R-MAT:%20A%20Recursive%20Model%20for%20Graph%20Mining

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
Graph Patterns
Power Laws
Network values vs Rank
Count vs Stress
Eigenvalue vs Rank
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Our Proposed Generator
Choose quadrant b
a
b
a0.4
b0.15
c
d
c0.15
d0.3
Initially
Choose quadrant c
and so on
..
Final cell chosen, drop an edge here.
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Our Proposed Generator
Communities
b
a
RedHat
Communities within communities
Linux guys
b
c
d
Mandrake
Windows guys
c
d
Cross-community links

• Shows a community effect

6
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
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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)
Hop-plot
Singular value vs Rank
Count vs Indegree
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-modeledby the DGX distri-bution
  Bi01
• Essentially fits aparabola insteadof a line to
  thelog-log plot.

Drifting surfers
Count
Devoted surfer
Degree
Clickstream data
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Our Proposed Generator

• R-MAT (Recursive MATrix) SIAM DM04

• Subdivide the adjacency matrix
• and choose one quadrant with probability
  (a,b,c,d)
• Recurse till we reach a 11 cell
• where we place an edge
• and repeat for all edges.

a 0.4
b 0.15
c 0.15
d 0.3