1. Graph Clustering based on Random Walk
Bing Lidong

2. Outline Background MCL MCL++ Graph Clustering Random Walks Basis
Inflation Operator
Algorithm
Convergence
MCL++
R-MCL
MLR-MCL

3. Outline Background MCL MCL++ Graph Clustering Random Walks Basis
Inflation Operator
Algorithm
Convergence
MCL++
R-MCL
MLR-MCL

4. Graph Clustering Clustering: group items naturally
Vector clustering Graph clustering
Many links within a cluster, and fewer links between clusters
Vectors are more likely to each other in the same cluster

5. Random Walk
Observation: If you start at a node, and then randomly travel to a connected node, you’re more likely to stay within a cluster than travel between.
This is what MCL based on.
Random walk on a graph is a Markov process, that means next state only depends on current state.

6. Outline Background MCL MCL++ Graph Clustering Random Walks Basis
Inflation Operator
Algorithm
Convergence
MCL++
R-MCL
MLR-MCL

7. Example 1 2 3 4 5 6 Transition matrix P P1000 What’s wrong??
What’s wrong??
P1000

8. What happened? 1 2 3 4 5 6
"Flow is easier within dense regions than across sparse boundaries, however, in the long run this effect disappears."
How to deal with it?
During the walking, we should encourage the intra-cluster communications and punish the inter-ones.

9. MCL Inflation MCL adjusting the transitions by columns.
For each vertex, the transition values are changed so that
Strong neighbors are further strengthened
Less popular neighbors are demoted.
This adjusting can be done by raising a single column to a non-negative power, and then re-normalizing.
This operation is named “Inflation” (the matrix powers is named “Expansion”)

10. Inflation operation

11. Inflation example
Strengthens strong flows, and weakens already weak flows
The inflation parameter, r, controls the extent of this strengthening / weakening. This influences the granularity of clusters.
Square, and
then normalize

12. MCL Algorithm Two processes are repeated alternately: Expansion
Inflation

13. Convergence
Convergence is not proven in the thesis, however it is shown experimentally that it often does occur.
In practice, the algorithm converges nearly always to a "doubly idempotent" matrix:
It's at steady state.
Every value in a single column has the same number

14. Example

15. Example (cont.)

16. Example (cont.)
How to interpret clusters?

17. MCL Interpreting Clusters
To interpret clusters, the vertices are split into two types. Attractors, which attract other vertices, and vertices that are being attracted by the attractors.
Attractors have at least one positive flow value within their corresponding row (in the steady state matrix).
Each attractor is attracting the vertices which have positive values within its row.
Attractors and the elements they attract are swept together into the same cluster.

18. Overlapping clusters
Only when a vertex is attracted exactly equally by more than one cluster
This occurs only when both clusters are isomorphic

19. Inflation parameter

20. MCL Analysis For clusters with large diameter, MCL has problems
Distributing flow across cluster needs long expansion and low inflation (otherwise the cluster will split).
Takes many iterations and causes MCL to be sensitive to small perturbations in the graph.

21. MCL Analysis (cont.) O(N3), where N is the number of vertices
N3 cost of one matrix multiplication on two matrices of dimension N.
Inflation can be done in O(N2) time
The number of steps to converge is not proven, but experimentally shown to be ~10 to 100 steps, and mostly consist of sparse matrices after the first few steps.
Speed can be improved through pruning
Inspect matrix and set small values directly to zero
Works well when the diameter of the clusters is small

22. Outline Background MCL MCL++ Graph Clustering Random Walks Basis
Inflation Operator
Algorithm
Convergence
MCL++
R-MCL
MLR-MCL

23. References
[1] S. V. Dongen. Graph Clustering by Flow Simulation. PhD Thesis, University of Utrecht, 2000.  archive.library.uu.nl/dissertations/ /inhoud.htm
[2] winter/MCL_Presentation2.pdf
[3] V. Satuluri and S. Parthasarathy. Scalable Graph Clustering Using Stochastic Flows: Applications to Community Discovery, KDD'09.
[4] atuluri_sgcusfacd/kdd09_satuluri_sgcusfacd_01.ppt