Local-global nested graph kernels using nested complexity traces

Abstract In this paper, we propose two novel local-global nested graph kernels, namely the nested aligned kernel and the nested reproducing kernel, drawing on depth-based complexity traces. Both of the nested kernels gauge the nested depth complexity trace through a family of K-layer expansion subgraphs rooted at the centroid vertex, i.e., the vertex with minimum shortest path length variance to the remaining vertices. Specifically, for a pair of graphs, we commence by computing the centroid depth-based complexity traces rooted at the centroid vertices. The first nested kernel is defined by measuring the global alignment kernel, which is based on the dynamic time warping framework, between the complexity traces. Since the required global alignment kernel incorporates the whole spectrum of alignment costs between the complexity traces, this nested kernel can provide rich statistic measures. The second nested kernel, on the other hand, is defined by measuring the basic reproducing kernel between the complexity traces. Since the associated reproducing kernel only requires time complexity O(1), this nested kernel has very low computational complexity. We theoretically show that both of the proposed nested kernels can simultaneously reflect the local and global graph characteristics in terms of the nested complexity traces. Experiments on standard graph datasets abstracted from bioinformatics and computer vision databases demonstrate the effectiveness and efficiency of the proposed graph kernels

Graph Convolutional Neural Networks based on Quantum Vertex Saliency

This paper proposes a new Quantum Spatial Graph Convolutional Neural Network (QSGCNN) model that can directly learn a classification function for graphs of arbitrary sizes. Unlike state-of-the-art Graph Convolutional Neural Network (GCNN) models, the proposed QSGCNN model incorporates the process of identifying transitive aligned vertices between graphs, and transforms arbitrary sized graphs into fixed-sized aligned vertex grid structures. In order to learn representative graph characteristics, a new quantum spatial graph convolution is proposed and employed to extract multi-scale vertex features, in terms of quantum information propagation between grid vertices of each graph. Since the quantum spatial convolution preserves the grid structures of the input vertices (i.e., the convolution layer does not change the original spatial sequence of vertices), the proposed QSGCNN model allows to directly employ the traditional convolutional neural network architecture to further learn from the global graph topology, providing an end-to-end deep learning architecture that integrates the graph representation and learning in the quantum spatial graph convolution layer and the traditional convolutional layer for graph classifications. We demonstrate the effectiveness of the proposed QSGCNN model in relation to existing state-of-the-art methods. The proposed QSGCNN model addresses the shortcomings of information loss and imprecise information representation arising in existing GCN models associated with the use of SortPooling or SumPooling layers. Experiments on benchmark graph classification datasets demonstrate the effectiveness of the proposed QSGCNN model

Entropic Dynamic Time Warping Kernels for Co-evolving Financial Time Series Analysis

In this work, we develop a novel framework to measure the similarity between dynamic financial networks, i.e., time-varying financial networks. Particularly, we explore whether the proposed similarity measure can be employed to understand the structural evolution of the financial networks with time. For a set of time-varying financial networks with each vertex representing the individual time series of a different stock and each edge between a pair of time series representing the absolute value of their Pearson correlation, our start point is to compute the commute time matrix associated with the weighted adjacency matrix of the network structures, where each element of the matrix can be seen as the enhanced correlation value between pairwise stocks. For each network, we show how the commute time matrix allows us to identify a reliable set of dominant correlated time series as well as an associated dominant probability distribution of the stock belonging to this set. Furthermore, we represent each original network as a discrete dominant Shannon entropy time series computed from the dominant probability distribution. With the dominant entropy time series for each pair of financial networks to hand, we develop a similarity measure based on the classical dynamic time warping framework, for analyzing the financial time-varying networks. We show that the proposed similarity measure is positive definite and thus corresponds to a kernel measure on graphs. The proposed kernel bridges the gap between graph kernels and the classical dynamic time warping framework for multiple financial time series analysis. Experiments on time-varying networks extracted through New York Stock Exchange (NYSE) database demonstrate the effectiveness of the proposed approach.Comment: Previously, the original version of this manuscript appeared as arXiv:1902.09947v2, that was submitted as a replacement by a mistake. Now, that article has been replaced to correct the error, and this manuscript is distinct from that articl