Presentation on Graph based approaches to Gene Expression Clustering.

1. GENE EXPRESSION
CLUSTERING
GRAPH BASED APPROACHES
A P R E S E N T A T I O N B Y GOVIND M (M120432CS)
MTECH COMPUTER SCIENCE AND ENGINEERING
N AT I O N A L I N S T I T U T E O F T E C H N O L O G Y C A L I C U T
govindmaheswaran@gmail.com

2. Clustering and Graph Theory
Using Graphs in
Clustering
Simple Graph Partitioning Outline
Spectral Graph Partitioning
Conclusion

3. Clustering
• Process of Grouping a set of data objects, in terms of similarity
• Same Cluster => Similar Objects and vice versa.
• Widely used in data mining, market analysis etc.
• Used to make sense of Bioinformatics data.
• Two major purposes, in Bioinformatics
• Find properties of genes ( Relationship among genes, deduce the functions of genes etc)
• Predict more relevant factors (eg. Clustering cancerous and non cancerous
genes, finding the effect of a medication)

4. Graphs
• Data Structure
• Used in multiple domains
• Key Terms
• Edge
• Vertex
• Weighted Graph

5. Some Graph Theory
• Cut
• Partitioning

6. Clustering using Graphs
Involves 3 steps
1. Preprocessing
◦ Convert data set into a graph
◦ Using Adjacency matrix and Degree Matrix representation
◦ Similarity between nodes can be taken as the weight of an edge.
2. Partitioning
◦ Partition the graph
3. Clustering
◦ Repeat until required number of clusters are obtained
◦ Alternatively, extra iterations followed by joinings may also be implemented.

7. Simple Graph Partitioning
• Weight of an edge = Similarity between the nodes
• Find Minimum Cut
• Edge Value decreases, cluster differs

8. Simple Graph Partitioning : The
Algorithm
Input : Graph G<V,E>, Number of Clusters k
Output: Cluster of Graphs
Repeat k-1 times
Low_val = infinity
For each edge e of the graph
Calculate Cut_Cost, cost of a CUT at that edge
if Cut_Cost < Low_val
Low_Val = cut_cost
Cut_Edge = e
Cut at edge e

9. Simple Graph Partitioning (cont..)
• Advantage
• Simple to implement
• Uses the concept of Min Cut.
• Disadvantage
• What about intra-cluster similarity..?

10. Spectral Graph Partitioning
• Is widely used
• Uses Eigen Vectors of Laplacian Matrix
• Recursive algorithm
• Qualitatively Good
• Computationally Better than SGP.

11. Some graph theory…
d1 = 7
• Degree : d2 = 3
d3 = 1
d4 = 0
0 2 5 0
• Affinity Matrix : 0 0 3 0
0 0 0 1
0 0 0 0
7 0 0 0
0 3 0 0
• Degree Matrix 0 0 1 0
0 0 0 0
-7 2 5 0
0 -3 3 0
• Laplacian Matrix : 0 0 -1 1
0 0 0 0

12. Some more Graph Theory…
• Spectrum : Eigen vectors, arranged in the order of magnitude of eigen values.
• Eigen Values of Graphs
• Calculated as Eigen values of Laplacian matrix of the graph
• Corresponidngly Eigen Vectors too
• Fiedler Theorm
• Correlation b/w eigen vectors and graph properties
• Principal Eigen Vectors. Kth Principal Eigen Vector.
• Principal Eigen Vector : Centrality of Vertices
• 2nd Principal Eigen Vector : algebraic connectivity
• Called Fiedler Vector
• Matrix of positive and negative values
• Partition is decided by the Sign of the value.

13. Spectral Graph Partitioning
Input : Graph G<V,E>
Output: Graphs G1< V1,E1>, G2< V2,E2>
Create the Laplacian Vector L, of the Graph G.
Calculate the Fiedler Vector F
for each vertex vi in G
if F[i]>0
V1.append(v)
else
V2.append(v)

14. SPG : Example
2nd Principal Vector = <0.415, 0.309, 0.069, −0.221, 0.221, −0.794>
2nd Principal Vector = <0.415, 0.309, -0.190, 0.169, >
(of 1235)

15. SGP : Bipartitioning Method
(contd.)
• Recursive Algorithm
• Although better than Simple Graph Partitioning, not optimum
• Multiple times bipartitioning.
• Can be improved by Multipartitioning
• Use more eigen vectors.

16. Conclusion
• Clustering is Based on simple concepts of graph theory
• Optimal results (Spectral methods)
• Can give better performance than traditional clustering.
• Preprocessing overhead.

17. References
1. Yanhua Chen; Ming Dong; Rege, M., "Gene Expression Clustering: a Novel Graph Partitioning
Approach," Neural Networks, 2007. IJCNN 2007. International Joint Conference on
, vol., no., pp.1542,1547, 12-17 Aug. 2007, doi: 10.1109/IJCNN.2007.4371187
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4371187&isnumber=4370
891
2. Hagen, L.; Kahng, A.B., "New spectral methods for ratio cut partitioning and clustering,"
Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on
, vol.11, no.9, pp.1074,1085, Sep 1992, doi: 10.1109/43.159993
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=159993&isnumber=4190
3. Donath, W.E.; Hoffman, A.J., "Lower Bounds for the Partitioning of Graphs," IBM Journal of
Research and Development, vol. 17, pp. 420-425, 1973.
4. Pavla Kabel´ıková , “Graph Partitioning Using Spectral Methods”, Thesis, VˇSB - Technical
University of Ostrava, 2006.
5. Chung, F.R.K., "Spectral Graph Theory," American Mathematical Society, 1997.