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Federated graph learning (FGL) has emerged as a promising distributed training paradigm for graph neural networks across multiple local systems without direct data sharing. This approach is particularly beneficial in privacy-sensitive scenarios and offers a new perspective on addressing scalability challenges in large-scale graph learning. Despite the proliferation of FGL, the diverse motivations from practical applications, spanning various research backgrounds and experimental settings, pose a significant challenge to fair evaluation. To fill this gap, we propose OpenFGL, a unified benchmark designed for the primary FGL scenarios: Graph-FL and Subgraph-FL. Specifically, OpenFGL includes 38 graph datasets from 16 application domains, 8 federated data simulation strategies that emphasize graph properties, and 5 graph-based downstream tasks. Additionally, it offers 18 recently proposed SOTA FGL algorithms through a user-friendly API, enabling a thorough comparison and comprehensive evaluation of their effectiveness, robustness, and efficiency. Empirical results demonstrate the ability of FGL while also revealing its potential limitations, offering valuable insights for future exploration in this thriving field., Comment: Under Review

Random feature attention (RFA) adopts random fourier feature (RFF) methods to approximate the softmax function, resulting in a linear time and space attention mechanism that enables the construction of an efficient Transformer. Inspired by RFA, we propose Macformer, a Transformer architecture that employs random Maclaurin features (RMF) to approximate various dot-product kernels, thereby accelerating attention computations for long sequence. Macformer consists of Random Maclaurin Feature Attention (RMFA) and pre-post Scaling Batch Normalization (ppSBN), the former is an unbiased approximation for dot-product kernelized attention and the later is a two-stage regularization mechanism guaranteeing the error of RMFA. We conducted toy experiments to demonstrate the efficiency of RMFA and ppSBN, and experiments on long range arena (LRA) benchmark to validate the acceleration and accuracy of Macformer with different dot-product kernels. Experiment results of Macformer are consistent with our theoretical analysis.

When using exploratory visual analysis to examine multivariate hierarchical data, users often need to query data to narrow down the scope of analysis. However, formulating effective query expressions remains a challenge for multivariate hierarchical data, particularly when datasets become very large. To address this issue, we develop a declarative grammar, HiRegEx (Hierarchical data Regular Expression), for querying and exploring multivariate hierarchical data. Rooted in the extended multi-level task topology framework for tree visualizations (e-MLTT), HiRegEx delineates three query targets (node, path, and subtree) and two aspects for querying these targets (features and positions), and uses operators developed based on classical regular expressions for query construction. Based on the HiRegEx grammar, we develop an exploratory framework for querying and exploring multivariate hierarchical data and integrate it into the TreeQueryER prototype system. The exploratory framework includes three major components: top-down pattern specification, bottom-up data-driven inquiry, and context-creation data overview. We validate the expressiveness of HiRegEx with the tasks from the e-MLTT framework and showcase the utility and effectiveness of TreeQueryER system through a case study involving expert users in the analysis of a citation tree dataset., Comment: 11 pages, 8 figures, accepted at IEEE VIS 2024

A signed graph is a graph where each edge receives a sign, positive or negative. The signed graph model has been used in many real applications, such as protein complex discovery and social network analysis. Finding cohesive subgraphs in signed graphs is a fundamental problem. A k-plex is a common model for cohesive subgraphs in which every vertex is adjacent to all but at most k vertices within the subgraph. In this paper, we propose the model of size-constrained antagonistic k-plex in a signed graph. The proposed model guarantees that the resulting subgraph is a k-plex and can be divided into two sub-k-plexes, both of which have positive inner edges and negative outer edges. This paper aims to identify all maximal antagonistic k-plexes in a signed graph. Through rigorous analysis, we show that the problem is NP-Hardness. We propose a novel framework for maximal antagonistic k-plexes utilizing set enumeration. Efficiency is improved through pivot pruning and early termination based on the color bound. Preprocessing techniques based on degree and dichromatic graphs effectively narrow the search space before enumeration. Extensive experiments on real-world datasets demonstrate our algorithm's efficiency, effectiveness, and scalability.

Given a hypergraph, influence maximization (IM) is to discover a seed set containing $k$ vertices that have the maximal influence. Although the existing vertex-based IM algorithms perform better than the hyperedge-based algorithms by generating random reverse researchable (RR) sets, they are inefficient because (i) they ignore important structural information associated with hyperedges and thus obtain inferior results, (ii) the frequently-used sampling methods for generating RR sets have low efficiency because of a large number of required samplings along with high sampling variances, and (iii) the vertex-based IM algorithms have large overheads in terms of running time and memory costs. To overcome these shortcomings, this paper proposes a novel approach, called \emph{HyperIM}. The key idea behind \emph{HyperIM} is to differentiate structural information of vertices for developing stratified sampling combined with highly-efficient strategies to generate the RR sets. With theoretical guarantees, \emph{HyperIM} is able to accelerate the influence spread, improve the sampling efficiency, and cut down the expected running time. To further reduce the running time and memory costs, we optimize \emph{HyperIM} by inferring the bound of the required number of RR sets in conjunction with stratified sampling. Experimental results on real-world hypergraphs show that \emph{HyperIM} is able to reduce the number of required RR sets and running time by orders of magnitude while increasing the influence spread by up to $2.73X$ on average, compared to the state-of-the-art IM algorithms., Comment: 15 pages,10figures

Embedding visual representations within original hierarchical tables can mitigate additional cognitive load stemming from the division of users' attention. The created hierarchical table visualizations can help users understand and explore complex data with multi-level attributes. However, because of many options available for transforming hierarchical tables and selecting subsets for embedding, the design space of hierarchical table visualizations becomes vast, and the construction process turns out to be tedious, hindering users from constructing hierarchical table visualizations with many data insights efficiently. We propose InsigHTable, a mixed-initiative and insight-driven hierarchical table transformation and visualization system. We first define data insights within hierarchical tables, which consider the hierarchical structure in the table headers. Since hierarchical table visualization construction is a sequential decision-making process, InsigHTable integrates a deep reinforcement learning framework incorporating an auxiliary rewards mechanism. This mechanism addresses the challenge of sparse rewards in constructing hierarchical table visualizations. Within the deep reinforcement learning framework, the agent continuously optimizes its decision-making process to create hierarchical table visualizations to uncover more insights by collaborating with analysts. We demonstrate the usability and effectiveness of InsigHTable through two case studies and sets of experiments. The results validate the effectiveness of the deep reinforcement learning framework and show that InsigHTable can facilitate users to construct hierarchical table visualizations and understand underlying data insights.

Chain-of-thought distillation is a powerful technique for transferring reasoning abilities from large language models (LLMs) to smaller student models. Previous methods typically require the student to mimic the step-by-step rationale produced by LLMs, often facing the following challenges: (i) Tokens within a rationale vary in significance, and treating them equally may fail to accurately mimic keypoint tokens, leading to reasoning errors. (ii) They usually distill knowledge by consistently predicting all the steps in a rationale, which falls short in distinguishing the learning order of step generation. This diverges from the human cognitive progression of starting with easy tasks and advancing to harder ones, resulting in sub-optimal outcomes. To this end, we propose a unified framework, called KPOD, to address these issues. Specifically, we propose a token weighting module utilizing mask learning to encourage accurate mimicry of keypoint tokens by the student during distillation. Besides, we develop an in-rationale progressive distillation strategy, starting with training the student to generate the final reasoning steps and gradually extending to cover the entire rationale. To accomplish this, a weighted token generation loss is proposed to assess step reasoning difficulty, and a value function is devised to schedule the progressive distillation by considering both step difficulty and question diversity. Extensive experiments on four reasoning benchmarks illustrate our KPOD outperforms previous methods by a large margin., Comment: Accepted by ICML 2024

Counting small cohesive subgraphs in a graph is a fundamental operation with numerous applications in graph analysis. Previous studies on cohesive subgraph counting are mainly based on the clique model, which aim to count the number of $k$-cliques in a graph with a small $k$. However, the clique model often proves too restrictive for practical use. To address this issue, we investigate a new problem of counting cohesive subgraphs that adhere to the hereditary property. Here the hereditary property means that if a graph $G$ has a property $\mathcal{P}$, then any induced subgraph of $G$ also has a property $\mathcal{P}$. To count these hereditary cohesive subgraphs (\hcss), we propose a new listing-based framework called \hcslist, which employs a backtracking enumeration procedure to count all \hcss. A notable limitation of \hcslist is that it requires enumerating all \hcss, making it intractable for large and dense graphs due to the exponential growth in the number of \hcss with respect to graph size. To overcome this limitation, we propose a novel pivot-based framework called \hcspivot, which can count most \hcss in a combinatorial manner without explicitly listing them. Two additional noteworthy features of \hcspivot is its ability to (1) simultaneously count \hcss of any size and (2) simultaneously count \hcss for each vertex or each edge, while \hcslist is only capable of counting a specific size of \hcs and obtaining a total count of \hcss in a graph. We focus specifically on two \hcs: $s$-defective clique and $s$-plex, with several non-trivial pruning techniques to enhance the efficiency. We conduct extensive experiments on 8 large real-world graphs, and the results demonstrate the high efficiency and effectiveness of our solutions.

In this paper, we present a collection of novel and scalable algorithms designed to tackle the challenges inherent in the $k$-clique densest subgraph problem (\kcdsp) within network analysis. We propose \psctl, a novel algorithm based on the Frank-Wolfe approach for addressing \kcdsp, effectively solving a distinct convex programming problem. \textcolor{black}{\psctl is able to approximate \kcdsp with near optimal guarantees.} The notable advantage of \psctl lies in its time complexity, which is independent of the count of $k$-cliques, resulting in remarkable efficiency in practical applications. Additionally, we present \spath, a sampling-based algorithm with the capability to handle networks on an unprecedented scale, reaching up to $1.8\times 10^9$ edges. By leveraging the \ccpath algorithm as a uniform $k$-clique sampler, \spath ensures the efficient processing of large-scale network data, accompanied by a detailed analysis of accuracy guarantees. Together, these contributions represent a significant advancement in the field of $k$-clique densest subgraph discovery. In experimental evaluations, our algorithms demonstrate orders of magnitude faster performance compared to the current state-of-the-art solutions.

(p,q)-clique enumeration on a bipartite graph is critical for calculating clustering coefficient and detecting densest subgraph. It is necessary to carry out subgraph enumeration while protecting users' privacy from any potential attacker as the count of subgraph may contain sensitive information. Most recent studies focus on the privacy protection algorithms based on edge LDP (Local Differential Privacy). However, these algorithms suffer a large estimation error due to the great amount of required noise. In this paper, we propose a novel idea of k-stars LDP and a novel k-stars LDP algorithm for (p, q)-clique enumeration with a small estimation error, where a k-stars is a star-shaped graph with k nodes connecting to one node. The effectiveness of edge LDP relies on its capacity to obfuscate the existence of an edge between the user and his one-hop neighbors. This is based on the premise that a user should be aware of the existence of his one-hop neighbors. Similarly, we can apply this premise to k-stars as well, where an edge is a specific genre of 1-stars. Based on this fact, we first propose the k-stars neighboring list to enable our algorithm to obfuscate the existence of k-stars with Warner' s RR. Then, we propose the absolute value correction technique and the k-stars sampling technique to further reduce the estimation error. Finally, with the two-round user-collector interaction mechanism, we propose our k-stars LDP algorithm to count the number of (p, q)-clique while successfully protecting users' privacy. Both the theoretical analysis and experiments have showed the superiority of our algorithm over the algorithms based on edge LDP.

End-to-end motion planning models equipped with deep neural networks have shown great potential for enabling full autonomous driving. However, the oversized neural networks render them impractical for deployment on resource-constrained systems, which unavoidably requires more computational time and resources during reference.To handle this, knowledge distillation offers a promising approach that compresses models by enabling a smaller student model to learn from a larger teacher model. Nevertheless, how to apply knowledge distillation to compress motion planners has not been explored so far. In this paper, we propose PlanKD, the first knowledge distillation framework tailored for compressing end-to-end motion planners. First, considering that driving scenes are inherently complex, often containing planning-irrelevant or even noisy information, transferring such information is not beneficial for the student planner. Thus, we design an information bottleneck based strategy to only distill planning-relevant information, rather than transfer all information indiscriminately. Second, different waypoints in an output planned trajectory may hold varying degrees of importance for motion planning, where a slight deviation in certain crucial waypoints might lead to a collision. Therefore, we devise a safety-aware waypoint-attentive distillation module that assigns adaptive weights to different waypoints based on the importance, to encourage the student to accurately mimic more crucial waypoints, thereby improving overall safety. Experiments demonstrate that our PlanKD can boost the performance of smaller planners by a large margin, and significantly reduce their reference time., Comment: Accepted by CVPR 2024

Scalable graph neural networks (GNNs) have emerged as a promising technique, which exhibits superior predictive performance and high running efficiency across numerous large-scale graph-based web applications. However, (i) Most scalable GNNs tend to treat all nodes in graphs with the same propagation rules, neglecting their topological uniqueness; (ii) Existing node-wise propagation optimization strategies are insufficient on web-scale graphs with intricate topology, where a full portrayal of nodes' local properties is required. Intuitively, different nodes in web-scale graphs possess distinct topological roles, and therefore propagating them indiscriminately or neglect local contexts may compromise the quality of node representations. This intricate topology in web-scale graphs cannot be matched by small-scale scenarios. To address the above issues, we propose \textbf{A}daptive \textbf{T}opology-aware \textbf{P}ropagation (ATP), which reduces potential high-bias propagation and extracts structural patterns of each node in a scalable manner to improve running efficiency and predictive performance. Remarkably, ATP is crafted to be a plug-and-play node-wise propagation optimization strategy, allowing for offline execution independent of the graph learning process in a new perspective. Therefore, this approach can be seamlessly integrated into most scalable GNNs while remain orthogonal to existing node-wise propagation optimization strategies. Extensive experiments on 12 datasets, including the most representative large-scale ogbn-papers100M, have demonstrated the effectiveness of ATP. Specifically, ATP has proven to be efficient in improving the performance of prevalent scalable GNNs for semi-supervised node classification while addressing redundant computational costs., Comment: Accepted by WWW 2024

Graph database engines play a pivotal role in efficiently storing and managing graph data across various domains, including bioinformatics, knowledge graphs, and recommender systems. Ensuring data accuracy within graph database engines is paramount, as inaccuracies can yield unreliable analytical outcomes. Current bug-detection approaches are confined to specific graph query languages, limiting their applicabilities when handling graph database engines that use various graph query languages across various domains. Moreover, they require extensive prior knowledge to generate queries for detecting bugs. To address these challenges, we introduces DGDB, a novel paradigm harnessing large language models(LLM), such as ChatGPT, for comprehensive bug detection in graph database engines. DGDB leverages ChatGPT to generate high-quality queries for different graph query languages. It subsequently employs differential testing to identify bugs in graph database engines. We applied this paradigm to graph database engines using the Gremlin query language and those using the Cypher query language, generating approximately 4,000 queries each. In the latest versions of Neo4j, Agensgraph, and JanusGraph databases, we detected 2, 5, and 3 wrong-result bugs, respectively.

Most existing graph neural networks (GNNs) are limited to undirected graphs, whose restricted scope of the captured relational information hinders their expressive capabilities and deployments in real-world scenarios. Compared with undirected graphs, directed graphs (digraphs) fit the demand for modeling more complex topological systems by capturing more intricate relationships between nodes, such as formulating transportation and financial networks. While some directed GNNs have been introduced, their inspiration mainly comes from deep learning architectures, which lead to redundant complexity and computation, making them inapplicable to large-scale databases. To address these issues, we propose LightDiC, a scalable variant of the digraph convolution based on the magnetic Laplacian. Since topology-related computations are conducted solely during offline pre-processing, LightDiC achieves exceptional scalability, enabling downstream predictions to be trained separately without incurring recursive computational costs. Theoretical analysis shows that LightDiC utilizes directed information to achieve message passing based on the complex field, which corresponds to the proximal gradient descent process of the Dirichlet energy optimization function from the perspective of digraph signal denoising, ensuring its expressiveness. Experimental results demonstrate that LightDiC performs comparably well or even outperforms other SOTA methods in various downstream tasks, with fewer learnable parameters and higher training efficiency. Notably, LightDiC is the first DiGNN to provide satisfactory results in the most representative large-scale database (ogbn-papers100M)., Comment: Accepted by VLDB 2024

With the rapid advancement of AI applications, the growing needs for data privacy and model robustness have highlighted the importance of machine unlearning, especially in thriving graph-based scenarios. However, most existing graph unlearning strategies primarily rely on well-designed architectures or manual process, rendering them less user-friendly and posing challenges in terms of deployment efficiency. Furthermore, striking a balance between unlearning performance and framework generalization is also a pivotal concern. To address the above issues, we propose \underline{\textbf{M}}utual \underline{\textbf{E}}volution \underline{\textbf{G}}raph \underline{\textbf{U}}nlearning (MEGU), a new mutual evolution paradigm that simultaneously evolves the predictive and unlearning capacities of graph unlearning. By incorporating aforementioned two components, MEGU ensures complementary optimization in a unified training framework that aligns with the prediction and unlearning requirements. Extensive experiments on 9 graph benchmark datasets demonstrate the superior performance of MEGU in addressing unlearning requirements at the feature, node, and edge levels. Specifically, MEGU achieves average performance improvements of 2.7\%, 2.5\%, and 3.2\% across these three levels of unlearning tasks when compared to state-of-the-art baselines. Furthermore, MEGU exhibits satisfactory training efficiency, reducing time and space overhead by an average of 159.8x and 9.6x, respectively, in comparison to retraining GNN from scratch., Comment: Accepted by AAAI 2024 Oral

Federated Graph Learning (FGL) is a distributed machine learning paradigm that enables collaborative training on large-scale subgraphs across multiple local systems. Existing FGL studies fall into two categories: (i) FGL Optimization, which improves multi-client training in existing machine learning models; (ii) FGL Model, which enhances performance with complex local models and multi-client interactions. However, most FGL optimization strategies are designed specifically for the computer vision domain and ignore graph structure, presenting dissatisfied performance and slow convergence. Meanwhile, complex local model architectures in FGL Models studies lack scalability for handling large-scale subgraphs and have deployment limitations. To address these issues, we propose Federated Graph Topology-aware Aggregation (FedGTA), a personalized optimization strategy that optimizes through topology-aware local smoothing confidence and mixed neighbor features. During experiments, we deploy FedGTA in 12 multi-scale real-world datasets with the Louvain and Metis split. This allows us to evaluate the performance and robustness of FedGTA across a range of scenarios. Extensive experiments demonstrate that FedGTA achieves state-of-the-art performance while exhibiting high scalability and efficiency. The experiment includes ogbn-papers100M, the most representative large-scale graph database so that we can verify the applicability of our method to large-scale graph learning. To the best of our knowledge, our study is the first to bridge large-scale graph learning with FGL using this optimization strategy, contributing to the development of efficient and scalable FGL methods., Comment: Accepted by VLDB 2024

Recently, Federated Graph Learning (FGL) has attracted significant attention as a distributed framework based on graph neural networks, primarily due to its capability to break data silos. Existing FGL studies employ community split on the homophilous global graph by default to simulate federated semi-supervised node classification settings. Such a strategy assumes the consistency of topology between the multi-client subgraphs and the global graph, where connected nodes are highly likely to possess similar feature distributions and the same label. However, in real-world implementations, the varying perspectives of local data engineering result in various subgraph topologies, posing unique heterogeneity challenges in FGL. Unlike the well-known label Non-independent identical distribution (Non-iid) problems in federated learning, FGL heterogeneity essentially reveals the topological divergence among multiple clients, namely homophily or heterophily. To simulate and handle this unique challenge, we introduce the concept of structure Non-iid split and then present a new paradigm called \underline{Ada}ptive \underline{F}ederated \underline{G}raph \underline{L}earning (AdaFGL), a decoupled two-step personalized approach. To begin with, AdaFGL employs standard multi-client federated collaborative training to acquire the federated knowledge extractor by aggregating uploaded models in the final round at the server. Then, each client conducts personalized training based on the local subgraph and the federated knowledge extractor. Extensive experiments on the 12 graph benchmark datasets validate the superior performance of AdaFGL over state-of-the-art baselines. Specifically, in terms of test accuracy, our proposed AdaFGL outperforms baselines by significant margins of 3.24\% and 5.57\% on community split and structure Non-iid split, respectively., Comment: Accepted by ICDE 2024

Recently, graph neural networks (GNNs) have shown prominent performance in semi-supervised node classification by leveraging knowledge from the graph database. However, most existing GNNs follow the homophily assumption, where connected nodes are more likely to exhibit similar feature distributions and the same labels, and such an assumption has proven to be vulnerable in a growing number of practical applications. As a supplement, heterophily reflects dissimilarity in connected nodes, which has gained significant attention in graph learning. To this end, data engineers aim to develop a powerful GNN model that can ensure performance under both homophily and heterophily. Despite numerous attempts, most existing GNNs struggle to achieve optimal node representations due to the constraints of undirected graphs. The neglect of directed edges results in sub-optimal graph representations, thereby hindering the capacity of GNNs. To address this issue, we introduce AMUD, which quantifies the relationship between node profiles and topology from a statistical perspective, offering valuable insights for Adaptively Modeling the natural directed graphs as the Undirected or Directed graph to maximize the benefits from subsequent graph learning. Furthermore, we propose Adaptive Directed Pattern Aggregation (ADPA) as a new directed graph learning paradigm for AMUD. Empirical studies have demonstrated that AMUD guides efficient graph learning. Meanwhile, extensive experiments on 16 benchmark datasets substantiate the impressive performance of ADPA, outperforming baselines by significant margins of 3.96., Comment: Accepted by ICDE 2024

Mining cohesive subgraphs in attributed graphs is an essential problem in the domain of graph data analysis. The integration of fairness considerations significantly fuels interest in models and algorithms for mining fairness-aware cohesive subgraphs. Notably, the relative fair clique emerges as a robust model, ensuring not only comprehensive attribute coverage but also greater flexibility in distributing attribute vertices. Motivated by the strength of this model, we for the first time pioneer an investigation into the identification of the maximum relative fair clique in large-scale graphs. We introduce a novel concept of colorful support, which serves as the foundation for two innovative graph reduction techniques. These techniques effectively narrow the graph's size by iteratively removing edges that do not belong to relative fair cliques. Furthermore, a series of upper bounds of the maximum relative fair clique size is proposed by incorporating consideration of vertex attributes and colors. The pruning techniques derived from these upper bounds can significantly trim unnecessary search space during the branch-and-bound procedure. Adding to this, we present a heuristic algorithm with a linear time complexity, employing both a degree-based greedy strategy and a colored degree-based greedy strategy to identify a larger relative fair clique. This heuristic algorithm can serve a dual purpose by aiding in branch pruning, thereby enhancing overall search efficiency. Extensive experiments conducted on six real-life datasets demonstrate the efficiency, scalability, and effectiveness of our algorithms.

The pseudo-inverse of a graph Laplacian matrix, denoted as $L^\dagger$, finds extensive application in various graph analysis tasks. Notable examples include the calculation of electrical closeness centrality, determination of Kemeny's constant, and evaluation of resistance distance. However, existing algorithms for computing $L^\dagger$ are often computationally expensive when dealing with large graphs. To overcome this challenge, we propose novel solutions for approximating $L^\dagger$ by establishing a connection with the inverse of a Laplacian submatrix $L_v$. This submatrix is obtained by removing the $v$-th row and column from the original Laplacian matrix $L$. The key advantage of this connection is that $L_v^{-1}$ exhibits various interesting combinatorial interpretations. We present two innovative interpretations of $L_v^{-1}$ based on spanning trees and loop-erased random walks, which allow us to develop efficient sampling algorithms. Building upon these new theoretical insights, we propose two novel algorithms for efficiently approximating both electrical closeness centrality and Kemeny's constant. We extensively evaluate the performance of our algorithms on five real-life datasets. The results demonstrate that our novel approaches significantly outperform the state-of-the-art methods by several orders of magnitude in terms of both running time and estimation errors for these two graph analysis tasks. To further illustrate the effectiveness of electrical closeness centrality and Kemeny's constant, we present two case studies that showcase the practical applications of these metrics.

Typical Convolutional Neural Networks (ConvNets) depend heavily on large amounts of image data and resort to an iterative optimization algorithm (e.g., SGD or Adam) to learn network parameters, which makes training very time- and resource-intensive. In this paper, we propose a new training paradigm and formulate the parameter learning of ConvNets into a prediction task: given a ConvNet architecture, we observe there exist correlations between image datasets and their corresponding optimal network parameters, and explore if we can learn a hyper-mapping between them to capture the relations, such that we can directly predict the parameters of the network for an image dataset never seen during the training phase. To do this, we put forward a new hypernetwork based model, called PudNet, which intends to learn a mapping between datasets and their corresponding network parameters, and then predicts parameters for unseen data with only a single forward propagation. Moreover, our model benefits from a series of adaptive hyper recurrent units sharing weights to capture the dependencies of parameters among different network layers. Extensive experiments demonstrate that our proposed method achieves good efficacy for unseen image datasets on two kinds of settings: Intra-dataset prediction and Inter-dataset prediction. Our PudNet can also well scale up to large-scale datasets, e.g., ImageNet-1K. It takes 8967 GPU seconds to train ResNet-18 on the ImageNet-1K using GC from scratch and obtain a top-5 accuracy of 44.65%. However, our PudNet costs only 3.89 GPU seconds to predict the network parameters of ResNet-18 achieving comparable performance (44.92%), more than 2,300 times faster than the traditional training paradigm., Comment: Accepted by IEEE TIP

Graph embedding has become a powerful tool for learning latent representations of nodes in a graph. Despite its superior performance in various graph-based machine learning tasks, serious privacy concerns arise when the graph data contains personal or sensitive information. To address this issue, we investigate and develop graph embedding algorithms that satisfy local differential privacy (LDP). We introduce a novel privacy-preserving graph embedding framework, named PrivGE, to protect node data privacy. Specifically, we propose an LDP mechanism to obfuscate node data and utilize personalized PageRank as the proximity measure to learn node representations. Furthermore, we provide a theoretical analysis of the privacy guarantees and utility offered by the PrivGE framework. Extensive experiments on several real-world graph datasets demonstrate that PrivGE achieves an optimal balance between privacy and utility, and significantly outperforms existing methods in node classification and link prediction tasks., Comment: to be published in CIKM 2024

Deep learning models are usually black boxes when deployed on machine learning platforms. Prior works have shown that the attributes ($e.g.$, the number of convolutional layers) of a target black-box neural network can be exposed through a sequence of queries. There is a crucial limitation: these works assume the dataset used for training the target model to be known beforehand and leverage this dataset for model attribute attack. However, it is difficult to access the training dataset of the target black-box model in reality. Therefore, whether the attributes of a target black-box model could be still revealed in this case is doubtful. In this paper, we investigate a new problem of Domain-agnostic Reverse Engineering the Attributes of a black-box target Model, called DREAM, without requiring the availability of the target model's training dataset, and put forward a general and principled framework by casting this problem as an out of distribution (OOD) generalization problem. In this way, we can learn a domain-agnostic model to inversely infer the attributes of a target black-box model with unknown training data. This makes our method one of the kinds that can gracefully apply to an arbitrary domain for model attribute reverse engineering with strong generalization ability. Extensive experimental studies are conducted and the results validate the superiority of our proposed method over the baselines.

Knowledge distillation (KD) has shown to be effective to boost the performance of graph neural networks (GNNs), where the typical objective is to distill knowledge from a deeper teacher GNN into a shallower student GNN. However, it is often quite challenging to train a satisfactory deeper GNN due to the well-known over-parametrized and over-smoothing issues, leading to invalid knowledge transfer in practical applications. In this paper, we propose the first Free-direction Knowledge Distillation framework via reinforcement learning for GNNs, called FreeKD, which is no longer required to provide a deeper well-optimized teacher GNN. Our core idea is to collaboratively learn two shallower GNNs to exchange knowledge between them. As we observe that one typical GNN model often exhibits better and worse performances at different nodes during training, we devise a dynamic and free-direction knowledge transfer strategy that involves two levels of actions: 1) node-level action determines the directions of knowledge transfer between the corresponding nodes of two networks; and then 2) structure-level action determines which of the local structures generated by the node-level actions to be propagated. Additionally, considering that different augmented graphs can potentially capture distinct perspectives of the graph data, we propose FreeKD-Prompt that learns undistorted and diverse augmentations based on prompt learning for exchanging varied knowledge. Furthermore, instead of confining knowledge exchange within two GNNs, we develop FreeKD++ to enable free-direction knowledge transfer among multiple GNNs. Extensive experiments on five benchmark datasets demonstrate our approaches outperform the base GNNs in a large margin. More surprisingly, our FreeKD has comparable or even better performance than traditional KD algorithms that distill knowledge from a deeper and stronger teacher GNN., Comment: arXiv admin note: substantial text overlap with arXiv:2206.06561

Maximal biclique enumeration is a fundamental problem in bipartite graph data analysis. Existing biclique enumeration methods mainly focus on non-attributed bipartite graphs and also ignore the \emph{fairness} of graph attributes. In this paper, we introduce the concept of fairness into the biclique model for the first time and study the problem of fairness-aware biclique enumeration. Specifically, we propose two fairness-aware biclique models, called \nonesidebc~and \ntwosidebc~respectively. To efficiently enumerate all {\nonesidebc}s, we first present two non-trivial pruning techniques, called fair $\alpha$-$\beta$ core pruning and colorful fair $\alpha$-$\beta$ core pruning, to reduce the graph size without losing accuracy. Then, we develop a branch and bound algorithm, called \onesideFBCEM, to enumerate all single-side fair bicliques on the reduced bipartite graph. To further improve the efficiency, we propose an efficient branch and bound algorithm with a carefully-designed combinatorial enumeration technique. Note that all of our techniques can also be extended to enumerate all bi-side fair bicliques. We also extend the two fairness-aware biclique models by constraining the ratio of the number of vertices of each attribute to the total number of vertices and present corresponding enumeration algorithms. Extensive experimental results on five large real-world datasets demonstrate our methods' efficiency, effectiveness, and scalability., Comment: Accepte by ICDE 2023

Deep Neural Networks (DNNs) have been ubiquitously adopted in internet of things and are becoming an integral of our daily life. When tackling the evolving learning tasks in real world, such as classifying different types of objects, DNNs face the challenge to continually retrain themselves according to the tasks on different edge devices. Federated continual learning is a promising technique that offers partial solutions but yet to overcome the following difficulties: the significant accuracy loss due to the limited on-device processing, the negative knowledge transfer caused by the limited communication of non-IID data, and the limited scalability on the tasks and edge devices. In this paper, we propose FedKNOW, an accurate and scalable federated continual learning framework, via a novel concept of signature task knowledge. FedKNOW is a client side solution that continuously extracts and integrates the knowledge of signature tasks which are highly influenced by the current task. Each client of FedKNOW is composed of a knowledge extractor, a gradient restorer and, most importantly, a gradient integrator. Upon training for a new task, the gradient integrator ensures the prevention of catastrophic forgetting and mitigation of negative knowledge transfer by effectively combining signature tasks identified from the past local tasks and other clients' current tasks through the global model. We implement FedKNOW in PyTorch and extensively evaluate it against state-of-the-art techniques using popular federated continual learning benchmarks. Extensive evaluation results on heterogeneous edge devices show that FedKNOW improves model accuracy by 63.24% without increasing model training time, reduces communication cost by 34.28%, and achieves more improvements under difficult scenarios such as large numbers of tasks or clients, and training different complex networks.

Tabular visualization techniques integrate visual representations with tabular data to avoid additional cognitive load caused by splitting users' attention. However, most of the existing studies focus on simple flat tables instead of hierarchical tables, whose complex structure limits the expressiveness of visualization results and affects users' efficiency in visualization construction. We present HiTailor, a technique for presenting and exploring hierarchical tables. HiTailor constructs an abstract model, which defines row/column headings as biclustering and hierarchical structures. Based on our abstract model, we identify three pairs of operators, Swap/Transpose, ToStacked/ToLinear, Fold/Unfold, for transformations of hierarchical tables to support users' comprehensive explorations. After transformation, users can specify a cell or block of interest in hierarchical tables as a TableUnit for visualization, and HiTailor recommends other related TableUnits according to the abstract model using different mechanisms. We demonstrate the usability of the HiTailor system through a comparative study and a case study with domain experts, showing that HiTailor can present and explore hierarchical tables from different viewpoints. HiTailor is available at https://github.com/bitvis2021/HiTailor.

Graph structured data often possess dynamic characters in nature. Recent years have witnessed the increasing attentions paid to dynamic graph neural networks for modelling graph data. However, almost all existing approaches operate under the assumption that, upon the establishment of a new link, the embeddings of the neighboring nodes should undergo updates to learn temporal dynamics. Nevertheless, these approaches face the following limitation: If the node introduced by a new connection contains noisy information, propagating its knowledge to other nodes becomes unreliable and may even lead to the collapse of the model. In this paper, we propose Ada-DyGNN: a robust knowledge Adaptation framework via reinforcement learning for Dynamic Graph Neural Networks. In contrast to previous approaches, which update the embeddings of the neighbor nodes immediately after adding a new link, Ada-DyGNN adaptively determines which nodes should be updated. Considering that the decision to update the embedding of one neighbor node can significantly impact other neighbor nodes, we conceptualize the node update selection as a sequence decision problem and employ reinforcement learning to address it effectively. By this means, we can adaptively propagate knowledge to other nodes for learning robust node embedding representations. To the best of our knowledge, our approach constitutes the first attempt to explore robust knowledge adaptation via reinforcement learning specifically tailored for dynamic graph neural networks. Extensive experiments on three benchmark datasets demonstrate that Ada-DyGNN achieves the state-of-the-art performance. In addition, we conduct experiments by introducing different degrees of noise into the dataset, quantitatively and qualitatively illustrating the robustness of Ada-DyGNN., Comment: 14 pages, 6 figures

Blind Face Restoration (BFR) aims to recover high-quality face images from low-quality ones and usually resorts to facial priors for improving restoration performance. However, current methods still suffer from two major difficulties: 1) how to derive a powerful network architecture without extensive hand tuning; 2) how to capture complementary information from multiple facial priors in one network to improve restoration performance. To this end, we propose a Face Restoration Searching Network (FRSNet) to adaptively search the suitable feature extraction architecture within our specified search space, which can directly contribute to the restoration quality. On the basis of FRSNet, we further design our Multiple Facial Prior Searching Network (MFPSNet) with a multi-prior learning scheme. MFPSNet optimally extracts information from diverse facial priors and fuses the information into image features, ensuring that both external guidance and internal features are reserved. In this way, MFPSNet takes full advantage of semantic-level (parsing maps), geometric-level (facial heatmaps), reference-level (facial dictionaries) and pixel-level (degraded images) information and thus generates faithful and realistic images. Quantitative and qualitative experiments show that MFPSNet performs favorably on both synthetic and real-world datasets against the state-of-the-art BFR methods. The codes are publicly available at: https://github.com/YYJ1anG/MFPSNet., Comment: We found some problems with the article and need to withdrawal it

Knowledge distillation (KD) has demonstrated its effectiveness to boost the performance of graph neural networks (GNNs), where its goal is to distill knowledge from a deeper teacher GNN into a shallower student GNN. However, it is actually difficult to train a satisfactory teacher GNN due to the well-known over-parametrized and over-smoothing issues, leading to invalid knowledge transfer in practical applications. In this paper, we propose the first Free-direction Knowledge Distillation framework via Reinforcement learning for GNNs, called FreeKD, which is no longer required to provide a deeper well-optimized teacher GNN. The core idea of our work is to collaboratively build two shallower GNNs in an effort to exchange knowledge between them via reinforcement learning in a hierarchical way. As we observe that one typical GNN model often has better and worse performances at different nodes during training, we devise a dynamic and free-direction knowledge transfer strategy that consists of two levels of actions: 1) node-level action determines the directions of knowledge transfer between the corresponding nodes of two networks; and then 2) structure-level action determines which of the local structures generated by the node-level actions to be propagated. In essence, our FreeKD is a general and principled framework which can be naturally compatible with GNNs of different architectures. Extensive experiments on five benchmark datasets demonstrate our FreeKD outperforms two base GNNs in a large margin, and shows its efficacy to various GNNs. More surprisingly, our FreeKD has comparable or even better performance than traditional KD algorithms that distill knowledge from a deeper and stronger teacher GNN., Comment: Accepted to KDD 2022

Blind Face Restoration (BFR) aims to construct a high-quality (HQ) face image from its corresponding low-quality (LQ) input. Recently, many BFR methods have been proposed and they have achieved remarkable success. However, these methods are trained or evaluated on privately synthesized datasets, which makes it infeasible for the subsequent approaches to fairly compare with them. To address this problem, we first synthesize two blind face restoration benchmark datasets called EDFace-Celeb-1M (BFR128) and EDFace-Celeb-150K (BFR512). State-of-the-art methods are benchmarked on them under five settings including blur, noise, low resolution, JPEG compression artifacts, and the combination of them (full degradation). To make the comparison more comprehensive, five widely-used quantitative metrics and two task-driven metrics including Average Face Landmark Distance (AFLD) and Average Face ID Cosine Similarity (AFICS) are applied. Furthermore, we develop an effective baseline model called Swin Transformer U-Net (STUNet). The STUNet with U-net architecture applies an attention mechanism and a shifted windowing scheme to capture long-range pixel interactions and focus more on significant features while still being trained efficiently. Experimental results show that the proposed baseline method performs favourably against the SOTA methods on various BFR tasks.

In this paper, we explore the problem of deep multi-view subspace clustering framework from an information-theoretic point of view. We extend the traditional information bottleneck principle to learn common information among different views in a self-supervised manner, and accordingly establish a new framework called Self-supervised Information Bottleneck based Multi-view Subspace Clustering (SIB-MSC). Inheriting the advantages from information bottleneck, SIB-MSC can learn a latent space for each view to capture common information among the latent representations of different views by removing superfluous information from the view itself while retaining sufficient information for the latent representations of other views. Actually, the latent representation of each view provides a kind of self-supervised signal for training the latent representations of other views. Moreover, SIB-MSC attempts to learn the other latent space for each view to capture the view-specific information by introducing mutual information based regularization terms, so as to further improve the performance of multi-view subspace clustering. To the best of our knowledge, this is the first work to explore information bottleneck for multi-view subspace clustering. Extensive experiments on real-world multi-view data demonstrate that our method achieves superior performance over the related state-of-the-art methods.

Domain adaptive semantic segmentation attempts to make satisfactory dense predictions on an unlabeled target domain by utilizing the supervised model trained on a labeled source domain. In this work, we propose Semantic-Guided Pixel Contrast (SePiCo), a novel one-stage adaptation framework that highlights the semantic concepts of individual pixels to promote learning of class-discriminative and class-balanced pixel representations across domains, eventually boosting the performance of self-training methods. Specifically, to explore proper semantic concepts, we first investigate a centroid-aware pixel contrast that employs the category centroids of the entire source domain or a single source image to guide the learning of discriminative features. Considering the possible lack of category diversity in semantic concepts, we then blaze a trail of distributional perspective to involve a sufficient quantity of instances, namely distribution-aware pixel contrast, in which we approximate the true distribution of each semantic category from the statistics of labeled source data. Moreover, such an optimization objective can derive a closed-form upper bound by implicitly involving an infinite number of (dis)similar pairs, making it computationally efficient. Extensive experiments show that SePiCo not only helps stabilize training but also yields discriminative representations, making significant progress on both synthetic-to-real and daytime-to-nighttime adaptation scenarios., Comment: Accepted by IEEE Transactions on Pattern Analysis and Machine Intelligence (T-PAMI). Code is available at https://github.com/BIT-DA/SePiCo

Finding all maximal $k$-plexes on networks is a fundamental research problem in graph analysis due to many important applications, such as community detection, biological graph analysis, and so on. A $k$-plex is a subgraph in which every vertex is adjacent to all but at most $k$ vertices within the subgraph. In this paper, we study the problem of enumerating all large maximal $k$-plexes of a graph and develop several new and efficient techniques to solve the problem. Specifically, we first propose several novel upper-bounding techniques to prune unnecessary computations during the enumeration procedure. We show that the proposed upper bounds can be computed in linear time. Then, we develop a new branch-and-bound algorithm with a carefully-designed pivot re-selection strategy to enumerate all $k$-plexes, which outputs all $k$-plexes in $O(n^2\gamma_k^n)$ time theoretically, where $n$ is the number of vertices of the graph and $\gamma_k$ is strictly smaller than 2. In addition, a parallel version of the proposed algorithm is further developed to scale up to process large real-world graphs. Finally, extensive experimental results show that the proposed sequential algorithm can achieve up to $2\times$ to $100\times$ speedup over the state-of-the-art sequential algorithms on most benchmark graphs. The results also demonstrate the high scalability of the proposed parallel algorithm. For example, on a large real-world graph with more than 200 million edges, our parallel algorithm can finish the computation within two minutes, while the state-of-the-art parallel algorithm cannot terminate within 24 hours.

Deep neural networks (DNNs) have become ubiquitous techniques in mobile and embedded systems for applications such as image/object recognition and classification. The trend of executing multiple DNNs simultaneously exacerbate the existing limitations of meeting stringent latency/accuracy requirements on resource constrained mobile devices. The prior art sheds light on exploring the accuracy-resource tradeoff by scaling the model sizes in accordance to resource dynamics. However, such model scaling approaches face to imminent challenges: (i) large space exploration of model sizes, and (ii) prohibitively long training time for different model combinations. In this paper, we present LegoDNN, a lightweight, block-grained scaling solution for running multi-DNN workloads in mobile vision systems. LegoDNN guarantees short model training times by only extracting and training a small number of common blocks (e.g. 5 in VGG and 8 in ResNet) in a DNN. At run-time, LegoDNN optimally combines the descendant models of these blocks to maximize accuracy under specific resources and latency constraints, while reducing switching overhead via smart block-level scaling of the DNN. We implement LegoDNN in TensorFlow Lite and extensively evaluate it against state-of-the-art techniques (FLOP scaling, knowledge distillation and model compression) using a set of 12 popular DNN models. Evaluation results show that LegoDNN provides 1,296x to 279,936x more options in model sizes without increasing training time, thus achieving as much as 31.74% improvement in inference accuracy and 71.07% reduction in scaling energy consumptions., Comment: 13 pages, 15 figures

Domain adaptation (DA) attempts to transfer the knowledge from a labeled source domain to an unlabeled target domain that follows different distribution from the source. To achieve this, DA methods include a source classification objective to extract the source knowledge and a domain alignment objective to diminish the domain shift, ensuring knowledge transfer. Typically, former DA methods adopt some weight hyper-parameters to linearly combine the training objectives to form an overall objective. However, the gradient directions of these objectives may conflict with each other due to domain shift. Under such circumstances, the linear optimization scheme might decrease the overall objective value at the expense of damaging one of the training objectives, leading to restricted solutions. In this paper, we rethink the optimization scheme for DA from a gradient-based perspective. We propose a Pareto Domain Adaptation (ParetoDA) approach to control the overall optimization direction, aiming to cooperatively optimize all training objectives. Specifically, to reach a desirable solution on the target domain, we design a surrogate loss mimicking target classification. To improve target-prediction accuracy to support the mimicking, we propose a target-prediction refining mechanism which exploits domain labels via Bayes' theorem. On the other hand, since prior knowledge of weighting schemes for objectives is often unavailable to guide optimization to approach the optimal solution on the target domain, we propose a dynamic preference mechanism to dynamically guide our cooperative optimization by the gradient of the surrogate loss on a held-out unlabeled target dataset. Extensive experiments on image classification and semantic segmentation benchmarks demonstrate the effectiveness of ParetoDA, Comment: Accepted in NeurIPS 2021

Unsupervised domain adaptation has recently emerged as an effective paradigm for generalizing deep neural networks to new target domains. However, there is still enormous potential to be tapped to reach the fully supervised performance. In this paper, we present a novel active learning strategy to assist knowledge transfer in the target domain, dubbed active domain adaptation. We start from an observation that energy-based models exhibit \textit{free energy biases} when training (source) and test (target) data come from different distributions. Inspired by this inherent mechanism, we empirically reveal that a simple yet efficient energy-based sampling strategy sheds light on selecting the most valuable target samples than existing approaches requiring particular architectures or computation of the distances. Our algorithm, Energy-based Active Domain Adaptation (EADA), queries groups of target data that incorporate both domain characteristic and instance uncertainty into every selection round. Meanwhile, by aligning the free energy of target data compact around the source domain via a regularization term, domain gap can be implicitly diminished. Through extensive experiments, we show that EADA surpasses state-of-the-art methods on well-known challenging benchmarks with substantial improvements, making it a useful option in the open world. Code is available at https://github.com/BIT-DA/EADA., Comment: Camera ready for AAAI 2022. Code is available at https://github.com/BIT-DA/EADA

Recently deep learning has been successfully applied to unsupervised active learning. However, current method attempts to learn a nonlinear transformation via an auto-encoder while ignoring the sample relation, leaving huge room to design more effective representation learning mechanisms for unsupervised active learning. In this paper, we propose a novel deep unsupervised Active Learning model via Learnable Graphs, named ALLG. ALLG benefits from learning optimal graph structures to acquire better sample representation and select representative samples. To make the learnt graph structure more stable and effective, we take into account $k$-nearest neighbor graph as a priori, and learn a relation propagation graph structure. We also incorporate shortcut connections among different layers, which can alleviate the well-known over-smoothing problem to some extent. To the best of our knowledge, this is the first attempt to leverage graph structure learning for unsupervised active learning. Extensive experiments performed on six datasets demonstrate the efficacy of our method.

Deep subspace clustering has attracted increasing attention in recent years. Almost all the existing works are required to load the whole training data into one batch for learning the self-expressive coefficients in the framework of deep learning. Although these methods achieve promising results, such a learning fashion severely prevents from the usage of deeper neural network architectures (e.g., ResNet), leading to the limited representation abilities of the models. In this paper, we propose a new deep subspace clustering framework, motivated by the energy-based models. In contrast to previous approaches taking the weights of a fully connected layer as the self-expressive coefficients, we propose to learn an energy-based network to obtain the self-expressive coefficients by mini-batch training. By this means, it is no longer necessary to load all data into one batch for learning, and it thus becomes a reality that we can utilize deeper neural network models for subspace clustering. Considering the powerful representation ability of the recently popular self-supervised learning, we attempt to leverage self-supervised representation learning to learn the dictionary. Finally, we propose a joint framework to learn both the self-expressive coefficients and dictionary simultaneously, and train the model in an end-to-end manner. The experiments are performed on three publicly available datasets, and extensive experimental results demonstrate our method can significantly outperform the other related approaches. For instance, on the three datasets, our method can averagely achieve $13.8\%$, $15.4\%$, $20.8\%$ improvements in terms of Accuracy, NMI, and ARI over SENet which is proposed very recently and obtains the second best results in the experiments., Comment: There are some errors in Table 1

Causal inference is to estimate the causal effect in a causal relationship when intervention is applied. Precisely, in a causal model with binary interventions, i.e., control and treatment, the causal effect is simply the difference between the factual and counterfactual. The difficulty is that the counterfactual may never been obtained which has to be estimated and so the causal effect could only be an estimate. The key challenge for estimating the counterfactual is to identify confounders which effect both outcomes and treatments. A typical approach is to formulate causal inference as a supervised learning problem and so counterfactual could be predicted. Including linear regression and deep learning models, recent machine learning methods have been adapted to causal inference. In this paper, we propose a method to estimate Causal Effect by using Variational Information Bottleneck (CEVIB). The promising point is that VIB is able to naturally distill confounding variables from the data, which enables estimating causal effect by using observational data. We have compared CEVIB to other methods by applying them to three data sets showing that our approach achieved the best performance. We also experimentally showed the robustness of our method.

Betweenness centrality, measured by the number of times a vertex occurs on all shortest paths of a graph, has been recognized as a key indicator for the importance of a vertex in the network. However, the betweenness of a vertex is often very hard to compute because it needs to explore all the shortest paths between the other vertices. Recently, a relaxed concept called ego-betweenness was introduced which focuses on computing the betweenness of a vertex in its ego network. In this work, we study a problem of finding the top-k vertices with the highest ego-betweennesses. We first develop two novel search algorithms equipped with a basic upper bound and a dynamic upper bound to efficiently solve this problem. Then, we propose local-update and lazy-update solutions to maintain the ego-betweennesses for all vertices and the top-k results when the graph is updated, respectively. In addition, we also present two efficient parallel algorithms to further improve the efficiency. The results of extensive experiments on five large real-life datasets demonstrate the efficiency, scalability, and effectiveness of our algorithms.

Cohesive subgraph mining on attributed graphs is a fundamental problem in graph data analysis. Existing cohesive subgraph mining algorithms on attributed graphs do not consider the fairness of attributes in the subgraph. In this paper, we, for the first time, introduce fairness into the widely-used clique model to mine fairness-aware cohesive subgraphs. In particular, we propose three novel fairness-aware maximal clique models on attributed graphs, called weak fair clique, strong fair clique and relative fair clique, respectively. To enumerate all weak fair cliques, we develop an efficient backtracking algorithm called WFCEnum equipped with a novel colorful k-core based pruning technique. We also propose an efficient enumeration algorithm called SFCEnum to find all strong fair cliques based on a new attribute-alternatively-selection search technique. To further improve the efficiency, we also present several non-trivial ordering techniques for both weak and strong fair clique enumerations. To enumerate all relative fair cliques, we design an enhanced colorful k-core based pruning technique for 2D attribute, and then develop two efficient search algorithms: RFCRefineEnum and RFCAlterEnum based on the ideas of WFCEnum and SFCEnum for arbitrary dimension attribute. The results of extensive experiments on four real-world graphs demonstrate the efficiency, scalability and effectiveness of the proposed algorithms.