Selection and diagnostics: Can we compute measures, such as Bayesian information criterion (BIC) and marginal likelihood, to quantify and compare the goodness of fit of different SBMs? Inference: Once the likelihood can be computed, how should we infer the group memberships and the block matrix? Are there efficient and scalable inference algorithms? Modelling: How should the SBM be structured or extended to realistically describe real-world networks, with or without additional information on the nodes or the edges? ![]() Subsequently, the usual statistical challenges arise: Therefore, the goal of fitting an SBM to a graph is to infer these two components simultaneously. However, in applications to real data, neither the group memberships nor the block matrix is observed or given. It will also be straightforward to evaluate the likelihood of data observed, for modelling purposes. Given the group memberships, the block matrix, and the assumptions of an SBM (to be detailed in “ Stochastic block models” section), it is straightforward to generate a synthetic network for simulation purposes, as has been done in the example. For two nodes in the same group, that is, of the same colour, the probability of an edge is 0.8, while for two nodes in different groups, the edge probability is 0.05. The probability of having such an edge or not is independent of that of any other pair of nodes. ![]() In fact, this model is generated by taking each pair of nodes at a time, and simulating an (undirected) edge between them. For example, compared to nodes 2 to 25, node 1 does not seem a lot more connected to other nodes, both within the same group or with another group. Moreover, the connectivity pattern is rather “uniform”. ![]() The nodes within the same group are more closely connected to each other, than with nodes in another group. ![]() The nodes are divided into 3 groups, with groups 1, 2 and 3 containing 25, 30 and 35 nodes, respectively. 1, in which the network consists of 90 nodes and 1192 edges. We introduce them by considering the example in Fig. They can be used to discover or understand the (latent) structure of a network, as well as for clustering purposes. Stochastic block models (SBMs) are an increasingly popular class of models in statistical analysis of graphs or networks.
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