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With the rapid expansion of graphs and networks and the growing magnitude ofdata from all areas of science, effective treatment and compression schemes ofcontext-dependent data is extremely desirable. A particularly interestingdirection is to compress the data while keeping the "structural information"only and ignoring the concrete labelings. Under this direction, Choi andSzpankowski introduced the structures (unlabeled graphs) which allowed them tocompute the structural entropy of the Erd\H{o}s--R\'enyi random graph model.Moreover, they also provided an asymptotically optimal compression algorithmthat (asymptotically) achieves this entropy limit and runs in expectation inlinear time.  In this paper, we consider the Stochastic Block Models with an arbitrarynumber of parts. Indeed, we define a partitioned structural entropy forStochastic Block Models, which generalizes the structural entropy for unlabeledgraphs and encodes the partition information as well. We then compute thepartitioned structural entropy of the Stochastic Block Models, and provide acompression scheme that asymptotically achieves this entropy limit.

Structural Entropy of the Stochastic Block Models