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R-MAT: A Recursive Model for Graph Mining
Deepayan Chakrabarti Yiping Zhan Christos Faloutsos
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Introduction Graphs are ubiquitous
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.
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“Network values” vs Rank
Graph Patterns Effective Diameter Count vs Indegree Count vs Outdegree Hop-plot Power Laws “Network values” vs Rank Count vs Stress Eigenvalue vs Rank
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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.
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Our Proposed Generator
Communities RedHat a b Communities within communities Linux guys b c d Mandrake Windows guys c d Cross-community links Shows a “community” effect
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Experiments (Epinions directed graph)
Effective Diameter Count vs Indegree Count vs Outdegree Hop-plot Count vs Stress Eigenvalue vs Rank “Network value” ►R-MAT matches directed graphs
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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
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Experiments (Epinions undirected graph)
Count vs Indegree Singular value vs Rank Hop-plot “Network value” Count vs Stress ►R-MAT matches undirected graphs
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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)
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The “DGX”/lognormal distribution
Deviations from power-laws have been observed [Pennock+ ’02] These are well-modeled by the DGX distri- bution [Bi+’01] Essentially fits a parabola instead of a line to the log-log plot. “Drifting” surfers Count “Devoted” surfer Degree Clickstream data
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Our Proposed Generator
R-MAT (Recursive MATrix) [SIAM DM’04] 2n 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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