Random walk based graph sampling has been recognized as a fundamental technique to collect uniform node samples from a large graph. In this paper, we first present a comprehensive analysis of the drawbacks of three widely-used random walk based graph sampling algorithms, called re-weighted random walk (RW) algorithm, Metropolis-Hastings random walk (MH) algorithm and maximum-degree random walk (MD) algorithm. Then, to address the limitations of these algorithms, we propose two general random walk based algorithms, named rejection-controlled Metropolis-Hastings (RCMH) algorithm and generalized maximum-degree random walk (GMD) algorithm. We show that RCMH balances the tradeoff between the limitations of RW and MH, and GMD balances the tradeoff between the drawbacks of RW and MD. To further improve the performance of our algorithms, we integrate the so-called delayed acceptance technique and the non-backtracking random walk technique into RCMH and GMD respectively. We conduct extensive experiments over four real-world datasets, and the results demonstrate the effectiveness of the proposed algorithms.
Given a huge real graph, how can we derive a representative sample? There are many known algorithms to compute interesting measures (shortest paths, centrality, betweenness, etc.), but several of them become impractical for large graphs. Thus graph sampling is essential.The natural questions to ask are (a) which sampling method to use, (b) how small can the sample size be, and (c) how to scale up the measurements of the sample (e.g., the diameter), to get estimates for the large graph. The deeper, underlying question is subtle: how do we measure success?.We answer the above questions, and test our answers by thorough experiments on several, diverse datasets, spanning thousands nodes and edges. We consider several sampling methods, propose novel methods to check the goodness of sampling, and develop a set of scaling laws that describe relations between the properties of the original and the sample.In addition to the theoretical contributions, the practical conclusions from our work are: Sampling strategies based on edge selection do not perform well; simple uniform random node selection performs surprisingly well. Overall, best performing methods are the ones based on random-walks and "forest fire"; they match very accurately both static as well as evolutionary graph patterns, with sample sizes down to about 15% of the original graph.
Graph representation learning has recently been applied to a broad spectrum of problems ranging from computer graphics and chemistry to high energy physics and social media. The popularity of graph neural networks has sparked interest, both in academia and in industry, in developing methods that scale to very large graphs such as Facebook or Twitter social networks. In most of these approaches, the computational cost is alleviated by a sampling strategy retaining a subset of node neighbors or subgraphs at training time. In this paper we propose a new, efficient and scalable graph deep learning architecture which sidesteps the need for graph sampling by using graph convolutional filters of different size that are amenable to efficient precomputation, allowing extremely fast training and inference. Our architecture allows using different local graph operators (e.g. motif-induced adjacency matrices or Personalized Page Rank diffusion matrix) to best suit the task at hand. We conduct extensive experimental evaluation on various open benchmarks and show that our approach is competitive with other state-of-the-art architectures, while requiring a fraction of the training and inference time. Moreover, we obtain state-of-the-art results on ogbn-papers100M, the largest public graph dataset, with over 110 million nodes and 1.5 billion edges.
Recent years have witnessed the emerging success of graph neural networks (GNNs) for modeling structured data. However, most GNNs are designed for homogeneous graphs, in which all nodes and edges belong to the same types, making them infeasible to represent heterogeneous structures. In this paper, we present the Heterogeneous Graph Transformer (HGT) architecture for modeling Web-scale heterogeneous graphs. To model heterogeneity, we design node- and edge-type dependent parameters to characterize the heterogeneous attention over each edge, empowering HGT to maintain dedicated representations for different types of nodes and edges. To handle dynamic heterogeneous graphs, we introduce the relative temporal encoding technique into HGT, which is able to capture the dynamic structural dependency with arbitrary durations. To handle Web-scale graph data, we design the heterogeneous mini-batch graph sampling algorithm---HGSampling---for efficient and scalable training. Extensive experiments on the Open Academic Graph of 179 million nodes and 2 billion edges show that the proposed HGT model consistently outperforms all the state-of-the-art GNN baselines by 9%--21% on various downstream tasks.
Background subtraction is a fundamental pre-processing task in computer vision. This task becomes challenging in real scenarios due to variations in the background for both static and moving camera sequences. Several deep learning methods for background subtraction have been proposed in the literature with competitive performances. However, these models show performance degradation when tested on unseen videos; and they require huge amount of data to avoid overfitting. Recently, graph-based algorithms have been successful approaching unsupervised and semi-supervised learning problems. Furthermore, the theory of graph signal processing and semi-supervised learning have been combined leading to new insights in the field of machine learning. In this paper, concepts of recovery of graph signals are introduced in the problem of background subtraction. We propose a new algorithm called Graph BackGround Subtraction (GraphBGS), which is composed of: instance segmentation, background initialization, graph construction, graph sampling, and a semi-supervised algorithm inspired from the theory of recovery of graph signals. Our algorithm has the advantage of requiring less labeled data than deep learning methods while having competitive results on both: static and moving camera videos. GraphBGS outperforms unsupervised and supervised methods in several challenging conditions on the publicly available Change Detection (CDNet2014), and UCSD background subtraction databases.
A serious challenge when finding influential actors in real-world social networks is the lack of knowledge about the structure of the underlying network. Current state-of-the-art methods rely on hand-crafted sampling algorithms; these methods sample nodes and their neighbours in a carefully constructed order and choose opinion leaders from this discovered network to maximize influence spread in the (unknown) complete network. In this work, we propose a reinforcement learning framework for network discovery that automatically learns useful node and graph representations that encode important structural properties of the network. At training time, the method identifies portions of the network such that the nodes selected from this sampled subgraph can effectively influence nodes in the complete network. The realization of such transferable network structure based adaptable policies is attributed to the meticulous design of the framework that encodes relevant node and graph signatures driven by an appropriate reward scheme. We experiment with real-world social networks from four different domains and show that the policies learned by our RL agent provide a 10-36% improvement over the current state-of-the-art method.
Graph Convolutional Networks (GCNs) are powerful models for learning representations of attributed graphs. To scale GCNs to large graphs, state-of-the-art methods use various layer sampling techniques to alleviate the "neighbor explosion" problem during minibatch training. We propose GraphSAINT, a graph sampling based inductive learning method that improves training efficiency and accuracy in a fundamentally different way. By changing perspective, GraphSAINT constructs minibatches by sampling the training graph, rather than the nodes or edges across GCN layers. Each iteration, a complete GCN is built from the properly sampled subgraph. Thus, we ensure fixed number of well-connected nodes in all layers. We further propose normalization technique to eliminate bias, and sampling algorithms for variance reduction. Importantly, we can decouple the sampling from the forward and backward propagation, and extend GraphSAINT with many architecture variants (e.g., graph attention, jumping connection). GraphSAINT demonstrates superior performance in both accuracy and training time on five large graphs, and achieves new state-of-the-art F1 scores for PPI (0.995) and Reddit (0.970).
The Graph Convolutional Network (GCN) model and its variants are powerful graph embedding tools for facilitating classification and clustering on graphs. However, a major challenge is to reduce the complexity of layered GCNs and make them parallelizable and scalable on very large graphs -- state-of the art techniques are unable to achieve scalability without losing accuracy and efficiency. In this paper, we propose novel parallelization techniques for graph sampling-based GCNs that achieve superior scalable performance on very large graphs without compromising accuracy. Specifically, our GCN guarantees work-efficient training and produces order of magnitude savings in computation and communication. To scale GCN training on tightly-coupled shared memory systems, we develop parallelization strategies for the key steps in training: For the graph sampling step, we exploit parallelism within and across multiple sampling instances, and devise an efficient data structure for concurrent accesses that provides theoretical guarantee of near-linear speedup with number of processing units. For the feature propagation step within the sampled graph, we improve cache utilization and reduce DRAM communication by data partitioning. We prove that our partitioning strategy is a 2-approximation for minimizing the communication time compared to the optimal strategy. We demonstrate that our parallel graph embedding outperforms state-of-the-art methods in scalability (with respect to number of processors, graph size and GCN model size), efficiency and accuracy on several large datasets. On a 40-core Xeon platform, our parallel training achieves $64\times$ speedup (with AVX) in the sampling step and $25\times$ speedup in the feature propagation step, compared to the serial implementation, resulting in a net speedup of $21\times$.
Empirical risk minimization is the main tool for prediction problems, but its extension to relational data remains unsolved. We solve this problem using recent ideas from graph sampling theory to (i) define an empirical risk for relational data and (ii) obtain stochastic gradients for this empirical risk that are automatically unbiased. This is achieved by considering the method by which data is sampled from a graph as an explicit component of model design. By integrating fast implementations of graph sampling schemes with standard automatic differentiation tools, we provide an efficient turnkey solver for the risk minimization problem. We establish basic theoretical properties of the procedure. Finally, we demonstrate relational ERM with application to two non-standard problems: one-stage training for semi-supervised node classification, and learning embedding vectors for vertex attributes. Experiments confirm that the turnkey inference procedure is effective in practice, and that the sampling scheme used for model specification has a strong effect on model performance. Code is available at https://github.com/wooden-spoon/relational-ERM.
Nowadays, how to effectively evaluate visual properties has become a popular topic for fine-grained visual comprehension. In this paper we study the problem of how to estimate such visual properties from a ranking perspective with the help of the annotators from online crowdsourcing platforms. The main challenges of our task are two-fold. On one hand, the annotations often contain contaminated information, where a small fraction of label flips might ruin the global ranking of the whole dataset. On the other hand, considering the large data capacity, the annotations are often far from being complete. What is worse, there might even exist imbalanced annotations where a small subset of samples are frequently annotated. Facing such challenges, we propose a robust ranking framework based on the principle of Hodge decomposition of imbalanced and incomplete ranking data. According to the HodgeRank theory, we find that the major source of the contamination comes from the cyclic ranking component of the Hodge decomposition. This leads us to an outlier detection formulation as sparse approximations of the cyclic ranking projection. Taking a step further, it facilitates a novel outlier detection model as Huber's LASSO in robust statistics. Moreover, simple yet scalable algorithms are developed based on Linearized Bregman Iteration to achieve an even less biased estimator. Statistical consistency of outlier detection is established in both cases under nearly the same conditions. Our studies are supported by experiments with both simulated examples and real-world data. The proposed framework provides us a promising tool for robust ranking with large scale crowdsourcing data arising from computer vision.
The aim of this paper is to propose a least mean squares (LMS) strategy for adaptive estimation of signals defined over graphs. Assuming the graph signal to be band-limited, over a known bandwidth, the method enables reconstruction, with guaranteed performance in terms of mean-square error, and tracking from a limited number of observations over a subset of vertices. A detailed mean square analysis provides the performance of the proposed method, and leads to several insights for designing useful sampling strategies for graph signals. Numerical results validate our theoretical findings, and illustrate the performance of the proposed method. Furthermore, to cope with the case where the bandwidth is not known beforehand, we propose a method that performs a sparse online estimation of the signal support in the (graph) frequency domain, which enables online adaptation of the graph sampling strategy. Finally, we apply the proposed method to build the power spatial density cartography of a given operational region in a cognitive network environment.
We present a new random sampling strategy for k-bandlimited signals defined on graphs, based on determinantal point processes (DPP). For small graphs, ie, in cases where the spectrum of the graph is accessible, we exhibit a DPP sampling scheme that enables perfect recovery of bandlimited signals. For large graphs, ie, in cases where the graph's spectrum is not accessible, we investigate, both theoretically and empirically, a sub-optimal but much faster DPP based on loop-erased random walks on the graph. Preliminary experiments show promising results especially in cases where the number of measurements should stay as small as possible and for graphs that have a strong community structure. Our sampling scheme is efficient and can be applied to graphs with up to $10^6$ nodes.
Graph-based methods have been quite successful in solving unsupervised and semi-supervised learning problems, as they provide a means to capture the underlying geometry of the dataset. It is often desirable for the constructed graph to satisfy two properties: first, data points that are similar in the feature space should be strongly connected on the graph, and second, the class label information should vary smoothly with respect to the graph, where smoothness is measured using the spectral properties of the graph Laplacian matrix. Recent works have justified some of these smoothness conditions by showing that they are strongly linked to the semi-supervised smoothness assumption and its variants. In this work, we reinforce this connection by viewing the problem from a graph sampling theoretic perspective, where class indicator functions are treated as bandlimited graph signals (in the eigenvector basis of the graph Laplacian) and label prediction as a bandlimited reconstruction problem. Our approach involves analyzing the bandwidth of class indicator signals generated from statistical data models with separable and nonseparable classes. These models are quite general and mimic the nature of most real-world datasets. Our results show that in the asymptotic limit, the bandwidth of any class indicator is also closely related to the geometry of the dataset. This allows one to theoretically justify the assumption of bandlimitedness of class indicator signals, thereby providing a sampling theoretic interpretation of graph-based semi-supervised classification.
Graph Signal Processing (GSP) is a promising framework to analyze multi-dimensional neuroimaging datasets, while taking into account both the spatial and functional dependencies between brain signals. In the present work, we apply dimensionality reduction techniques based on graph representations of the brain to decode brain activity from real and simulated fMRI datasets. We introduce seven graphs obtained from a) geometric structure and/or b) functional connectivity between brain areas at rest, and compare them when performing dimension reduction for classification. We show that mixed graphs using both a) and b) offer the best performance. We also show that graph sampling methods perform better than classical dimension reduction including Principal Component Analysis (PCA) and Independent Component Analysis (ICA).
The goal of this paper is to propose novel strategies for adaptive learning of signals defined over graphs, which are observed over a (randomly time-varying) subset of vertices. We recast two classical adaptive algorithms in the graph signal processing framework, namely, the least mean squares (LMS) and the recursive least squares (RLS) adaptive estimation strategies. For both methods, a detailed mean-square analysis illustrates the effect of random sampling on the adaptive reconstruction capability and the steady-state performance. Then, several probabilistic sampling strategies are proposed to design the sampling probability at each node in the graph, with the aim of optimizing the tradeoff between steady-state performance, graph sampling rate, and convergence rate of the adaptive algorithms. Finally, a distributed RLS strategy is derived and is shown to be convergent to its centralized counterpart. Numerical simulations carried out over both synthetic and real data illustrate the good performance of the proposed sampling and reconstruction strategies for (possibly distributed) adaptive learning of signals defined over graphs.
Applied researchers often construct a network from a random sample of nodes in order to infer properties of the parent network. Two of the most widely used sampling schemes are subgraph sampling, where we sample each vertex independently with probability $p$ and observe the subgraph induced by the sampled vertices, and neighborhood sampling, where we additionally observe the edges between the sampled vertices and their neighbors. In this paper, we study the problem of estimating the number of motifs as induced subgraphs under both models from a statistical perspective. We show that: for any connected $h$ on $k$ vertices, to estimate $s=\mathsf{s}(h,G)$, the number of copies of $h$ in the parent graph $G$ of maximum degree $d$, with a multiplicative error of $\epsilon$, (a) For subgraph sampling, the optimal sampling ratio $p$ is $\Theta_{k}(\max\{ (s\epsilon^2)^{-\frac{1}{k}}, \; \frac{d^{k-1}}{s\epsilon^{2}} \})$, achieved by Horvitz-Thompson type of estimators. (b) For neighborhood sampling, we propose a family of estimators, encompassing and outperforming the Horvitz-Thompson estimator and achieving the sampling ratio $O_{k}(\min\{ (\frac{d}{s\epsilon^2})^{\frac{1}{k-1}}, \; \sqrt{\frac{d^{k-2}}{s\epsilon^2}}\})$. This is shown to be optimal for all motifs with at most $4$ vertices and cliques of all sizes. The matching minimax lower bounds are established using certain algebraic properties of subgraph counts. These results quantify how much more informative neighborhood sampling is than subgraph sampling, as empirically verified by experiments on both synthetic and real-world data. We also address the issue of adaptation to the unknown maximum degree, and study specific problems for parent graphs with additional structures, e.g., trees or planar graphs.