Deep depth-based representations of graphs through deep learning networks

Graph-based representations are powerful tools in structural pattern recognition and machine learning. In this paper, we propose a framework of computing the deep depth-based representations for graph structures. Our work links the ideas of graph complexity measures and deep learning networks. Specifically, for a set of graphs, we commence by computing depth-based representations rooted at each vertex as vertex points. In order to identify an informative depth-based representation subset, we employ the well-known k-means method to identify M dominant centroids of the depth-based representation vectors as prototype representations. To overcome the burdensome computation of using depth-based representations for all graphs, we propose to use the prototype representations to train a deep autoencoder network, that is optimized using Stochastic Gradient Descent together with the Deep Belief Network for pretraining. By inputting the depth-based representations of vertices over all graphs to the trained deep network, we compute the deep representation for each vertex. The resulting deep depth-based representation of a graph is computed by averaging the deep representations of its complete set of vertices. We theoretically demonstrate that the deep depth-based representations of graphs not only reflect both the local and global characteristics of graphs through the depth-based representations, but also capture the main structural relationship and information content over all graphs under investigations. Experimental evaluations demonstrate the effectiveness of the proposed method.