Large Uncertain Graphs Research Articles

Core Decomposition on Uncertain Graphs Revisited

Core decomposition on uncertain graphs is a fundamental problem in graph analysis. Given an uncertain graph G, the core decomposition problem is to determine all (k, \eta)-cores in G, where a (k, \eta)-core is a maximal subgraph of G such that each node has an \eta-degree no less than k within the subgraph. The state-of-the-art algorithm for solving this problem is based on a peeling technique which iteratively removes nodes with the smallest \eta-degrees and also dynamically updates their neighbors' \eta-degrees. Unfortunately, we find that such a peeling algorithm with the dynamical \eta-degree updating technique is incorrect due to the inaccuracy of the recursive floating-point number division operations involved in the dynamical updating procedure. To solve this problem, we propose a bottom-up algorithm based on an on-demand computational strategy. To further improve the efficiency, we also develop a more-efficient top-down algorithm with several nontrivial optimization techniques. Both of our algorithms do not involve any floating-point number division operations, thus the correctness can be guaranteed. We conduct extensive experiments to evaluate our algorithms using five large real-life datasets. The results show that our algorithms are at least three orders of magnitude faster than the existing exact algorithms on large uncertain graphs.

Computing K-Cores in Large Uncertain Graphs: An Index-based Optimal Approach

Uncertain graph management and analysis have attracted many research attentions. Among them, computing k-cores in uncertain graphs (aka, (k,)-cores) is an important problem and has emerged in many applications. Given an uncertain graph, the (k,)-cores can be derived by iteratively removing the vertex with an -degree of less than k and updating the -degrees of its neighbors. However, the results heavily depend on the two input parameters k and, and the settings for these parameters are unique to the specific graph structure and the user's subjective requirements. Additionally, computing and updating the -degree for each vertex is costly. To overcome these drawbacks, we have developed an index-based solution for computing (k,)-cores in this paper. The size of the index is well bounded by O(m), where m is the number of edges in the graph. Based on this index, queries can be answered in optimal time. We propose an algorithm for index construction with several different optimizations. We also propose a new algorithm for index construction in external memory, when the uncertain graph cannot be entirely loaded in memory. We conduct extensive experiments on eight real-world datasets to practically evaluate the performance of all the proposed algorithms.

Efficient clustering of large uncertain graphs using neighborhood information

This work addresses the problem of clustering large uncertain graphs. The data is represented as a graph where the proposed solution uses the neighborhood information for the purpose of clustering. The proposed approach converts an uncertain graph to a certain graph by predicting about the existence of the edges in the uncertain graph. For the purpose of prediction, a classifier is used. The proposed approach is compared with baseline approaches for clustering graphs having uncertainties over the edges; uncertain k-means (UK-Mean) and Fuzzy-DBSCAN (FDBSCAN). Additionally, the results are also compared with two state-of-the-art approaches namely, CUDAP (clustering algorithm for uncertain data based on approximate backbone) and PEEDR (partially expected edit distance reduction). Experiments are conducted using two natively uncertain and nine synthetically converted uncertain benchmark datasets. The results are compared with the baseline and the state-of-the-art methods using Davies–Bouldin index, Dunn index and Silhouette coefficient, widely used cluster validity indices. The results show that the proposed approach performs better than the other four methods.

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.

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.

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.

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...

Graph similarity search on large uncertain graph databases

Many studies have been conducted on seeking an efficient solution for graph similarity search over certain (deterministic) graphs due to its wide application in many fields, including bioinformatics, social network analysis, and Resource Description Framework data management. All prior work assumes that the underlying data is deterministic. However, in reality, graphs are often noisy and uncertain due to various factors, such as errors in data extraction, inconsistencies in data integration, and for privacy-preserving purposes. Therefore, in this paper, we study similarity graph containment search on large uncertain graph databases. Similarity graph containment search consists of subgraph similarity search and supergraph similarity search. Different from previous works assuming that edges in an uncertain graph are independent of each other, we study uncertain graphs where edges' occurrences are correlated. We formally prove that subgraph or supergraph similarity search over uncertain graphs is $$\#$&#P-hard; thus, we employ a filter-and-verify framework to speed up these two queries. For the subgraph similarity query, in the filtering phase, we develop tight lower and upper bounds of subgraph similarity probability based on a probabilistic matrix index (PMI). PMI is composed of discriminative subgraph features associated with tight lower and upper bounds of subgraph isomorphism probability. Based on PMI, we can filter out a large number of uncertain graphs and maximize the pruning capability. During the verification phase, we develop an efficient sampling algorithm to validate the remaining candidates. For the supergraph similarity query, in the filtering phase, we propose two pruning algorithms, one lightweight and the other strong, based on maximal common subgraphs of query graph and data graph. We run the two pruning algorithms against a probabilistic index that consists of powerful graph features. In the verification, we design an approximate algorithm based on the Horvitz---Thompson estimator to fast validate the remaining candidates. The efficiencies of our proposed solutions to the subgraph and supergraph similarity search have been verified through extensive experiments on real uncertain graph datasets.

Probabilistic SimRank computation over uncertain graphs

Existing studies on graph-based analysis focus on standard graphs that are precise and complete. However, in many prevalent application domains, such as social, biological, and mobile networks, the existence of an edge in a graph is best captured by a likelihood measure or a probability. This paper investigates the problem of node similarity computation on large uncertain graphs. We first extend the well-known SimRank to define meaningful similarity measure on probabilistic graphs, and then propose a probabilistic framework to compute it. Based on the observation that the probabilistic transition probabilities computed on the whole graph equal to those computed on multiple sub-graphs, we design a SubG algorithm to compute the probabilistic transition matrix. Furthermore, we propose an efficient dynamic programming algorithm to degrade the time complexity from exponential to polynomial. We give the corresponding theoretical justification and extend the proposed algorithm to incremental dynamic programming algorithm, which further improves the efficiency and can be applied to dynamic graphs. Extensive experiments on synthetic and real data sets verify that the proposed methods are efficient and effective.

Mining Top-k Maximal Cliques from Large Uncertain Graphs

Mining Top-k Maximal Cliques from Large Uncertain Graphs

Efficient subgraph search over large uncertain graphs

Retrieving graphs containing a query graph from a large graph database is a key task in many graph-based applications, including chemical compounds discovery, protein complex prediction, and structural pattern recognition. However, graph data handled by these applications is often noisy, incomplete, and inaccurate because of the way the data is produced. In this paper, we study subgraph queries over uncertain graphs. Specifically, we consider the problem of answering threshold-based probabilistic queries over a large uncertain graph database with the possible world semantics. We prove that problem is #P-complete, therefore, we adopt a filtering-and-verification strategy to speed up the search. In the filtering phase, we use a probabilistic inverted index, PIndex, based on subgraph features obtained by an optimal feature selection process. During the verification phase, we develop exact and bound algorithms to validate the remaining candidates. Extensive experimental results demonstrate the effectiveness of the proposed algorithms.

Mining Frequent Subgraph Patterns from Uncertain Graph Data

In many real applications, graph data is subject to uncertainties due to incompleteness and imprecision of data. Mining such uncertain graph data is semantically different from and computationally more challenging than mining conventional exact graph data. This paper investigates the problem of mining uncertain graph data and especially focuses on mining frequent subgraph patterns on an uncertain graph database. A novel model of uncertain graphs is presented, and the frequent subgraph pattern mining problem is formalized by introducing a new measure, called expected support. This problem is proved to be NP-hard. An approximate mining algorithm is proposed to find a set of approximately frequent subgraph patterns by allowing an error tolerance on expected supports of discovered subgraph patterns. The algorithm uses efficient methods to determine whether a subgraph pattern can be output or not and a new pruning method to reduce the complexity of examining subgraph patterns. Analytical and experimental results show that the algorithm is very efficient, accurate, and scalable for large uncertain graph databases. To the best of our knowledge, this paper is the first one to investigate the problem of mining frequent subgraph patterns from uncertain graph data.