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

Graph summarization is the problem of producing smaller graph representations of an input graph dataset, in such a way that the smaller compressed graphs capture relevant structural information for downstream tasks. There is a recent graph summarization method that formulates an optimal transport-based framework that allows prior information about node, edge, and attribute importance (never defined in that work) to be incorporated into the graph summarization process. However, very little is known about the statistical properties of this framework. To elucidate this question, we consider the problem of supervised graph summarization, wherein by using information theoretic measures we seek to preserve relevant information about a class label. To gain a theoretical perspective on the supervised summarization problem itself, we first formulate it in terms of maximizing the Shannon mutual information between the summarized graph and the class label. We show an NP-hardness of approximation result for this problem, thereby constraining what one should expect from proposed solutions. We then propose a summarization method that incorporates mutual information estimates between random variables associated with sample graphs and class labels into the optimal transport compression framework. We empirically show performance improvements over previous works in terms of classification accuracy and time on synthetic and certain real datasets. We also theoretically explore the limitations of the optimal transport approach for the supervised summarization problem and we show that it fails to satisfy a certain desirable information monotonicity property.