Graph alignment, the task of identifying corresponding nodes across different graphs, is crucial for applications ranging from social network analysis to bioinformatics. Although most existing methods leverage graph neural networks (GNNs) to learn node embeddings for attributed graphs and match them based on node similarity, they often rely on objectives designed for node classification or link prediction. These approaches preserve node proximity within individual graphs but fail to capture cross-graph correspondence knowledge, leading to suboptimal alignment performance.

Graph-level Contrastive Learning (GCL) has evolved as a powerful technique to derive representations from contrastive view pairs. Without access to labeled data, GCL typically takes two views augmented from the same graph as a positive pair and embeds them in nearby locations, while treating views from different graphs as negative pairs and pushing away their representations. Since the construction of contrastive pairs plays an important role in GCL, considerable attention has been paid to informative pairs mining. However, existing informative pairs mining methods suffer from the following two challenges: 1) Previous studies merely pay attention to the informative negative pairs while neglecting the informative positive pairs. Nevertheless, most augmentation methods require random perturbations, which may destroy the critical semantics of a graph, leading to false positive pairs (uninformative positives). 2) For informative negatives mining, most existing studies either overly emphasize hard negatives despite their potential unreliability, or rely on precise clustering pseudo-labels, which are error-prone especially in the early training stage. To solve the above challenges, we propose an informative sample-aware progressive graph contrastive learning framework, which filters both uninformative positives and negatives. In particular, we first present a progressive views sampler to evaluate the learning hardness of each view via clustering. Then, we feed model views with appropriate hardness, meaning those that aren't too challenging for the current model to assign pseudo labels confidently. Furthermore, we propose two samplers to filter out uninformative positives and negatives, respectively. Empirical results demonstrate the efficacy of our method IsGCL, which outperforms baselines by a margin of 2.5% on both MUTAG and PTC-MR in unsupervised learning settings. Furthermore, IsGCL maintains competitive training efficiency11Code available at https://github.com/jxzeng-git/IsGCL.

Tabular data is prevalent in many fields. In practice, tabular data classification may encounter severe challenges due to class imbalance, i.e., some majority classes overwhelm minority ones. Such imbalance could lead to biased prediction tendency of trained classifiers towards majority classes. Oversampling minority classes is an essential solution due to its generality and independence of downstream tasks. Recent years have witnessed the advantages of generative adversarial networks (GANs) in synthetic data generation, favored for their ability to generate quasi-realistic samples. However, challenges arise when the size of minority classes is too small to provide sufficient information for learning real data distributions. Furthermore, the generated minority-class samples could exacerbate the class overlap problem, i.e., some generated samples unexpectedly overlap with partial majority-class samples. To address these challenges, this paper presents B2BGAN, a novel GAN-based approach for oversampling imbalanced tabular data. To capture the real data distribution in a fine-grained manner, we propose a novel backbone-to-branches neural network for the generator to fit the majority and minority classes simultaneously. The backbone network fits the whole distribution of the entire data, while each branch network grasps the distinctive characteristics of individual classes. To alleviate the class overlap problem of generated samples, we develop a prototype-guided loss function to ensure that generated samples are closer to the corresponding class prototypes. We evaluate the effectiveness of B2BGAN on six real-world datasets using six metrics. Experimental results demonstrate that our method outperforms state-of-the-art models by 5.38% in AUC and 10.19% in AP.

Legal case retrieval aims to automatically scour comparable legal cases based on a given query, which is crucial for offering relevant precedents to support the judgment in intelligent legal systems. Due to similar goals, it is often associated with a similar case matching task. To address them, a daunting challenge is assessing the uniquely defined legal-rational similarity within the judicial domain, which distinctly deviates from the semantic similarities in general text retrieval. Past works either tagged domain-specific factors or incorporated reference laws to capture legal-rational information. However, their heavy reliance on expert or unrealistic assumptions restricts their practical applicability in real-world scenarios. In this article, we propose an end-to-end model named LCM-LAI to solve the above challenges. Through meticulous theoretical analysis, LCM-LAI employs a dependent multi-task learning framework to capture legal-rational information within legal cases by a law article prediction sub-task, without any additional assumptions in inference. In addition, LCM-LAI proposes an article-aware attention mechanism to evaluate the legal-rational similarity between across-case sentences based on the law distribution, which is more effective than semantic similarity. We perform a series of exhaustive experiments that include two different tasks that involving four real-world datasets. The results demonstrate that LCM-LAI achieves state-of-the-art performance.

Legal Judgment Prediction (LJP) aims to automatically predict a law case’s judgment results based on the text description of its facts. In practice, the confusing law articles (or charges) problem frequently occurs, reflecting that the law cases applicable to similar articles (or charges) tend to be misjudged. Although some recent works based on prior knowledge solve this issue well, they ignore that confusion also occurs between law articles with a high posterior semantic similarity due to the data imbalance problem instead of only between the prior highly similar ones, which is this work’s further finding. This article proposes an end-to-end model named D-LADAN to solve the above challenges. On the one hand, D-LADAN constructs a graph among law articles based on their text definition and proposes a graph distillation operator (GDO) to distinguish the ones with a high prior semantic similarity. On the other hand, D-LADAN presents a novel momentum-updated memory mechanism to dynamically sense the posterior similarity between law articles (or charges) and a weighted GDO to adaptively capture the distinctions for revising the inductive bias caused by the data imbalance problem. We perform extensive experiments to demonstrate that D-LADAN significantly outperforms state-of-the-art methods in accuracy and robustness.

Computing set difference cardinalities is a critical task in database optimization, network management, and anomaly detection. Due to the limited computational and mem-ory resources, exactly calculating set difference cardinalities becomes impractical in real-world applications. To solve this issue, sketch methods such as Odd sketch, Tug-of-War sketch, and HyperLogLog sketch can be extended to provide approximate estimations of set difference cardinalities. They use a family of hash functions to compress all elements in a set into a compact data structure. Unfortunately, Odd sketch suffers from limited estimation range, while Tug-of-War sketch and HyperLogLog sketch unavoidably face the problems of large estimation errors and high computational costs. In this paper, we design a novel data structure of bit array GXBits to fast and accurately estimate set difference cardinalities in a large range. In GXBits, the prob-ability of each bit recording its corresponding elements follows a variant of geometric distributions and varies across different bits. We conduct extensive experiments on synthetic datasets and real-world datasets. Experimental results demonstrate that our method GXBits is more computationally and memory efficient, and significantly increases the estimation accuracy of existing methods by up to 221.3 times.

Graph retrieval aims to find the most similar graphs in a graph database given a query graph, which is a fundamental problem with many real-world applications in chemical engineering, code analysis, etc. To date, existing neural graph retrieval methods generally fall into two categories: Embedding Based Paradigm (Ebp) and Matching Based Paradigm (Mbp). The Ebp models learn an individual vectorial representation for each graph and the retrieval process can be accelerated by pre-computing these representations. The Mbp models learn a neural matching function to compare graphs on a pair-by-pair basis, in which the fine-grained pairwise comparison leads to higher retrieval accuracy but severely degrades retrieval efficiency. In this paper, to combine the advantage of Ebp in retrieval efficiency with that of Mbp in retrieval accuracy, we propose a novel Graph RetrievAl framework via KNowledge Distillation, namely GRAND. The key point is to leverage the idea of knowledge distillation to transfer the fine-grained graph comparison knowledge from an Mbp model to an Ebp model, such that the Ebp model can generate better graph representations and thus yield higher retrieval accuracy. At the same time, we can still pre-compute and index the improved graph representations to retain the retrieval speed of Ebp. Towards this end, we propose to perform knowledge distillation from three perspectives: score, node, and subgraph levels. In addition, we propose to perform mutual two-way knowledge transfer between Mbp and Ebp, such that Mbp and Ebp complement and benefit each other. Extensive experiments on three real-world datasets show that GRAND improves the performance of Ebp by a large margin and the improvement is consistent for different combinations of Ebp and Mbp models. For example, GRAND achieves performance gains of mostly more than 10% and up to 16.88% in terms of Recall@K on different datasets.

Graph few-shot learning (GFSL) has achieved great success in node classification tasks with rare labels. However, graph few-shot classification (GFSC) models often encounter the problem of classifying test samples with unobserved (or unknown) classes due to the rareness of labels. We formulate this problem as out-of-distribution (OOD) sample detection in inductive graph few-shot learning. This paper presents SMUG, a novel GFSL framework that can detect unobserved classes. Since we have no ground-truth OOD samples in a practical training dataset, it is challenging for the GFSC model to retrieve knowledge about unknown classes from labeled samples. To address this difficulty, we propose a sand mixing scheme to introduce observed classes as artificial OOD samples into meta-tasks. We also develop two unsupervised OOD discriminators to identify OOD samples. Thus, we can assess the performance of OOD discriminators since we know the true classes of these artificial OOD samples. Subsequently, we design a novel training procedure to optimize the encoder based on the performance of the OOD discriminators and the GFSC model. It not only enables the GFSL model to distinguish OOD samples but also promotes the classification accuracy of normal samples. We conduct extensive experiments to evaluate the effectiveness of SMUG based on four benchmark datasets. Experimental results demonstrate that SMUG achieves superior performance over state-of-the-art approaches in OOD detection and node classification. The source code of this paper is available at https://github.com/Memepp/SMUG.

Kernel density estimation is a useful method for estimating the probability distribution of data. It is a challenge to achieve efficient kernel density estimation, especially for large-scale and high-dimension stream data. We proposerotation kernel, a novel kernel function for density estimation. The rotation kernel density can be fast estimated by a data structure namedRotation Kernel Density Sketch(RKDS). RKDS is a time- and memory-efficient method for kernel density estimation, even over data streams and distributed systems. RKDS is applicable for estimating density at specific points and also for representing data distribution. We provide theoretical analysis for rotation kernel and RKDS. Furthermore, we apply RKDS to outlier detection, concept drift detection, and personalized federated learning. Experiments show that our method improves time efficiency by up to$3\times 10^{3}$times compared with baselines. RKDS also provides comparable detecting precision and better delay on outlier detection and concept drift detection tasks.

Graph alignment aims to find correspondent nodes between two graphs. Most existing algorithms assume that correspondent nodes in different graphs have similar local structures. However, this principle may not apply to some real-world application scenarios when two graphs have different densities. Some correspondent node pairs may have very different local structures in these cases. Nevertheless, correspondent nodes are expected to have similar importance, inspiring us to exploit global topology consistency for graph alignment. This paper presents GTCAlign, an unsupervised graph alignment framework based on global topology consistency. An indicating matrix is calculated to show node pairs with consistent global topology based on a comprehensive centrality metric. A graph convolutional network (GCN) encodes local structural and attributive information into low-dimensional node embeddings. Then, node similarities are computed based on the obtained node embeddings under the guidance of the indicating matrix. Moreover, a pair of nodes are more likely to be aligned if most of their neighbors are aligned, motivating us to develop an iterative algorithm to refine the alignment results recursively. We conduct extensive experiments on real-world and synthetic datasets to evaluate the effectiveness of GTCAlign. The experimental results show that GTCAlign outperforms state-of-the-art graph alignment approaches.

Network embedding algorithms learn a mapping from the discrete representation of nodes to continuous vector spaces that preserve node proximity. Despite recent efforts to design novel models, little attention has been given to understanding the instability of network embedding. In this paper, we define the stability of node embeddings as the invariance of the nearest neighbors of nodes in different instantiations. We find that existing embedding approaches have significant amounts of instability. In addition, network structures and algorithm models influence the stability of node embeddings significantly. We also examine the implications of embedding instability for downstream tasks and find remarkable impacts on performance.

Counting the number of distinct elements distributed over multiple data holders is a fundamental problem with many real-world applications ranging from crowd counting to network monitoring. Although a number of space and computationally efficient sketch methods (e.g., the Flajolet-Martin sketch and the HyperLogLog sketch) for cardinality estimation have been proposed to solve the above problem, these sketch methods are insecure when considering privacy concerns related to the use of each data holder's personal dataset. Despite a recently proposed protocol that successfully implements the well-known Flajolet-Martin (FM) sketch on a secret-sharing based multiparty computation (MPC) framework for solving the problem of private distributed cardinality estimation (PDCE), we observe that this MPC-FM protocol is not differentially private. In addition, the MPC-FM protocol is computationally expensive, which limits its applications to data holders with limited computation resources. To address the above issues, in this paper we propose a novel protocol DP-DICE, which is computationally efficient and differentially private for solving the problem of PDCE. Experimental results show that our DP-DICE achieves orders of magnitude speedup and reduces the estimation error by several times in comparison with state-of-the-arts under the same security requirements.

The well-known Gumbel-Max Trick for sampling elements from a categorical distribution (or more generally a non-negative vector) and its variants have been widely used in areas such as machine learning and information retrieval. To sample a random element$i$in proportion to its positive weight$v_{i}$, the Gumbel-Max Trick first computes a Gumbel random variable$g_{i}$for each positive weight element$i$, and then samples the element$i$with the largest value of$g_{i}+\ln v_{i}$. Recently, applications including similarity estimation and weighted cardinality estimation require to generate$k$independent Gumbel-Max variables from high dimensional vectors. However, it is computationally expensive for a large$k$(e.g., hundreds or even thousands) when using the traditional Gumbel-Max Trick. To solve this problem, we propose a novel algorithm,FastGM, which reduces the time complexity from$O(kn^+)$to$O(k \ln k + n^+)$, where$n^+$is the number of positive elements in the vector of interest. FastGM stops the procedure of Gumbel random variables computing for many elements, especially for those with small weights. We perform experiments on a variety of real-world datasets and the experimental results demonstrate that FastGM is orders of magnitude faster than state-of-the-art methods without sacrificing accuracy or incurring additional expenses.

Graph Neural Networks (GNNs) have been success-fully applied to a variety of graph analysis tasks. Some recent studies have demonstrated that decoupling neighbor aggregation and feature transformation helps to scale GNNs to large graphs. However, very large graphs, with billions of nodes and millions of features, are still beyond the capacity of most existing GNNs. In addition, when we are only interested in a small number of nodes (called target nodes) in a large graph, it is inefficient to use the existing GNNs to infer the labels of these few target nodes. The reason is that they need to propagate and aggregate either node features or predicted labels over the whole graph, which incurs high additional costs relative to the few target nodes. To solve the above challenges, in this paper we propose a novel scalable and effective GNN framework COSAL. In COSAL, we substitute the expensive aggregation with an efficient proximate node selection mechanism, which picks out the most important$K$nodes for each target node according to the graph topology. We further propose a fine-grained neighbor importance quantification strategy to enhance the expressive power of COSAL. Empirical results demonstrate that our COSAL achieves superior performance in accuracy, training speed, and partial inference efficiency. Remarkably, in terms of node classification accuracy, our model COSAL outperforms baselines by significant margins of 2.22%, 2.23%, and 3.95% on large graph datasets Amazon2M, MAG-Scholar-C, and ogbn-papers100M, respectively.11Code available at https://github.com/joyce-x/COSAL.

Many real-world datasets are given in the stream of entity-identifier pairs, and measuring data distribution on these datasets is fundamental for applications such as privacy protection. In this paper, we study the problem of computing the cumulative distribution for different cardinalities (i.e., the number of distinct entities owning the same identifier). However, previous sketch-based methods cost large memory space especially when there are a large number of identifiers, and sampling-based methods require much time for cardinality estimation. A recent work KHyperLogLog combines both sketch and sampling methods but it is wasteful to separately build a HyperLogLog sketch of large size for identifiers with small cardinalities. To address these challenges, we propose a memory-efficient method EV-HLL, which designs a shared structure to store all sampled identifiers and their entities and utilizes additional sketches to track value updates during the sampling procedure. Meanwhile, EV-HLL provides real-time unbiased estimations according to value changes whenever a new entity-identifier pair arrives. We evaluate the performance of EV-HLL and other state-of-the-arts on real-world available datasets. Experimental results demonstrate that comparing to other methods, EV-HLL effectively reduces their memory usage with the same estimation accuracy and has higher accuracy with the same memory usage.

Online monitoring user cardinalities in graph streams is fundamental for many applications such as anomaly detection. These graph streams may contain edge duplicates and have a large number of user-item pairs, which makes it infeasible to exactly compute user cardinalities due to limited computational and memory resources. Existing methods are designed to approximately estimate user cardinalities, but their accuracy highly depends on complex parameters and they cannot provide anytime-available estimation. To address these problems, we develop novel bit/register sharing algorithms, which use a bit/register array to build a compact sketch of all users’ connected items. Our algorithms exploit the dynamic properties of the bit/register arrays (e.g., the fraction of zero bits in the bit array) to significantly improve the estimation accuracy, and have low time complexity$O(1)$to update the estimations for a new user-item pair. In addition, our algorithms are simple and easy to use, without requirements to tune any parameter. Furthermore, we extend our methods to detect super spreaders with large cardinalities in real-time. We evaluate the performance of our methods on real-world datasets. The experimental results demonstrate that our methods are several times more accurate and faster than state-of-the-art methods using the same amount of memory.

Bipartite graphs widely exist in real-world scenarios and model binary relations like host-website, author-paper, and user-product. In bipartite graphs, a butterfly (i.e.,$2\times 2$bi-clique) is the smallest non-trivial cohesive structure and plays an important role in applications such as anomaly detection. Considerable efforts focus on counting butterflies in static bipartite graphs. However, they suffer from high time and space complexity when the bipartite graph of interest is given as a stream of edges. Although there are methods for approximately counting butterflies from bipartite graph streams, they suffer from either low accuracy or high time complexity. Therefore, it is still a challenge to accurately estimate butterfly counts from bipartite graph streams in a short time. To address this issue, we develop novel algorithms by exploiting the bipartite nature, which subtly integrates sampling and sketching techniques. We provide accurate estimators for butterfly counts and derive simple yet exact formulas for bounding their errors. We also conduct extensive experiments on a variety of real-world large bipartite graphs. Experimental results demonstrate that our algorithms are up to 20.0 times more accurate and up to 286.3 times faster than state-of-the-art methods under the same memory usage.

With numerous nodes on online heterogeneous networks, how to reach and extract target nodes of our specific interests is a pressing problem. In this paper, we propose a novel heterogeneous network crawler,MCrawl. It addresses the problem via iterative online heterogeneous network crawling by navigating its available APIs, starting from a set of target nodes, i.e., seed nodes. We are facing two challenges towards addressing the problem. First, to navigate within a vast network, how do we start from a small set of target nodes? In other words, which nodes in the “current frontier” and which direction shall we expand, to reach promising target nodes quickly? We propose motif-based crawling to exploit the complex structures and rich semantics of heterogeneous networks. Second, in many scenarios, we do not have a classifier to assess the quality of the harvested nodes and thus the motifs to expand. We develop a probabilistic inference framework to estimate the yield and harvest rates of motifs, achieving principled bootstrapping for crawling. Our experiment on real networks of MCrawl achieves significant margins over baselines.

Network embedding algorithms learn a mapping from the discrete representation of nodes to continuous vector spaces that preserve the proximities of nodes. The techniques have a wide range of applications in various downstream tasks such as node classification, link prediction, and network alignment. Despite recent efforts to the design of novel models, little attention has been paid to understanding the instability of network embedding. In this paper, we fill this gap by investigating several state-of-the-art network embedding methods. Node embeddings form a geometric shape in the latent space. Characterizing the geometry is critical to figure out the variance of network embedding. Hence, we define two metrics to characterize the geometric properties and find that node embeddings tremble in different instantiations of an embedding space. Then, we formally define the stability of node embeddings as the invariance of the nearest neighbors of nodes. Experimental results show that existing embedding approaches have significant amounts of instability. We explore the influence factors that affect the stability of different methods and find that both network structure and algorithm models affect the stability of node embeddings significantly. Finally, we examine the implications of embedding instability for downstream tasks and find remarkable impacts on the performance.

Many real-world datasets are given in the format of data streams, and processing these data streams is fundamental for many applications such as anomaly detection. In this paper, we study the problem of computing item frequencies, finding topk hot items, and detecting heavy changes. However, the widelyused sketches cost large memory usage and their performance is easily affected by the unbalanced distribution of data streams. To solve this issue, a novel method Cold Filter (CF) is proposed to split cold items and hot items, and use a separate structure to record the frequencies of hot items. Typically, CF has a small filter range and is only effective for filtering cold items with small frequencies. For some real-world applications, however, the cold items' frequencies may also be greater than hundreds or even tens of thousands. To solve the above challenges, we exploit the “LogLog” structure and develop a memory-efficient method LogLog Filter (LLF) to accurately estimate the above three metrics. LLF builds a register array where each register approximately counts the sum of item frequencies hashed into it. Our method remarkably enlarges the filter range of CF with fewer bits and only requires 4 bits to filter cold items with frequencies up to 224. We conduct extensive experiments on real-world and synthetic datasets, and the experimental results demonstrate the efficiency and effectiveness of our method.

Kernel density estimation method is a powerful tool and is widely used in many important real-world applications such as anomaly detection and statistical learning. Unfortunately, current kernel methods suffer from high computational or space costs when dealing with large-scale, high-dimensional datasets, especially when the datasets of interest are given in a stream fashion. Although there are sketch methods designed for kernel density estimation over data streams, they still suffer from high computational costs. To address this problem, in this paper, we propose a novel Rotation Kernel. The Rotation Kernel is based on a Rotation Hash method and is much faster to compute. To achieve memory-efficient kernel density estimation over data streams, we design a method, RKD-Sketch, which compresses high dimensional data streams into a small array of integer counters. We conduct extensive experiments on both synthetic and real-world datasets, and experimental results demonstrate that our RKD-Sketch saves up to 216 times computational resources and up to 104 times space resources than state-of-the-arts. Furthermore, we apply our Rotation Kernel in active learning. Results show that our method achieves up to 256 times speedup and saves up to 13 times space to achieve the same accuracy as the baseline methods.

As an efficient tool for approximate similarity computation and search, Locality Sensitive Hashing (LSH) has been widely used in many research areas including databases, data mining, information retrieval, and machine learning. Classical LSH methods typically require to perform hundreds or even thousands of hashing operations when computing the LSH sketch for each input item (e.g., a set or a vector); however, this complexity is still too expensive and even impractical for applications requiring processing data in real-time. To address this issue, several fast methods such as OPH and BCWS have been proposed to efficiently compute the LSH sketches; however, these methods may generate many sketches with empty bins, which may introduce large errors for similarity estimation and also limit their usage for fast similarity search. To solve this issue, we propose a novel densification method, i.e., BiDens. Compared with existing densification methods, our BiDens is more efficient to fill a sketch's empty bins with values of its non-empty bins in either the forward or backward directions. Furthermore, it also densifies empty bins to satisfy the densification principle (i.e., the LSH property). Theoretical analysis and experimental results on similarity estimation, fast similarity search, and kernel linearization using real-world datasets demonstrate that our BiDens is up to 106 times faster than state-of-the-art methods while achieving the same or even better accuracy.

Estimating set similarity and detecting highly similar sets are fundamental problems in areas such as databases and machine learning. MinHash is a well-known technique for approximating Jaccard similarity of sets and has been successfully used for many applications. Its two compressed versions, b-bit MinHash and Odd Sketch, can significantly reduce the memory usage of the MinHash, especially for estimating high similarities (i.e., similarities around 1). Although MinHash can be applied to static sets as well as streaming sets, of which elements are given in a streaming fashion, unfortunately, b-bit MinHash and Odd Sketch fail to deal with streaming data. To solve this problem, we previously designed a memory-efficient sketch method, MaxLogHash, to accurately estimate Jaccard similarities in streaming sets. Compared with MinHash, our method uses smaller sized registers (each register consists of less than 7 bits) to build a compact sketch for each set. In this paper, we further develop a faster method, MaxLogOPH++. Compared with MaxLogHash, MaxLogOPH++ reduces the time complexity for updating each coming element from O(k) with a small additional memory. We conduct experiments on a variety of datasets, and experimental results demonstrate the efficiency and effectiveness of our methods.

The well-known Gumbel-Max Trick for sampling elements from a categorical distribution (or more generally a nonnegative vector) and its variants have been widely used in areas such as machine learning and information retrieval. To sample a random element i (or a Gumbel-Max variable i) in proportion to its positive weight vi, the Gumbel-Max Trick first computes a Gumbel random variable gi for each positive weight element i, and then samples the element i with the largest value of gi + ln vi. Recently, applications including similarity estimation and graph embedding require to generate k independent Gumbel-Max variables from high dimensional vectors. However, it is computationally expensive for a large k (e.g., hundreds or even thousands) when using the traditional Gumbel-Max Trick. To solve this problem, we propose a novel algorithm, FastGM, that reduces the time complexity from O(kn+) to O(kln k + n+), where n+ is the number of positive elements in the vector of interest. Instead of computing k independent Gumbel random variables directly, we find that there exists a technique to generate these variables in descending order. Using this technique, our method FastGM computes variables gi + ln vi for all positive elements i in descending order. As a result, FastGM significantly reduces the computation time because we can stop the procedure of Gumbel random variables computing for many elements especially for those with small weights. Experiments on a variety of real-world datasets show that FastGM is orders of magnitude faster than state-of-the-art methods without sacrificing accuracy and incurring additional expenses.

Many real-world networks such as Twitter and YouTube are given as fully dynamic graph streams represented as sequences of edge insertions and deletions. (e.g., users can subscribe and unsubscribe to channels on YouTube). Existing similarity estimation methods such as MinHash and OPH are customized to static graphs. We observe that they are indeed sampling methods and exhibit a sampling bias when applied to fully dynamic graph streams, which results in large estimation errors. To solve this challenge, we develop a fast and accurate sketch method VOS. VOS processes each edge in the graph stream of interest with small time complexity O(1) and uses small memory space to build a compact sketch of the dynamic graph stream over time. Based on the sketch built on-the-fly, we develop a method to estimate user similarities over time. We conduct extensive experiments and the experimental results demonstrate the efficiency and efficacy of our method.

Recently, considerable efforts have been devoted to approximately computing the global and local (i.e., incident to each node) triangle counts of a large graph stream represented as a sequence of edges. Existing approximate triangle counting algorithms rely on sampling techniques to reduce the computational cost. However, their estimation errors are significantly determined by the covariance between sampled triangles. Moreover, little attention has been paid to developing parallel one-pass streaming algorithms that can be used to fast and approximately count triangles on a multi-core machine or a cluster of machines. To solve these problems, we develop a novel parallel method REPT to significantly reduce the covariance (even completely eliminate the covariance for some cases) between sampled triangles. We theoretically prove that REPT is more accurate than parallelizing existing triangle count estimation algorithms in a direct manner. In addition, we also conduct extensive experiments on a variety of real-world graphs, and the results demonstrate that our method REPT is several times more accurate than state-of-the-art methods.

Online monitoring user cardinalities (or degrees) in graph streams is fundamental for many applications. For example in a bipartite graph representing user-website visiting activities, user cardinalities (the number of distinct visited websites) are monitored to report network anomalies. These real-world graph streams may contain user-item duplicates and have a huge number of distinct user-item pairs, therefore, it is infeasible to exactly compute user cardinalities when memory and computation resources are limited. Existing methods are designed to approximately estimate user cardinalities, whose accuracy highly depends on parameters that are not easy to set. Moreover, these methods cannot provide anytime-available estimation, as the user cardinalities are computed at the end of the data stream. Realtime applications such as anomaly detection require that user cardinalities are estimated on the fly. To address these problems, we develop novel bit and register sharing algorithms, which use a bit array and a register array to build a compact sketch of all users' connected items respectively. Compared with previous bit and register sharing methods, our algorithms exploit the dynamic properties of the bit and register arrays (e.g., the fraction of zero bits in the bit array at each time) to significantly improve the estimation accuracy, and have low time complexity (O(1)) to update the estimations each time they observe a new useritem pair. In addition, our algorithms are simple and easy to use, without requirements to tune any parameter. We evaluate the performance of our methods on real-world datasets. The experimental results demonstrate that our methods are several times more accurate and faster than state-of-the-art methods using the same amount of memory.

Estimating set similarity and detecting highly similar sets are fundamental problems in areas such as databases, machine learning, and information retrieval. MinHash is a well-known technique for approximating Jaccard similarity of sets and has been successfully used for many applications such as similarity search and large scale learning. Its two compressed versions, b-bit MinHash and Odd Sketch, can significantly reduce the memory usage of the original MinHash method, especially for estimating high similarities (i.e., similarities around 1). Although MinHash can be applied to static sets as well as streaming sets, of which elements are given in a streaming fashion and cardinality is unknown or even infinite, unfortunately, b-bit MinHash and Odd Sketch fail to deal with streaming data. To solve this problem, we design a memory efficient sketch method, MaxLogHash, to accurately estimate Jaccard similarities in streaming sets. Compared to MinHash, our method uses smaller sized registers (each register consists of less than 7 bits) to build a compact sketch for each set. We also provide a simple yet accurate estimator for inferring Jaccard similarity from MaxLogHash sketches. In addition, we derive formulas for bounding the estimation error and determine the smallest necessary memory usage (i.e., the number of registers used for a MaxLogHash sketch) for the desired accuracy. We conduct experiments on a variety of datasets, and experimental results show that our method MaxLogHash is about 5 times more memory efficient than MinHash with the same accuracy and computational cost for estimating high similarities.

People participate in multiple online social networks, e.g., Facebook, Twitter, and Linkedin, and these social networks with heterogeneous social content and user relationship are named as heterogeneous social networks. Group structure widely exists in heterogeneous social networks, which reveals the evolution of human cooperation. Detecting similar groups in heterogeneous networks has a great significance for many applications, such as recommendation system and spammer detection, using the wealth of group information. Although promising, this novel problem encounters a variety of technical challenges, including incomplete data, high time complexity, and ground truth. To address the research gap and technical challenges, we take advantage of a ratio-cut optimization function to model this novel problem by the linear mixed-effects method and graph spectral theory. Based on this model, we propose an efficient algorithm called D igger to detect the similar groups in the large graphs. D igger consists of three steps, including measuring user similarity, construct a matching graph, and detecting similar groups. We adopt several strategies to lower the computational cost and detail the basis of labeling the ground truth. We evaluate the effectiveness and efficiency of our algorithm on five different types of online social networks. The extensive experiments show that our method achieves 0.693, 0.783, and 0.735 in precision, recall, and F1-measure, which significantly surpass the state-of-arts by 24.4%, 15.3%, and 20.7%, respectively. The results demonstrate that our proposal can detect similar groups in heterogeneous networks effectively.

Monitoring user behaviors over high speed links is important for applications such as network anomaly detection. Previous work focuses on monitoring anomalies such as extremely frequent users occurring in a short timeslot such as 1 minute. Little attention has been paid to detect users with stealthy behaviors (e.g., persistent, co-occurrence, anti-co-occurrence, and periodic behaviors) over a long period of time at the timeslot granularity. Due to limited computation and storage resources on routers, it is prohibitive to collect massive network traffic in a long period of time. We develop an end-to-end method for solving challenges in both long-term online traffic collection and offline user behavior analysis. We conduct extensive experiments on a variety of real-world traffic to evaluate the performance of detecting persistent, co-occurrence, anti-co-occurrence, and periodic behaviors, and the results demonstrate that our method significantly outperforms state-of-the-art methods.

Despite recent efforts in counting 3-node and 4-node graphlets, little attention has been paid to characterizing 5-node graphlets. In this paper, we develop a computationally efficient sampling method to estimate 5-node graphlet counts. We not only provide a fast sampling method and unbiased estimators of graphlet counts, but also derive simple yet exact formulas for the variances of the estimators which are of great value in practice-the variances can be used to bound the estimates' errors and determine the smallest necessary sampling budget for a desired accuracy. We conduct experiments on a variety of real-world datasets, and the results show that our method is several orders of magnitude faster than the state-of-the-art methods with the same accuracy.

Counting 3-, 4-, and 5-node graphlets in graphs is important for graph mining applications such as discovering abnormal/ evolution patterns in social and biology networks. In addition, it is recently widely used for computing similarities between graphs and graph classification applications such as protein function prediction and malware detection. However, it is challenging to compute these graphlet counts for a large graph or a large set of graphs due to the combinatorial nature of the problem. Despite recent efforts in counting 3-node and 4-node graphlets, little attention has been paid to characterizing 5-node graphlets. In this paper, we develop a computationally efficient sampling method to estimate 5-node graphlet counts. We not only provide a fast sampling method and unbiased estimators of graphlet counts, but also derive simple yet exact formulas for the variances of the estimators which are of great value in practice-the variances can be used to bound the estimates' errors and determine the smallest necessary sampling budget for a desired accuracy. We conduct experiments on a variety of real-world datasets, and the results show that our method is several orders of magnitude faster than the state-of-the-art methods with the same accuracy.

Counting triangles in a large graph is important for detecting network anomalies such as spam web pages and suspicious accounts (e.g., fraudsters and advertisers) on online social networks. However, it is challenging to compute the number of triangles in a large graph represented as a stream of edges with a low computational cost when given a limited memory. Recently, several effective sampling-based approximation methods have been developed to solve this problem. However, they assume the graph stream of interest contains no duplicate edges, which does not hold in many real-world graph streams (e.g., phone calling networks). In this paper, we observe that these methods exhibit a large estimation error or computational cost even when modified to deal with duplicate edges using deduplication techniques such as Bloom filter and hash-based sampling. To solve this challenge, we design a one-pass streaming algorithm for uniformly sampling distinct edges at a high speed. Compared to state-of-the-art algorithms, our algorithm reduces the sampling cost per edge from O (log k ) ( k is the maximum number of sampled edges determined by the available memory space) to O (1) without using any additional memory space. Based on sampled edges, we develop a simple yet accurate method to infer the number of triangles in the original graph stream. We conduct extensive experiments on a variety of real-world large graphs, and the results demonstrate that our method is several times more accurate and faster than state-of-the-art methods with the same memory usage.

Random walk-based graph sampling methods have become increasingly popular and important for characterizing large-scale complex networks. While powerful, they are known to exhibit problems when the graph is loosely connected, which slows down the convergence of a random walk and can result in poor estimation accuracy. In this work, we observe that many graphs under study, called target graphs, usually do not exist in isolation. In many situations, a target graph is often related to an auxiliary graph and an affiliation graph, and the target graph becomes better connected when viewed from these three graphs as a whole, or what we called a hybrid social-affiliation network. This viewpoint brings extra benefits to the graph sampling framework, e.g., when directly sampling a target graph is difficult or inefficient, we can efficiently sample it with the assistance of auxiliary and affiliation graphs. We propose three sampling methods on such a hybrid social-affiliation network to estimate target graph characteristics, and conduct extensive experiments on both synthetic and real datasets, to demonstrate the effectiveness of these new sampling methods.

Characterizing user pair relationships is important for applications such as friend recommendation and interest targeting in online social networks (OSNs). Due to the large-scale nature of such networks, it is infeasible to enumerate all user pairs and thus sampling is used. In this article, we show that it is a great challenge for OSN service providers to characterize user pair relationships, even when they possess the complete graph topology. The reason is that when sampling techniques (i.e., uniform vertex sampling (UVS) and random walk (RW)) are naively applied, they can introduce large biases, particularly for estimating similarity distribution of user pairs with constraints like existence of mutual neighbors, which is important for applications such as identifying network homophily. Estimating statistics of user pairs is more challenging in the absence of the complete topology information, as an unbiased sampling technique like UVS is usually not allowed and exploring the OSN graph topology is expensive. To address these challenges, we present unbiased sampling methods to characterize user pair properties based on UVS and RW techniques. We carry out an evaluation of our methods to show their accuracy and efficiency. Finally, we apply our methods to three OSNs—Foursquare, Douban, and Xiami—and discover that significant homophily is present in these networks.

Exploring statistics of locally connected subgraph patterns (also known as network motifs) has helped researchers better understand the structure and function of biological and Online Social Networks (OSNs). Nowadays, the massive size of some critical networks—often stored in already overloaded relational databases—effectively limits the rate at which nodes and edges can be explored, making it a challenge to accurately discover subgraph statistics. In this work, we propose sampling methods to accurately estimate subgraph statistics from as few queried nodes as possible. We present sampling algorithms that efficiently and accurately estimate subgraph properties of massive networks. Our algorithms require no precomputation or complete network topology information. At the same time, we provide theoretical guarantees of convergence. We perform experiments using widely known datasets and show that, for the same accuracy, our algorithms require an order of magnitude less queries (samples) than the current state-of-the-art algorithms.

Characterizing user pair relationships is important for applications such as friend recommendation and interest targeting in online social networks (OSNs). Due to the large scale nature of such networks, it is infeasible to enumerate all user pairs and so sampling is used. In this paper, we show that it is a great challenge even for OSN service providers to characterize user pair relationships even when they possess the complete graph topology. The reason is that when sampling techniques (i.e., uniform vertex sampling (UVS) and random walk (RW)) are naively applied, they can introduce large biases, in particular, for estimating similarity distribution of user pairs with constraints such as existence of mutual neighbors, which is important for applications such as identifying network homophily. Estimating statistics of user pairs is more challenging in the absence of the complete topology information, since an unbiased sampling technique such as UVS is usually not allowed, and exploring the OSN graph topology is expensive. To address these challenges, we present asymptotically unbiased sampling methods to characterize user pair properties based on UVS and RW techniques respectively. We carry out an evaluation of our methods to show their accuracy and efficiency. Finally, we apply our methods to two Chinese OSNs, Doudan and Xiami, and discover significant homophily is present in these two networks.