The document discusses graph-based clustering methods. It describes how graphs can be used to represent real-world networks from domains like biology, technology, social networks, and economics. It introduces the idea of using minimal spanning trees and hierarchical clustering to identify clusters in graph data. Two common algorithms for finding minimal spanning trees are described: Prim's algorithm and Kruskal's algorithm. Different strategies for iteratively deleting branches from the minimal spanning tree are also summarized to form clusters, such as deleting the branch with the maximum weight or inconsistent branches based on a reference value.

1. Summer School
“Achievements and Applications of Contemporary Informatics,
Mathematics and Physics” (AACIMP 2011)
August 8-20, 2011, Kiev, Ukraine
Graph Based Clustering
Erik Kropat
University of the Bundeswehr Munich
Institute for Theoretical Computer Science,
Mathematics and Operations Research
Neubiberg, Germany

2. Real World Networks
• Biological Networks
− Gene regulatory networks
− Metabolic networks
− Neural networks
− Food webs
food web
• Technological Networks
− Telecommunication networks
− Internet
− Power grids
power grid

3. Real World Networks
• Social Networks
− Communication networks
− Organizational networks
− Social media
− Online communities
social networks
• Economic Networks
− Financial market networks
− Trade networks
− Collaboration networks
economic networks
Source: Frank Schweitzer et al., “Economic Networks: The New Challenges,” Science 325, no. 5939 (July 24, 2009): 422-425.

4. 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

5. 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.

6. 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

7. Idea
graph
minimal spanning tree clusters

8. Graph Based Clustering
Hierarchical method
(1) Determine a minimal spanning tree (MST)
(2) Delete branches iteratively
New connected components = Cluster
4
6 5
1 8
3

9. Minimal Spanning Trees

10. 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.
c c
4 4
6 5 6 5
b b
1 8 1 8
a 3 d a 3 d
graph G = (V, E) minimal spanning tree

11. Minimal spanning trees can be calculated with...
(1) Prim’s algorithm.
(2) Kruskal’s algorithm.
c
4
6 5
b
1 8
a 3 d

12. Example – Prims’s Algorithm
Set VT = {a}, ET = { } Choose an edge (x,y) with minimal weight
such that x ∈ VT and y ∉ VT.
VT = {a,b} and ET = { (a,b) }.
c c
4 4
6 5 6 5
b b
1 8 1 8
a 3 d a 3 d

13. Example– Prims’s Algorithm
Choose an edge (x,y) with minimal weight Choose an edge (x,y) with minimal weight
such that x ∈ VT and y ∉ VT. such that x ∈ VT and y ∉ VT.
VT = {a,b,d} and ET = { (a,b), (a,d) }. VT = {a,b,c,d} and ET = { (a,b), (a,d),(b,c) }.
c c
4 4
6 5 6 5
b b
1 8 1 8
c c
a 3 d a 3 d

14. 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

15. 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

16. Branch Deletion

17. Delete Branches - Different Strategies
(1) Delete the branch with maximum weight.
(2) Delete inconsistent branches.
(3) Delete by analysis of weights.

18. (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.
2
2 2
4
2 3
6
2

19. Example: Delete the branch with maximum weight
2
2 2
4
2 3
Minimum spanning tree
6
2
Ordered weights of branches: 6, 4, 3, 2, 2, 2, 2, 2.

20. Example: Delete the branch with maximum weight
2
2 2
4
2 3
6
2
Ordered weights of branches: 6, 4, 3, 2, 2, 2, 2, 2.
Step 1: Delete branch (weight 6) ⇒ 2 clusters

21. Example: Delete the branch with maximum weight
2
2 2
4
2 3
6
2
Ordered weights of branches: 6, 4, 3, 2, 2, 2, 2, 2.
Step 1: Delete branch (weight 6) ⇒ 2 clusters
Step 2: Delete branch (weight 4) ⇒ 3 clusters

22. (2) Delete inconsistent branches
• A branch e is inconsistent, if the corresponding weight de
_
is (much) larger than a reference value de .
_
• The reference value de can be defined by the average weight
of all branches adjacent to e.
_
3+2+1
de = _________ = 2
3
1
e 3
6 _
2
d e = 6 > 2 = de
⇒ e inconsistent

23. (3) Delete by analysis of weights
• Perform an “analysis” of all weights of branches in the MST.
Determine a threshold S.
• The threshold can be estimated by
histograms on the weights of branches (= length of branches).
• Delete a branches, if the corresponding weight higher than the threshold S.
Number
Number
S
weight of branch weight of branch
(length of branch)

24. Exercise d
3 20
5
e c
9 8
1 4
15 g 6
12
f b
10 2
a
Find a minimal spanning tree and provide a clustering of the graph
by deleting all inconsistent branches.

25. Example
Set VT = {a}, ET = { } Choose an edge (x,y) with minimal weight
such that x ∈ VT and y ∉ VT.

26. Example
Choose an edge (x,y) with minimal weight Choose an edge (x,y) with minimal weight
such that x ∈ VT and y ∉ VT. such that x ∈ VT and y ∉ VT.

27. Example
Choose an edge (x,y) with minimal weight Choose an edge (x,y) with minimal weight
such that x ∈ VT and y ∉ VT. such that x ∈ VT and y ∉ VT.

28. Example
Choose an edge (x,y) with minimal weight
such that x ∈ VT and y ∉ VT.
minimal spanning tree

29. Example
For each branch calculate the reference value
(average weight of adjacent branches)
d
3
(3) (4.5) 5
e c
1 (3) (4) 4
g 6
(3.6)
f b
(5)
2
a

30. Example
Delete inconsistent branches
(weight is larger than the reference value)
d
2 clusters
3
(3)
e c
1 (3) (4) 4
g
f b
Noise?
a

31. Summary

32. Summary
• In graph based clustering objects are represented as nodes
in a complete or connected graph.
• The distance between two objects is given by the weight
of the corresponding branch.
• Hierarchical method
(1) Determine a minimal spanning tree (MST)
(2) Delete branches iteratively
• Visualization of information in large datasets.

33. Literature
• V. Kumar, M. Steinbach, P.-N. Tan
Introduction to Data Mining.
Addison Wesley, 2005.
Other work mentioned in the presentation
• J.A. Dunne, R.J. Williams, N.D. Martinez, R.A. Wood, D.H. Erwin
Compilation and Network Analyses of Cambrian Food Webs.
PLoS Biol 6(4): e102. doi:10.1371/journal.pbio.0060102
• F. Schweitzer, G. Fagiolo, D. Sornette, F. Vega-Redondo,
A. Vespignani, D.R. White
Economic Networks: The New Challenges.
Science 325, no. 5939 (July 24, 2009): 422-425.

34. Thank you very much!