Graphs are an important tool for modeling data in many diverse domains. Recent increase in sensor technology and deployment, the adoption of online services, and the scale of VLSI circuits has caused the size of these graphs to skyrocket. Finding clusters of highly connected vertices within these graphs is a critical part of their analysis. In this paper we apply the multilevel paradigm to the modularity graph clustering problem. We improve upon the state of the art by introducing new efficient methods for coarsening graphs, creating initial clusterings, and performing local refinement on the resulting clusterings. We establish that for a graph with n vertices and m edges, these algorithms have an O(m+n) runtime complexity and an O(m+n) space complexity, and show that in practice they are extremely fast. We present shared-memory parallel formulations of these algorithms to take full advantage of modern architectures, which we show have a parallel runtime of O(m/p+n/p+k), where p is the number of threads and k is the number of clusters. Finally, we present the product of this research, the clustering tool Nerstrand.1 In serial mode, Nerstrand runs in a fraction of the time of current methods and produces results of equal quality. When run in parallel mode, Nerstrand exhibits significant speedup with less than one percent degradation of clustering quality. Nerstrand works well on large graphs, clustering a graph with over 105 million vertices and 3.3 billion edges in 90 s.
Bibliographical noteFunding Information:
This work was supported in part by NSF ( IOS-0820730 , IIS-0905220 , OCI-1048018 , CNS-1162405 , and IIS-1247632 ) and the Digital Technology Center at the University of Minnesota . Access to research and computing facilities was provided by the Digital Technology Center and the Minnesota Supercomputing Institute.
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- Graph clustering
- Multilevel paradigm
- Shared-memory parallel