  In this paper, we introduce a method called graph fusion embedding, designed
for multi-graph embedding with shared vertex sets. Under the framework of
supervised learning, our method exhibits a remarkable and highly desirable
synergistic effect: for sufficiently large vertex size, the accuracy of vertex
classification consistently benefits from the incorporation of additional
graphs. We establish the mathematical foundation for the method, including the
asymptotic convergence of the embedding, a sufficient condition for asymptotic
optimal classification, and the proof of the synergistic effect for vertex
classification. Our comprehensive simulations and real data experiments provide
compelling evidence supporting the effectiveness of our proposed method,
showcasing the pronounced synergistic effect for multiple graphs from disparate
sources.
