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:
We study large-scale, distributed graph clustering. Given an undirected graph, our objective is to partition the nodes into disjoint sets called clusters. A cluster should contain many internal edges while being sparsely connected to other clusters. In the context of a social network, a cluster could be a group of friends. Modularity and map equation are established formalizations of this internally-dense-externally-sparse principle. We present two versions of a simple distributed algorithm to optimize both measures. They are based on Thrill, a distributed big data processing framework that implements an extended MapReduce model. The algorithms for the two measures, DSLM-Mod and DSLM-Map, differ only slightly. Adapting them for similar quality measures is straight-forward. We conduct an extensive experimental study on real-world graphs and on synthetic benchmark graphs with up to 68 billion edges. Our algorithms are fast while detecting clusterings similar to those detected by other sequential, parallel and distributed clustering algorithms. Compared to the distributed GossipMap algorithm, DSLM-Map needs less memory, is up to an order of magnitude faster and achieves better quality.

Cited Abstract(s):
[1]: This paper presents a study of graph partitioning schemes for parallel graph community detection on distributed memory machines. We investigate the relationship between graph structure and parallel clustering effectiveness, and develop a heuristic partitioning algorithm suitable for modularity-based algorithms. We demonstrate the accuracy and scalability of our approach using several real-world large graph datasets compared with state-of-the-art parallel algorithms on the Cray XK7 supercomputer at Oak Ridge National Laboratory. Given the ubiquitous graph model, we expect this high-performance solution will help lead to new insights in numerous fields.

[2]: In this paper we present a parallel formulation of the multilevel graph partitioning and sparse matrix ordering algorithm. A key feature of our parallel formulation (that distinguishes it from other proposed parallel formulations of multilevel algorithms) is that it partitions the vertices of the graph intopparts while distributing the overall adjacency matrix of the graph among allpprocessors. This mapping results in substantially smaller communication than one-dimensional distribution for graphs with relatively high degree, especially if the graph is randomly distributed among the processors. We also present a parallel algorithm for computing a minimal cover of a bipartite graph which is a key operation for obtaining a small vertex separator that is useful for computing the fill reducing ordering of sparse matrices. Our parallel algorithm achieves a speedup of up to 56 on 128 processors for moderate size problems, further reducing the already moderate serial run time of multilevel schemes. Furthermore, the quality of the produced partitions and orderings are comparable to those produced by the serial multilevel algorithm that has been shown to outperform both spectral partitioning and multiple minimum degree.

[3]: Detecting community structures in graphs is a well studied problem in graph data analytics. Unprecedented growth in graph structured data due to the development of the world wide web and social networks in the past decade emphasizes the need for fast graph data analytics techniques. In this paper we present a simple yet efficient approach to detect communities in large scale graphs by modifying the sequential Louvain algorithm for community detection. The proposed distributed memory parallel algorithm targets the costly first iteration of the initial method by parallelizing it. Experimental results on a MPI setup with 128 parallel processes shows that up to ≈5× performance improvement is achieved as compared to the sequential version while not compromising the correctness of the final result.
The first is to partition the graph into a subgraph per machine. Each subgraph is then clustered independently on one machine. Then, all nodes of each cluster are merged summing up weights of parallel edges. The resulting coarser graph is clustered on a single machine. This assumes the coarsened graph fits in the memory of a single machine. [1] , for the partitioning, the input node ID range is chunked into equally sized parts. This can work well, but is problematic if input node IDs do not reflect the graph structure. [3] , the input graph is partitioned using the non-distributed, parallel graph partitioning algorithm ParMETIS [2] . While this is independent of node IDs, it requires that the graph fits into the memory of one machine for the partitioning.