大规模时序图中持续性稠密子图搜索算法研究

时序图是一种边上带有时间戳的图结构,其中边上的时间戳表示该边出现时间,即图随时间变化不断变化.图数据中的稠密子图挖掘问题具有非常强烈的现实意义.目前,时序图中大多数现有的工作都集中在稠密子图检测问题,该问题目标是找到时序图中所有的目标子图.然而,当时序图的规模过大时,这一问题将变得极其复杂且收效甚微.旨在研究在时序图中...     

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 social networks. Specifically, as a fundamental problem, subgraph similarity all-matching is widely applied in exploratory data analysis. The purpose of subgraph similarity all-matching is to find all the similarity occurrences of the query graph in a large data graph. Numerous algorithms and pruning methods have been developed for the subgraph matching problem over a certain graph. However, insufficient efforts are devoted to subgraph similarity all-matching over an uncertain data graph, which is quite challenging due to high computation costs. In this paper, we define the problem of subgraph similarity maximal all-matching over a large uncertain data graph and propose a framework to solve this problem. To further improve the efficiency, several speed-up techniques are proposed such as the partial graph evaluation, the vertex pruning, the calculation model transformation, the incremental evaluation method and the probability upper bound filtering. Finally, comprehensive experiments are conducted on real graph data to test the performance of our framework and optimization methods. The results verify that our solutions can outperform the basic approach by orders of magnitudes in efficiency.     

层次序列索引的大规模动态标签图子图查询

标签图常用于智能交通网、生物信息网等新兴领域的建模。子图查询作为图数据分析的关键问题，引起了研究者的广泛关注。对现有子图查询算法的研究发现，随着图数据规模增大且频繁更新，传统子图查询算法普遍存在查询效率低，存储开销大，忽略顶点标签信息等问题。为此，提出了一种支持大规模动态标签图子图查询的层次序列索引（Dynamic Hierarchical Sequence，DHS），该索引提取数据图中带有顶点编号的层次拓扑序列关系以实现子图查询；针对图的动态变化，提出了更新点拓扑扩展式索引维护策略，仅从局部变化顶点及边开始进行增量式更新，大大降低了重建索引造成的巨大开销；提出了基于DHS索引的子图查询方法，仅需将查询图与数据图的层次序列进行匹配即可获得候选集，并在其上利用关系匹配策略获得最终查询结果。实验证明提出的方法在保证高效查询的同时降低了索引的创建及维护时间，提高了子图查询效率。     

An Efficient Framework for Multiple Subgraph Pattern Matching Models

With the popularity of storing large data graph in cloud, the emergence of subgraph pattern matching on a remote cloud has been inspired. Typically, subgraph pattern matching is defined in terms of subgraph isomorphism, which is an NP-complete problem and sometimes too strict to find useful matches in certain applications. And how to protect the privacy of data graphs in subgraph pattern matching without undermining matching results is an important concern. Thus, we propose a novel framework to achieve the privacy-preserving subgraph pattern matching in cloud. In order to protect the structural privacy in data graphs, we firstly develop a k-automorphism model based method. Additionally, we use a cost-model based label generalization method to protect label privacy in both data graphs and pattern graphs. During the generation of the k-automorphic graph, a large number of noise edges or vertices might be introduced to the original data graph. Thus, we use the outsourced graph, which is only a subset of a k-automorphic graph, to answer the subgraph pattern matching. The efficiency of the pattern matching process can be greatly improved in this way. Extensive experiments on real-world datasets demonstrate the high efficiency of our framework.

     

On Independent Vertex Sets in Subclasses of Apple-Free Graphs

In a finite undirected graph, an apple consists of a chordless cycle of length at least 4, and an additional vertex which is not in the cycle and sees exactly one of the cycle vertices. A graph is apple-free if it contains no induced subgraph isomorphic to an apple. Apple-free graphs are a common generalization of chordal graphs, claw-free graphs and cographs and occur in various papers. The Maximum Weight Independent Set (MWS) problem is efficiently solvable on chordal graphs, on cographs as well as on claw-free graphs. In this paper, we obtain partial results on some subclasses of apple-free graphs where our results show that the MWS problem is solvable in polynomial time. The main tool is a combination of clique separators with modular decomposition. Our algorithms are robust in the sense that there is no need to recognize whether the input graph is in the given graph class; the algorithm either solves the MWS problem correctly or detects that the input graph is not in the given class.     

Querying massive graph data: A compress and search approach

Querying graph data is a fundamental problem that witnesses an increasing interest especially for massive graph databases which come as a promising alternative to relational databases for big data modeling. In this paper, we study the problem of subgraph isomorphism search which consists to enumerate the embedding of a query graph in a data graph. The most known solutions of this NP-complete problem are backtracking-based and result in a high computational cost when we deal with massive graph databases. We address this problem and its challenges via graph compression with modular decomposition. In our approach, subgraph isomorphism search is performed on compressed graphs without decompressing them yielding substantial reduction of the search space and consequently a significant saving in processing time as well as in storage space for the graphs. We evaluated our algorithms on nine real-word datasets. The experimental results show that our approach is efficient and scalable.     

Fast processing of graph queries on a large database of small and medium-sized data graphs

We propose a new way of indexing a large database of small and medium-sized graphs and processing exact subgraph matching (or subgraph isomorphism) and approximate (full) graph matching queries. Rather than decomposing a graph into smaller units (e.g., paths, trees, graphs) for indexing purposes, we represent each graph in the database by its graph signature, which is essentially a multiset. We construct a disk-based index on all the signatures via bulk loading. During query processing, a query graph is also mapped into its signature, and this signature is searched using the index by performing multiset operations. To improve the precision of exact subgraph matching, we develop a new scheme using the concept of line graphs. Through extensive evaluation on real and synthetic graph datasets, we demonstrate that our approach provides a scalable and efficient disk-based solution for a large database of small and medium-sized graphs.     

Fast graph query processing with a low-cost index

This paper studies the problem of processing supergraph queries, that is, given a database containing a set of graphs, find all the graphs in the database of which the query graph is a supergraph. Existing works usually construct an index and performs a filtering-and-verification process, which still requires many subgraph isomorphism testings. There are also significant overheads in both index construction and maintenance. In this paper, we design a graph querying system that achieves both fast indexing and efficient query processing. The index is constructed by a simple but fast method of extracting the commonality among the graphs, which does not involve any costly operation such as graph mining. Our query processing has two key techniques, direct inclusion and filtering. Direct inclusion allows partial query answers to be included directly without candidate verification. Our filtering technique further reduces the candidate set by operating on a much smaller projected database. Experimental results show that our method is significantly more efficient than the existing works in both indexing and query processing, and our index has a low maintenance cost.     

Structure and attribute index for approximate graph matching in large graphs

The increasing popularity of graph data in various domains has lead to a renewed interest in developing efficient graph matching techniques, especially for processing large graphs. In this paper, we study the problem of approximate graph matching in a large attributed graph. Given a large attributed graph and a query graph, we compute a subgraph of the large graph that best matches the query graph. We propose a novel structure-aware and attribute-aware index to process approximate graph matching in a large attributed graph. We first construct an index on the similarity of the attributed graph, by partitioning the large search space into smaller subgraphs based on structure similarity and attribute similarity. Then, we construct a connectivity-based index to give a concise representation of inter-partition connections. We use the index to find a set of best matching paths. From these best matching paths, we compute the best matching answer graph using a greedy algorithm. Experimental results on real datasets demonstrate the efficiency of both index construction and query processing. We also show that our approach attains high-quality query answers.     

Incremental <Emphasis Type="Italic">k</Emphasis>-core decomposition: algorithms and evaluation

A k-core of a graph is a maximal connected subgraph in which every vertex is connected to at least k vertices in the subgraph. k-core decomposition is often used in large-scale network analysis, such as community detection, protein function prediction, visualization, and solving NP-hard problems on real networks efficiently, like maximal clique finding. In many real-world applications, networks change over time. As a result, it is essential to develop efficient incremental algorithms for dynamic graph data. In this paper, we propose a suite of incremental k-core decomposition algorithms for dynamic graph data. These algorithms locate a small subgraph that is guaranteed to contain the list of vertices whose maximum k-core values have changed and efficiently process this subgraph to update the k-core decomposition. We present incremental algorithms for both insertion and deletion operations, and propose auxiliary vertex state maintenance techniques that can further accelerate these operations. Our results show a significant reduction in runtime compared to non-incremental alternatives. We illustrate the efficiency of our algorithms on different types of real and synthetic graphs, at varying scales. For a graph of 16 million vertices, we observe relative throughputs reaching a million times, relative to the non-incremental algorithms.     

Maximal Biclique Subgraphs and Closed Pattern Pairs of the Adjacency Matrix: A One-to-One Correspondence and Mining Algorithms

Maximal biclique (also known as complete bipartite) subgraphs can model many applications in Web mining, business, and bioinformatics. Enumerating maximal biclique subgraphs from a graph is a computationally challenging problem, as the size of the output can become exponentially large with respect to the vertex number when the graph grows. In this paper, we efficiently enumerate them through the use of closed patterns of the adjacency matrix of the graph. For an undirected graph G without self-loops, we prove that 1) the number of closed patterns in the adjacency matrix of G is even, 2) the number of the closed patterns is precisely double the number of maximal biclique subgraphs of G, and 3) for every maximal biclique subgraph, there always exists a unique pair of closed patterns that matches the two vertex sets of the subgraph. Therefore, the problem of enumerating maximal bicliques can be solved by using efficient algorithms for mining closed patterns, which are algorithms extensively studied in the data mining field. However, this direct use of existing algorithms causes a duplicated enumeration. To achieve high efficiency, we propose an O(mn) time delay algorithm for a nonduplicated enumeration, in particular, for enumerating those maximal bicliques with a large size, where m and n. are the number of edges and vertices of the graph, respectively. We evaluate the high efficiency of our algorithm by comparing it to state- of-the-art algorithms on three categories of graphs: randomly generated graphs, benchmarks, and a real-life protein interaction network. In this paper, we also prove that if self-loops are allowed in a graph, then the number of closed patterns in the adjacency matrix is not necessarily even, but the maximal bicliques are exactly the same as those of the graph after removing all the self-loops.     

大型网络中近似子图匹配研究

为降低噪声对近似子图匹配准确率的影响,提出一种改进的近似子图匹配方法。在预处理阶段,利用k-近邻顶点集为数据图中的每个顶点建立标签-权重向量索引。在查询过程中,基于单个近邻标签的权重距离和所有近邻标签的整体匹配程度进行两级过滤,生成顶点候选集,采用生成树匹配和图匹配的方式确定查询图在大型网络中的位置。在真实数据集上的实验结果表明,该方法具有较高的执行效率和匹配准确率。     

The lexicographically first maximal subgraph problems:P-completeness andNC algorithms

The lexicographically first maximal (lfm) subgraph problem for a property is to compute the lfm vertex set whose induced subgraph satisfies . The main contribution of this paper is theP-completeness of the lfm subgraph problem for any nontrivial hereditary property. We also observe that most of the lfm subgraph problems are stillP-complete even if the instances are restricted to graphs with degree 3. However, some exceptions are found. For example, it is shown that the lfm 4-cycle free subgraph problem is inNC ² for graphs with degree 3 but turns out to beP-complete for graphs with degree 4. Further, we analyze the complexity of the lfmedge-induced subgraph problem for some graph properties and show that it has a different complexity feature.This work was done while the author visited Universität-GH-Paderborn and a part of this paper was presented at the 14th ICALP, Karlsruhe, 1987.     

Lindex: a lattice-based index for graph databases

Subgraph querying has wide applications in various fields such as cheminformatics and bioinformatics. Given a query graph, q, a subgraph-querying algorithm retrieves all graphs, D(q), which have q as a subgraph, from a graph database, D. Subgraph querying is costly because it uses subgraph isomorphism tests, which are NP-complete. Graph indices are commonly used to improve the performance of subgraph querying in graph databases. Subgraph-querying algorithms first construct a candidate answer set by filtering out a set of false answers and then verify each candidate graph using subgraph isomorphism tests. To build graph indices, various kinds of substructure (subgraph, subtree, or path) features have been proposed with the goal of maximizing the filtering rate. Each of them works with a specifically designed index structure, for example, discriminative and frequent subgraph features work with gIndex, δ-TCFG features work with FG-index, etc. We propose Lindex, a graph index, which indexes subgraphs contained in database graphs. Nodes in Lindex represent key-value pairs where the key is a subgraph in a database and the value is a list of database graphs containing the key. We propose two heuristics that are used in the construction of Lindex that allows us to determine answers to subgraph queries conducting less subgraph isomorphism tests. Consequently, Lindex improves subgraph-querying efficiency. In addition, Lindex is compatible with any choice of features. Empirically, we demonstrate that Lindex used in conjunction with subgraph indexing features proposed in previous works outperforms other specifically designed index structures. As a novel index structure, Lindex (1) is effective in filtering false graphs (2) provides fast index lookups, (3) is fast with respect to index construction and maintenance, and (4) can be constructed using any set of substructure index features. These four properties result in a fast and scalable subgraph-querying infrastructure. We substantiate the benefits of Lindex and its disk-resident variation Lindex+ theoretically and empirically.     

A user parameter-free approach for mining robust sequential classification rules

Recent years have witnessed extensive studies of graph classification due to the rapid increase in applications involving structural data and complex relationships. To support graph classification, all existing methods require that training graphs should be relevant (or belong) to the target class, but cannot integrate graphs irrelevant to the class of interest into the learning process. In this paper, we study a new universum graph classification framework which leverages additional “non-example” graphs to help improve the graph classification accuracy. We argue that although universum graphs do not belong to the target class, they may contain meaningful structure patterns to help enrich the feature space for graph representation and classification. To support universum graph classification, we propose a mathematical programming algorithm, ugBoost, which integrates discriminative subgraph selection and margin maximization into a unified framework to fully exploit the universum. Because informative subgraph exploration in a universum setting requires the search of a large space, we derive an upper bound discriminative score for each subgraph and employ a branch-and-bound scheme to prune the search space. By using the explored subgraphs, our graph classification model intends to maximize the margin between positive and negative graphs and minimize the loss on the universum graph examples simultaneously. The subgraph exploration and the learning are integrated and performed iteratively so that each can be beneficial to the other. Experimental results and comparisons on real-world dataset demonstrate the performance of our algorithm.     

gMLC: a multi-label feature selection framework for graph classification

Graph classification has been showing critical importance in a wide variety of applications, e.g. drug activity predictions and toxicology analysis. Current research on graph classification focuses on single-label settings. However, in many applications, each graph data can be assigned with a set of multiple labels simultaneously. Extracting good features using multiple labels of the graphs becomes an important step before graph classification. In this paper, we study the problem of multi-label feature selection for graph classification and propose a novel solution, called gMLC, to efficiently search for optimal subgraph features for graph objects with multiple labels. Different from existing feature selection methods in vector spaces that assume the feature set is given, we perform multi-label feature selection for graph data in a progressive way together with the subgraph feature mining process. We derive an evaluation criterion to estimate the dependence between subgraph features and multiple labels of graphs. Then, a branch-and-bound algorithm is proposed to efficiently search for optimal subgraph features by judiciously pruning the subgraph search space using multiple labels. Empirical studies demonstrate that our feature selection approach can effectively boost multi-label graph classification performances and is more efficient by pruning the subgraph search space using multiple labels.     

A new force-directed graph drawing method based on edge–edge repulsion

The conventional force-directed methods for drawing undirected graphs are based on either vertex–vertex repulsion or vertex–edge repulsion. In this paper, we propose a new force-directed method based on edge–edge repulsion to draw graphs. In our framework, edges are modelled as charged springs, and a final drawing can be generated by adjusting positions of vertices according to spring forces and the repulsive forces, derived from potential fields, among edges. Different from the previous methods, our new framework has the advantage of overcoming the problem of zero angular resolution, guaranteeing the absence of any overlapping of edges incident to the common vertex. Given graph layouts probably generated by previous algorithms as the inputs to our algorithm, experimental results reveal that our approach produces promising drawings not only preserving the original properties of a high degree of symmetry and uniform edge length, but also preventing zero angular resolution and usually having larger average angular resolution. However, it should be noted that exhibiting a higher degree of symmetry and larger average angular resolution does not come without a price, as the new approach might result in the increase in undesirable overlapping of vertices as some of our experimental results indicate. To ease the problem of node overlapping, we also consider a hybrid approach which takes into account both edge–edge and vertex–vertex repulsive forces in drawing a graph.     

A subgraph matching algorithm based on subgraph index for knowledge graph

The problem of subgraph matching is one fundamental issue in graph search, which is NP-Complete problem. Recently, subgraph matching has become a popular research topic in the field of knowledge graph analysis, which has a wide range of applications including question answering and semantic search. In this paper, we study the problem of subgraph matching on knowledge graph. Specifically, given a query graph q and a data graph G, the problem of subgraph matching is to conduct all possible subgraph isomorphic mappings of q on G. Knowledge graph is formed as a directed labeled multi-graph having multiple edges between a pair of vertices and it has more dense semantic and structural features than general graph. To accelerate subgraph matching on knowledge graph, we propose a novel subgraph matching algorithm based on subgraph index for knowledge graph, called as F G q_T-Match. The subgraph matching algorithm consists of two key designs. One design is a subgraph index of matching-driven flow graph ( F G q_T), which reduces redundant calculations in advance. Another design is a multi-label weight matrix, which evaluates a near-optimal matching tree for minimizing the intermediate candidates. With the aid of these two key designs, all subgraph isomorphic mappings are quickly conducted only by traversing F G q_T. Extensive empirical studies on real and synthetic graphs demonstrate that our techniques outperform the state-of-the-art algorithms.     

Feedback vertex sets in star graphs

In a graph G=(V,E), a subset F⊂V(G) is a feedback vertex set of G if the subgraph induced by V(G)?F is acyclic. In this paper, we propose an algorithm for finding a small feedback vertex set of a star graph. Indeed, our algorithm can derive an upper bound to the size of the feedback vertex set for star graphs. Also by applying the properties of regular graphs, a lower bound can easily be achieved for star graphs.     

面向大规模二部图的分布式Tip分解算法

Tip分解作为图数据管理领域的热点研究问题,已被广泛应用于文档聚类和垃圾邮件组检测等实际场景中.随着图数据规模的爆炸式增长,单机内存已无法满足其存储需求,亟需研究分布式环境下Tip分解技术.现有分布式图计算系统的通信模式无法适用于二部图,为此,首先提出一种基于中继的通信模式,以实现分布式环境下处理二部图时消息的有效传递...