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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.

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.

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.

Uncertain graph has been widely used to represent graph data with inherent uncertainty in structures. Reliability search is a fundamental problem in uncertain graph analytics. This paper investigates on a new problem with broad real-world applications, the top-k reliability search problem on uncertain graphs, that is, finding the k vertices v with the highest reliabilities of connections from a source vertex s to v. Note that the existing algorithm for the threshold-based reliability search problem is inefficient for the top-k reliability search problem. We propose a new algorithm to efficiently solve the top-k reliability search problem. The algorithm adopts two important techniques, namely the BFS sharing technique and the offline sampling technique. The BFS sharing technique exploits overlaps among different sampled possible worlds of the input uncertain graph and performs a single BFS on all possible worlds simultaneously. The offline sampling technique samples possible worlds offline and stores them using a compact structure. The algorithm also takes advantages of bit vectors and bitwise operations to improve efficiency. In addition, we generalize the top-k reliability search problem from single-source case to the multi-source case and show that the multi-source case of the problem can be equivalently converted to the single-source case of the problem. Moreover, we define two types of the reverse top-k reliability search problems with different semantics on uncertain graphs. We propose appropriate solutions for both of them. Extensive experiments carried out on both real and synthetic datasets verify that the optimized algorithm outperforms the baselines by 1---2 orders of magnitude in execution time while achieving comparable accuracy. Meanwhile, the optimized algorithm exhibits linear scalability with respect to the size of the input uncertain graph.

The k-truss of a graph is the largest edge-induced subgraph such that every edge is contained in at least k triangles within the subgraph, where a triangle is a cycle consisting of three vertices. As a new notion of cohesive subgraphs, truss has recently attracted a lot of research attentions in the database and data mining fields. At the same time, uncertainty is an intrinsic property of massive graph data, and truss decomposition (i.e., finding all k-trusses of a graph) has become a key primitive on uncertain graphs. In this paper, we study the truss decomposition problem on uncertain graphs, that is, finding all highly probable k-trusses of an uncertain graph. We first give an formal statement of the truss decomposition problem on uncertain graphs. Then, we prove that the truss decomposition of an uncertain graph attains two elegant properties, namely uniqueness and hierarchy. We show that the truss decomposition of an uncertain graph can be found in $$O(m^{1.5}Q)$$O(m1.5Q) time by proposing an in-memory algorithm called $$\mathtt {TD_{mem}}$$TDmem, where m is the number of edges of the uncertain graph, and Q is at most the maximum number of common neighbors of the endpoints of an edge. When an uncertain graph is too large to fit into main memory, we propose an external-memory algorithm $$\mathtt {TD_{I/O}}$$TDI/O to find the truss decomposition of the uncertain graph. Extensive experiments have been carried out to evaluate the practical performance of the proposed algorithms. The experimental results verify that both $$\mathtt {TD_{mem}}$$TDmem and $$\mathtt {TD_{I/O}}$$TDI/O are efficient when an uncertain graph is small enough to fit into main memory, and that $$\mathtt {TD_{I/O}}$$TDI/O is much faster than $$\mathtt {TD_{mem}}$$TDmem when the graph is too large to fit into main memory.

Data cube is the foundation of on-line analytical processing (OLAP), which can provide users with data views from different perspectives and granularities. Heterogeneous information networks consist of multiple types of nodes and edges which represent different semantic relations. With the rapid development of social networks and knowledge graphs, heterogeneous information networks have become increasingly popular. In heterogeneous information networks, cube is the set of aggregate graphs and cube query is required for supporting OLAP. The existing research mainly studies aggregate graph query on homogeneous networks, but only considers the attributes of nodes. To overcome these challenges, this paper investigates cube query problem on heterogeneous information networks. (1) A novel cube model for heterogeneous information networks is proposed, which captures both the attribute and structure semantics. (2) Because the total number of aggregate graphs is huge, computing and storing them cost plenty of time and storage. The problem of partial cube materialization on heterogeneous information networks is investigated. Given a fixed size of memory space, select a subset of aggregate graphs in cube, to minimize the computing cost of the whole cube. This optimization problem is proved to be NP-complete and there is no $$n^{1-{\epsilon }}$$n1-∈ approximation algorithm unless P $$=$$= NP. (3) A greedy algorithm is proposed for partial cube materialization based on two interesting dependencies between aggregate graphs, attribute dependence and path dependence. (4) Experiments on real world data sets show the cube definition is meaningful, and the partial cube materialization algorithm is efficient.

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.

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.