Particles Competition and

Particles Competition and
Cooperation in Networks for
Semi
-
Supervised Learning

Fabricio

Breve

Department of Electrical & Computer Engineering

University of Alberta


Seminar
-

October 09, 2009

Contents


Introduction

◦
Semi
-
Supervised Learning


Model Description

◦
Initial Configuration

◦
Node and Particle Dynamics

◦
Random
-
Deterministic Walk

◦
Algorithm


Computer Simulations

◦
Synthetic Data Sets

◦
Real
-
World Data Sets

◦
Fuzzy Output and Outlier Detection


Conclusions

Introduction


Data sets under processing are becoming
larger

◦
In many situations only a small subset of items
can be effectively labeled


Labeling process is often:


Expensive


Time consuming


Requires intensive human involvement


Introduction


Supervised Learning

◦
Only labeled data items are used for training


Unsupervised Learning

◦
All data items are unlabeled


Semi
-
Supervised Learning

◦
Combines a few labeled data items with a
large number of unlabeled data to produce
betters classifiers

X. Zhu, “Semi
-
supervised
learning literature
survey,” Computer
Sciences, University of
Wisconsin
-
Madison,
Tech. Rep. 1530, 2005
.

Semi
-
Supervised Learning:

Graph
-
Based Methods


X. Zhu, Z.
Ghahramani
, and J. Lafferty, “Semi
-
supervised learning using
gaussian

fields
and harmonic functions,” in
Proceedings of the Twentieth International Conference on
Machine Learning
, 2003, pp. 912
–
919.


D. Zhou, O.
Bousquet
, T. N.
Lal
, J. Weston, and B.
Schölkopf
, “Learning with local and
global consistency,” in
Advances in Neural Information Processing Systems
, vol. 16. MIT
Press, 2004, pp. 321
–
328. [Online]. Available:
http://www.kyb.tuebingen.mpg.de/bs/people/weston/localglobal.pdf


X. Zhu and Z.
Ghahramani
, “Learning from labeled and unlabeled data with label
propagation,” Carnegie Mellon University, Pittsburgh, Tech. Rep. CMU
-
CALD
-
02
-
107,
2002. [Online]. Available:
http://citeseer.ist.psu.edu/581346.html


F. Wang and C. Zhang, “Label propagation through linear neighborhoods,”
IEEE
Transactions on Knowledge and Data Engineering
, vol. 20, no. 1, pp. 55
–
67, Jan. 2008.


A. Blum and S.
Chawla
, “Learning from labeled and unlabeled data using graph
mincuts
,” in
Proceedings of the Eighteenth International Conference on Machine Learning
.
San Francisco: Morgan Kaufmann, 2001, pp. 19
–
26.


M.
Belkin
, I.
Matveeva
, and P.
Niyogi
, “Regularization and
semisupervised

learning on
large graphs,” in
Conference on Learning Theory.

Springer, 2004, pp. 624
–
638.


M.
Belkin
, N. P., and V.
Sindhwani
, “On manifold regularization,” in
Proceedings of the
Tenth International Workshop on Artificial Intelligence and Statistics (AISTAT 2005).
New
Jersey: Society for Artificial Intelligence and Statistics, 2005, pp. 17
–
24.


T.
Joachims
, “
Transductive

learning via spectral graph partitioning,” in
Proceedings of
International Conference on Machine Learning
. AAAI Press, 2003, pp. 290
–
297.




Graph
-
Based Methods


Advantage of identifying many different
class distributions


Most of them share the regularization
framework, differing only in the particular
choice of the loss function and the
regularizer


Most of them have high order of
computational complexity (
O
(
n
3
)), making
their applicability limited to small or
middle size data sets.

X. Zhu, “Semi
-
supervised learning literature survey,” Computer

Sciences, University of Wisconsin
-
Madison, Tech. Rep. 1530, 2005.

Particle

Competition


M. G.
Quiles
, L. Zhao, R. L. Alonso, and R. A. F.
Romero, “Particle competition for complex
network community detection,” Chaos, vol.
18, no. 3, p. 033107, 2008. [Online]. Available:
http://link.aip.org/link/?CHAOEH/18/033107/
1

◦
Particles walk in the network and compete with
each other in such a way that each of them tries
to possess as many nodes as possible.

◦
Each particle prevents other particles to invade
its territory.

◦
Finally, each particle is confined inside a network
community.

Illustration of the
community detection
process by competitive
particle walking. The
total number if nodes is
N=128, the number of

communities is
M=4. The
proportion of out links is
z
out

/ k=0.2, and the
average node degree is
k=16. (a) Initial
configuration. Four
particles, represented

by yellow the lightest
gray, cyan the second
lightest gray, orange the
third lightest gray, and
blue the second darkest
gray, are randomly put
in the network. Red the
darkest gray represents
free nodes. (b) A
snapshot at iteration
250. (c) A snapshot at
iteration 3500. (d) A
snapshot at iteration
7000.

Proposed Method


Particles competition and cooperation in
networks

◦
Competition for possession of nodes of the
network

◦
Cooperation among particles from the same
team (label)


Each team of particles tries to dominate as many
nodes as possible in a cooperative way and at the
same time prevent intrusion of particles of other
teams.

◦
Random
-
deterministic walk



Initial

Configuration


A particle is generated for each labeled node of
the network

◦
Particle’s
home node


Particles with same label play for the same team


Nodes have an ownership vector

◦
Labeled nodes have ownership set to their respective
teams.


Ex: [ 1 0 0 0 ] (4 classes, node labeled as class A)

◦
Unlabeled nodes have levels set equally for each team


Ex: [ 0.25 0.25 0.25 0.25 ] (4 classes, unlabeled node)


Particles initial position is set to their respective
home nodes.


0
1
0
1
Node and Particle Dynamics


Node Dynamics

◦
When a particle selects a neighbor to visit:


It decreases the domination level of the other
teams in this same node


It increases the domination level of its team in the
target node


Exception:


Labeled nodes domination levels are fixed


0
1
0
1
t

t+1

Node and Particle Dynamics


Particle Dynamics

◦
A particle will get:


stronger when it is targeting a node being
dominated by its team


weaker when it is targeting a node dominated by
other teams


0.8

0.2

0.2

0.8

0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
4

2

Node and Particle Dynamics


Distance table

◦
Keep the particle aware of
how far it is from its home
node


Prevents the particle from losing
all its strength when walking into
enemies neighborhoods


Keep them around to protect
their own neighborhood.

◦
Updated dynamically with local
information


Does not require any prior
calculation


0

1

1

2

3

3

4

4

?

Particles Walk


Shocks

◦
A particle really visits a target
node only if the domination
level of its team is higher
than others;

◦
otherwise, a shock happens
and the particle stays at the
current node until next
iteration.


How a particle chooses a
neighbor node to target?

◦
Random walk

◦
Deterministic walk


0.6

0.4

0.3

0.7

Random
-
deterministic walk


Random walk


The particle randomly
chooses any neighbor
to visit with no
concern about
domination levels or
distance



Deterministic walk


The particle will prefer
visiting nodes that its
team already
dominates and nodes
that are closer to their
home nodes


The particles must exhibit both movements in order to achieve
an equilibrium between exploratory and defensive behavior

0.8

0.2

0.6

0.4

0.3

0.7

Deterministic Moving Probabilities

Random Moving Probabilities

35%

18%

47%

33%

33%

33%

v
1

v
2

v
3

v
4

v
2

v
3

v
4

v
2

v
3

v
4

Algorithm

1)
Build the adjacency matrix,

2)
Set nodes domination levels,

3)
Set initial positions of particles at their corresponding
home nodes. Set particle strength and distance,

4)
Repeat steps 5 to 9 until convergence or until a predefined
number of steps has been achieved,

5)
For each particle, complete steps 6 to 9,

6)
Select the target node by using the combined random
-
deterministic rule,

7)
Update target node domination levels,

8)
Update particle strength,

9)
Update particle distance table,

10)
Label each unlabeled data item by the team of maximum
level of domination.

SYNTHETIC DATA SETS

Computer Simulations

Fig. 1. Classification of the
banana
-
shaped patterns. (a)
toy data set with 2; 000
samples divided in two classes,
20 samples are pre
-
labeled
(red circles and blue squares).
(b) classification achieved by
the proposed algorithm.

Fig. 3. Time series for different
values of
p
det
: (a) correct
detection rate (b) nodes’
maximum domination level (c)
average particle strength. Each
point is the average of 200
realizations using a banana
-
shaped toy data set

REAL
-
WORLD DATA
SETS

Computer Simulations

Fuzzy Output and Outlier
Detection


There are common cases where some
nodes in a network can belong to more
than one community

◦
Example: In a social network of friendship,
individuals often belong to several
communities: their families, their colleagues,
their classmates, etc

◦
These are called
overlap nodes

◦
Most known community detection algorithms
do not have a mechanism to detect them

Fuzzy Output and Outlier
Detection


Particle’s standard algorithm

◦
Final ownership levels define nodes labels


Very volatile under certain conditions


In overlap nodes the dominating team changes frequently


Levels do not correspond to overlap measures


Particle’s modified algorithm

◦
New variable: temporal averaged domination level
for each team at each node


Weighted by particle strength


Considers only the random movements


Now the champion is not the team who have won the last
games, but rather the team who have won more games in
the whole championship

Fig. 9. Fuzzy classification of two banana
-
shaped classes generated with different
variance parameters: (a)
s

= 0.6 (b)
s

= 0.8 (c)
s

= 1.0. Nodes size and colors
represent their respective overlap index detected by the proposed
method
.

Fig. 9. Fuzzy classification of two banana
-
shaped classes generated with different
variance parameters: (a)
s

= 0.6 (b)
s

= 0.8 (c)
s

= 1.0. Nodes size and colors
represent their respective overlap index detected by the proposed
method
.

Fig. 9. Fuzzy classification of two banana
-
shaped classes generated with different
variance parameters: (a)
s

= 0.6 (b)
s

= 0.8 (c)
s

= 1.0. Nodes size and colors
represent their respective overlap index detected by the proposed
method
.

Fig. 10. Classification of normally distributed classes (Gaussian distribution). (a) toy
data set with 1,000 samples divided in four classes, 20 samples are labeled,

5 from each class (red squares, blue triangles, green lozenges and purple stars). (b)
nodes size and colors represent their respective overlap index detected by the
proposed method.

Fig. 10. Classification of normally distributed classes (Gaussian distribution). (a) toy
data set with 1,000 samples divided in four classes, 20 samples are labeled,

5 from each class (red squares, blue triangles, green lozenges and purple stars). (b)
nodes size and colors represent their respective overlap index detected by the
proposed method.

Fig. 11. Comparative between the standard and the modified models: (a)
artificial data set with some wrongly labeled nodes (b) classification by the
standard particles method (c) classification by the modified particles method

Fig. 11. Comparative between the standard and the modified models: (a)
artificial data set with some wrongly labeled nodes (b) classification by the
standard particles method (c) classification by the modified particles method

Fig. 11. Comparative between the standard and the modified models: (a)
artificial data set with some wrongly labeled nodes (b) classification by the
standard particles method (c) classification by the modified particles method

Fig. 12. The karate club network. Nodes size and colors represent their
respective overlap index detected by the
proposed

method
.

Conclusions


The main contributions of the proposed model can
be outlined in the following way:

◦
unlike most other graph
-
based models, it does not rely on
loss functions or
regularizers
;

◦
it can classify data with many different distribution,
including linearly non
-
separable data;

◦
it has a lower order of complexity than other graph
-
based
models, thus it can be used to classify large data sets;

◦
it can achieve better classification rate than other classical
graph
-
based methods;

◦
it can detect overlap nodes and provide a fuzzy output for
each of them;

◦
it can be used to detect outliers and, consequently, to stop
error propagation.