Block modelling in dynamic networks with non-homogeneous Poisson processes and exact ICL

We develop a model in which interactions between nodes of a dynamic network are counted by non homogeneous Poisson processes. In a block modelling perspective, nodes belong to hidden clusters (whose number is unknown) and the intensity functions of the counting processes only depend on the clusters of nodes. In order to make inference tractable we move to discrete time by partitioning the entire time horizon in which interactions are observed in fixed-length time sub-intervals. First, we derive an exact integrated classification likelihood criterion and maximize it relying on a greedy search approach. This allows to estimate the memberships to clusters and the number of clusters simultaneously. Then a maximum-likelihood estimator is developed to estimate non parametrically the integrated intensities. We discuss the over-fitting problems of the model and propose a regularized version solving these issues. Experiments on real and simulated data are carried out in order to assess the proposed methodology

Community Detection in Complex Networks

The stochastic block model is a powerful tool for inferring community structure from network topology. However, the simple block model considers community structure as the only underlying attribute for forming the relational interactions among the nodes, this makes it prefer a Poisson degree distribution within each community, while most real-world networks have a heavy-tailed degree distribution. This is essentially because the simple assumption under the traditional block model is not consistent with some real-world circumstances where factors other than the community memberships such as overall popularity also heavily affect the pattern of the relational interactions. The degree-corrected block model can accommodate arbitrary degree distributions within communities by taking nodes\u27 popularity or degree into account. But since it takes the vertex degrees as parameters rather than generating them, it cannot use them to help it classify the vertices, and its natural generalization to directed graphs cannot even use the orientations of the edges. We developed several variants of the block model with the best of both worlds: they can use vertex degrees and edge orientations in the classification process, while tolerating heavy-tailed degree distributions within communities. We show that for some networks, including synthetic networks and networks of word adjacencies in English text, these new block models achieve a higher accuracy than either standard or degree-corrected block models. Another part of my work is to develop even more generalized block models, which incorporates other attributes of the nodes. Many data sets contain rich information about objects, as well as pairwise relations between them. For instance, in networks of websites, scientific papers, patents and other documents, each node has content consisting of a collection of words, as well as hyperlinks or citations to other nodes. In order to perform inference on such data sets, and make predictions and recommendations, it is useful to have models that are able to capture the processes which generate the text at each node as well as the links between them. Our work combines classic ideas in topic modeling with a variant of the mixed-membership block model recently developed in the statistical physics community. The resulting model has the advantage that its parameters, including the mixture of topics of each document and the resulting overlapping communities, can be inferred with a simple and scalable expectation- maximization algorithm. We test our model on three data sets, performing unsupervised topic classification and link prediction. For both tasks, our model outperforms several existing state-of-the-art methods, achieving higher accuracy with significantly less computation, analyzing a data set with 1.3 million words and 44 thousand links in a few minutes

A generative model for reciprocity and community detection in networks

We present a probabilistic generative model and efficient algorithm to model reciprocity in directed networks. Unlike other methods that address this problem such as exponential random graphs, it assigns latent variables as community memberships to nodes and a reciprocity parameter to the whole network rather than fitting order statistics. It formalizes the assumption that a directed interaction is more likely to occur if an individual has already observed an interaction towards her. It provides a natural framework for relaxing the common assumption in network generative models of conditional independence between edges, and it can be used to perform inference tasks such as predicting the existence of an edge given the observation of an edge in the reverse direction. Inference is performed using an efficient expectation-maximization algorithm that exploits the sparsity of the network, leading to an efficient and scalable implementation. We illustrate these findings by analyzing synthetic and real data, including social networks, academic citations and the Erasmus student exchange program. Our method outperforms others in both predicting edges and generating networks that reflect the reciprocity values observed in real data, while at the same time inferring an underlying community structure. We provide an open-source implementation of the code online