Graph Based Clustering Summer School “Achievements and Applications of Contemporary Informatics, Mathematics and Physics” (AACIMP 2011) August 8-20, 2011, Kiev, Ukraine Erik Kropat University of the Bundeswehr Munich Institute for Theoretical Computer Science, Mathematics and Operations Research Neubiberg, Germany
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Graph Based Clustering
Summer School
“Achievements and Applications of Contemporary Informatics,
Mathematics and Physics” (AACIMP 2011)
August 8-20, 2011, Kiev, Ukraine
Erik Kropat
University of the Bundeswehr Munich Institute for Theoretical Computer Science,
Mathematics and Operations Research
Neubiberg, Germany
Real World Networks
• Biological Networks
− Gene regulatory networks
− Metabolic networks
− Neural networks
− Food webs
• Technological Networks
− Telecommunication networks
− Internet
− Power grids
food web
power grid
Real World Networks
• Social Networks
− Communication networks
− Organizational networks
− Social media
− Online communities
• Economic Networks
− Financial market networks
− Trade networks
− Collaboration networks
social networks
economic networks
Source: Frank Schweitzer et al., “Economic Networks: The New Challenges,” Science 325, no. 5939 (July 24, 2009): 422-425.
Graph-Theory
• Graph theory can provide more detailed information about the inner structure of the data set in terms of
− cliques (subsets of nodes where each pair of elements is connected)
− clusters (highly connected groups of nodes)
− centrality (important nodes, hubs)
− outliers . . . (unimportant nodes)
• Applications
− social network analysis
− diffusion of information
− spreading of diseases or rumours
⇒ marketing campaigns, viral marketing, social network advertising
Graph-Based Clustering
• Collection of a wide range of very popular clustering algorithms
that are based on graph-theory.
• Organize information in large datasets to facilitate users
for faster access to required information.
Idea
• Objects are represented as nodes in a complete or connected graph.
• Assign a weight to each branch between the two nodes x and y.
The weight is defined by the distance d(x,y) between the nodes.
Clustering Distance between
clusters Distance between objects
Idea
minimal spanning tree
graph
clusters
Graph Based Clustering
Hierarchical method
(1) Determine a minimal spanning tree (MST)
(2) Delete branches iteratively
New connected components = Cluster
1
3
5
8
4
6
Minimal Spanning Trees
Minimal Spanning Tree
A minimal spanning tree of a connected graph G = (V,E)
is a connected subgraph with minimal weight
that contains all nodes of G and has no cycles.
1
3
5
8
4
6
a
1
3
5
8
4
6
c
d
b
a
c
d
b
minimal spanning tree graph G = (V, E)
Minimal spanning trees can be calculated with...
(1) Prim’s algorithm.
(2) Kruskal’s algorithm.
a
1
3
5
8
4
6
c
d
b
Example – Prims’s Algorithm
1
3
5
8
4
6
a
b
c
d
Set VT = {a}, ET = { }
1
3
5
8
4
6
a
b
c
d
Choose an edge (x,y) with minimal weight such that x ∈ VT and y ∉ VT.
VT = {a,b} and ET = { (a,b) }.
Example– Prims’s Algorithm
c
1
3
5
8
4
6
a
b
c
d
Choose an edge (x,y) with minimal weight such that x ∈ VT and y ∉ VT.
VT = {a,b,d} and ET = { (a,b), (a,d) }.
Choose an edge (x,y) with minimal weight such that x ∈ VT and y ∉ VT.
VT = {a,b,c,d} and ET = { (a,b), (a,d),(b,c) }.
c
1
3
5
8
4
6
a
c
d
b
Prim’s Algorithm
INPUT: Weighted graph G = (V, E), undirected + connected
OUTPUT: Minimal spanning tree T = (VT, ET) (1) Set VT = {v}, ET = { }, where v is an arbitrary node from V (starting point).
(2) REPEAT
(3) Choose an edge (a,b) with minimal weight, such that a ∈ VT and b ∉ VT.
(4) Set VT = VT ∪ {b} and ET = ET ∪ { (a,b) }.
(5) UNTIL VT = V
Kruskal’s Algorithm
INPUT: Weighted graph G = (V, E), undirected + connected
OUTPUT: Minimal spanning tree T = (VT, ET) (1) Set VT = V, ET = { }, H = E.
(2) Initialize a queue to contain all edges in G, using the weights in ascending order as keys.
(3) WHILE H ≠ { }
(4) Choose an edge e ∈ H with minimal weight.
(5) Set H = H \ {e}.
(6) If (VT, ET ∪ {e}) has no cycles, then ET = ET ∪ {e} .
(7) END
Branch Deletion
Delete Branches - Different Strategies
(1) Delete the branch with maximum weight.
(2) Delete inconsistent branches.
(3) Delete by analysis of weights.
(1) Delete the branch with maximum weight
• In each step, create two new clusters by deleting the branch with maximum weight.
• Repeat until the given number of clusters is reached.