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Approximate Counting of Cycles in Streams

Approximate Counting of Cycles in Streams. He Sun Max Planck Institute for Informatics Joint work with Madhusudan Manjunath , Kurt Mehlhorn and Konstantinos Panagiotou. Sub-graph Counting. Given a graph G=(V,E) with n nodes and m edges and a small graph H with

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Approximate Counting of Cycles in Streams

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  1. Approximate Counting of Cycles in Streams He Sun Max Planck Institute for Informatics Joint work with MadhusudanManjunath, Kurt Mehlhorn and KonstantinosPanagiotou

  2. Sub-graph Counting • Given a graph G=(V,E) with n nodes and m edges and a small graph H with • constant k edges, count the number of occurrences of H in G.

  3. Motivation Evaluate network connectivity Detect network motifs, e.g. biological networks Community detection Graph databases

  4. Data Streaming Model • Massive data sets • Limited working space • Sub-linear or poly-log space • Fast updating time • Desired approximation

  5. Related Works Problem: Can not deal with the dynamic case!! Cash Register Model:each item Counting triangles Bar-Yossef et al. 2002, Jowhari et al. 2005, Buriol et al. 2006 Counting K3,3 Bordino et al. 2008 Counting any sub-graph of three and four nodes Bordino et al. 2008

  6. Related Works Previous work can only count sub-graphs up to 6 edges. Open: Which sub-graphs can be counted in the data streaming model? Cash Register Model:each item Counting triangles Bar-Yossef et al. 2002, Jowhari et al. 2005, Buriol et al. 2006 Counting K3,3 Bordino et al. 2008 Counting any sub-graph of three and four nodes Bordino et al. 2008 Turnstile Model:each item Counting Triangles Jowhari et al. 2005

  7. Our Results We give an unbiased estimator for counting regular graphs with arbitrary size. We present the first algorithm for counting regular graphs with arbitrary size in the turnstile model. We study the roles of complex-valued hash functions in graph counting.

  8. Our Results Let G be a graph with n nodes and m edges. For anyk, there is an algorithm with space to -approximate the number of Ck In G. Moreover, the algorithm works in the turnstile model. We give an unbiased estimator for counting regular graphs with arbitrary size. We present the first algorithm for counting regular graphs with arbitrary size in the turnstile model. We study the roles of complex-valued hash functions in graph counting.

  9. Warm up: Counting Triangles Let be a 12-wise independent hash function. Let For every coming edge , let

  10. Algorithm Framework Every node is associates with a 8k-wise independent hash function 8k-wise independent hash function Give every edge an arbitrary orientation and each corresponds to one variable .

  11. Algorithm Framework (contd.) Update Step For every coming edge , do the following for each

  12. Algorithm Framework (contd.) Update Step For every coming edge , do the following for each Query Step Output the real part of , where is a constant and

  13. Observation • Need to control • vertex number • degree sequence • multi-edge • connectivity

  14. Unbiased Estimator The required random variables can be constructed within constant time and space.

  15. Proof Sketch By definition, we have Let be an arbitrary orientation of . Define Then

  16. Proof Sketch (contd.) By the properties of and , we get

  17. Open Questions Thank you • What else can be counted? • Unbiased estimator for counting any sub-graph in the turnstile model was known. • Reduce the variance. • Investigate the role of Clifford algebra in graph counting.

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