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R-MAT: A Recursive Model for Graph Mining

R-MAT: A Recursive Model for Graph Mining. Deepayan Chakrabarti Yiping Zhan Christos Faloutsos. Introduction. Protein Interactions [genomebiology.com]. Internet Map [lumeta.com]. Food Web [Martinez ’91]. Graphs are ubiquitous “Patterns”  regularities that occur in many graphs

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R-MAT: A Recursive Model for Graph Mining

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  1. R-MAT: A Recursive Model for Graph Mining Deepayan Chakrabarti Yiping Zhan Christos Faloutsos

  2. Introduction Protein Interactions [genomebiology.com] Internet Map [lumeta.com] Food Web [Martinez ’91] • Graphs are ubiquitous • “Patterns”  regularities that occur in many graphs • We want a realistic and efficient graph generator • which matches many patterns • and would be very useful for simulation studies.

  3. Effective Diameter Hop-plot Count vs Outdegree Count vs Indegree Graph Patterns Power Laws “Network values” vs Rank Count vs Stress Eigenvalue vs Rank

  4. Our Proposed Generator Choose quadrant b a b a=0.4 b=0.15 c d c=0.15 d=0.3 Initially Choose quadrant c and so on ….. Final cell chosen, “drop” an edge here.

  5. Our Proposed Generator Communities b RedHat a Communities within communities Linux guys b c d Mandrake Windows guys c d Cross-community links Shows a “community” effect

  6. Effective Diameter Experiments (Epinions directed graph) Count vs Indegree Count vs Outdegree Hop-plot Count vs Stress Eigenvalue vs Rank “Network value” ►R-MAT matches directed graphs

  7. Experiments (Clickstream bipartite graph) Count vs Indegree Count vs Outdegree Hop-plot Singular value vs Rank Left “Network value” Right “Network value” ►R-MAT matches bipartite graphs

  8. Experiments (Epinions undirected graph) Singular value vs Rank Hop-plot Count vs Indegree “Network value” Count vs Stress ►R-MAT matches undirected graphs

  9. Conclusions The R-MAT graph generator • matches the patterns mentioned before • along with DGX/lognormal degree distributions  can be shown theoretically • exhibits a “Community” effect • generates undirected, directed, bipartite and weighted graphs with ease • requires only 3 parameters (a,b,c), • and, is fast and scalable  O(E logN)

  10. The “DGX”/lognormal distribution • Deviations from power-laws have been observed [Pennock+ ’02] • These are well-modeledby the DGX distri-bution [Bi+’01] • Essentially fits aparabola insteadof a line to thelog-log plot. “Drifting” surfers Count “Devoted” surfer Degree Clickstream data

  11. 2n 2n Our Proposed Generator • R-MAT (Recursive MATrix) [SIAM DM’04] • Subdivide the adjacency matrix • and choose one quadrant with probability (a,b,c,d) • Recurse till we reach a 1*1 cell • where we place an edge • and repeat for all edges. a = 0.4 b = 0.15 c = 0.15 d = 0.3

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