We present, GeniePath, a scalable approach for learning adaptive receptive fields of neural networks defined on permutation invariant graph data. In GeniePath, we propose an adaptive path layer consists of two complementary functions designed for breadth and depth exploration respectively, where the former learns the importance of different sized neighborhoods, while the latter extracts and filters signals aggregated from neighbors of different hops away. Our method works in both transductive and inductive settings, and extensive experiments compared with competitive methods show that our approaches yield state-of-the-art results on large graphs.

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We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S 4 or a quotient of S 3 × R. This gives a clean classification result removing the earlier additional assumptions in [13] by Wallach and the second author.

In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC 1 , without any curvature bound. New results on ancient solutions for the Ricci and Kähler-Ricci flow are also obtained. The main new feature is that no curvature upper bound is assumed.2010 Mathematics Subject Classification. 53C44, 53C55.

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