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Finding Communities on Complex Networks via

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Finding Communities on Complex Networks via Max-flow, Min-cut Theorem Seung-Woo Son, Hawoong Jeong 1 and Jae Dong Noh 2 1 Dept. Physics, Korea Advanced ... – PowerPoint PPT presentation

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Title: Finding Communities on Complex Networks via


1
Finding Communities on Complex Networks via
Max-flow, Min-cut Theorem
  • Seung-Woo Son, Hawoong Jeong 1 and Jae Dong Noh 2
  • 1 Dept. Physics, Korea Advanced Institute Science
    and Technology (KAIST)
  • 2 Dept. Physics, Chungnam National University,
    Daejeon, KOREA

2
Networks Communities
Symbolic structure which contains a set of
nodes(vertices) and links(edges).
Modular and community structures have functional
meaning in biological network or social network
sometimes. Finding community structures in
network level with unknown their function or
social relation is a key issue in network study.
E.Ravaz, A.L.Somera, D.A.Mongru, Z.N.Oltvai,
A.-L.Barabasi, Science, vol. 297 (2002)
3
Flow Problems in Networks
Traffic, flow of internet packets, information
flow (news), transport, energy flow etc.
4
Example of minimum cut on simple network with
maximum flow
simple network Edge e is undirected and has
unit capacity. ( Ce1 )
maximum flow 2
Max flow min cut theorem
5
Max-flow, Min-cut Theorem
st-cut (S,S) partition of the node set V
st-cut interface
st-cut capacity interface(cut) energy
min-cut minimum energy interface bottle neck
6
Communities Finding Method
Can we use the min-cut to shear off a community?
the number of nodes separated by min-cut (
close to source node i )
Multiplication Index
min-cuts of node i-j pairs when Mij gt M
Available(valid) min-cut
7
Three Possible Case
fried egg diagram )
weak source and strong target
weak source and weak target
two strong nodes
Cant find community structure
Community !!
Threshold
8
I. BA model
No community structure
BA model has just one hub and has no modular
structure.
9
II. Zachary Karate Club Network
There is community structure !
M
It has two community structures and core nodes
are 1,33,34.
10
III. Collaboration Network
Statistical Physics and Structure of RNA
It shows community structure.
M
N
Agent-based Models and Mathematical Ecology
11
IV. Applied to weighted network
Networks in economy
We can separate the strongly correlated stock
indexes.
DATA daily close prices of the selected 137
stocks in the US stock market for the 21-yr
period 1983-2003 (freely available at
http//finance.yahoo.com)
D.-H. Kim, et al., in preparation.
12
Summary
  • We study the maximum flow and minimum cut theorem
    on networks and properties of min-cuts.
  • We introduce the multiplication of cluster size
    Mij and develop the method to find the community
    structure.
  • We test the new community finding method on
    several networks and our method finds their
    community structures well.

13
Zachary Karate Club N34
14
Application to Random Fields Ising Magnetic Model
15
(No Transcript)
16
Application to Economic Problem
17
Discussion
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