Pattern Discovery from Graph-Structured Data - A Data Mining Perspective

Graph-structured databases have a wide range of emerging applications, e.g., the Semantic Web, eXtensible Markup Language (XML), biological databases and network topologies. To-date, there has already been voluminous real-world (possibly cyclic and schemaless) graph-structured data. Therefore, data engineering in graph-structured databases has recently received a lot of attention, where there are limitations as well as scope for significant developments. In these databases, there exist many different indexes and different query languages, e.g., XQuery, regular expressions, Web Ontology Langauge and subgraph isomorphism, while there are few graphical user interfaces for effectively querying subgraphs. In this paper, we examine and evaluate the current stateof- the-art in graph-structured databases with respect to (i) query languages, (ii) dynamic aspects, (iii) data mining, (iv) graphical user interfaces, and (v) modern computer architecture on graph-structured data. In addition, the incremental maintenance of graph indexes/views will be addressed

Subgraph Isomorphism is a fundamental problem in graph theory. It has many applications in social network analysis, molecular investigations, knowledge graphs, etc. Given a Query Graph and a Data Graph, the target of Subgraph Isomorphism, i.e., Subgraph Matching, is to determine if this Query Graph is isomorphic to any subgraph of the Data Graph. This work proposes a new type of Query Graph, combined with multiple general Query Graphs. We call it Compulsory-Optional Query Graph (CO Query Graph). This new type of Query Graph contains all the vertices in the combined general Query Graph, and each vertex corresponds to a search priority. Based on CO Query Graph, the previous multiple match processes can be reduced to one. It tremendously improves search efficiency. The Subgraph Isomorphism based on this new kind of Query Graph is an extension and improvement of the previous Subgraph Isomorphism studies. We propose a backtracking-pruning-based CO solver (BPC). This algorithm builds on the backtracking-pruning framework. BPC modifies the output criterion and matching conditions to satisfy the CO query context. A case study of real-world graph data illustrates that BPC built on CO Query Graph is more efficient than conventional Query Graphs. To verify the effectiveness of our method, we conducted experiments on the synthetic graph and real-world data. The results show that the BPC can significantly reduce the search space and improve the search efficiency in the recursive calls and the response time. Experiments resulting from synthetic graph data analysis allow us to primarily identify the critical factor that affects the efficiency of the BPC primarily.

Recently, due to the rapid growth of electronic data having graph structures such as HTML and XML texts and chemical compounds, many researchers have been interested in data mining and machine learning techniques for finding useful patterns from graph-structured data (graph data). Since graph data contain a huge number of substructures and it tends to be computationally expensive to decide whether or not such data have given structural features, graph mining problems face computational difficulties. Let ${\mathcal{C}}$ be a graph class which satisfies a connected hereditary property and contains infinitely many different biconnected graphs, and for which a special kind of the graph isomorphism problem can be computed in polynomial time. In this paper, we consider learning and mining problems for ${\mathcal{C}}$ . Firstly, we define a new graph pattern, which is called a block preserving graph pattern (bp-graph pattern) for ${\mathcal{C}}$ . Secondly, we present a polynomial time algorithm for deciding whether or not a given bp-graph pattern matches a given graph in ${\mathcal{C}}$ . Thirdly, by giving refinement operators over bp-graph patterns, we present a polynomial time algorithm for finding a minimally generalized bp-graph pattern for ${\mathcal{C}}$ . Outerplanar graphs are planar graphs which can be embedded in the plane in such a way that all of vertices lie on the outer boundary. Many pharmacologic chemical compounds are known to be represented by outerplanar graphs. The class of connected outerplanar graphs ${\mathcal{O}}$ satisfies the above conditions for ${\mathcal{C}}$ . Next, we propose two incremental polynomial time algorithms for enumerating all frequent bp-graph patterns with respect to a given finite set of graphs in ${\mathcal{O}}$ . Finally, by reporting experimental results obtained by applying the two graph mining algorithms to a subset of the NCI dataset, we evaluate the performance of the two graph mining algorithms.

Graph matching, as one of the most fundamental problems in graph database, has a wide range of applications. Due to the large scale of graph database and the hardness of graph matching, it is an attractive alternative to make use of the cloud to store massive data graphs and conduct the complex graph matching. To protect the privacy, data graphs are usually encrypted before being outsourced to the cloud. A few schemes have been proposed to support graph matching query over encrypted graphs. However, none of them can realize efficient subgraph extraction when the matched subgraph needs to be exactly located at the data graph. The graph user has to perform the complex subgraph isomorphism operation to extract the isomorphic subgraph from the matched data graph in state-of-the-art schemes. The time complexity of judging subgraph isomorphism is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(n!n^{2})$</tex-math></inline-formula> at most, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of vertices in the data graph. In order to solve this problem, we propose a privacy-preserving graph matching query scheme supporting quick subgraph extraction in this paper. In our design, two non-colluding cloud servers are adopted to accomplish the matching operation jointly. Neither of them can infer the plaintexts of graphs. The first cloud server can prune some impossible matched vertices from the data graph, and get a possible matched matrix to represent which vertices in the data graph might match with the vertices in the query graph. The second cloud server performs the related matrix operations based on its stored information and the information sent by the first cloud server. The two cloud servers jointly verify the correctness of the matched matrix. The graph user can directly and quickly extract the matrix of the subgraph isomorphic to the query graph from the data graph matrix based on the matched matrix, and further recover this subgraph. No subgraph isomorphism operation is involved in the procedure of subgraph extraction for the graph user. The time complexity of subgraph extraction is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(m^{2})$</tex-math></inline-formula> in our scheme, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> is the number of vertices in the query graph. The extensive experiments with real-world database demonstrate the efficiency of the proposed privacy-preserving graph matching scheme.

Pattern discovery via constraint programming

The field of data mining increasingly adapts methods and algorithms from advanced matrix computations, graph theory and optimization. In these methods, the data is described using matrix representations (graphs are represented by their adjacency matrices) and the data mining problem is formulated as an optimization problem with matrix variables. With these, the data mining task becomes a process of minimizing or maximizing a desired objective function of matrix variables. Prominent examples include spectral clustering, non-negative matrix factorization, Principal Component Analysis (PCA) and Singular Value Decomposition (SVD) related clustering and dimension reduction, tensor analysis and L-1 regularization. These matrix-formulated optimization-centric methodologies are rapidly becoming a significant part of data mining, and have evolved into a popular and rapidlyexpanding research area for solving challenging data mining problems. First, they are amenable to vigorous analysis and benefit from the well-established knowledge in linear algebra, graph theory and optimization accumulated through centuries. Second, they can be efficiently computed thanks to the mature software tools developed by scientific computing communities. Third, they are simple to implement and easy to

Top‐k node selection in graph data is an essential problem in computer science and applications. In view of an important issue in the field of graph data, subgraph matching issue, we define the problem and propose its method for the top‐k key node query w.r.t. the subgraph matching. Unlike the general top‐k query problem, we aim to find out k nodes that make the matching subgraphs in data graph G that are covered by the k nodes as more as possible. This is a problem of the maximum coverage of subgraph matching, which belongs to the NP‐hard problem. We study the problem based on a greedy algorithm and give an intuitive solution. Considering the characteristics of the top‐k problem, we propose an improved and more efficient greedy algorithm. Experiments on real social network graph data set (Twitter) show that the related results represent the key nodes that can better reveal the essential characteristics of the query graph in the data graph G. The key node query problem in subgraph matching proposed in this article may have extensive applications in reality, such as the assessment of the influence of specific group members in social network, the detection of abnormal communication in a computer communication network, the road traffic evaluation and load balance problem in a road traffic network, and so on.

Knowledge discovery in databases and data mining aim at semiautomatic tools for the analysis of large data sets. We consider some methods used in data mining, concentrating on levelwise search for all frequently occurring patterns. We show how this technique can be used in various applications. We also discuss possibilities for compiling data mining queries into algorithms, and look at the use of sampling in data mining. We conclude by listing several open research problems in data mining and knowledge discovery.

In this paper, we study a new graph learning problem: learning to count subgraph isomorphisms. Different from other traditional graph learning problems such as node classification and link prediction, subgraph isomorphism counting is NP-complete and requires more global inference to oversee the whole graph. To make it scalable for large-scale graphs and patterns, we propose a learning framework that augments different representation learning architectures and iteratively attends pattern and target data graphs to memorize intermediate states of subgraph isomorphism searching for global counting. We develop both small graphs (<= 1,024 subgraph isomorphisms in each) and large graphs (<= 4,096 subgraph isomorphisms in each) sets to evaluate different representation and interaction modules. A mutagenic compound dataset, MUTAG, is also used to evaluate neural models and demonstrate the success of transfer learning. While the learning based approach is inexact, we are able to generalize to count large patterns and data graphs in linear time compared to the exponential time of the original NP-complete problem. Experimental results show that learning based subgraph isomorphism counting can speed up the traditional algorithm, VF2, 10-1,000 times with acceptable errors. Domain adaptation based on fine-tuning also shows the usefulness of our approach in real-world applications.

Subgraph matching is a basic operation in graph analytics, finding all occurrences of a query graph Q in a data graph G. A common approach is to first filter out non-candidate vertices in G, and then order the vertices in Q to enumerate results. Recent work has started to utilize the GPU to accelerate subgraph matching. However, the effectiveness of current GPU-based filtering and ordering methods is limited, and the result enumeration often runs out of memory quickly. To address these problems, we propose EGSM, an efficient approach to GPU-based subgraph matching. Specifically, we design a data structure Cuckoo trie to support dynamic maintenance of candidates for filtering, and order query vertices based on estimated numbers of candidate vertices on the fly. Furthermore, we perform a hybrid breadth-first and depth-first search with memory management for result enumeration. Consequently, EGSM significantly outperforms the state-of-the-art GPU-accelerated algorithms, including GSI and CuTS.

Over the past few decades, numerous optimization-based methods have been proposed for solving the classification problem in data mining. Classic optimization-based methods do not consider attribute interactions toward classification. Thus, a novel learning machine is needed to provide a better understanding on the nature of classification when the interaction among contributions from various attributes cannot be ignored. The interactions can be described by a non-additive measure while the Choquet integral can serve as the mathematical tool to aggregate the values of attributes and the corresponding values of a non-additive measure. As a main part of this research, a new nonlinear classification method with non-additive measures is proposed. Experimental results show that applying non-additive measures on the classic optimization-based models improves the classification robustness and accuracy compared with some popular classification methods. In addition, motivated by well-known Support Vector Machine approach, we transform the primal optimization-based nonlinear classification model with the signed non-additive measure into its dual form by applying Lagrangian optimization theory and Wolfes dual programming theory. As a result, 2n – 1 parameters of the signed non-additive measure can now be approximated with m (number of records) Lagrangian multipliers by applying necessary conditions of the primal classification problem to be optimal. This method of parameter approximation is a breakthrough for solving a non-additive measure practically when there are relatively small number of training cases available (mn－1). Furthermore, the kernel-based learning method engages the nonlinear classifiers to achieve better classification accuracy. The research produces practically deliverable nonlinear models with the non-additive measure for classification problem in data mining when interactions among attributes are considered.

As a result of the huge amounts of data available in any subject of research, data mining technologies confront considerable obstacles. Due to the large volume of data, most existing data mining methods are inapplicable to many real-world problems. Data mining methods become ineffective when the problem size becomes too large. Scalability is an issue that must be addressed in data mining algorithms in order to construct high-performance, efficient, and scalable data mining algorithms. A new scalability technique is proposed and applied to several data mining problems in this research. The proposed scaling methodology is developed using a cascade approach. The approach begins with the collection of a large data set from several sources, which is then preprocessed. Once the dataset has been preprocessed, decompose it into smaller data sets of equal size subsets. Then apply a data mining approach to each subset, with the identical data mining method stated for each subset. The findings of the data mining approach on all subsets are pooled and aggregated for the final output. The performance of the suggested algorithm is assessed using a number of criteria, including accuracy, precision, recall, F-score, and execution time. Social advertising and bank marketing are the two datasets on which the proposed method is tested. The suggested approach's performance is compared with non-scale data mining methods, and it is found that the scaling method outperformed non-scale data mining methods on all measures.

The subgraph searching is a fundamental operation for the analysis and exploration of graphs. Nowadays, molecular databases are nearing close to one hundred million molecules. Since finding all the data graphs in a graph database that contain the query graph using subgraph isomorphism is an NP-complete problem, indexes are built and processed. Further, to assist the formulation of the query by a user, the visual exploratory subgraph query paradigm proposes a graphical user interface and leverages exploration time to reduce query processing time. However, state-of-the-art approaches need to scale better to dynamic graph databases and suffer from efficiency problems. In addition, the existing Summarisation-based frequent subgraph mining for visual exploratory subgraph searching (SuMExplorer) is lacking implementation and evaluation study for handling visual subgraph similarity search and modify operations. In this paper, we present a novel index structure, which aids the subgraph searching using the summarised-based weighted frequent subgraph mining on data graphs. By the structure-preserving, we exploit the indexes to support similarity and modify operations. We conduct extensive performance studies on both real-world and synthetic datasets to evaluate the overall performance of the extended SuMExplorer to the recent visual exploratory FERRARI and traditional subgraph search algorithms (such as the gIndex and the GRAPES-DD). Our results showed that our indexes can query up to 3 times faster in comparison to the FERRARI while reducing the storage footprint by 2 orders of magnitude.

With the popularity of the Internet and the development of information technology, hundreds of millions of information and data that are so close to our human kind are recorded. In recent decades, as an interdisciplinary, data science has had a vigorous development. This paper mainly discusses two core problems in data science, that are data analysis and data mining. Beginning with the data analysis based on statistics, this paper discusses and reviews data mining based on traditional machine learning and that based on neural network. Then it discusses the common problems in data mining. Each part summarizes the corresponding application and research. Through the discussion of the basic contents in data analysis and data mining, this paper provides a powerful help for the future research in this field.

Subgraph matching is a key operation on graph data. Social network (SN) providers may want to find all subgraphs within their social network that match certain query graph patterns. Unfortunately, subgraph matching is NP-complete, making its application to massive SNs a major challenge. Past work has shown how to implement subgraph matching on a single processor when the graph has 10-25M edges. In this paper, we show how to use cloud computing in conjunction with such existing single processor methods to efficiently match complex subgraphs on graphs as large as 778M edges. A cloud consists of one master compute node and k slave compute nodes. We first develop a probabilistic method to estimate probabilities that a vertex will be retrieved by a random query and that a pair of vertices will be successively retrieved by a random query. We use these probability estimates to define edge weights in an SN and to compute minimal edge cuts to partition the graph amongst k slave nodes. We develop algorithms for both master and slave nodes that try to minimize communication overhead. The resulting COSI system can answer complex queries over real-world SN data containing over 778M edges very efficiently.