Graph mining is one of the most important categories of graph algorithms. However, exploring the subgraphs of an input graph produces a huge amount of intermediate data. The 'think like a vertex' programming paradigm, pioneered by Pregel, cannot readily formulate mining problems, which is designed to produce graph computation problems like PageRank. Existing mining systems like Arabesque and RStream need large amounts of computing and memory resources. In this paper, we present Kaleido, an efficient single machine, out-of-core graph mining system which treats disks as an extension of memory. Kaleido treats intermediate data in graph mining tasks as a tensor and adopts a succinct data structure for the intermediate data. Kaleido utilizes the eigenvalue of the adjacency matrix of a subgraph to efficiently solve the subgraph isomorphism problems with an acceptable constraint that the vertex number of a subgraph is less than 9. Kaleido implements half-memory-half-disk storage for storing large intermediate data, which treats the disk as an extension of the memory. Comparing with two state-of-the-art mining systems, Arabesque and RStream, Kaleido outperforms them by a GeoMean 12.3 @math and 40.0 @math respectively.
gSpan @cite_34 is an efficient frequent subgraph mining algorithm designed for mining multiple input graphs. However, gSpan is designed for multiple graphs of mining problems. If we have a single input graph, we have to find multiple instances in the same graph, therefore it complexes the problem. Michihiro @cite_24 first proposed algorithms to mine patterns from a single graph. They use an expensive anti-monotonic definition of support based on the maximal independent set to find edge-disjoint embeddings. GraMi @cite_30 proposes an effective method in the single large graph and presents an extended version with supporting structural constraints and an approximate version. Pr z ulj @cite_10 introduces the motif counting problem. Ribeiro @cite_6 presents G-Tries which is an effective approach for storing and finding the frequency of motifs. Apar ' cio @cite_13 designs and implements a parallel version of G-Tries. Maximal clique is well studied problem.
@cite_30 @cite_13 @cite_6 @cite_24 @cite_34 @cite_10
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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. 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. 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. 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.