In this study, we formulate the concept of "mining maximal-size frequent subgraphs" in the challenging domain of visual data (images and videos). In general, visual knowledge can usually be modeled as attributed relational graphs (ARGs) with local attributes representing local parts and pairwise attributes describing the spatial relationship between parts. Thus, from a practical perspective, such mining of maximal-size subgraphs can be regarded as a general platform for discovering and modeling the common objects within cluttered and unlabeled visual data. Then, from a theoretical perspective, visual graph mining should encode and overcome the great fuzziness of messy data collected from complex real-world situations, which conflicts with the conventional theoretical basis of graph mining designed for tabular data. Common subgraphs hidden in these ARGs usually have soft attributes, with considerable inter-graph variation. More importantly, we should also discover the latent pattern space, including similarity metrics for the pattern and hidden node relations, during the mining process. In this study, we redefine the visual subgraph pattern that encodes all of these challenges in a general way, and propose an approximate but efficient solution to graph mining. We conduct five experiments to evaluate our method with different kinds of visual data, including videos and RGB RGB-D images. These experiments demonstrate the generality of the proposed method.
In the field of graph mining (reviewed in @cite_27 ), conventional ideas of mining maximal subgraph patterns have been realized as maximal frequent subgraph (MFS) extraction @cite_21 @cite_16 @cite_8 @cite_32 and the mining of maximal cliques @cite_7 @cite_41 . MFS extraction approaches @cite_21 @cite_11 @cite_16 usually require the graphs to contain distinct node edge labels or use local consistency to determine a set of node correspondence candidates between different graphs. Moreover, the graphs must have distinguishing structures. Thus, these methods define MFSs using graph isomorphisms. They mainly enumerate the nodes from different graphs to search the subgraphs with isomorphic (or similar) structures and labels. The distinct labels are used to prune the search range and avoid the NP-hard computation in the worst case. Similarly, the mining of maximal cliques @cite_7 @cite_41 @cite_42 @cite_20 mainly extracts dense cliques that maintain geometric consistency. Subgraph patterns for ARGs have been defined @cite_12 @cite_40 , and the softness of clique patterns has been formulated @cite_26 @cite_17 .
@cite_26 @cite_7 @cite_8 @cite_41 @cite_21 @cite_42 @cite_32 @cite_17 @cite_27 @cite_40 @cite_16 @cite_12 @cite_20 @cite_11
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Most previously proposed frequent graph mining algorithms are intended to find the complete set of all frequent, closed subgraphs. However, in many cases only a subset of the frequent subgraphs with a certain topology is of special interest. Thus, the method of mining the complete set of all frequent subgraphs is not suitable for mining these frequent subgraphs of special interest as it wastes considerable computing power and space on uninteresting subgraphs. In this paper we develop a new algorithm, CLAN, to mine the frequent closed cliques, the most coherent structures in the graph setting. By exploring some properties of the clique pattern, we can simplify the canonical label design and the corresponding clique (or subclique) isomorphism testing. Several effective pruning methods are proposed to prune the search space, while the clique closure checking scheme is used to remove the non-closed clique patterns. Our empirical results show that CLAN is very efficient for large dense graph databases with which the traditional graph mining algorithms fail. The novelty of our method is further demonstrated by the application of CLAN in mining highly correlated stocks from large stock market data. Numerous graph mining applications rely on detecting subgraphs which are large near-cliques. Since formulations that are geared towards finding large near-cliques are hard and frequently inapproximable due to connections with the Maximum Clique problem, the poly-time solvable densest subgraph problem which maximizes the average degree over all possible subgraphs "lies at the core of large scale data mining" [10]. However, frequently the densest subgraph problem fails in detecting large near-cliques in networks. In this work, we introduce the k-clique densest subgraph problem, k ≥ 2. This generalizes the well studied densest subgraph problem which is obtained as a special case for k=2. For k=3 we obtain a novel formulation which we refer to as the triangle densest subgraph problem: given a graph G(V,E), find a subset of vertices S* such that τ(S*)=max limitsS ⊆ V t(S) |S|, where t(S) is the number of triangles induced by the set S. On the theory side, we prove that for any k constant, there exist an exact polynomial time algorithm for the k-clique densest subgraph problem . Furthermore, we propose an efficient 1 k-approximation algorithm which generalizes the greedy peeling algorithm of Asahiro and Charikar [8,18] for k=2. Finally, we show how to implement efficiently this peeling framework on MapReduce for any k ≥ 3, generalizing the work of Bahmani, Kumar and Vassilvitskii for the case k=2 [10]. On the empirical side, our two main findings are that (i) the triangle densest subgraph is consistently closer to being a large near-clique compared to the densest subgraph and (ii) the peeling approximation algorithms for both k=2 and k=3 achieve on real-world networks approximation ratios closer to 1 rather than the pessimistic 1 k guarantee. An interesting consequence of our work is that triangle counting, a well-studied computational problem in the context of social network analysis can be used to detect large near-cliques. Finally, we evaluate our proposed method on a popular graph mining application. Extracting dense subgraphs from large graphs is a key primitive in a variety of graph mining applications, ranging from mining social networks and the Web graph to bioinformatics [41]. In this paper we focus on a family of poly-time solvable formulations, known as the k-clique densest subgraph problem (k-Clique-DSP) [57]. When k=2, the problem becomes the well-known densest subgraph problem (DSP) [22, 31, 33, 39]. Our main contribution is a sampling scheme that gives densest subgraph sparsifier, yielding a randomized algorithm that produces high-quality approximations while providing significant speedups and improved space complexity. We also extend this family of formulations to bipartite graphs by introducing the (p,q)-biclique densest subgraph problem ((p,q)-Biclique-DSP), and devise an exact algorithm that can treat both clique and biclique densities in a unified way. As an example of performance, our sparsifying algorithm extracts the 5-clique densest subgraph --which is a large-near clique on 62 vertices-- from a large collaboration network. Our algorithm achieves 100 accuracy over five runs, while achieving an average speedup factor of over 10,000. Specifically, we reduce the running time from ∼2 107 seconds to an average running time of 0.15 seconds. We also use our methods to study how the k-clique densest subgraphs change as a function of time in time-evolving networks for various small values of k. We observe significant deviations between the experimental findings on real-world networks and stochastic Kronecker graphs, a random graph model that mimics real-world networks in certain aspects. We believe that our work is a significant advance in routines with rigorous theoretical guarantees for scalable extraction of large near-cliques from networks. As data mining techniques are being increasingly applied to non-traditional domains, existing approaches for finding frequent itemsets cannot be used as they cannot model the requirement of these domains. An alternate way of modeling the objects in these data sets is to use graphs. Within that model, the problem of finding frequent patterns becomes that of discovering subgraphs that occur frequently over the entire set of graphs.The authors present a computationally efficient algorithm for finding all frequent subgraphs in large graph databases. We evaluated the performance of the algorithm by experiments with synthetic datasets as well as a chemical compound dataset. The empirical results show that our algorithm scales linearly with the number of input transactions and it is able to discover frequent subgraphs from a set of graph transactions reasonably fast, even though we have to deal with computationally hard problems such as canonical labeling of graphs and subgraph isomorphism which are not necessary for traditional frequent itemset discovery. This paper reformulates the theory of graph mining on the technical basis of graph matching, and extends its scope of applications to computer vision. Given a set of attributed relational graphs (ARGs), we propose to use a hierarchical And-Or Graph (AoG) to model the pattern of maximal-size common subgraphs embedded in the ARGs, and we develop a general method to mine the AoG model from the unlabeled ARGs. This method provides a general solution to the problem of mining hierarchical models from unannotated visual data without exhaustive search of objects. We apply our method to RGB RGB-D images and videos to demonstrate its generality and the wide range of applicability. The code will be available at https: sites.google.com site quanshizhang mining-and-or-graphs. We categorize this research in terms of its contribution to both graph theory and computer vision. From the theoretical perspective, this study can be considered as the first attempt to formulate the idea of mining maximal frequent subgraphs in the challenging domain of messy visual data, and as a conceptual extension to the unsupervised learning of graph matching. We define a soft attributed pattern (SAP) to represent the common subgraph pattern among a set of attributed relational graphs (ARGs), considering both their structure and attributes. Regarding the differences between ARGs with fuzzy attributes and conventional labeled graphs, we propose a new mining strategy that directly extracts the SAP with the maximal graph size without applying node enumeration. Given an initial graph template and a number of ARGs, we develop an unsupervised method to modify the graph template into the maximal-size SAP. From a practical perspective, this research develops a general platform for learning the category model (i.e., the SAP) from cluttered visual data (i.e., the ARGs) without labeling “what is where,” thereby opening the possibility for a series of applications in the era of big visual data. Experiments demonstrate the superior performance of the proposed method on RGB RGB-D images and videos. Over the years, frequent itemset discovery algorithms have been used to find interesting patterns in various application areas. However, as data mining techniques are being increasingly applied to nontraditional domains, existing frequent pattern discovery approaches cannot be used. This is because the transaction framework that is assumed by these algorithms cannot be used to effectively model the data sets in these domains. An alternate way of modeling the objects in these data sets is to represent them using graphs. Within that model, one way of formulating the frequent pattern discovery problem is that of discovering subgraphs that occur frequently over the entire set of graphs. We present a computationally efficient algorithm, called FSG, for finding all frequent subgraphs in large graph data sets. We experimentally evaluate the performance of FSG using a variety of real and synthetic data sets. Our results show that despite the underlying complexity associated with frequent subgraph discovery, FSG is effective in finding all frequently occurring subgraphs in data sets containing more than 200,000 graph transactions and scales linearly with respect to the size of the data set. One fundamental challenge for mining recurring subgraphs from semi-structured data sets is the overwhelming abundance of such patterns. In large graph databases, the total number of frequent subgraphs can become too large to allow a full enumeration using reasonable computational resources. In this paper, we propose a new algorithm that mines only maximal frequent subgraphs, i.e. subgraphs that are not a part of any other frequent subgraphs. This may exponentially decrease the size of the output set in the best case; in our experiments on practical data sets, mining maximal frequent subgraphs reduces the total number of mined patterns by two to three orders of magnitude.Our method first mines all frequent trees from a general graph database and then reconstructs all maximal subgraphs from the mined trees. Using two chemical structure benchmarks and a set of synthetic graph data sets, we demonstrate that, in addition to decreasing the output size, our algorithm can achieve a five-fold speed up over the current state-of-the-art subgraph mining algorithms. We investigate new approaches for frequent graph-based pattern mining in graph datasets and propose a novel algorithm called gSpan (graph-based substructure pattern mining), which discovers frequent substructures without candidate generation. gSpan builds a new lexicographic order among graphs, and maps each graph to a unique minimum DFS code as its canonical label. Based on this lexicographic order gSpan adopts the depth-first search strategy to mine frequent connected subgraphs efficiently. Our performance study shows that gSpan substantially outperforms previous algorithms, sometimes by an order of magnitude. In this paper, we propose a general framework for graph matching which is suitable for different problems of pattern recognition. The pattern representation we assume is at the same time highly structured, like for classic syntactic and structural approaches, and of subsymbolic nature with real-valued features, like for connectionist and statistic approaches. We show that random walk based models, inspired by Google's PageRank, give rise to a spectral theory that nicely enhances the graph topological features at node level. As a straightforward consequence, we derive a polynomial algorithm for the classic graph isomorphism problem, under the restriction of dealing with Markovian spectrally distinguishable graphs (MSD), a class of graphs that does not seem to be easily reducible to others proposed in the literature. The experimental results that we found on different test-beds of the TC-15 graph database show that the defined MSD class "almost always" covers the database, and that the proposed algorithm is significantly more efficient than top scoring VF algorithm on the same data. Most interestingly, the proposed approach is very well-suited for dealing with partial and approximate graph matching problems, derived for instance from image retrieval tasks. We consider the objects of the COIL-100 visual collection and provide a graph-based representation, whose node's labels contain appropriate visual features. We show that the adoption of classic bipartite graph matching algorithms offers a straightforward generalization of the algorithm given for graph isomorphism and, finally, we report very promising experimental results on the COIL-100 visual collection. Frequent coherent subgraphs can provide valuable knowledge about the underlying internal structure of a graph database, and mining frequently occurring coherent subgraphs from large dense graph databases has been witnessed several applications and received considerable attention in the graph mining community recently. In this paper, we study how to efficiently mine the complete set of coherent closed quasi-cliques from large dense graph databases, which is an especially challenging task due to the downward-closure property no longer holds. By fully exploring some properties of quasi-cliques, we propose several novel optimization techniques, which can prune the unpromising and redundant sub-search spaces effectively. Meanwhile, we devise an efficient closure checking scheme to facilitate the discovery of only closed quasi-cliques. We also develop a coherent closed quasi-clique mining algorithm, Cocain1 Thorough performance study shows that Cocain is very efficient and scalable for large dense graph databases. The complexity of the subgraph isomorphism problem where the pattern graph is of fixed size is well known to depend on the topology of the pattern graph. For instance, the larger the maximum independent set of the pattern graph is the more efficient algorithms are known. The situation seems to be substantially different in the case of induced subgraph isomorphism for pattern graphs of fixed size. We present two results which provide evidence that no topology of an induced subgraph of fixed size can be easier to detect or count than an independent set of related size. We show that: Any fixed pattern graph that has a maximum independent set of size k that is disjoint from other maximum independent sets is not easier to detect as an induced subgraph than an independent set of size k. It follows in particular that an induced path on k vertices is not easier to detect than an independent set on ⌈k 2 ⌉ vertices, and that an induced even cycle on k vertices is not easier to detect than an independent set on k 2 vertices. In view of linear time upper bounds on induced paths of length three and four, our lower bound is tight. Similar corollaries hold for the detection of induced complete bipartite graphs and induced complete split graphs. For an arbitrary pattern graph H on k vertices with no isolated vertices, there is a simple subdivision of H, resulting from splitting each edge into a path of length four and attaching a distinct path of length three at each vertex of degree one, that is not easier to detect or count than an independent set on k vertices, respectively. Finally, we show that the so called diamond, paw and C 4 are not easier to detect as induced subgraphs than an independent set on three vertices. Graph mining methods enumerate frequently appearing subgraph patterns, which can be used as features for subsequent classification or regression. However, frequent patterns are not necessarily informative for the given learning problem. We propose a mathematical programming boosting method (gBoost) that progressively collects informative patterns. Compared to AdaBoost, gBoost can build the prediction rule with fewer iterations. To apply the boosting method to graph data, a branch-and-bound pattern search algorithm is developed based on the DFS code tree. The constructed search space is reused in later iterations to minimize the computation time. Our method can learn more efficiently than the simpler method based on frequent substructure mining, because the output labels are used as an extra information source for pruning the search space. Furthermore, by engineering the mathematical program, a wide range of machine learning problems can be solved without modifying the pattern search algorithm. Given a query graph q and a data graph g, the subgraph isomorphism search finds all occurrences of q in g and is considered one of the most fundamental query types for many real applications. While this problem belongs to NP-hard, many algorithms have been proposed to solve it in a reasonable time for real datasets. However, a recent study has shown, through an extensive benchmark with various real datasets, that all existing algorithms have serious problems in their matching order selection. Furthermore, all algorithms blindly permutate all possible mappings for query vertices, often leading to useless computations. In this paper, we present an efficient and robust subgraph search solution, called TurboISO, which is turbo-charged with two novel concepts, candidate region exploration and the combine and permute strategy (in short, Comb Perm). The candidate region exploration identifies on-the-fly candidate subgraphs (i.e, candidate regions), which contain embeddings, and computes a robust matching order for each candidate region explored. The Comb Perm strategy exploits the novel concept of the neighborhood equivalence class (NEC). Each query vertex in the same NEC has identically matching data vertices. During subgraph isomorphism search, Comb Perm generates only combinations for each NEC instead of permutating all possible enumerations. Thus, if a chosen combination is determined to not contribute to a complete solution, all possible permutations for that combination will be safely pruned. Extensive experiments with many real datasets show that TurboISO consistently and significantly outperforms all competitors by up to several orders of magnitude.