Uncertain Graph Research Articles

Uncertain maximal frequent subgraph mining algorithm based on adjacency matrix and weight

How to mine many interesting subgraphs in uncertain graph has become an important research field in data mining. In this paper, a novel algorithm Uncertain Maximal Frequent Subgraph Mining Algorithm Based on Adjacency Matrix and Weight (UMFGAMW) is proposed. The definition of the adjacency matrix and the standard matrix coding for uncertain graph are presented. The correspondence between the adjacency matrix and uncertain graph is established. A new vertex ordering policy for computing the standard coding of uncertain graph adjacency matrix is designed. The complexity of uncertain graph standard coding is reduced, and the matching speed of uncertain subgraph standard coding is improved. The definition of the weight of uncertain graph and the mean weight of uncertain edge is proposed. The importance of the uncertain subgraphs that meet the minimum support threshold in the graph dataset is fully considered. Finally, a depth-first search weighted uncertain maximal frequent subgraph mining algorithm is discussed. According to the limiting condition of the uncertain maximum frequent subgraph and weighed uncertain edge, the number of mining results is reduced effectively. Experimental results demonstrate that the UMFGAMW algorithm has higher efficiency and better scalability.

On efficiently finding reverse k-nearest neighbors over uncertain graphs

Reverse k-nearest neighbor ($$\hbox {R}k\hbox {NN}$$RkNN) query on graphs returns the data objects that take a specified query object q as one of their k-nearest neighbors. It has significant influence in many real-life applications including resource allocation and profile-based marketing. However, to the best of our knowledge, there is little previous work on $$\hbox {R}k\hbox {NN}$$RkNN search over uncertain graph data, even though many complex networks such as traffic networks and protein---protein interaction networks are often modeled as uncertain graphs. In this paper, we systematically study the problem of reversek-nearest neighbor search on uncertain graphs ($$\hbox {UG-R}k\hbox {NN}$$UG-RkNN search for short), where graph edges contain uncertainty. First, to address $$\hbox {UG-R}k\hbox {NN}$$UG-RkNN search, we propose three effective heuristics, i.e., GSP, EGR, and PBP, which minimize the original large uncertain graph as a much smaller essential uncertain graph, cut down the number of possible graphs via the newly introduced graph conditional dominance relationship, and reduce the validation cost of data nodes in order to improve query efficiency. Then, we present an efficient algorithm, termed as SDP, to support $$\hbox {UG-R}k\hbox {NN}$$UG-RkNN retrieval by seamlessly integrating the three heuristics together. In view of the high complexity of $$\hbox {UG-R}k\hbox {NN}$$UG-RkNN search, we further present a novel algorithm called TripS, with the help of an adaptive stratified sampling technique. Extensive experiments using both real and synthetic graphs demonstrate the performance of our proposed algorithms.

Enumeration of Maximal Cliques from an Uncertain Graph

We consider the enumeration of dense substructures (maximal cliques) from an uncertain graph. For parameter 0 <; α <; 1, we define the notion of an a-maximal clique in an uncertain graph. We present matching upper and lower bounds on the number of a-maximal cliques possible within a (uncertain) graph. We present an algorithm to enumerate a-maximal cliques whose worst-case runtime is near-optimal, and an experimental evaluation showing the practical utility of the algorithm.

An uncertain chromatic number of an uncertain graph based on $$\alpha $$ α -cut coloring

An uncertain graph is a graph in which the edges are indeterminate and the existence of edges are characterized by belief degrees which are uncertain measures. This paper aims to bring graph coloring and uncertainty theory together. A new approach for uncertain graph coloring based on an $$\alpha $$ -cut of an uncertain graph is introduced in this paper. Firstly, the concept of $$\alpha $$ -cut of uncertain graph is given and some of its properties are explored. By means of $$\alpha $$ -cut coloring, we get an $$\alpha $$ -cut chromatic number and examine some of its properties as well. Then, a fact that every $$\alpha $$ -cut chromatic number may be a chromatic number of an uncertain graph is obtained, and the concept of uncertain chromatic set is introduced. In addition, an uncertain chromatic algorithm is constructed. Finally, a real-life decision making problem is given to illustrate the application of the uncertain chromatic set and the effectiveness of the uncertain chromatic algorithm.

DistR: A Distributed Method for the Reachability Query over Large Uncertain Graphs

Among uncertain graph queries, reachability, i.e., the probability that one vertex is reachable from another, is likely the most fundamental one. Although this problem has been studied within the field of network reliability, solutions are implemented on a single computer and can only handle small graphs. However, as the size of graph applications continually increases, the corresponding graph data can no longer fit within a single computer's memory and must therefore be distributed across several machines. Furthermore, the computation of probabilistic reachability queries is #P-complete making it very expensive even on small graphs. In this paper, we develop an efficient distributed strategy, called DistR , to solve the problem of reachability query over large uncertain graphs. Specifically, we perform the task in two steps: distributed graph reduction and distributed consolidation . In the distributed graph reduction step, we find all of the maximal subgraphs of the original graph, whose reachability probabilities can be calculated in polynomial time, compute them and reduce the graph accordingly. After this step, only a small graph remains. In the distributed consolidation step, we transform the problem into a relational join process and provide an approximate answer to the #P-complete reachability query. Extensive experimental studies show that our distributed approach is efficient in terms of both computational and communication costs, and has high accuracy.

On Computing the Edge-Connectivity of an Uncertain Graph

Uncertain graphs are employed to describe graph models with imprecise expert data. In uncertain graphs, due to the existence of uncertain edges, edge-connectivity is essentially an uncertain variable. Different from that in a deterministic graph, it is more suitable to investigate the possibility (uncertain measure) that an uncertain graph is $k$ -edge-connected, which is the main aim of this paper. We first deduce a theoretical formula to calculate the uncertain measure, and on this basis, we then propose an algorithm, which is derived from maximum flow algorithms, to numerically calculate the uncertain measure. The proposed algorithm is also proved to be a polynomial time algorithm, and its effectiveness and efficiency are illustrated by numerical examples.

Towards efficient top-k reliability search on uncertain graphs

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.

Truss decomposition of uncertain graphs

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.

Recursive Stratified Sampling: A New Framework for Query Evaluation on Uncertain Graphs

Uncertain graph management has been recognized as an important research topic in recent years. In this paper, we first introduce two types of query evaluation problems on uncertain graphs, named expectation query evaluation and threshold query evaluation. Most previous solutions for these problems are based on naive Monte-Carlo ( NMC ) sampling, which typically result in large variances. To reduce the variance of NMC , we propose two efficient estimators, called RSS-I and RSS-II estimators, based on the idea of recursive stratified sampling ( RSS ). To further reduce the variances of RSS-I and RSS-II , we propose a recursive cut-set based stratified sampling estimator for a particular kind of query evaluation problem. We show that all the proposed estimators are unbiased and their variances are significantly smaller than that of NMC . Moreover, the time complexity of all the proposed estimators are the same as that of NMC under a mild assumption. In addition, we develop an elegant graph simplification technique to further improve the accuracy and running time of our estimators. We also apply the proposed estimators to three different uncertain graph query evaluation problems. Finally, we conduct extensive experiments to evaluate the proposed estimators, and the results show the accuracy, efficiency, and scalability of our estimators.

Efficient pattern matching on big uncertain graphs

A significant amount of research has been devoted to seeking efficient solutions to the problem of pattern matching over graphs. This interest is largely due to the many applications that require such efficient solutions, including protein complex prediction, social network analysis, and structural pattern recognition. However, in many real applications, the graph data are often noisy, incomplete, and inaccurate. In other words, there exist many uncertain graphs. Therefore, in this paper, we study pattern matching in the context of large uncertain graphs. Specifically, we want to retrieve all qualified matches of a query pattern in the uncertain graph. Though pattern matching over uncertain graphs is NP-hard, we employ a filtering-and-verification framework to speed up the search. In the filtering phase, we propose a probabilistic matching tree (PM-tree) built from match cuts obtained by a cut selection process. Based on the PM-tree, we devise a collective pruning strategy to prune a large number of unqualified matches. During the verification phase, we develop an efficient sampling algorithm to validate the remaining candidates. Extensive experimental results demonstrate the effectiveness and efficiency of the proposed algorithms. Finally, we show how our solution can be applied to querying knowledge graphs.

Uncertain Graph Processing through Representative Instances

Data in several applications can be represented as an uncertain graph whose edges are labeled with a probability of existence. Exact query processing on uncertain graphs is prohibitive for most applications, as it involves evaluation over an exponential number of instantiations. Thus, typical approaches employ Monte-Carlo sampling, which (i) draws a number of possible graphs (samples), (ii) evaluates the query on each of them, and (iii) aggregates the individual answers to generate the final result. However, this approach can also be extremely time consuming for large uncertain graphs commonly found in practice. To facilitate efficiency, we study the problem of extracting a single representative instance from an uncertain graph. Conventional processing techniques can then be applied on this representative to closely approximate the result on the original graph. In order to maintain data utility, the representative instance should preserve structural characteristics of the uncertain graph. We start with representatives that capture the expected vertex degrees, as this is a fundamental property of the graph topology. We then generalize the notion of vertex degree to the concept of n -clique cardinality, that is, the number of cliques of size n that contain a vertex. For the first problem, we propose two methods: Average Degree Rewiring (ADR), which is based on random edge rewiring, and Approximate B-Matching (ABM), which applies graph matching techniques. For the second problem, we develop a greedy approach and a game-theoretic framework. We experimentally demonstrate, with real uncertain graphs, that indeed the representative instances can be used to answer, efficiently and accurately, queries based on several metrics such as shortest path distance, clustering coefficient, and betweenness centrality.

On uncertain graphs modeling and queries

Large-scale, highly-interconnected networks pervade both our society and the natural world around us. Uncertainty, on the other hand, is inherent in the underlying data due to a variety of reasons, such as noisy measurements, lack of precise information needs, inference and prediction models, or explicit manipulation, e.g., for privacy purposes. Therefore, uncertain, or probabilistic, graphs are increasingly used to represent noisy linked data in many emerging application scenarios, and they have recently become a hot topic in the database research community. While many classical graph algorithms such as reachability and shortest path queries become # P -complete, and hence, more expensive in uncertain graphs; various complex queries are also emerging over uncertain networks, such as pattern matching, information diffusion, and influence maximization queries. In this tutorial, we discuss the sources of uncertain graphs and their applications, uncertainty modeling, as well as the complexities and algorithmic advances on uncertain graphs processing in the context of both classical and emerging graph queries. We emphasize the current challenges and highlight some future research directions.

English

English

An Approach for Keyword Searching inUncertain Graph Data

Keyword searching is generally used for retrieving the relevant data from the database. For input query, the related data is retrieved. But it is tedious task to search keyword on uncertain graph. In this paper, the keyword searching technique over uncertain graph is introduced. The Keyword routing method is used to route the keywords to relevant source. In this approach two methods are included. The keyword relationship graph deduces the relationship between keywords and the element mentioning them. The scoring mechanism computes the score of keywords at each level which reduces the ambiguity. The result will include the subtree of the entire graph which includes all keywords of input query having high score and in addition it retrieves the most relevant data . Effective results are derived from employed method.

Subgraph similarity maximal all-matching over a large uncertain graph

Recently, uncertain graph data management and mining techniques have attracted significant interests and research efforts due to potential applications such as protein interaction networks and soci...

Threshold-Based Shortest Path Query over Large Correlated Uncertain Graphs

With the popularity of uncertain data, queries over uncertain graphs have become a hot topic in the database community. As one of the important queries, the shortest path query over an uncertain graph has attracted much attention of researchers due to its wide applications. Although there are some efficient solutions addressing this problem, all existing models ignore an important property existing in uncertain graphs: the correlation among the edges sharing the same vertex. In this paper, we apply Markov network to model the hidden correlation in uncertain graphs and compute the shortest path. Unfortunately, calculating the shortest path and corresponding probability over uncertain graphs modeled by Markov networks is a #P-hard problem. Thus, we propose a filtering-and-verification framework to accelerate the queries. In the filtering phase, we design a probabilistic shortest path index based on vertex cuts and blocks of a graph. We find a series of upper bounds and prune the vertices and edges whose upper bounds of the shortest path probability are lower than the threshold. By carefully picking up the blocks and vertex cuts, the index is optimized to have the maximum pruning capability, so that we can filter a large number of vertices which make no contribution to the final shortest path query results. In the verification phase, we develop an efficient sampling algorithm to determine the final query answers. Finally, we verify the efficiency and effectiveness of our solutions with extensive experiments.

Probabilistic Path Queries in Path Networks: An Effective and Efficient Clustering Methods

Efficiently processing shortest path (SP) queries over stochastic networks attracted a lot of research attention as such queries are very popular in the emerging real world applications such as Intelligent Transportation Systems and communication networks whose edge weights can be modeled as a random variable. Some pervious works aim at finding the most likely SP (the path with largest probability to be SP), and others search the least-expected-weight path. In all these works, the definitions of the shortest path query are based on simple probabilistic models which can be converted into the multi-objective optimal issues on a weighted graph. Challenging problem, two algorithms, the PEEDR and the CPGS clustering algorithm. Reliable clusters are those which are not likely to be disconnected in the context of different instantiations of the uncertain graph. we provide a generalized reliability measurement from two basic intuitions (purity and size balance) to overcome the challenges from standard reliability criterion, and develop a novel k-means algorithm to solve the uncertain clustering problem

Path Optimality Conditions for Minimum Spanning Tree Problem with Uncertain Edge Weights

This paper investigates the uncertain minimum spanning tree (UMST) problem where the edge weights are assumed to be uncertain variables. In order to propose effective solving methods for the UMST problem, path optimality conditions as well as some equivalent definitions for two commonly used types of UMST, namely, uncertain expected minimum spanning tree (expected UMST) and uncertain α-minimum spanning tree (α-UMST), are discussed. It is shown that both the expected UMST problem and the α-UMST problem can be transformed into an equivalent classical minimum spanning tree problem on a corresponding deterministic graph, which leads to effective algorithms with low computational complexity. Furthermore, the notion of uncertain most minimum spanning tree (most UMST) is initiated for an uncertain graph, and then the equivalent relationship between the α-UMST and the most UMST is proved. Numerical examples are presented as well for illustration.

Tree index of uncertain graphs

With the development of system complexity, a variety of nondeterministic information is frequently encountered in networks. To deal with the nondeterministic information, uncertainty theory is proposed as a powerful tool. Within the framework of uncertainty theory, in this paper, a concept of tree index of uncertain graph is proposed. It also gives a method to calculate tree index of uncertain graphs. Moreover, the tree index of uncertain cycle and the path index of uncertain graphs are proposed and analyzed in a similar way.

Reliability Evaluation of the Minimum Spanning Tree on Uncertain Graph

Minimum spanning tree is a minimum-cost spanning tree connecting the whole network, but it couldn't be directly obtained on uncertain graph. In this paper, we define the reliability as the existence probability of all minimum spanning trees and present an algorithm for evaluating reliability of the minimum spanning tree on uncertain graph. The time complexity of the algorithm is O(Nmn), where n, m and N stand for the number of vertices, edges and minimum spanning trees, respectively. Because this algorithm spends more time finding minimum spanning tree, we propose an improved algorithm whose time complexity is O(Nm). The improved algorithm uses disjoint set data structure so that the average time complexity on finding a new minimum spanning tree is O(m/n). The two algorithms are analyzed in detail and the experiment results agree with theoretical analysis.