Community Detection : A Simple Example Joon Ho Park, Yumlembam - - PowerPoint PPT Presentation

Community Detection : A Simple Example

Joon Ho Park, Yumlembam Hemajit and Ki-Ho Lee

Project Motivation

• To understand the basics of community det

ection

• To apply the ideas on traditional methods fo

r community detection to known system

• To figure out the clustering of proteins from t

he analogy of the project

Quick Review of Community Detection

• Traditional Methods of Clustering

– Graph Partitioning

• Dividing vertices in groups of predefined size
• Minimizing cut size (# edges running between clusters)

– Hierarchical clustering

• Including small clusters in larger clusters according to similarity
• Agglomerative (bottom-up) or divisive (top-down) algorithms

– Partitional clustering

• Distance between vertices = dissimilarity between vertices
• E.g., k-means clustering: minimizing the total intra-cluster distance

– Spectral clustering

• Clustering by eigenvectors of matrices (e.g., similarity matrix)

Mountain Top Valley Top Valley Top Mountain Hub Mountain Con dominium & S ki Valley Condo minium & Ski

# of vertices : 16 # of edges : 27 Z1 VT V VH Z2 MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2

Graph Partitioning

• Simplest conditions

– Dividing into two groups of equal size – Minimal # of edges between two groups – Maximal # of edges inside the modules – Kernighan-Ling algorithm

• Maximizing Q
• Q = (# of edges inside the modules) – (# of edges lying between them)

Z1 VT V VH Z2 MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 Cut Size : 11 Q = 4

Z2 Z1 VT V VH MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 Cut Size : 9 Q = 9

Z2 Z1 VT V VH MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 Cut Size : 8 Q = 10

Z2 Z1 VT V VH MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 Cut Size : 6 Q = 15

AT1 V Z2 Z1 VT VH MT H AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 Cut Size : 5 Q = 17

AT1 V Z2 Z1 VT VH MT H AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 Cut Size : 5 Q = 17

AT1 V Z2 Z1 VT VH MT H AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 Cut Size : 5 Q = 17

AT1 V Z2 Z1 VT VH MT H AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 Cut Size : 5 Q = 17

Hierarchical Clustering

• Simplest conditions

– Divisive algorithm

• Clusters are iteratively split by removing edges conn

ecting vertices with low similarity

– Vertex similarity

• Defined by the # of edge-(or vertex-) independent pa

ths between two vertices

• Independent paths do not share any edge (vertex).

Z1 VT V VH Z2 MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 4 2 3 3 3 3 2 2 3 2 3 2 3 3 3 3 3 3 2 4 3 3 3 2 2 2 3

Z1 VT V VH Z2 MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 2 3 3 3 3 3 3 3 2

Z1 VT V VH Z2 MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 2 3 3 3 3

Z1 VT V VH Z2 MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 3 3 3 3 3

Z1 VT V VH Z2 MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 2 3 3 3 2

1 2 3 4 6 5 7 9 10 11 14 12 13 8 15 16

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − = 2 ... ... ... ... ... ... ... ... 3 1 ... 4 1 ... 1 3 1 ... 1 3 L

size cut 4 1

T

= Ls s

1 2 3 4 6 5 7 9 10 11 14 12 13 8 15 16

size cut 4 1

T

= Ls s

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − − − = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 S

5

Z1 VT V VH Z2 MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 4 2 3 3 3 3 2 2 3 2 3 2 3 3 3 3 3 3 2 4 3 3 3 2 2 2 3

Z1 VT V VH Z2 MT H AT1 AP1 MH AT2 MC&S AP3 Z3 VC&S AP2 2 3 3 3 3

Not good enough !

Conclusions

• Graph partitioning proposes a basic idea for com

munity detection

• The concept of similarity is adopted to hierarchic

al, partitional and spectral clustering

• We’ve realized that the community detection can

be used for the clustering of protein databases if the similarity is replaced by the score (TM-score

• r RMSD, etc)