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Graph summarization is the problem of producing smaller graph representationsof an input graph dataset, in such a way that the smaller compressed graphscapture relevant structural information for downstream tasks. There is a recentgraph summarization method that formulates an optimal transport-based frameworkthat allows prior information about node, edge, and attribute importance (neverdefined in that work) to be incorporated into the graph summarization process.However, very little is known about the statistical properties of thisframework. To elucidate this question, we consider the problem of supervisedgraph summarization, wherein by using information theoretic measures we seek topreserve relevant information about a class label. To gain a theoreticalperspective on the supervised summarization problem itself, we first formulateit in terms of maximizing the Shannon mutual information between the summarizedgraph and the class label. We show an NP-hardness of approximation result forthis problem, thereby constraining what one should expect from proposedsolutions. We then propose a summarization method that incorporates mutualinformation estimates between random variables associated with sample graphsand class labels into the optimal transport compression framework. Weempirically show performance improvements over previous works in terms ofclassification accuracy and time on synthetic and certain real datasets. Wealso theoretically explore the limitations of the optimal transport approachfor the supervised summarization problem and we show that it fails to satisfy acertain desirable information monotonicity property.

Promise and Limitations of Supervised Optimal Transport-Based Graph Summarization via Information Theoretic Measures