Abstract: Local structural information can increase the adaptability of graph convolutional networks to large graphs with heterogeneous topology. Existing methods only use relatively simplistic topological information, such as node degrees. We present a novel approach leveraging advanced topological information, i.e., persistent homology, which measures the information flow efficiency at different parts of the graph. To fully exploit such structural information in real world graphs, we propose a new network architecture which learns to use persistent homology information to reweight messages passed between graph nodes during convolution. For node classification tasks, our network outperforms existing ones on a broad spectrum of graph benchmarks.