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

2. 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.

3. “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

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
RedHat a b
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)
Effective Diameter
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)
Count vs Indegree
Singular value vs Rank
Hop-plot
“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-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

11. 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