A stochastic block model assigns each vertex of a graph to a class and, conditional on those classes, makes distinct edges independent random variables whose probabilities depend only on the classes of their endpoints.
For independent Bernoulli distribution upper-triangular entries of a symmetric zero-diagonal adjacency matrix of a graph, put . For a unit vector , has sub-Gaussian variance proxy at most , by the Hoeffding lemma. Hence . The volumetric bound for Euclidean metric nets gives a -metric net of the unit sphere with at most points. The quadratic form net bound gives on that net. The union bound gives ; integrating this tail proves the displayed expectation bound.
An additive class-effect logistic network model gives an edge joining classes and the log odds . A further coefficient multiplying represents a common within-class log-odds effect.
For independent binary edges with log odds , the natural statistics are the sum of the vertex degrees in each class and the total number of within-class edges. They form a minimal sufficient statistic whenever the natural parameter space contains an open subset of .
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The Stochastic Block Model (SBM) is a generative model used in network science to represent and analyze the structure of networks or graphs, particularly in relation to community detection and clustering. It is a way to simulate the interactions within a network based on the assumption that nodes belong to communities or blocks, which influence how they are connected.