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...of any k ′ -classes, for k ′ < k, in order to enhance the I/O and CPU performance for computing the k ′ -classes. Note that the above framework is similar to the top-down core decomposition framework =-=[9]-=-. However, the definition of k-truss is based on triangles instead of simple vertex degree as in k-core. Thus, the detailed operations in the algorithm are considerably different. In particular, the t...

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... patterns. Our τ -visible summary may be applied to visualize a large graph [15] using local substructures, visualizing a large graph can also use global structures such as k-trusses [23] and k-cores =-=[7]-=-. 3. MOTIVATION AND PROBLEM DEFINITION We first briefly review the classic recursive backtracking procedure that exhaustively enumerates all maximal cliques in an undirected graph G. We identify the d...

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...s approximation [18] use k-core decomposition as a subroutine. In terms of algorithms specific to finding k-core decompositions, an external-memory algorithm for k-core decomposition is introduced in =-=[9]-=-. There are also studies about k-core decomposition on directed [16] and weighted [17] networks. To the best of our knowledge, there is no study on incremental algorithms for k-core decomposition, whi...

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...ve also verified in Section 5 that our algorithm is significantly more efficient. Other related works that also avoid random disk access by extracting small subgraphs include core/truss decomposition =-=[11, 32]-=-, triangle listing [14, 15]. However, the subgraphs used in these works cannot guarantee the maximality of the cliques computed and their algorithm frameworks are also not suitable for the task of MCE...

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...works [12], as well as to study percolation on complex networks [13]. From the algorithmic perspective, efficient and scalable algorithms for k-core decomposition in a static graph already exist [4], =-=[14]-=-, [15]. However, in many real-world applications, such as online social network and the Internet, the network evolves over time. In such dynamic networks, many applications require to maintain the cor...

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...run in O(max(m,n)), which equals O(m) for connected networks. BZ algorithm requires random access to the whole graph, which should therefore be kept in the main memory for the sake of performance. In =-=[23]-=- the authors address the case in which the graph cannot – due to size constraints – be kept in the main memory, but has to be accessed through a (slow) external memory. They propose a modification of ...

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...longs to this model and we will compare it with the above two concepts in Section 3 in details. Recently, database researchers try to scale up the disk based cohesive subgraph discovery. Cheng et al. =-=[6]-=- propose a partition based solution for massive k-core mining. They also develop a disk based triangulation method [7] as a fundamental operation for cohesive subgraph discovery. Differently, we store...

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... Due to high number of random accesses to the graph, the algorithm can run efficiently only when the entire graph can fit into main memory of a single machine. To tackle this problem, Cheng et al. in =-=[9]-=- proposed an external-memory solution which can spill into disk when the graph is too large to fit into main memory. The proposed algorithm, however, does not consider any distributed scenario where t...

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...ı́yüce et al. [11] described ways to update the core numbers of vertices in a dynamic graph without recomputing the full core decomposition each time a vertex or edge is added. Finally, Cheng et al. =-=[12]-=- gave an alternate implementation of the core decomposition for systems with insufficient memory to store the entire graph at once. III. LOCAL ESTIMATION AND THEORY In order to estimate core numbers e...

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...angle counting cannot be applied for rectangle counting. In unipartite graphs, scalable algorithms for listing other important substructures such as maximal cliques and core subgraphs were studied in =-=[18]-=-, [19], [20], [21], [22], [23]. For uni-partite graphs, counting length-k cycles, for any k ≤ 7, takes O(|VG|ω) time [16], where ω < 2.37 is the exponent of matrix multiplication. For bipartite graphs...