Bounded Arboricity to Determine the Local Structure of Sparse Graphs
A known approach of detecting dense subgraphs (communities) in large sparse graphs involves first computing the probability vectors for short random walks on the graph, and then using these probability vectors to detect the communities, see Latapy and Pons . In this paper we focus on the first part of such an approach i.e. the computation of the probability vectors for the random walks, and propose a more efficient algorithm for computing these vectors in time complexity that is linear in the size of the output, in case the input graphs are restricted to a family of graphs of bounded arboricity. Such classes of graphs cover a large number of cases of interest, e.g all minor closed graph classes (planar graphs, graphs of bounded treewidth etc) and random graphs within the preferential attachment model, see Barabási and Albert . Our approach is extensible to other models of computation (PRAM, BSP or out-of-core computation) and also w.h.p. stays within the same complexity bounds for Erdős Renyi graphs.
KeywordsRandom Walk Random Graph Probability Vector Input Graph Sparse Graph
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- Aiello, W., Chung, F., Lu, L.: Random evolution in massive graphs. In: Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, pp. 510–519 (2001), URL: http://www.citeseer.ifi.unizh.ch/article/aiello01random.html
- Dehne, F.K.H.A., Dittrich, W., Hutchinson, D.: Efficient external memory algorithms by simulating coarse-grained parallel algorithms. In: ACM Symposium on Parallel Algorithms and Architectures, pp. 106–115 (1997)Google Scholar
- Fortune, S., Wyllie, J.: Parallelism in random access machines. In: 10th ACM Symposium on Theory of Computing, May 1978, pp. 114–118 (1978)Google Scholar
- Gebremedhin, A.H., Lassous, I.G., Gustedt, J., Telle, J.A.: PRO: a model for parallel resource-optimal computation. In: 16th Annual International Symposium on High Performance Computing Systems and Applications, pp. 106–113. IEEE (The Institute of Electrical and Electronics Engineers), Los Alamitos (2002)Google Scholar
- Gustedt, J.: Minimum spanning trees for minor-closed graph classes in parallel. In: Symposium on Theoretical Aspects of Computer Science, pp. 421–431 (1998)Google Scholar
- Kannan, S., Naor, M., Rudich, S.: Implicit representation of graphs. SIAM Journal On Discrete Mathematics 5, 596–603 (1992), URL: http://www.citeseer.ifi.unizh.ch/kannan92implicit.html MATHCrossRefMathSciNetGoogle Scholar