Graph Convolutional Neural Network

The benefit of localized features within the regular domain has given rise to the use of Convolutional Neural Networks (CNNs) in machine learning, with great proficiency in the image classification. The use of CNNs becomes problematic within the irregular spatial domain due to design and convolution of a kernel filter being non-trivial. One so- lution to this problem is to utilize graph signal processing techniques and the convolution theorem to perform convolutions on the graph of the irregular domain to obtain feature map responses to learnt filters. We propose graph convolution and pooling operators analogous to those in the regular domain. We also provide gradient calculations on the input data and spectral filters, which allow for the deep learning of an irregular spatial do- main problem. Signal filters take the form of spectral multipliers, applying convolution in the graph spectral domain. Applying smooth multipliers results in localized convo- lutions in the spatial domain, with smoother multipliers providing sharper feature maps. Algebraic Multigrid is presented as a graph pooling method, reducing the resolution of the graph through agglomeration of nodes between layers of the network. Evaluation of performance on the MNIST digit classification problem in both the regular and irregu- lar domain is presented, with comparison drawn to standard CNN. The proposed graph CNN provides a deep learning method for the irregular domains present in the machine learning community, obtaining 94.23% on the regular grid, and 94.96% on a spatially irregular subsampled MNIST

Parallel Graph Partitioning for Complex Networks

Processing large complex networks like social networks or web graphs has recently attracted considerable interest. In order to do this in parallel, we need to partition them into pieces of about equal size. Unfortunately, previous parallel graph partitioners originally developed for more regular mesh-like networks do not work well for these networks. This paper addresses this problem by parallelizing and adapting the label propagation technique originally developed for graph clustering. By introducing size constraints, label propagation becomes applicable for both the coarsening and the refinement phase of multilevel graph partitioning. We obtain very high quality by applying a highly parallel evolutionary algorithm to the coarsened graph. The resulting system is both more scalable and achieves higher quality than state-of-the-art systems like ParMetis or PT-Scotch. For large complex networks the performance differences are very big. For example, our algorithm can partition a web graph with 3.3 billion edges in less than sixteen seconds using 512 cores of a high performance cluster while producing a high quality partition -- none of the competing systems can handle this graph on our system.Comment: Review article. Parallelization of our previous approach arXiv:1402.328