You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
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.

Cited Abstract(s):
[1]: Mining frequent subgraphs is an important operation on graphs; it is defined as finding all subgraphs that appear frequently in a database according to a given frequency threshold. Most existing work assumes a database of many small graphs, but modern applications, such as social networks, citation graphs, or protein-protein interactions in bioinformatics, are modeled as a single large graph. In this paper we present GraMi, a novel framework for frequent subgraph mining in a single large graph. GraMi undertakes a novel approach that only finds the minimal set of instances to satisfy the frequency threshold and avoids the costly enumeration of all instances required by previous approaches. We accompany our approach with a heuristic and optimizations that significantly improve performance. Additionally, we present an extension of GraMi that mines frequent patterns. Compared to subgraphs, patterns offer a more powerful version of matching that captures transitive interactions between graph nodes (like friend of a friend) which are very common in modern applications. Finally, we present CGraMi, a version supporting structural and semantic constraints, and AGraMi, an approximate version producing results with no false positives. Our experiments on real data demonstrate that our framework is up to 2 orders of magnitude faster and discovers more interesting patterns than existing approaches.

[2]: Computing the frequency of small subgraphs on a large network is a computationally hard task. This is, however, an important graph mining primitive, with several applications, and here we present a novel multicore parallel algorithm for this task. At the core of our methodology lies a state-of-the-art data structure, the g-trie, which represents a collection of subgraphs and allows for a very efficient sequential search. Our implementation was done using Pthreads and can run on any multicore personal computer. We employ a diagonal work sharing strategy to dynamically and effectively divide work among threads during the execution. We assess the performance of our Pthreads implementation on a set of representative networks from various domains and with diverse topological features. For most networks, we obtain a speedup of over 50 for 64 cores and an almost linear speedup up to 32 cores, showcasing the flexibility and scalability of our algorithm. This paves the way for the usage of such counting algorithms on larger subgraph and network sizes without the obligatory access to a cluster.

[3]: The ability to find and count subgraphs of a given network is an important non trivial task with multidisciplinary applicability. Discovering network motifs or computing graphlet signatures are two examples of methodologies that at their core rely precisely on the subgraph counting problem. Here we present the g-trie, a data-structure specifically designed for discovering subgraph frequencies. We produce a tree that encapsulates the structure of the entire graph set, taking advantage of common topologies in the same way a prefix tree takes advantage of common prefixes. This avoids redundancy in the representation of the graphs, thus allowing for both memory and computation time savings. We introduce a specialized canonical labeling designed to highlight common substructures and annotate the g-trie with a set of conditional rules that break symmetries, avoiding repetitions in the computation. We introduce a novel algorithm that takes as input a set of small graphs and is able to efficiently find and count them as induced subgraphs of a larger network. We perform an extensive empirical evaluation of our algorithms, focusing on efficiency and scalability on a set of diversified complex networks. Results show that g-tries are able to clearly outperform previously existing algorithms by at least one order of magnitude.

[4]: 

[5]: 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.

[6]: Motivation: Analogous to biological sequence comparison, comparing cellular networks is an important problem that could provide insight into biological understanding and therapeutics. For technical reasons, comparing large networks is computationally infeasible, and thus heuristics, such as the degree distribution, clustering coefficient, diameter, and relative graphlet frequency distribution have been sought. It is easy to demonstrate that two networks are different by simply showing a short list of properties in which they differ. It is much harder to show that two networks are similar, as it requires demonstrating their similarity in all of their exponentially many properties. Clearly, it is computationally prohibitive to analyze all network properties, but the larger the number of constraints we impose in determining network similarity, the more likely it is that the networks will truly be similar. Results: We introduce a new systematic measure of a network's local structure that imposes a large number of similarity constraints on networks being compared. In particular, we generalize the degree distribution, which measures the number of nodes 'touching' k edges, into distributions measuring the number of nodes 'touching' k graphlets, where graphlets are small connected non-isomorphic subgraphs of a large network. Our new measure of network local structure consists of 73 graphlet degree distributions of graphlets with 2--5 nodes, but it is easily extendible to a greater number of constraints (i.e. graphlets), if necessary, and the extensions are limited only by the available CPU. Furthermore, we show a way to combine the 73 graphlet degree distributions into a network 'agreement' measure which is a number between 0 and 1, where 1 means that networks have identical distributions and 0 means that they are far apart. Based on this new network agreement measure, we show that almost all of the 14 eukaryotic PPI networks, including human, resulting from various high-throughput experimental techniques, as well as from curated databases, are better modeled by geometric random graphs than by Erdos--Reny, random scale-free, or Barabasi--Albert scale-free networks. Availability: Software executables are available upon request. Contact: natasha@ics.uci.edu
gSpan [5] 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 [4] 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 [1] 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 [6] introduces the motif counting problem. Ribeiro [3] presents G-Tries which is an effective approach for storing and finding the frequency of motifs. Apar ' cio [2] designs and implements a parallel version of G-Tries. Maximal clique is well studied problem.