A well-known problem in data science and machine learning is linear regression , which is recently extended to dynamic graphs. Existing exact algorithms for updating the solution of dynamic graph regression problem require at least a linear time (in terms of @math : the size of the graph). However, this time complexity might be intractable in practice. In the current paper, we utilize subsampled randomized Hadamard transform and CountSketch to propose the first randomized algorithms. Suppose that we are given an @math matrix embedding @math of the graph, where @math . Let @math be the number of samples required for a guaranteed approximation error, which is a sublinear function of @math . Our first algorithm reduces time complexity of pre-processing to @math . Then after an edge insertion or an edge deletion, it updates the approximate solution in @math time. Our second algorithm reduces time complexity of pre-processing to @math , where @math is the number of nonzero elements of @math . Then after an edge insertion or an edge deletion or a node insertion or a node deletion, it updates the approximate solution in @math time, with @math .
Abstract Let G be a simple graph with vertices 1, 2, …, n . We consider representations of G by nonzero vectors x 1 ,x 2 ,…,x n ϵ R d such that for i ≠ j the inner product x i · x j is negative or zero according as vertex i is adjacent to or not adjacent to vertex j . The least dimension d necessary for such representations is studied as a function of G and of various restrictions placed upon the coordinates of the vectors and the values of the inner products. In particular, for some d such that d n 2 , there always exists a representation in which all coordinates are from −1,1 and all negative inner products have the same value. Various types of vector representations of graphs are related to Hadamard matrices, families of sets with prescribed intersections, and other combinatorial problems. Numerous important problems can be framed as learning from graph data. We propose a framework for learning convolutional neural networks for arbitrary graphs. These graphs may be undirected, directed, and with both discrete and continuous node and edge attributes. Analogous to image-based convolutional networks that operate on locally connected regions of the input, we present a general approach to extracting locally connected regions from graphs. Using established benchmark data sets, we demonstrate that the learned feature representations are competitive with state of the art graph kernels and that their computation is highly efficient. Prediction tasks over nodes and edges in networks require careful effort in engineering features used by learning algorithms. Recent research in the broader field of representation learning has led to significant progress in automating prediction by learning the features themselves. However, present feature learning approaches are not expressive enough to capture the diversity of connectivity patterns observed in networks. Here we propose node2vec, an algorithmic framework for learning continuous feature representations for nodes in networks. In node2vec, we learn a mapping of nodes to a low-dimensional space of features that maximizes the likelihood of preserving network neighborhoods of nodes. We define a flexible notion of a node's network neighborhood and design a biased random walk procedure, which efficiently explores diverse neighborhoods. Our algorithm generalizes prior work which is based on rigid notions of network neighborhoods, and we argue that the added flexibility in exploring neighborhoods is the key to learning richer representations. We demonstrate the efficacy of node2vec over existing state-of-the-art techniques on multi-label classification and link prediction in several real-world networks from diverse domains. Taken together, our work represents a new way for efficiently learning state-of-the-art task-independent representations in complex networks. We present a semi-supervised learning framework based on graph embeddings. Given a graph between instances, we train an embedding for each instance to jointly predict the class label and the neighborhood context in the graph. We develop both transductive and inductive variants of our method. In the transductive variant of our method, the class labels are determined by both the learned embeddings and input feature vectors, while in the inductive variant, the embeddings are defined as a parametric function of the feature vectors, so predictions can be made on instances not seen during training. On a large and diverse set of benchmark tasks, including text classification, distantly supervised entity extraction, and entity classification, we show improved performance over many of the existing models.