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Temporal graphs are graphs whose edges are associated with timestamps. Subgraph matching on temporal graphs retrieves temporal subgraphs whose edge timestamps satisfy user-specified temporal orders. In this paper, a temporal query pattern is composed of a query graph with an arbitrary structure and a partial order on the edge set of the query graph. Moreover, the time span between the minimum and the maximum edge timestamps in a matching is required to be less than or equal to a specified threshold. This paper proposes a temporal subgraph matching algorithm based on two key techniques. First, a memory-efficient index structure called TO-tree is designed to compactly store all necessary information required for finding all temporal subgraph matchings. The TO-tree constructed for a temporal query pattern is much smaller than the temporal graph because unnecessary information is mostly excluded from the TO-tree by three powerful filters. The second technique is a temporal subgraph matching enumeration method that runs on the TO-tree instead of on the temporal graph. This enumeration method expands temporal subgraph matchings in an edge-by-edge manner. Since the TO-tree can fit into the main memory, the enumeration method runs very fast on the TO-tree. An extensive experimental evaluation has been carried out. The experimental results show that the TO-tree index structure is memory-efficient, which in turn enables a fast temporal subgraph matching enumeration. Overall, our algorithm is at least 3X faster than the baseline algorithm adapted from the state-of-the-art non-temporal subgraph matching algorithm CECI and is at least 4X faster than the temporal subgraph matching algorithm HASSE .

When characterizing the cohesion of a subgraph, many cohesive subgraph models impose various constraints on the relationship between the minimum degree of vertices and the number of vertices in the subgraph. However, the constraints imposed by the clique, quasi-clique and k-core models are so rigid that they cannot simultaneously characterize cohesive subgraphs of various scales in a graph. This paper characterizes the flexibility of a cohesive subgraph model by the constraint it imposes on this relationship and proposes f -constraint core (f -CC), a new model that can flexibly characterize cohesive subgraphs of different scales. We formalize the FlexCS problem that finds r representative f -CCs in a graph that are most diversified. This problem is NP-hard and is even NP-hard to be approximated within |V |1−𝜖 for any 𝜖 > 0. To solve the problem efficiently, a divide-and-conquer algorithm is proposed together with some effective techniques including f -CC generation, sub-problem filtering, early termination, and index-based connected component retrieval. Extensive experiments verify the flexibility of the f -CC model and the effectiveness of the proposed algorithm.

SPARQL 1.1 offers a type of navigational query for RDF systems, called regular path query (RPQ). A regular path query allows for retrieving node pairs with the paths between them satisfying regular expressions. Regular path queries are always difficult to be evaluated efficiently because of the possible large search space. Thus there has been no scalable and practical solution so far. In this paper, we present Leon+, an in-memory distributed framework, to address the RPQ problem in the context of the knowledge graph. To reduce search space and mitigate mounting communication costs, Leon+ takes advantage of join-ahead pruning via a novel RDF summarization technique together with a path partitioning strategy. We also develop a subtle cost model to devise query plans to achieve high efficiency for complex RPQs. As there has been no available RPQ benchmark, we create micro-benchmarks on both synthetic and real-world datasets. A thorough experimental evaluation is presented between our approach and the state-of-the-art RDF stores. The results show that our approach outperforms 5x faster than the competitors on single RPQ. For query workload, it saves up to 1/2 time and 2/3 communication overheads over the baseline method.

Mining dense subgraphs on multi-layer graphs is an interesting problem, which has witnessed lots of applications in practice. To overcome the limitations of the quasi-clique-based approach, we propose d-coherent core (d-CC), a new notion of dense subgraph on multi-layer graphs, which has several elegant properties. We formalize the diversified coherent core search (DCCS) problem, which finds kd-CCs that can cover the largest number of vertices. We propose a greedy algorithm with an approximation ratio of $$1 - 1/e$$ and two search algorithms with an approximation ratio of 1/4. Furthermore, we propose some optimization techniques to further speed up the algorithms. The experiments verify that the search algorithms are faster than the greedy algorithm and produce comparably good results as the greedy algorithm in practice. As opposed to the quasi-clique-based approach, our DCCS algorithms can fast detect larger dense subgraphs that cover most of the quasi-clique-based results.

Interactive graph search leverages human intelligence to categorize target labels in a hierarchy, which is useful for image classification, product categorization, and database search. However, many existing interactive graph search studies aim at identifying a single target optimally, and suffer from the limitations of asking too many questions and not being able to handle multiple targets. To address these two limitations, in this paper, we study a new problem of budget constrained interactive graph search for multiple targets called kBM-IGS problem. Specifically, given a set of multiple targets T in a hierarchy and two parameters k and b , the goal is to identify a k -sized set of selections S , such that the closeness between selections S and targets T is as small as possible, by asking at most a budget of b questions. We theoretically analyze the updating rules and design a penalty function to capture the closeness between selections and targets. To tackle the kBM-IGS problem, we develop a novel framework to ask questions using the best vertex with the largest expected gain, which provides a balanced trade-off between target probability and benefit gain. Based on the kBM-IGS framework, we first propose an efficient algorithm STBIS to handle the SingleTarget problem, which is a special case of kBM-IGS. Then, we propose a dynamic programming based method kBM-DP to tackle the MultipleTargets problem. To further improve efficiency, we propose two heuristic but efficient algorithms, kBM-Topk and kBM-DP+. Experiments on large real-world datasets with ground-truths verify both the effectiveness and efficiency of our algorithms.

The existing literatures of the query processing on knowledge graphs focus on an exhaustive enumeration of all matches, which is time-consuming. Users are often interested in diversified top-k matches, rather than the entire match set. Motivated by these, this paper formalizes the diversified top-k querying (DTQ) problem in the context of RDF/SPARQL and proposes a diversification function to balance importance and diversity. We first prove that the decision problem of DTQ is NP-complete, and give a baseline algorithm with an approximation ratio of 2. Secondly, an index-based algorithm with the early termination property is proposed. The index is adept in parallel diversified top-k selection in multicore architectures. Using real-world and synthetic data, we experimentally verify that our algorithms are efficient and effective in computing meaningful diversified top-k matches.

SimRank is a similarity measure between vertices in a graph. Recently, many algorithms have been proposed to efficiently evaluate SimRank similarities. However, the existing algorithms either overlook uncertainty in graph structures or depends on an unreasonable assumption. In this paper, we study SimRank on uncertain graphs. Following the random-walk-based formulation of SimRank on deterministic graphs and the possible world model of uncertain graphs, we first define random walks on uncertain graphs and show that our definition of random walks satisfies Markov’s property. We formulate our SimRank measure based on random walks on uncertain graphs. We discover a critical difference between random walks on uncertain graphs and random walks on deterministic graphs, which makes all existing SimRank computation algorithms on deterministic graphs inapplicable to uncertain graphs. For SimRank computation, we consider computing both single-pair SimRank and single-source top- $K$ SimRank. We propose three algorithms, namely the sampling algorithm with high efficiency, the two-phase algorithm with comparable efficiency and higher accuracy, and a speeding-up algorithm with much higher efficiency. Meanwhile, we present an optimized algorithm for efficient computing the single-source top- $K$ SimRank. The experimental results verify the effectiveness of our SimRank measure and the efficiency of the proposed SimRank computation algorithms.

Structural holes in social networks are vertices that serve as gateways for information exchange between communities. Although many algorithms have been proposed to detect structural holes, they are not scalable to big graphs. This paper proposes a structural hole detection algorithm ESH based on distributed parallel graph processing frameworks. Instead of using substructures in social networks, the algorithm exploits a factor diffusion process in structural hole detection. The algorithm naturally fits the vertex-centric programming models and can be easily implemented on the graph-parallel processing frameworks. Extensive experiments show that ESH can handle social networks with billions of links and produce structural holes of higher quality than the existing algorithms.

Making massive graph data easily understandable by people is a demanding task in a variety of real applications. Graph compression is an effective approach to reducing the size of graph data as well as its complexity in structures. This paper proposes a simple yet effective graph compression method called the star-based graph compression. This method compresses a graph by shrinking a collection of disjoint subgraphs called stars. Compressing a graph into the optimal star-based compressed graph with the highest compression ratio is shown to be NP-complete. We propose a greedy compression algorithm called StarZip. We experimentally verify that StarZip achieves compression ratios of 3.8–45.7 and 2.9–241.6 in terms of vertex count and edge count, respectively. Besides, we study the shortest path queries on compressed graphs. On the real graphs, the StarSSSP algorithm for processing shortest path queries on compressed graphs is 4X–20X faster than Dijkstra’s algorithm running on original graphs. The average absolute error between the query results of StarSSSP and the exact shortest distances is about 1. On the synthetic graphs, StarSSSP is up to 313X faster than Dijkstra’s algorithm, and the average absolute error is also about 1.

As similar queries keep springing up in real query logs, few RDF systems address this problem. In this paper, we propose Leon, a distributed RDF system, which can also deal with multi-query problem. First, we apply a characteristic-set-based partitioning scheme. This scheme (i) supports the fully parallel processing of join within characteristic sets; (ii) minimizes data communication by applying direct transmission of intermediate results instead of broadcasting. Then, Leon revisits the classical problem of multi-query optimization in the context of RDF/SPARQL. In light of the NP-hardness of the multi-query optimization for SPARQL, we propose a heuristic algorithm that partitions the input batch of queries into groups, and discover the common sub-query of multiple SPARQL queries. Our MQO algorithm incorporates with a subtle cost model to generate execution plans.

Rectangle has been recognized as an essential motif in a large number of real-world networks. Counting rectangles in a network plays an important role in network analysis. This paper comprehensively studies the rectangle counting problem on large networks. We propose a novel counting paradigm called the wedge-centric counting, where a wedge is a simple path consisting of three vertices. Unlike the traditional edge-centric counting, the wedge-centric counting uses wedges instead of edges as building blocks of rectangles. The main advantage of the wedge-centric counting is that it does not need to access two-hop neighbors. Based on this paradigm, we develop a collection of rectangle counting algorithms, including an in-memory algorithm with lower time complexity, an external-memory algorithm with the optimal I/O complexity, and two randomized algorithms with provable error bounds. The experimental results on a variety of real networks verify the effectiveness and the efficiency of the proposed wedge-centric rectangle counting algorithms.

Masses of large-scale knowledge graphs on various domains have sprung up in recent years. They are no longer able to be managed on a single machine. The distributed RDF systems intervene in the scalability issue using partitioning techniques. However, most of these systems are unaware of query workload and employ static partitioning. As diverse and dynamic workloads keep emerging in the knowledge graph applications, they cannot consistently provide good performance. To address the problem, we propose a workload-aware partitioning framework WISE, which could be deployed on any initial partitioning. It encodes the incoming SPARQL queries in a novel structure called query span and periodically examines the query span to identify the frequent query patterns. The triples of a frequent query pattern are moved to the same partition, aiming at improving the response time in the future. Our experiments on various RDF datasets and workloads indicate that WISE achieves dramatic communication reduction and considerable performance improvement over the baseline method. The migration overhead only accounts for a small portion of the total runtime.

Mining dense subgraphs on multi-layer graphs is an interesting problem, which has witnessed lots of applications in practice. To overcome the limitations of the quasi-clique-based approach, we propose d-coherent core (d-CC), a new notion of dense subgraph on multi-layer graphs, which has several elegant properties. We formalize the diversified coherent core search (DCCS) problem, which finds k d-CCs that can cover the largest number of vertices. We propose a greedy algorithm with an approximation ratio of 1 - 1/e and two search algorithms with an approximation ratio of 1/4. The experiments verify that the search algorithms are faster than the greedy algorithm and produce comparably good results as the greedy algorithm in practice. As opposed to the quasi-clique-based approach, our DCCS algorithms can fast detect larger dense subgraphs that cover most of the quasi-clique-based results., Comment: Full Version, 17 Pages

This paper studies a novel approach to processing massive uncertain graph data. In this approach, we propose a new framework to simultaneously process a query on a set of randomly sampled possible worlds of an uncertain graph. Based on this framework, we develop a series of algorithms to analyze massive uncertain graphs, including breadth-first search, shortest distance queries, triangle counting, and core decomposition. We implement this approach based on GraphLab, one of the stateof-the-art graph processing frameworks. By sharing fine-grained internal processing steps on common substructures of sampled possible worlds, the new approach achieves tens to hundreds of times speedup in execution time on a cluster of 20 servers.

The emerging non-volatile memory (NVM) technologies have attracted much attention due to its advantages over the existing DRAM technology such as non-volatility, byte-addressability and high storage density. These promising features make NVM a promising replacement of DRAM. Although the reading cost of NVM is close to that of DRAM, the writing cost is significantly higher than that of DRAM. Existing algorithms designed on DRAM treat read and write equally and thus are not applicable to NVM. In this paper, we investigate efficient algorithms for subgraph matching, a fundamental problem in graph databases, on NVM. We first give a detailed evaluation on several existing subgraph matching algorithms by experiments and theoretical analysis. Then, we propose our write-limited subgraph matching algorithm based on the analysis. We also extend our algorithm to answer subgraph matching on dynamic graphs. Experiments on an NVM simulator demonstrate a significant improvement in efficiency against the existing algorithms.

SimRank is a similarity measure between vertices in a graph, which has become a fundamental technique in graph analytics. Recently, many algorithms have been proposed for efficient evaluation of SimRank similarities. However, the existing SimRank computation algorithms either overlook uncertainty in graph structures or is based on an unreasonable assumption (Du et al). In this paper, we study SimRank similarities on uncertain graphs based on the possible world model of uncertain graphs. Following the random-walk-based formulation of SimRank on deterministic graphs and the possible worlds model of uncertain graphs, we define random walks on uncertain graphs for the first time and show that our definition of random walks satisfies Markov's property. We formulate the SimRank measure based on random walks on uncertain graphs. We discover a critical difference between random walks on uncertain graphs and random walks on deterministic graphs, which makes all existing SimRank computation algorithms on deterministic graphs inapplicable to uncertain graphs. To efficiently compute SimRank similarities, we propose three algorithms, namely the baseline algorithm with high accuracy, the sampling algorithm with high efficiency, and the two-phase algorithm with comparable efficiency as the sampling algorithm and about an order of magnitude smaller relative error than the sampling algorithm. The extensive experiments and case studies verify the effectiveness of our SimRank measure and the efficiency of our SimRank computation algorithms., 14 pages, under review by ICDE'16

In recent years, lots of data in various domain can be represented and described by uncertain graph model, such as protein interaction networks, social networks, wireless sensor networks, etc. This paper investigates the most reliable minimum spanning tree problem, which aims to find the minimum spanning tree MST with largest probability among all possible MSTs on uncertain graphs. In fact, the most reliable MST is an optimal choice between stability and cost. Therefore it has wide applications in practice, for example, it can serve as the basic constructs in a telecommunication network, the link of which can be unreliable and may fail with certain probability. A brute-force method needs to enumerate all possible MSTs and the time consumption grows exponentially with edge size. Hence we put forward an approximate algorithm in $$Od^{2}|V|^{2}$$Od2|V|2, where d is the largest vertex degree and |V| is vertex size. We point out that the algorithm can achieve exact solution with expected probability at least $$1-\frac{1}{2}^{d+1/2}^{|V|-1}$$1-12d+1/2|V|-1 and the expected approximation ratio is at least $$\frac{1}{2}^{d|V|}$$12d|V| when edge probability is uniformly distributed. Our extensive experimental results show that our proposed algorithm is both efficient and effective.

Heterogeneous information networks contain heterogeneous types of nodes and edges, e.g., social networks and knowledge graphs. A meta-path is a path connecting nodes through a sequence of heterogeneous edges, representing different kinds of semantic relations among nodes. Meta-paths are good mechanisms to improve the quality of graph analysis on heterogeneous information networks. This paper presents an iceberg cube framework for heterogeneous information networks based on meta-paths. To the best of our knowledge, there is no such proposal in the past. 1 We use meta-paths to measure the similarities of nodes, and prove the problem is NP-hard. 2 An optimal solution is proposed for the strict case. We develop the variant of slice tree to aggregate networks hierarchically. 3 To improve the scalability, a general approximate algorithm is provided for fast aggregation, where random walk on meta-paths is employed to measure the similarities. 4 Two pruning strategies are designed for reducing search space when the aggregate function is monotonic. 5 Experiments on both real-world and synthetic networks demonstrate the effectiveness and efficiency of the algorithms.

Uncertain graph data has been collected, processed and analyzed in a wide range of applications. Under the possible world model, an uncertain graph represents a probability distribution over all its possible worlds. Each possible world is a deterministic graph in which the uncertain graph may be present in practice. To deal with the hardness of computations on uncertain graphs, Parchas et al. first proposed the concept of a degree-based representative possible world. This approach is distinguished from the sampling approach in that it computes on one representative possible world instead of on a large number of possible worlds sampled at random. However, the degree-based representative possible world only tries to preserve vertex degrees. In this paper, we are motivated by the fact that motif structures such as triangles can affect the structural properties of graphs and propose the concept of a triangle-based representative possible world. We also develop an algorithm for finding the triangle-based representative possible world of an uncertain graph. We conducted extensive experimental evaluations and show that the triangle-based representative possible world outperforms the degree-based representative possible world in preserving the structural characteristics of an uncertain graph in expectation.

In this paper, we propose bitruss, a new notion of a dense subgraph of a bipartite graph. Specifically, the k-bitruss of a bipartite graph is the largest edge-induced subgraph H such that every edge of H is contained in at least k rectangles within H. The bitruss decomposition of a bipartite graph is the set of all nonempty k-bitrusses of the bipartite graph for $$k \ge 0$$ki¾ź0. In this paper, we show that the bitruss decomposition of a bipartite graph have three important properties. First, the bitruss decomposition is unique. Second, the bitruss decomposition is hierarchical, that is, the $$k + 1$$k+1-bitruss is a subgraph of the k-bitruss for all $$k \ge 0$$ki¾ź0. Third, the bitruss decomposition can be computed in polynomial time. These three interesting properties make bitruss a promising notion of dense bipartite subgraphs.

Frequent subgraph mining has been extensively studied on certain graph data. However, uncertainty is intrinsic in graph data in practice, but there is very few work on mining uncertain graph data. This paper focuses on mining frequent subgraphs over uncertain graph data under the probabilistic semantics. Specifically, a measure called $${\varphi}$$ -frequent probability is introduced to evaluate the degree of recurrence of subgraphs. Given a set of uncertain graphs and two real numbers $${0 < \varphi, \tau < 1}$$ , the goal is to quickly find all subgraphs with $${\varphi}$$ -frequent probability at least ?. Due to the NP-hardness of the problem and to the #P-hardness of computing the $${\varphi}$$ -frequent probability of a subgraph, an approximate mining algorithm is proposed to produce an $${(\varepsilon, \delta)}$$ -approximate set ? of "frequent subgraphs", where $${0 < \varepsilon < \tau}$$ is error tolerance, and 0 < ? < 1 is a confidence bound. The algorithm guarantees that (1) any frequent subgraph S is contained in ? with probability at least ((1 ? ?) /2) s , where s is the number of edges in S; (2) any infrequent subgraph with $${\varphi}$$ -frequent probability less than $${\tau - \varepsilon}$$ is contained in ? with probability at most ?/2. The theoretical analysis shows that to obtain any frequent subgraph with probability at least 1 ? Δ, the input parameter ? of the algorithm must be set to at most $${1 - 2 (1 - \Delta)^{1 / \ell_{\max}}}$$ , where 0 < Δ < 1, and ? max is the maximum number of edges in frequent subgraphs. Extensive experiments on real uncertain graph data verify that the proposed algorithm is practically efficient and has very high approximation quality. Moreover, the difference between the probabilistic semantics and the expected semantics on mining frequent subgraphs over uncertain graph data has been discussed in this paper for the first time.

Heterogeneous computing (HC) environment utilizes diverse resources with different computational capabilities to solve computing-intensive applications having diverse computational requirements and constraints. The task assignment problem in HC environment can be formally defined as for a given set of tasks and machines, assigning tasks to machines to achieve the minimum makespan. In this paper we propose a new task scheduling heuristic, high standard deviation first (HSTDF), which considers the standard deviation of the expected execution time of a task as a selection criterion. Standard deviation of the expected execution time of a task represents the amount of variation in task execution time on different machines. Our conclusion is that tasks having high standard deviation must be assigned first for scheduling. A large number of experiments were carried out to check the effectiveness of the proposed heuristic in different scenarios, and the comparison with the existing heuristics (Max-min, Sufferage, Segmented Min-average, Segmented Min-min, and Segmented Max-min) clearly reveals that the proposed heuristic outperforms all existing heuristics in terms of average makespan.

With the advent of heterogeneous information networks that consist of multi-type, interconnected nodes, such as bibliographic networks and knowledge graphs, it is important to study flexible aggregation in such networks. In this paper, we investigate the flexible aggregation problem on heterogeneous information networks, which is defined on multi-type of nodes and relations. We develop an efficient heuristic algorithm for aggregation in two phases: informational aggregation and structural aggregation. Extensive experiments on real world data sets demonstrate the effectiveness and efficiency of the proposed algorithm.

In the era of the Big Data, how to analyze such a vast quantity of data is a challenging problem, and conducting a multi-way theta-join query is one of the most time consuming operations. MapReduce has been mentioned most in the massive data processing area and some join algorithms based on it have been raised in recent years. However, MapReduce paradigm itself may not be suitable to some scenarios and multi-way theta-join seems to be one of them. Many multi- way theta-join algorithms on traditional parallel database have been raised for many years, but no algorithm has been mentioned on the CMD (coordinate modulo distribution) storage method, although some algorithms on equal-join have been proposed. In this paper, we proposed a multi-way theta-join method based on CMD, which takes the advantage of the CMD storage method. Experiments suggest that it’s a valid and efficient method which achieves significant improvement compared to those applied on the MapReduce.

Existing studies on graph mining focus on exact graphs that are precise and complete. However, graph data tends to be uncertain in practice due to noise, incompleteness and inaccuracy. This paper investigates the problem of finding top-k maximal cliques in an uncertain graph. A new model of uncertain graphs is presented, and an intuitive measure is introduced to evaluate the significance of vertex sets. An optimized branch-and-bound algorithm is developed to find top-k maximal cliques, which adopts efficient pruning rules, a new searching strategy and effective preprocessing methods. The extensive experimental results show that the proposed algorithm is very efficient on real uncertain graphs, and the top-k maximal cliques are very useful for real applications, e.g. protein complex prediction.

Graph data are subject to uncertainties in many applications due to incompleteness and imprecision of data. Mining uncertain graph data is semantically different from and computationally more challenging than mining exact graph data. This paper investigates the problem of mining frequent subgraph patterns from uncertain graph data. The frequent subgraph pattern mining problem is formalized by designing a new measure called expected support. An approximate mining algorithm is proposed to find an approximate set of frequent subgraph patterns by allowing an error tolerance on the expected supports of the discovered subgraph patterns. The algorithm uses an efficient approximation algorithm to determine whether a subgraph pattern can be output or not. The analytical and experimental results show that the algorithm is very efficient, accurate and scalable for large uncertain graph databases.

In recent years, large amount of data modeled by graphs, namely graph data, have been collected in various domains. Efficiently processing queries on graph databases has attracted a lot of research attentions. Supergraph query is a kind of new and important queries in practice. A supergraph query, q, on a graph database D is to retrieve all graphs in D such that q is a supergraph of them. Because the number of graphs in databases is large and subgraph isomorphism testing is NP-complete, efficiently processing such queries is a big challenge. This paper first proposes an optimal compact method for organizing graph databases. Common subgraphs of the graphs in a database are stored only once in the compact organization of the database, in order to reduce the overall cost of subgraph isomorphism testings from stored graphs to queries during query processing. Then, an exact algorithm and an approximate algorithm for generating significant feature set with optimal order are proposed to construct indices on graph databases. The optimal order on the feature set is to reduce the number of subgraph isomorphism testings during query processing. Based on the compact organization of graph databases, a novel algorithm of testing subgraph isomorphisms from multiple graphs to one graph is presented. Finally, based on all these techniques, a query processing method is proposed. Analytical and experimental results show that the proposed algorithms outper-form the existing similar algorithms by one to two orders of magnitude.

Heterogeneous computing (HC) environment consists of different resources connected with high-speed links to provide a variety of computational capabilities for computing-intensive applications having multifarious computational requirements. The problem of optimal assignment of tasks to machines in HC environment is proven to be NP-complete requiring use of heuristics to find the near optimal solution. In this work we conduct a performance study of task scheduling heuristics in HC environment. Overall we have implemented 16 heuristics, among them 7 are proposed in this paper. Based on experimental results we specify the circumstances under which one heuristic will outperform the others.