Ustica island in the Mediterranean Sea. Preliminary results show the effectiveness of the dynamic approach in improving the static solution with respect to almost all the system performance metrics considered