Infinite-degree-corrected stochastic block model

Reconstructing Sparse Multiplex Networks with Application to Covert Networks

Network structure provides critical information for understanding the dynamic behavior of complex systems. However, the complete structure of real-world networks is often unavailable, thus it is crucially important to develop approaches to infer a more complete structure of networks. In this paper, we integrate the configuration model for generating random networks into an Expectation-Maximization-Aggregation (EMA) framework to reconstruct the complete structure of multiplex networks. We validate the proposed EMA framework against the Expectation-Maximization (EM) framework and random model on several real-world multiplex networks, including both covert and overt ones. It is found that the EMA framework generally achieves the best predictive accuracy compared to the EM framework and the random model. As the number of layers increases, the performance improvement of EMA over EM decreases. The inferred multiplex networks can be leveraged to inform the decision-making on monitoring covert networks as well as allocating limited resources for collecting additional information to improve reconstruction accuracy. For law enforcement agencies, the inferred complete network structure can be used to develop more effective strategies for covert network interdiction.

The Bayesian Cut

An important task in the analysis of graphs is separating nodes into densely connected groups with little interaction between each other. Prominent methods here include flow based graph cutting procedures as well as statistical network modeling approaches. However, adequately accounting for this, the so-called community structure, in complex networks remains a major challenge. We present a novel generic Bayesian probabilistic model for graph cutting in which we derive an analytical solution to the marginalization of nuisance parameters under constraints enforcing community structure. As a part of the solution a large scale approximation for integrals involving multiple incomplete gamma functions is derived. Our multiple cluster solution presents a generic tool for Bayesian inference on Poisson weighted graphs across different domains. Applied on three real world social networks as well as three image segmentation problems our approach shows on par or better performance to existing spectral graph cutting and community detection methods, while learning the underlying parameter space. The developed procedure provides a principled statistical framework for graph cutting and the Bayesian Cut source code provided enables easy adoption of the procedure as an alternative to existing graph cutting methods.

Using connectomics for predictive assessment of brain parcellations

The organization of the human brain remains elusive, yet is of great importance to the mechanisms of integrative brain function. At the macroscale, its structural and functional interpretation is conventionally assessed at the level of cortical units. However, the definition and validation of such cortical parcellations are problematic due to the absence of a true gold standard. We propose a framework for quantitative evaluation of brain parcellations via statistical prediction of connectomics data. Specifically, we evaluate the extent in which the network representation at the level of cortical units (defined as parcels) accounts for high-resolution brain connectivity. Herein, we assess the pertinence and comparative ranking of ten existing parcellation atlases to account for functional (FC) and structural connectivity (SC) data based on data from the Human Connectome Project (HCP), and compare them to data-driven as well as spatially-homogeneous geometric parcellations including geodesic parcellations with similar size distributions as the atlases. We find substantial discrepancy in parcellation structures that well characterize FC and SC and differences in what well represents an individual's functional connectome when compared against the FC structure that is preserved across individuals. Surprisingly, simple spatial homogenous parcellations generally provide good representations of both FC and SC, but are inferior when their within-parcellation distribution of individual parcel sizes is matched to that of a valid atlas. This suggests that the choice of fine grained and coarse representations used by existing atlases are important. However, we find that resolution is more critical than the exact border location of parcels.

Overlapping community detection for count-value networks

Detecting network overlapping community has become a very hot research topic in the literature. However, overlapping community detection for count-value networks that naturally arise and are pervasive in our modern life, has not yet been thoroughly studied. We propose a generative model for count-value networks with overlapping community structure and use the Indian buffet process to model the community assignment matrix Z; thus, provide a flexible nonparametric Bayesian scheme that can allow the number of communities K to increase as more and more data are encountered instead of to be fixed in advance. Both collapsed and uncollapsed Gibbs sampler for the generative model have been derived. We conduct extensive experiments on simulated network data and real network data, and estimate the inference quality on single variable parameters. We find that the proposed model and inference procedure can bring us the desired experimental results.

Dynamic degree-corrected blockmodels for social networks: A nonparametric approach

A nonparametric approach to the modelling of social networks using degree-corrected stochastic blockmodels is proposed. The model for static network consists of a stochastic blockmodel using a probit regression formulation, and popularity parameters are incorporated to account for degree heterogeneity. We specify a Dirichlet process prior to detect community structure as well as to induce clustering in the popularity parameters. This approach is flexible yet parsimonious as it allows the appropriate number of communities and popularity clusters to be determined automatically by the data. We further discuss and implement extensions of the static model to dynamic networks. In a Bayesian framework, we perform posterior inference through MCMC algorithms. The models are illustrated using several real-world benchmark social networks.

Sparse graphs using exchangeable random measures

Statistical network modelling has focused on representing the graph as a discrete structure, namely the adjacency matrix. When assuming exchangeability of this array-which can aid in modelling, computations and theoretical analysis-the Aldous-Hoover theorem informs us that the graph is necessarily either dense or empty. We instead consider representing the graph as an exchangeable random measure and appeal to the Kallenberg representation theorem for this object. We explore using completely random measures (CRMs) to define the exchangeable random measure, and we show how our CRM construction enables us to achieve sparse graphs while maintaining the attractive properties of exchangeability. We relate the sparsity of the graph to the Lévy measure defining the CRM. For a specific choice of CRM, our graphs can be tuned from dense to sparse on the basis of a single parameter. We present a scalable Hamiltonian Monte Carlo algorithm for posterior inference, which we use to analyse network properties in a range of real data sets, including networks with hundreds of thousands of nodes and millions of edges.

The missing link: Predicting connectomes from noisy and partially observed tract tracing data

Our understanding of the wiring map of the brain, known as the connectome, has increased greatly in the last decade, mostly due to technological advancements in neuroimaging techniques and improvements in computational tools to interpret the vast amount of available data. Despite this, with the exception of the C. elegans roundworm, no definitive connectome has been established for any species. In order to obtain this, tracer studies are particularly appealing, as these have proven highly reliable. The downside of tract tracing is that it is costly to perform, and can only be applied ex vivo. In this paper, we suggest that instead of probing all possible connections, hitherto unknown connections may be predicted from the data that is already available. Our approach uses a 'latent space model' that embeds the connectivity in an abstract physical space. Regions that are close in the latent space have a high chance of being connected, while regions far apart are most likely disconnected in the connectome. After learning the latent embedding from the connections that we did observe, the latent space allows us to predict connections that have not been probed previously. We apply the methodology to two connectivity data sets of the macaque, where we demonstrate that the latent space model is successful in predicting unobserved connectivity, outperforming two baselines and an alternative model in nearly all cases. Furthermore, we show how the latent spatial embedding may be used to integrate multimodal observations (i.e. anterograde and retrograde tracers) for the mouse neocortex. Finally, our probabilistic approach enables us to make explicit which connections are easy to predict and which prove difficult, allowing for informed follow-up studies.