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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. 

Breakthroughs in machine learning are rapidly changing science and society, yet our fundamental understanding of this technology has lagged far behind. Indeed, one of the central tenets of the field, the bias–variance trade-off, appears to be at odds with the observed behavior of methods used in modern machine-learning practice. The bias–variance trade-off implies that a model should balance underfitting and overfitting: Rich enough to express underlying structure in data and simple enough to avoid fitting spurious patterns. However, in modern practice, very rich models such as neural networks are trained to exactly fit (i.e., interpolate) the data. Classically, such models would be considered overfitted, and yet they often obtain high accuracy on test data. This apparent contradiction has raised questions about the mathematical foundations of machine learning and their relevance to practitioners. In this paper, we reconcile the classical understanding and the modern practice within a unified performance curve. This “double-descent” curve subsumes the textbook U-shaped bias–variance trade-off curve by showing how increasing model capacity beyond the point of interpolation results in improved performance. We provide evidence for the existence and ubiquity of double descent for a wide spectrum of models and datasets, and we posit a mechanism for its emergence. This connection between the performance and the structure of machine-learning models delineates the limits of classical analyses and has implications for both the theory and the practice of machine learning.