Mining Uncertain Graphs: An Overview

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.

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.

Community search (CS) on graphs returns the largest densely connected vertex subset containing a query vertex, namely k-community, where every vertex's degree in the induced subgraph is not less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> . It has significant influence in many real-life applications including event organization and friend recommendation. Many complex networks such as social networks and protein-protein interaction (PPI) networks are often modeled as uncertain graphs. In this paper, we identify and study the problem of reliable community search on uncertain graphs (UCS for short). Given an uncertain graph, a query vertex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$q$</tex> , a positive integer <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> and a probability threshold θ, the reliable community, viz., (k, θ) -community, of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$q$</tex> is the largest vertex subset, so that the probability of every vertex to be in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$q$</tex> 's k-community is not less than θ. We prove that it is a NP-hard problem. We propose two novel pruning strategies to reduce the candidate set to a much smaller size. We develop an efficient index, namely CD-index, with which the pruning process can be done in optimal time. We also present efficient sampling algorithms on top of stratified sampling and lazy sampling to accelerate the search under accuracy guarantees. Extensive experiments using four real-world datasets demonstrate the superior performance of proposed algorithms to the state-of-the-art approaches.

Uncertain graph models are widely used in real-world applications such as knowledge graphs and social networks. To capture the uncertainty, each edge in an uncertain graph is associated with an existential probability that signifies the likelihood of the existence of the edge. One notable issue of querying uncertain graphs is that the results are sometimes uninformative because of the edge uncertainty. In this paper, we consider probabilistic reachability queries, which are one of the fundamental classes of graph queries. To make the results more informative, we adopt a crowdsourcing-based approach to clean the uncertain edges. However, considering the time and monetary cost of crowdsourcing, it is a problem to efficiently select a limited set of edges for cleaning that maximizes the quality improvement. We prove that the edge selection problem is #P-hard. In light of the hardness of the problem, we propose a series of edge selection algorithms, followed by a number of optimization techniques and pruning heuristics for reducing the computation time. Our experimental results demonstrate that our proposed techniques outperform a random selection by up to 27 times in terms of the result quality improvement and the brute-force solution by up to 60 times in terms of the elapsed time.

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.

Cohesive subgraph mining is a fundamental problem in the field of graph data analysis. Many existing cohesive graph mining algorithms are mainly tailored to deterministic graphs. Real-world graphs, however, are often not deterministic, but uncertain in nature. Applications of such uncertain graphs include protein-protein interactions networks with experimentally inferred links and sensor networks with uncertain connectivity links. In this article, we study the problem of mining cohesive subgraphs from an uncertain graph. Specifically, we introduce a new <inline-formula><tex-math notation="LaTeX">$(\alpha,\gamma)$</tex-math></inline-formula> -quasi-clique model to model the cohesive subgraphs in an uncertain graph, and propose a basic enumeration algorithm to find all maximal <inline-formula><tex-math notation="LaTeX">$(\alpha,\gamma)$</tex-math></inline-formula> -quasi-cliques. We also develop an advanced enumeration algorithm based on several novel pruning rules, including early termination and candidate set reduction. To further improve the efficiency, we propose several optimization techniques. Extensive experiments on five real-world datasets demonstrate that our solutions are almost three times faster than the baseline approach.

Maximum weight independent sets and cliques in intersection graphs of filaments

With the exponential growth of graph‐structured data, single‐machine efficient analysis has become increasingly impractical, making high‐performance distributed graph computing systems indispensable. The efficacy of these systems hinges critically on high‐quality graph partitioning. The edges of many graphs stemmed from real applications are uncertain, but many existing graph partitioning algorithms are only for deterministic graphs without considering uncertainty. This paper presents a novel partitioning algorithm, PAUG (Partitioning Algorithm for Uncertain Graphs), tailored for uncertain graphs. First, it formalizes the partitioning problem as an optimization task to minimize the cut‐edge ratio while balancing load. Second, it introduces probabilistic similarity to quantify vertex relationships under uncertainty. Finally, it details the PAUG algorithm which consists of initial partition phase and score‐function‐guided refinement strategy. Experimental results shows that PAUG achieves an average 23.2% reduction in cut‐edge ratio and a 26.2% improvement in load balance over state‐of‐the‐art algorithms.

The undirected capacitated arc routing problem with profits

The authors propose a method for solving the stereo correspondence problem. The method consists of extracting local image structures and matching similar such structures between two images. Linear edge segments are extracted from both the left and right images. Each segment is characterized by its position and orientation in the image as well as its relationships with the nearby segments. A relational graph is thus built from each image. For each segment in one image as set of potential assignments is represented as a set of nodes in a correspondence graph. Arcs in the graph represent compatible assignments established on the basis of segment relationships. Stereo matching becomes equivalent to searching for sets of mutually compatible nodes in this graph. Sets are found by looking for maximal cliques. The maximal clique best suited to represent a stereo correspondence is selected using a benefit function. Numerous results obtained with this method are shown.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In recent years, large-scale studies have been undertaken to describe, at least partially, protein-protein interaction maps, or interactomes, for a number of relevant organisms, including human. However, current interactomes provide a somehow limited picture of the molecular details involving protein interactions, mostly because essential experimental information, especially structural data, is lacking. Indeed, the gap between structural and interactomics information is enlarging and thus, for most interactions, key experimental information is missing. We elaborate on the observation that many interactions between proteins involve a pair of their constituent domains and, thus, the knowledge of how protein domains interact adds very significant information to any interactomic analysis. In this work, we describe a novel use of the neighborhood cohesiveness property to infer interactions between protein domains given a protein interaction network. We have shown that some clustering coefficients can be extended to measure a degree of cohesiveness between two sets of nodes within a network. Specifically, we used the meet/min coefficient to measure the proportion of interacting nodes between two sets of nodes and the fraction of common neighbors. This approach extends previous works where homolog coefficients were first defined around network nodes and later around edges. The proposed approach substantially increases both the number of predicted domain-domain interactions as well as its accuracy as compared with current methods.

The k-core of a graph is the largest subgraph in which every vertex is connected to at least k other vertices within the subgraph. Core decomposition finds the k-core of the graph for every possible k. Past studies have shown important applications of core decomposition such as in the study of the properties of large networks (e.g., sustainability, connectivity, centrality, etc.), for solving NP-hard problems efficiently in real networks (e.g., maximum clique finding, densest subgraph approximation, etc.), and for large-scale network fingerprinting and visualization. The k-core is a well accepted concept partly because there exists a simple and efficient algorithm for core decomposition, by recursively removing the lowest degree vertices and their incident edges. However, this algorithm requires random access to the graph and hence assumes the entire graph can be kept in main memory. Nevertheless, real-world networks such as online social networks have become exceedingly large in recent years and still keep growing at a steady rate. In this paper, we propose the first external-memory algorithm for core decomposition in massive graphs. When the memory is large enough to hold the graph, our algorithm achieves comparable performance as the in-memory algorithm. When the graph is too large to be kept in the memory, our algorithm requires only O(k max ) scans of the graph, where k max is the largest core number of the graph. We demonstrate the efficiency of our algorithm on real networks with up to 52.9 million vertices and 1.65 billion edges.

We consider an optimization problem arising in the design of optical networks. We are given a bipartite graph G = (L,R,E) over the node set L ∪ R where the edge set is E ⊆ {[u, v] : u ∈ L, v ∈ R}, and implicitly a collection of all four-nodes cycles in the complete graph over V. The goal is to find a minimum size sub-collection of graphs G1,G2, . . . , Gp where for each i Gi is isomorphic to a cycle over four nodes, and such that the edge set E is contained in the union (over all i) of the edge sets of Gi. Noting that every four edge cycle can be a part of the solution, this covering problem is a special case of the unweighted 4-set cover problem. This specialization allows us to obtain an improved approximation guarantee. Whereas the currently best known approximation algorithm for the general unweighted 4-set cover problem has an approximation ratio of H4 - 196/390 ≅ 1.58077 (where Hp denotes the p-th harmonic number), we show that for every Ɛ > 0 there is a polynomial time (13/10 + Ɛ)-approximation algorithm for our problem. Our analysis of the greedy algorithm shows that when applied to covering a bipartite graph using copies of Kq,q bicliques, it returns a feasible solution whose cost is at most (Hq2 -Hq +1)OPT +1 where OPT denotes the optimal cost, thus improving the approximation bound by a factor of almost 2.

The k-core is commonly used as a measure of importance and well connectedness for nodes in diverse applications in social networks and bioinformatics. Since network data is commonly noisy and incomplete, a fundamental issue is to understand how robust the core decomposition is to noise. Further, in many settings, such as online social media networks, usually only a sample of the network is available. Therefore, a related question is: How robust is the top core set under such sampling?We find that, in general, the top core is quite sensitive to both noise and sampling; we quantify this in terms of the Jaccard similarity of the set of top core nodes between the original and perturbed/sampled graphs. Most importantly, we find that the overlap with the top core set varies non-monotonically with the extent of perturbations/sampling. We explain some of these empirical observations by rigorous analysis in simple network models. Our work has important implications for the use of the core decomposition and nodes in the top cores in network analysis applications, and suggests the need for a more careful characterization of the missing data and sensitivity to it.KeywordsRandom GraphNoise ModelCitation NetworkLink PredictionJaccard IndexThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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