Recent research on pattern discovery has progressed from mining frequent patterns and sequences to mining structured patterns, such as trees and graphs. Graphs as general data structure can model complex relations among data with wide applications in web exploration and social networks. However, the process of mining large graph patterns is a challenge due to the existence of large number of subgraphs. In this paper, we aim to mine only frequent complete graph patterns. A graph g in a database is complete if every pair of distinct vertices is connected by a unique edge. Grid Complete Graph (GCG) is a mining algorithm developed to explore interesting pruning techniques to extract maximal complete graphs from large spatial dataset existing in Sloan Digital Sky Survey (SDSS) data. Using a divide and conquer strategy, GCG shows high efficiency especially in the presence of large number of patterns. In this paper, we describe GCG that can mine not only simple co-location spatial patterns but also complex ones. To the best of our knowledge, this is the first algorithm used to exploit the extraction of maximal complete graphs in the process of mining complex co-location patterns in large spatial dataset.
@cite_0 used cliques as a co-location pattern (subgraphs), but in our research we used complete graphs instead. Similar to our approach, they separated the clique mining from the pattern mining stages. However, they did not use maximal complete graph. They treated each clique as a transaction and used an Apriori based technique for mining association rules. Since they used cliques (rather than maximal complete graphs) as their transactions, the counting of pattern instances is very different. They considered complex relationships within the pattern mining stage. However, their definition of negative patterns is very different -- they used infrequent types while we base our definition on the concept of absence in . They also used a different measure, namely, maxPI.
@cite_0
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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. In this paper, we present a clique-based method for mining fuzzy graph patterns of money laundering and financing terrorism. The method will contribute to a new generation of intelligent anti-money laundering systems that incorporate comprehensive information from various information sources as well as from human subject matter experts. A fuzzy degree of confidence can therefore be associated to each relation between any two actors in a graph of transactions. We reduce the problem of fuzzy subgraph isomorphism to the problem of finding a fuzzy set of maximal cliques in a fuzzy extension of a mathematical construct that is similar to Vizing's modular product of graphs. 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. Several real-world applications require mining of bicliques, as they represent correlated pairs of data clusters. However, the mining quality is adversely affected by missing and noisy data. Moreover, some applications only require strong interactions between data members of the pairs, but bicliques are pairs that display complete interactions. We address these two limitations by proposing maximal quasi-bicliques. Maximal quasi-bicliques tolerate erroneous and missing data, and also relax the interactions between the data members of their pairs. Besides, maximal quasi-bicliques do not suffer from skewed distribution of missing edges that prior quasi-bicliques have. We develop an algorithm MQBminer, which mines the complete set of maximal quasi-bicliques from either bipartite or non-bipartite graphs. We demonstrate the versatility and effectiveness of maximal quasi-bicliques to discover highly correlated pairs of data in two diverse real-world datasets. First, we propose to solve a novel financial stocks analysis problem using maximal quasi-bicliques to co-cluster stocks and financial ratios. Results show that the stocks in our co-clusters usually have significant correlations in their price performance. Second, we use maximal quasi-bicliques on a mining protein network problem and we show that pairs of protein groups mined by maximal quasi-bicliques are more significant than those mined by maximal bicliques. Copyright © 2009 Wiley Periodicals, Inc., A Wiley Company