For complex networks such as the Web, the Network Dismantling (ND) problem, which asks for the minimum-cost removal of nodes that destroys the giant connected component in the network, is significant in system robustness and misinformation containment. In this paper, we propose a heuristic algorithm, IG+, which is based on reverse dismantling and incorporates novel optimizations. Besides, we design two dynamic algorithms, CCRT-ins and CCRT-rem, employing tree-like indexes to update dismantling results efficiently. Experiments show that our methods outperform state-of-the-art approaches in both effectiveness and efficiency, and can dismantle 10-million-scale networks at arbitrary granularity in a few minutes.

Traffic prediction serves as a cornerstone for systems and network services such as the Web of Vehicles (WoV), online navigation, and smart city applications. Despite the proliferation of model architectures in recent years, existing approaches often suffer from highly customized structures and weak transferability, making it difficult to cope with increasing task heterogeneity and modeling complexity. To address these challenges, we propose ST-LEGO, a modular assembly framework driven by large language models (LLMs) that supports flexible structural composition and automated code generation. ST-LEGO employs a multi-agent collaborative system comprising a Prompt Agent, Assemble Agent, and Code Agent, which are responsible for understanding task requirements, dynamically assembling structural modules, and automatically generating executable PyTorch code. By introducing a standardized module library and an intermediate structural description language (DSL), the framework enables controllable generation, reusable composition, and cross-task generalization of model architectures. Empirical results on multiple real-world traffic datasets demonstrate that models generated by ST-LEGO achieve superior accuracy, structural diversity, and convergence compared to a wide range of manually designed baselines. These results highlight the unique potential and scalability of LLMs as structural architects for traffic prediction, offering a new paradigm for integrating language models into web-interactive intelligent transportation systems.

Timely detection of anomalous interactions between different entities is crucial for the security and stability of Web-related systems, which are often modeled as streaming graphs. A major issue in existing anomaly detection methods is the effectiveness and the corresponding interpretability. In this paper, we introduce a novel evaluation metric, namely Interaction Willingness, to measure the propensity for entity interactions. Based on this metric, we design two efficient anomaly detection algorithms, AnoD and AnoC, tailored for real-time detection of two prevalent types of anomalous interaction behaviors (dense-type and counter-type), respectively. Notably, we adopt and extend the High-Order Count-Min Sketch for the implicit storage of streaming graphs, and ensure that both the space cost and the time cost of AnoD and AnoC for processing each new streaming edge remain constant and user-controllable, with estimation guarantees. Experimental evaluations on 6 real-world datasets demonstrate that the proposed evaluation metric and the associated algorithms achieve superior detection performance with clear interpretability for detecting anomalous interactions.

The$k$-truss is a popular cohesive subgraph model for graph analysis, which requires each edge in the subgraph to be contained in at least$k-2$triangles, each consists three pairwisely connected edges. In this paper, we study the$k$-truss breaking problem (TBP) that aims to find the smallest set of edges whose removal makes the graph free of$k$-truss. The problem has been formulated in the literature with applications in community deception, critical connection identification, etc. However, existing solutions cannot scale to large graphs. We observe that chosen edges in a high-quality solution usually have high triangle support, while most share triangles with a significant number of easy-breaking edges (i.e., low-support edges). Motivated by these, we propose the Easy-Breaking Heuristic (EBH) that prioritizes the candidate edges based on their impact on easy-breaking edges. We also design several optimizations to further enhance the performance of EBH. Additionally, we extend our framework to efficiently handle the$k$-truss minimization problem (TMP), which aims to identify a set of at most$b$edges whose removal minimizes the size of the remaining k-truss. Extensive experiments demonstrate that our proposed algorithm outperforms state-of-the-art approaches by up to three orders of magnitude in efficiency when solving TBP, while maintaining comparable effectiveness. Additionally, our proposed algorithm achieves up to four orders of magnitude improvement in efficiency for TMP, along with generally better effectiveness.

Fine-grained traffic management and prediction are fundamental to key applications such as autonomous driving, lane change guidance, and traffic signal control. However, obtaining lane-level traffic data has become a critical bottleneck for data-driven models due to limitations in the types and number of sensors and issues with the accuracy of tracking algorithms. To address this, we propose the Fine-grained Road Traffic Inference (FRTI) task, which aims to generate more detailed lane-level traffic information using limited road data, providing a more energy-efficient and cost-effective solution for precise traffic management. This task is abstracted as the first scene of the spatio-temporal graph node generation problem. We designed a two-stage framework-RoadDiff-to solve the FRTI task. This framework leverages the Road-Lane Correlation Autoencoder-Decoder and the Lane Diffusion Module to fully utilize the limited spatio-temporal dependencies and distribution relationships of road data to accurately infer fine-grained lane traffic states. Based on existing research, we designed several baseline models with the potential to solve the FRTI task and conducted extensive experiments on six datasets representing different road conditions to validate the effectiveness of the RoadDiff model in addressing the FRTI task. The relevant datasets and code are available at https://github.com/ShuhaoLii/RoadDiff.

All known AridViper tools including MICROPSIA

With the increasing complexity of urban transportation systems and the growing demand for dynamic, real-time responsiveness, Time-Dependent Minimum Travel Time Queries (TD-MTTQs) in time-dependent road networks have become a core challenge in intelligent transportation system research. To address the trade-offs between preprocessing complexity and query efficiency in existing index-based methods for large-scale road network applications, this paper proposes a 4-hop index method, TD-TNR-CH. The core methodology involves establishing local indexes from each node to its nearest critical nodes (named transit nodes) through strategic critical node selection, while simultaneously constructing query tables associated with candidate sets between these critical nodes. This architecture enables rapid computation of medium-to-long distance queries through efficient index lookups, while ensuring high responsiveness for short-distance queries via TCH-based local searches. Extensive experimental results on large-scale real-world road networks demonstrate that our method exhibits superior scalability, achieving query efficiency of up to 103 times that of the fastest existing algorithms. Furthermore, it shows exceptional stability across queries of varying distances. Additionally, leveraging its parallelized architecture, TD-TNR-CH requires only approximately 30 minutes of preprocessing time for large-scale networks, significantly outperforming comparable methods in terms of preprocessing efficiency.

Traffic prediction has long been a focal and pivotal area in research, witnessing both significant strides from city-level to road-level predictions in recent years. With the advancement of Vehicle-to-Everything (V2X) technologies, autonomous driving, and large-scale models in the traffic domain, lane-level traffic prediction has emerged as an indispensable direction. However, further progress in this field is hindered by the absence of comprehensive and unified evaluation standards, coupled with limited public availability of data and code. In this paper, we present the first systematic classification framework for lane-level traffic prediction, offering a structured taxonomy and analysis of existing methods. We construct three representative datasets from two real-world road networks, covering both regular and irregular lane configurations, and make them publicly available to support future research. We further establishes a unified spatial topology structure and prediction task formulation, and proposes a simple yet effective baseline model, GraphMLP, based on graph structure and MLP networks. This unified framework enables consistent evaluation across datasets and modeling paradigms. We also reproduce previously unavailable code from existing studies and conduct extensive experiments to assess a range of models in terms of accuracy, efficiency, and applicability, providing the first benchmark that jointly considers predictive performance and training cost for lane-level traffic scenarios. All datasets and code are released athttps://github.com/ShuhaoLii/LaneLevel-Traffic-Benchmark.

Lane-level traffic prediction is crucial for refined smart city applications, yet the scarcity and quality issues of datasets hinder its development. To overcome these challenges, this study introduces a novel M ulti- c hannel g raph-structured V ariational A uto E ncoder model, McgVAE. This model integrates road-level information to provide a global perspective for lane prediction and performs integrated tasks through three interconnected channels: the road-level channel ensures accurate prediction of road traffic states and communicates closely with the data quality channel to share historical and predicted road information; the data quality channel leverages road-level information to identify and correct missing and noisy data; and finally, the lane channel uses the aforementioned information for lane-level traffic prediction. After extensive experimental comparisons with multiple baseline models across three datasets, the McgVAE model demonstrated outstanding predictive performance and the ability to handle data missingness and noise

With the widespread application of intelligent transportation systems and navigation software, traffic prediction should be modeled in finer granularity to facilitate lane-changing guidance and congestion mitigation. However, existing studies divide the road network into continuous segments which assumes different lanes share the same spatio-temporal patterns. This paper proposes a novel lightweight, attention-based, fully convolutional model, named the Spatio-Temporal Attention- Based Convolutional network (ST-ABC), where lane segments are treated as graph nodes and dynamically models the adjacent spatial dependencies using local attention graph convolution. The attention-based dilated convolutions can process longer sequence periods in parallel, and a global attention layer allows individual nodes to be associated with the global context. By setting a target window, it can further reduce unnecessary computations and improve the prediction effect for the targeted area. Further-more, the ST-ABC model facilitates the simultaneous integration of spatio-temporal information and relational distance metrics among lane segments, enriching the granularity of multi-scaled spatial prediction. Empirical evaluations conducted on two real-world datasets substantiate the augmented efficacy of the STABC model in comparison to established models, with a marked prominence in long-term prediction scenarios.

Unsupervised Outlier Detection (UOD) is an important data mining task. With the advance of deep learning, deep Outlier Detection (OD) has received broad interest. Most deep UOD models are trained exclusively on clean datasets to learn the distribution of the normal data, which requires huge manual efforts to clean the real-world data if possible. Instead of relying on clean datasets, some approaches directly train and detect on unlabeled contaminated datasets, leading to the need for methods that are robust to such challenging conditions. Ensemble methods emerged as a superior solution to enhance model robustness against contaminated training sets. However, the training time is greatly increased by the ensemble mechanism.

Network resilience is a critical ability of a network to maintain its functionality against disturbances. A network is resilient/robust when a large portion of the nodes are to be better engaged in the network, i.e., they are less likely to leave given the changes on the network. Existing studies validate that the engagement of a node can be well captured by its coreness on network topology. Therefore, it is promising to maximize the number of nodes with increasing coreness values. In this paper, we propose and study thefollower maximization problem: maximizing the resilience gain (the number of coreness-increased vertices) via anchoring a set of vertices within a given budget. We prove that the problem is NP-hard and W[2]-hard, and it is NP-hard to approximate within an O(n^1-ε ) factor. We first propose an advanced greedy approach, followed by a time-dependent framework designed to quickly find high-quality results. The framework is initialized by the advanced greedy algorithm and incorporates novel techniques for optimizing the search space. The effectiveness and efficiency of our solution are verified with extensive experiments on 8 real-life datasets. Our source codes are available at https://github.com/Tsyxxxka/Follower-Maximization.

As the volume and ubiquity of graphs increase, a compact graph representation becomes essential for enabling efficient storage, transfer, and processing of graphs. Given a graph, the graph summarization problem asks for a compact representation that consists of a summary graph and the corrections, such that we can recreate the original graph from the representation exactly. Although this problem has been studied extensively, the existing works either trade summary compactness for efficiency, or vice versa. In particular, a well-known greedy method provides the most compact summary but incurs prohibitive time cost, while the state-of-the-art algorithms with practical overheads are more than 20% behind in summary compactness in our comparison with the greedy method. This paper presents Mags and Mags-DM, two algorithms that aim to bridge the compactness and efficiency in graph summarization. Mags adopts the existing greedy paradigm that provides state-of-the-art compactness, but significantly improves its efficiency with a novel algorithm design. Meanwhile, Mags-DM follows a different paradigm with practical efficiency and overcomes its limitations in compactness. Moreover, both algorithms can support parallel computing environments. We evaluate Mags and Mags-DM on graphs up to billion-scale and demonstrate that they achieve state-of-the-art in both compactness and efficiency, rather than in one of them. Compared with the method that offers state-of-the-art compactness, Mags and Mags-DM have a small difference (< 0.1% and < 2.1%) in compactness. For efficiency, Mags is on average 11.1x and 4.2x faster than the two state-of-the-art algorithms with practical overheads, while Mags-DM can further reduce the running time by 13.4x compared with Mags. This shows that graph summarization algorithms can be made practical while still offering a compact summary.

The biclique is a fundamental model of bipartite cohesive subgraphs. To analyze a bipartite graph, many existing works seek the maximum biclique, that is, the biclique with the largest number of edges. However, our finding is that the most similar biclique (i.e., the biclique whose vertices are the most similar to each other) can be a good alternative for understanding the network. Using the model, we can detect meaningful communities with high similarity and avoid unnecessary searches based on vertex similarity. In particular, we aim to find (i) local most similar biclique : the biclique that contains a query node q and the similarity between vertices is the highest, and (ii) global most similar biclique : the biclique with the highest similarity between vertices. Despite the NP-hardness of the problems, this paper presents two efficient algorithms, Mosib and Mosib-GloApp. Specifically, our Mosib is an exact algorithm for the most similar biclique search. The algorithm incorporates three novel graph reduction rules that can reduce the size of the bipartite graph while preserving the most similar biclique, as well as two similarity-first search rules that can prioritize the bicliques with high similarity in the search. These techniques can significantly improve the practical efficiency of the algorithm. Meanwhile, our Mosib-GloApp is an approximate algorithm that adopts a novel MinHash-based dividing method, and it can further improve the efficiency of the global most similar biclique search. We experimentally evaluate our algorithms on real-world networks, and show that the most similar biclique models can find meaningful results while being computed efficiently.

There are two fundamental problems in regular simple path queries (RSPQs). One is the reachability problem which asks whether there exists a simple path between the source and the target vertex matching the given regular expression, and the other is the enumeration problem which aims to find all the matched simple paths. As an important computing component of graph databases, RSPQs are supported in many graph database query languages such as PGQL and openCypher. However, answering RSPQs is known to be NP-hard, making it challenging to design scalable solutions to support a wide range of expressions. In this paper, we first introduce the class of transitive restricted expression , which covers more than 99% of real-world queries. Then, we propose an efficient algorithm framework to support both reachability and enumeration problems under transitive restricted expression constraints. To boost the performance, we develop novel techniques for reachability detection, the search of candidate vertices, and the reduction of redundant path computation. Extensive experiments demonstrate that our exact method can achieve comparable efficiency to the state-of-the-art approximate approach, and outperforms the state-of-the-art exact methods by up to 2 orders of magnitude.

Competitive viral marketing considers the product competition of multiple companies, where each user may adopt one product and propagate the product to other users. Existing studies focus on a traditional seeding strategy where a company only selects seeds from the users with no adopted product to maximize its influence (i.e., the number of users who will adopt its product). However, influential users are often rare, and the gain from traditional seeding will degrade as the number of seeds increases. Therefore, in this paper, we study the promisingcounteringstrategy which is to counter some users who initially use other products s.t. they will turn to adopting the target product and recommending it to others. We propose the problem ofinfluence countering: given a graph, a budgetb, a target companyCt, and a setSof the seeds adopting different companies (where each seed adopts one company), we counterbusers inSwho do not adoptCtto turn to adoptCts.t. the expected number of users who eventually adoptCtin the influence diffusion is maximized. Following existing studies, we formalize the diffusion process by the Multi-Campaigner Independent Cascade model. We prove the influence countering problem is #P-complete and its influence computation is #P-hard. Then, we propose two novel algorithmsMICandMIC+to address the problem. In general,MICestimates seed influence by its empirical average influence in multiple graph samplings, whileMIC+improvesMICby reducing the cost of influence estimation and the required number of samples. Given pre-setεandl, both algorithms return a (1 -ε)-approximate solution with at least 1 -n-lprobability. We also design an index forMIC+to efficiently process graphs that are frequently updated. The experiments on 8 real-world datasets show that our algorithms are efficient in practice while offering strong result quality.

The mode of$k$-core and its hierarchical decomposition have been applied in many areas, such as sociology, the world wide web, and biology. Algorithms on related studies often need an input value of parameter$k$, while there is no existing solution other than manual selection. In this paper, given a graph and a scoring metric, we aim to find the best value of$k$such that the score of the$k$-core (or$k$-core set) is the highest. The problem is challenging because there are various community scoring metrics and the computation is costly on large datasets. With the well-designed vertex ordering, we propose time-and-space-optimal algorithms to compute the best$k$, which are applicable to most community metrics. As real-world networks are often fast-evolving, we also design a novel framework to maintain the best$k$-core (set) against graph dynamics. We prove the dynamic algorithms are bounded, i.e., the update cost is decided by the changes of input and output. The proposed algorithms can benefit the solutions to$k$-core-related problems and their dynamic counterparts. Extensive experiments are conducted on 10 real-world networks with size up to billion-scale, which validates the efficiency of our algorithms and the effectiveness of the resulting$k$-cores.

Computing Multi-source Shortest Path query with Assembly points ($\mathsf {MSPA}$) is a fundamental graph problem. The$\mathsf {MSPA}$problem locates a set of assembly points to minimize the overall distance for transporting objects from different sources to a destination, where we can assemble objects at assembly points to reduce the total cost. We prove that the$\mathsf {MSPA}$problem is NP-hard. The intuitive method for computing the optimal set of assembly points and the corresponding set of paths is by Branch-and-Bound. However, the combination of different assembly points is exponential. By analyzing the structure of the path set based on the proposed distance graph, we find that the used paths can be combined into a tree. Hence, by defining the state of subtrees and the state transition equation, we propose a dynamic programming (DP) algorithm by pruning the redundant computation of subtrees. The experiment shows that the DP algorithm can achieve three orders of magnitude speedup in query processing time compared with the optimized Branch-and-Bound algorithm. Moreover, we reduce the transition candidates of the DP algorithm from the entire vertex set to certain neighbors. Extensive experiments are conducted on different types of real-world networks to demonstrate the performance of our DP algorithm.

As a fundamental graph problem, community search is applied in various areas, e.g., social networks, the world wide web, and biology. A common requirement from real applications is to return a community with a bounded size while most existing solutions do not constrain community size. Recent studies on size-constrained community search still have some critical issues, e.g., the existence of a better cohesiveness objective, some queries returning empty results, and inefficiency on partial queries. Thus, in this paper, we study the size-constrained truss community search (STCS). Given a graph$G$, a query vertex$q$, and size constraint$[l,h]$, the STCS problem aims to find a subgraph containing$q$with the largest min-support among all connected subgraphs having at least$l$and at most$h$vertices. We prove the STCS problem is NP-hard and APX-hard unless P = NP. An effective heuristic is proposed to quickly find a high-quality initial result. Then, a branch and bound algorithm is introduced to find the exact result, with novel optimizations, e.g., budget-cost-based bounding and branching strategies. Extensive experiments verify that the community quality returned by our algorithm is better and our algorithm is faster by up to 5 orders of magnitude, compared with the state-of-the-art.

The rapid rise and prevalence of social platforms have created great demands on effective schemes to limit the influence of negative information, e.g., blocking key vertices for influence minimization. However, there is currently no system providing practical schemes to solve the negative influence minimization problem with a blocking budget effectively and efficiently in the literature. In this demo, we present IMinimize, the first interactive system that provides audiences with vertex-blocking schemes over different budgets and demonstrates via visualization for comparison vividly and directly, aiming to help minimize the negative influence spreading in networks. Our IMinimize system applies an advanced greedy algorithm to select blocked vertices with both high efficiency and effectiveness. Furthermore, we extend IMinimize to the application of epidemic controlling and prevention and show the usability of IMinimize through two case studies of real-life applications.

Information cascade in online social networks can be rather negative, e.g., the spread of rumors may trigger panic. To limit the influence of misinformation in an effective and efficient manner, the influence minimization (IMIN) problem is studied in the literature: given a graph G and a seed set S, blocking at most b vertices such that the influence spread of the seed set is minimized. In this paper, we are the first to prove the IMIN problem is NP-hard and hard to approximate. Due to the hardness of the problem, existing works resort to greedy solutions and use Monte-Carlo Simulations to solve the problem. However, they are cost-prohibitive on large graphs since they have to enumerate all the candidate blockers and compute the decrease of expected spread when blocking each of them. To improve the efficiency, we propose the AdvancedGreedy algorithm (AG) based on a new graph sampling technique that applies the dominator tree structure, which can compute the decrease of the expected spread of all candidate blockers at once. Besides, we further propose the GreedyReplace algorithm (GR) by considering the relationships among candidate blockers. Extensive experiments on 8 real-life graphs demonstrate that our AG and GR algorithms are significantly faster than the state-of-the-art by up to 6 orders of magnitude, and GR can achieve better effectiveness with its time cost close to AG.

A large number of studies on Graph Outlier Detection (GOD) have emerged in recent years due to its wide applications, in which Unsupervised Node Outlier Detection (UNOD) on attributed networks is an important area. UNOD focuses on detecting two kinds of typical outliers in graphs: the structural outlier and the contextual outlier. Most existing works conduct experiments based on datasets with injected outliers. However, we find that the most widely-used outlier injection approach has a serious data leakage issue. By only utilizing such data leakage, a simple approach can achieve state-of-the-art performance in detecting outliers. In addition, we observe that existing algorithms have a performance drop with the mitigated data leakage issue. The other major issue is on balanced detection performance between the two types of outliers, which has not been considered by existing studies.In this paper, we analyze the cause of the data leakage issue in depth since the injection approach is a building block to advance UNOD. Moreover, we devise a novel variance-based model to detect structural outliers, which outperforms existing algorithms significantly and is more robust at kinds of injection settings. On top of this, we propose a new framework, Variance-based Graph Outlier Detection (VGOD), which combines our variance-based model and attribute reconstruction model to detect outliers in a balanced way. Finally, we conduct extensive experiments to demonstrate the effectiveness and efficiency of VGOD. The results on 5 real-world datasets validate that VGOD achieves not only the best performance in detecting outliers but also a balanced detection performance between structural and contextual outliers.

Hop-constrained s-t simple path (k-st path) enumeration is a fundamental problem in graph databases and plays an important role in many real-world applications. Given a dynamic graph G, a source-target pair s-t, and a hop constraint k, we aim to efficiently compute k-st paths: list all simple paths within length k from s to t, and then continuously maintain the results against edge updates. Although the k-st path enumeration has been well studied in static setting, the existing works on static graphs cannot be applied or adapted to handle dynamic graphs efficiently. To address the challenges on dynamic computation, we propose a partial path-based index structure and an efficient enumeration algorithm based on the index. We also propose several well-designed techniques to efficiently maintain the index and locate the affected results with graph updates. Comprehensive experiments verify that our proposed CPEupdatealgorithm outperforms the state-of-the-art methods by up to 4 orders of magnitude on dynamic graphs. The experiment results also show that the time cost of our initialization step CPEstartup(including index construction) is similar to the state-of-the-art static method.

Quantifying node importance on engagement dynamics is critical to support network stability. We can motivate or retain the users in a social platform according to their importance s.t. the network is more sustainable. Existing studies validate that the coreness of a node is the "best practice" on network topology to estimate the engagement of the node. In this paper, the importance of a node is the effect on the engagement of other nodes when its engagement is strengthened or weakened. Specifically, the importance of a node is quantified via two novel concepts: the anchor power to measure the engagement effect of node strengthening (i.e., the overall coreness gain) and the collapse power to measure the engagement effect of node weakening (i.e., the overall coreness loss). We find the computation of the two concepts can be naturally integrated into a shell component-based framework, and propose a unified static algorithm to compute both the anchored and collapsed followers. For evolving networks, efficient maintenance techniques are designed to update the follower sets of each node, which is faster than redoing the static algorithm by around 3 orders of magnitude. Extensive experiments on real-life data demonstrate the effectiveness of our model and the efficiency of our algorithms.

It is a challenge to generate an accurate machine learning model in a distributed network due to the increased concern in data privacy and high cost in gathering all raw data. This paper presents an adaptive asynchronous distributed clustering algorithm for agents in wireless network to learn the global models, while the privacy is protected. Moreover, the communication cost and clustering quality can be adaptively balanced. The proposed clustering algorithm does not require the number of clusters to be pre-defined. To improve the accuracy of the global model, we propose a bounding boxes based method to fully utilize the shape information of clusters. In addition, we consider different knowledge levels of agent and different requirements about the global model. In experiments on randomly generated network topologies, we demonstrate that methods which do more extensive clustering in each cycle, and which exchange descriptions of cluster shape and density instead of just centroids and data counts, achieve more consistent clustering, in significantly shorter elapsed time. We also show that the proposed methods can learn the same number of clusters as the ground truth when clusters are well separated from each other.

The model of k-core discovers a novel hierarchical structure of a network, which has been widely applied in various areas, e.g., sociology, biology, and brain science. Based on the containment relations of k-cores with different$k$, the hierarchical core decomposition (HCD) of a graph formalizes the hierarchy of all k-cores for each possible$k$• HCD is effective in locating high-quality subgraphs (e.g., densest subgraph search) and exploring particular network phenomena (e.g., user engagement study). However, existing solutions of HCD are still not efficient enough, for both the hierarchy construction and the subgraph search on the hierarchy. In this paper, we propose the first parallel construction algorithm PHCD for HCD, using a new union-find-based paradigm, and the first parallel algorithm PBKS to search high-quality subgraphs from the hierarchy with respect to various community scoring metrics. We prove the problem of hierarchy construction is$\mathcal{P}$-complete (difficult to parallelize effectively). Despite the negative result, our PHCD has a near-linear time cost, and PBKS is time-optimal in score computation for most community metrics. Extensive experiments are conducted on 10 real-world networks, where our proposed parallel algorithms significantly outperform the existing solutions, for both the hierarchy construction and the subgraph search.

Multilayer graphs are very powerful in representing the multiplex relationships among entities. The truss decomposition on single-layer graphs is a well-studied problem which divides a graph into a hierarchy structure, and has a series of applications. However, little attention has been paid to the truss decomposition on multilayer graphs. In addition, truss decomposition on multilayer graphs derives new challenges compared to the scenario on single-layer graphs. In this paper, we devise an efficient algorithm to compute the truss decomposition on multilayer graphs. Extensive experiments on 5 real-life datasets validate the effectiveness and efficiency of our methods.

The studies of k-truss based community search demonstrated that it can find high-quality personalized com-munities with good properties such as high connectivity and bounded diameter. Motivated by natural restrictions from real applications, in this paper, we investigate the search of triangle-connected k-truss with size constraint (denoted by SCkT) in a graph G: given a size constraint s, an integer k, and query set Q, SCkT search aims to find a triangle-connected k-truss H containing the vertices in Q and with size (i.e., total number of vertices in H) not exceeding s. We prove that the SCkT search problem is NP-hard. To tame the hardness, we fully exploit the properties of triangle-connected k-truss subgraphs s.t. a practically-efficient exact solution for SCkT search is developed. A novel and effective lower bound is proposed to early terminate unpromising search branches and narrow down the search space. Two search strategies, expansion and shrinking, are investigated to tailor for efficient support of SCkT search. A hybrid search method is proposed combining the expansion and shrinking strategies, where a score function is used to guide the search order. Our extensive experiments on real-life and synthetic graphs demonstrate the effectiveness of the SCkT model and the efficiency of the proposed techniques.

The model of k -core and its decomposition have been applied in various areas, such as social networks, the world wide web, and biology. A graph can be decomposed into an elegant k -core hierarchy to facilitate cohesive subgraph discovery and network analysis. As many real-life graphs are fast evolving, existing works proposed efficient algorithms to maintain the coreness value of every vertex against structure changes. However, the maintenance of the k -core hierarchy in existing studies is not complete because the connections among different k -cores in the hierarchy are not considered. In this paper, we study hierarchical core maintenance which is to compute the k -core hierarchy incrementally against graph dynamics. The problem is challenging because the change of hierarchy may be large and complex even for a slight graph update. In order to precisely locate the area affected by graph dynamics, we conduct in-depth analyses on the structural properties of the hierarchy, and propose well-designed local update techniques. Our algorithms significantly outperform the baselines on runtime by up to 3 orders of magnitude, as demonstrated on 10 real-world large graphs.

In this paper, we propose and study a novel cohesive subgraph model, named (k,p)-core, which is a maximal subgraph where each vertex has at least k neighbours and at least p fraction of its neighbours in the subgraph. The model is motivated by the finding that each user in a community should have at least a certain fraction p of neighbors inside the community to ensure user engagement, especially for users with large degrees. Meanwhile, the uniform degree constraint k, as applied in the k-core model, guarantees a minimum level of user engagement in a community, and is especially effective for users with small degrees. We propose an O(m) algorithm to compute a (k,p)-core with given k and p, and an O(dm) algorithm to decompose a graph by (k,p)-core, where m is the number of edges in the graph G and d is the degeneracy of G. A space efficient index is designed for time-optimal (k,p)-core query processing. Novel techniques are proposed for the maintenance of (k,p)-core index against graph dynamic. Extensive experiments on 8 reallife datasets demonstrate that our (k,p)-core model is effective and the algorithms are efficient.

The mode of k-core and its hierarchical decomposition have been applied in many areas, such as sociology, the world wide web, and biology. Algorithms on related studies often need an input value of parameter k, while there is no existing solution other than manual selection. In this paper, given a graph and a scoring metric, we aim to efficiently find the best value of k such that the score of the k-core (or k-core set) is the highest. The problem is challenging because there are various community scoring metrics and the computation is costly on large datasets. With the well-designed vertex ordering techniques, we propose time and space optimal algorithms to compute the best k, which are applicable to most community metrics. The proposed algorithms can compute the score of every k-core (set) and can benefit the solutions to other k-core related problems. Extensive experiments are conducted on 10 real-world networks with size up to billion-scale, which validates both the efficiency of our algorithms and the effectiveness of the resulting k-cores.

The stability of a social network has been widely studied as an important indicator for both the network holders and the participants. Existing works on reinforcing networks focus on a local view, e.g., the anchored k-core problem aims to enlarge the size of the k-core with a fixed input k. Nevertheless, it is more promising to reinforce a social network in a global manner: considering the engagement of every user (vertex) in the network. Since the coreness of a user has been validated as the "best practice" for capturing user engagement, we propose and study the anchored coreness problem in this paper: anchoring a small number of vertices to maximize the coreness gain (the total increment of coreness) of all the vertices in the network. We prove the problem is NP-hard and show it is more challenging than the existing local-view problems. An efficient heuristic algorithm is proposed with novel techniques on pruning search space and reusing the intermediate results. Extensive experiments on real-life data demonstrate that our model is effective for reinforcing social networks and our algorithm is efficient.

In social networks, the leave of critical users may significantly break network engagement, i.e., lead a large number of other users to drop out. A popular model to measure social network engagement is k-core, the maximal subgraph in which every vertex has at least k neighbors. To identify critical users, we propose the collapsed k-core problem: given a graph G, a positive integer k and a budget b, we aim to find b vertices in G such that the deletion of the b vertices leads to the smallest k-core. We prove the problem is NP-hard and in approximate. An efficient algorithm is proposed, which significantly reduces the number of candidate vertices. We also study the user leave towards the model of k-truss which further considers tie strength by conducting additional computation w.r.t. k-core. We prove the corresponding collapsed k-truss problem is also NP-hard and in approximate. An efficient algorithm is proposed to solve the problem. The advantages and disadvantages of the two proposed models are experimentally compared. Comprehensive experiments on nine real-life social networks demonstrate the effectiveness and efficiency of our proposed methods.

In many applications, graphs often involve the nodes with multi-dimensional numerical attributes, and it is desirable to retrieve a group of nodes that are both highly connected (e.g., clique) and optimal according to some ranking functions. It is well known that the skyline returns candidates for the optimal objects when ranking functions are not specified. Motivated by this, in this paper we formulate the novel model of skyline k-cliques over multi-valued attributed graphs and develop efficient algorithms to conduct the computation. To verify the group based dominance between two k-cliques, we make use of maximum bipartite matching and develop a set of optimization techniques to improve the verification efficiency. Then, a progressive computation algorithm is developed which enumerates the k-cliques in an order such that a k-clique is guaranteed not to be dominated by those generated after it. Novel pruning and early termination techniques are developed to exclude unpromising nodes or cliques by investigating the structural and attribute properties of the multi-valued attributed graph. Empirical studies on four real datasets demonstrate the effectiveness of the skyline k-clique model and the efficiency of the novel computing techniques.

Graph decomposition has been widely used to analyze real-life networks from different perspectives. Recent studies focus on the hierarchical graph decomposition methods to handle big graphs in many real-life applications such as community detection, network analysis, network visualization, internet topology analysis and protein function prediction. In this tutorial, we first highlight the importance of hierarchical graph decomposition in a variety of applications and the unique challenges that need to be addressed. Subsequently, we provide an overview of the existing models and the computation algorithms under different computing environments. Then we discuss the integration of existing models with other approaches to better capture the cohesiveness of subgraphs in real-life scenarios. Finally, we discuss the future research directions in this important and growing research area.

As one of the most representative cohesive subgraph models,k-core model has recently received significant attention in the literature. In this paper, we investigate the problem of the minimumk-core search: given a graphG, an integerkand a set of query verticesQ= {q}, we aim to find the smallestk-core subgraph containing every query vertexqϵQ.It has been shown that this problem is NP-hard with a huge search space, and it is very challenging to find the optimal solution. There are several heuristic algorithms for this problem, but they rely on simple scoring functions and there is no guarantee as to the size of the resulting subgraph, compared with the optimal solution. Our empirical study also indicates that the size of their resulting subgraphs may be large in practice. In this paper, we develop an effective and efficient progressive algorithm, namelyPSA, to provide a good trade-off between the quality of the result and the search time. Novel lower and upper bound techniques for the minimumk-core search are designed. Our extensive experiments on 12 real-life graphs demonstrate the effectiveness and efficiency of the new techniques.

User engagement and tie strength are fundamental and important components in social networks. The model of k-truss not only captures actively engaged users, but also ensures strong tie strength among these users. It motivates us to utilize the model of k-truss in preventing network unraveling, which simultaneously considers both of the basic components. In this paper, we propose and investigate the anchored k-truss problem to reinforce a network by anchoring critical users who can significantly stop the unraveling. We prove the problem is NP-hard for k ≥ 4. A fast edge deletion order based algorithm, named AKT, is proposed with efficient candidate exploration and pruning techniques based on the order. Comprehensive experiments on 10 real-life graphs demonstrate the effectiveness of our model and the efficiency of our methods.

In this paper, we investigate the problem of ( k,r )-core which intends to find cohesive subgraphs on social networks considering both user engagement and similarity perspectives. In particular, we adopt the popular concept of k -core to guarantee the engagement of the users (vertices) in a group (subgraph) where each vertex in a ( k,r )-core connects to at least k other vertices. Meanwhile, we consider the pairwise similarity among users based on their attributes. Efficient algorithms are proposed to enumerate all maximal ( k,r )-cores and find the maximum ( k,r )-core, where both problems are shown to be NP-hard. Effective pruning techniques substantially reduce the search space of two algorithms. A novel ( k,k' )-core based ( k,r )-core size upper bound enhances performance of the maximum ( k,r )-core computation. We also devise effective search orders for two mining algorithms where search priorities for vertices are different. Comprehensive experiments on real-life data demonstrate that the maximal/maximum ( k,r )-cores enable us to find interesting cohesive subgraphs, and performance of two mining algorithms is effectively improved by proposed techniques.

In this paper, we study the problem of the anchored k -core. Given a graph G , an integer k and a budget b , we aim to identify b vertices in G so that we can determine the largest induced subgraph J in which every vertex, except the b vertices, has at least k neighbors in J . This problem was introduced by Bhawalkar and Kleinberg e t al. in the context of user engagement in social networks, where a user may leave a community if he/she has less than k friends engaged. The problem has been shown to be NP-hard and inapproximable. A polynomial-time algorithm for graphs with bounded tree-width has been proposed. However, this assumption usually does not hold in real-life graphs, and their techniques cannot be extended to handle general graphs. Motivated by this, we propose an efficient algorithm, namely onion-layer based anchored k-core (OLAK), for the anchored k -core problem on large scale graphs. To facilitate computation of the anchored k -core, we design an onion layer structure, which is generated by a simple onion-peeling-like algorithm against a small set of vertices in the graph. We show that computation of the best anchor can simply be conducted upon the vertices on the onion layers , which significantly reduces the search space. Based on the well-organized layer structure, we develop efficient candidates exploration, early termination and pruning techniques to further speed up computation. Comprehensive experiments on 10 real-life graphs demonstrate the effectiveness and efficiency of our proposed methods.