A Ca-Svm Based Monte Carlo Approach for Evaluating Complex Network Reliability

jamaicacooperativeΤεχνίτη Νοημοσύνη και Ρομποτική

17 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

92 εμφανίσεις




Abstract



M
any real
-
world complex systems can be
modeled as networks.
Evaluation of network
reliability plays a
n

important role in engineering
applications.
W
hen evaluating the
S
-
T

complex
network reliability, the traditional approaches may
bring abo
ut
the problems of increasing
computational
complexity or

decreasing the calculation
accuracy
.
T
his paper proposes a CA
-
SVM based Monte Carlo
approach based on the drawbacks of traditional
approaches.

Support Vector Machine

(SVM)

is a fast
and efficient al
gorithm to ascertain the network
connectivity in simulation process.

Cellular automata
(CA)

is used for creating training data points
, which
speeds up the computing process. P
article swam
optimization (PSO) is used for parameters selection of
SVM
, which in
creases the accuracy of the result.
A
n
example is shown to illustrate the proposed approach.


Keywords



C
omplex network reliability
,
support

vector
machine
,
cellular automata, particle swarm optimization



I.
I
NTRODUCTION



M
any real
-
world complex system
s such as
communication systems

[
1
]
, power transmission and
distribution systems

[
2
]

and transportation systems

[
3
]

can
be modeled as networks.
The traditional
approaches

[
4
]

to
eval
uate the reliability of network

are based

on

the
minimal cut sets or minim
al path sets
.
N
evertheless these
approaches will lead to NP
-
hard problems due to the
incre
asing complexity of the network

[
4
]
.
A
s a result, some
works put forth

Monte
Carlo Simulation (MCS) based
approaches, which
are

now recognized as playing an
important

role in the evaluation of network reliability

[
4,

5
]
.


A
scertaining the connectivity of
S
-
T

network

is a
n

important
problem

under the above mentioned MCS
based approaches.
T
he
approaches frequently used before
are based on depth
-
first procedure or breadth
-
first
procedure

[
5
]
.
W
ith the increa
sing complexity of
the
network
, the abo
ve mentioned approaches will be

time
consuming.
C
ellular automata
(CA)

based

approach
,
which can ov
ercome the above difficulty, is proposed

to
the application in

network

reliabilit
y evaluation

[
6
,

7
]
.

R
ecently, s
ome works extend
the use of CA to problems
of computing the av
ailability of
the
renewable network

[
8
]
,
evaluating the
K
-
termin
al reliability of
the
network

[
9
]

and



T
his paper is supported by
National Natural Science Foundation of
China (70931004)
.

c
omputing the maximum unsplittable flow

of
the
network

[
9
]
.


Since
MCS based approaches require a large number
of connecti
vity evaluations of
S
-
T

network
, it may be
convenient to replace this evaluation by a
n

approximated
but
fast algorithm.

B
ecause

the
S
-
T

network has

two states
(operating or failed), i
t

s a two
-
c
ategory
classification

problem
to ascert
ain the connectivity of
the
network
.

Support Vector Machine

(SVM), as a artificial
intelligence technique, was developed by Vapnik

[
10
]
.
B
ecause of the superiority of
structural

risk minimization
principle

(SRM) over

empirical risk minimization
principle

(ERM), SVM has been widely used for
network

reliability evaluation

[
11,

12
]
.



The
quality of SVM classification models depends on
a proper setting of the parameters (SVM hyper
-
parameters
and SVM kernel parameters), s
o the main issue to apply
SVM is how to set these parameters.
T
he works published
before
chose

grid search method to set parameters

[
11,

12
]
.

H
owever, when using this m
ethod, one should increase
the
search range or dec
r
ease the step size to make the
optima
l solution accurate, which may result in a highly
time
-
consuming search process

[
13
]
.
T
o overcome this
problem, we substitute the grid search method with
particle swarm optimization (PSO)
.


In this work, we
proposed a CA
-
SVM based
Monte
Carlo
approach to e
valuate the complex network
reliability
.
F
irstly,
establish the tr
aining data with CA;
secondly, train the SVM
with PSO and cross
-
validation;
lastly,

ascertain the
connectivity of each simulation with
the SVM trained and compute the
S
-
T

network reliability
.



S
ection
II

briefly

describes CA
algorithm
. Section
III

briefly describes SVM and parameters selection using
PSO, then proposes

a CA
-
SVM
based Monte Carlo
approach

for evaluating

complex network reliability.

A
n
example

is presented in section
IV

to ill
ustrate the
proposed approach.




II.

ASCERTAINMENT OF THE

C
ONNECTIVITY OF
S
-
T NETWORK U
SING
CA



A.

D
escription of

CA


C
ellular a
utomata
(CA), a kind of
approach to
simul
ate
the
behavior

of d
ynamic discrete systems
, was
originally

conceived by Ulam and Von N
eumann in the
1950s.

CA consists of some cells, usually assumed to be
homogeneous

and with limited discrete states
.
E
ach cell

s
action at a given time
t

relies

on
its state at the time
t
-
1,
A
Ca
-
Svm Based Monte Carlo Approach for Evaluating Com
p
lex
Network

Reliability


Yuan
-
pen
g Ruan
1
,
Zhen

He
1

1

School of Management,

Tianjin University,

Tianjin,

China

(ruanyuanpeng@sina.com)






those

of its neighborhood
at the time
t
-
1

and

a
transition

rule
.
A
s shown in Fig. 1,
CA can be mainly classified into

one
-
dimensional CA and two
-
dimensional CA
based on
its cells


dimension
s
.

Two
-
dimensional CA also can be
classified into two categories: Von Neumann
neighborhood and Moore neighborhood

[
14
]
.









Fig.
1
.
Types of CA


B.

CA

A
lgorithm

of
A
scertain
ing

t
he
C
onnectivity



L
et
G
=(
N
,
A
) be a network graph, where
N

is the set of
n nodes,
A

is the set of directed arcs.
T
he
S
-
T

ne
twork
connectivity evaluation refers to finding if there is a path
from a source node
S

to a terminal node
T
.

It

s
assumed

that
E
i

is the neighborhood of the node
i
, defined as
E
i

=

{
j


N

s
.
t
.

(
j
,

i
)


A
}

and
w

(
i
,
t
)

is the state of node
i

at the
time
t
. T
he state
w

(
i
,
t
) of each node is binary, assuming
the value of 1 when node
i

is active

and of 0 when
passive
.



E
ach node
i

follows an OR
Boolean
transition
function

w
(
i,t+1
)
=
OR(
w
(
j,

t
)
,

,w
(
k,

t
)
,w
(
i,

t
))
,
j,

,k

E
i


(1)


As some works have shown
,
S
-
T

connectivity can be
computed in O(
n
) time using CA, which is an advantage
over the traditional approaches

[
6,9
].



T
he basic algorithm proceeds as follows:

1.

t

=

0

2.

Set all the cells state values to
0

3.

Set
w
(
S
,

0
)

=

1

4.

t

=

t

+

1

5.

Update all ce
lls states by
function

(1)

6.

If
w

(
T
,

t
)

=

1
, then stop
:
c
=1

and there is a path
between
S

and
T

7.

If
t

<

n



1

go to step 4. Else

8.

c

=

0

and there is no path between
S

and
T




III.
RELIABILITY EVALUATI
ON OF COMPLEX
NETWORK USING CA
-
SVM BASED MONTE CA
RLO
APPROACH


A.
Description

of SVM
C
lassifier



Support vector machine (SVM), which is desired to
find a separator to partition data
-
set as far as possible,
provides a novel approach to the two
-
category
classification problem
.


S
uppose a set of
N

train
ing data points {(
X
1
,
y
1
),

(
X
2
,
y
2
),


,

(
X
N
,
y
N
)}, where
y
i

= {1,
-
1}.
F
or linear SVM, as
shown in Fig.2, c
onsider the separating hyperplane






Fig.
2
.
Linear SVM


H
:
y

=
w





(1)

where
w

is normal to the hyperplane
H
.

The
two
hyperplanes

H
1
:

y

=
w

and

H
2
:

y

=
w



(2)

i
s
parallel to
H

and the data po
ints closest to the two
parallel

are called support vectors
.


T
o partition the
two groups completely, the optimal
separator can be obtained by a constrained
optimization

formulation

[
15
]
:





(3)

s.t.


H
owever, som
etimes it is impossible to separate the

training data points linearly.
T
o solve this problem,
imperfect

separation should be considered and the
formulation (
3
) will be
transformed

as fol
lows:




(4)

s.t.



T
he Lagrangian formulation for the dual problem of
formulation (
4
) is as follows:




(5)

s.t.


where

represents
the
Lagrangian multiplier of
X
i

and
C

is
the
penalty parameter.


After the solution has been obt
ained, the decision
function for new

X
i

is as follows:





(6)



F
or non
-
linear SVM, the decision function for new
X
i

can be obtained through
replacing


with kernel
function. There are ma
ny kernel functions which can be
used
[
15
]
.
A
mong them, the commonest are Gaussian

radial
basis

function and polynomial function, which is as
follows:


, (7)

.



(8)



B
ecause the reliability evaluation is a non
-
linear problem
and the parameter of Gaussian
radial basis
function

is
continuous, which is easy to be tuned
[
13
]
,
this paper
selects Gaussian
radial basis

function to train SVM
.


B.

Parameter O
ptimization U
sing PSO



S
ince the SVM generalization performance heavily
depends on the setting of
C

and

σ
, these parameters
should be set properly. P
article
swarm o
ptimization (PSO),
a population based optimization
algorithm
, was firs
t
introduced by
Kennedy and Eberhart
[
16
]
.
I
n PSO, each
particle represents a potential solution to the

optimization

problem.
T
he
performance
of each particle depends on the
pre
-
defined fitness function
.
E
ach particle flies

according
to its own experience
and the experience of its
neighboring particles

with a certain velocity
.


T
he flying velocity of each particle can be
updated
during each

iteration

with the equation



(
9
)

and
each particle will po
sition with the equation



(
10
)

T
he

and

respectively represent the best previous
position of each particle and the best particle in the whole

swarm
within

the iteration
t
.


and


determine
the weights of two parts.
c
1

and
c
2

are learning rates
which are nonnegative constants.
r
1

and
r
2

are
generated
random numbers
in the interval [0, 1].
w

is the ine
rtia
coefficient which is a constant in the interval [0, 1].


C. Proposed Approach




The proposed approach for reliability evaluation of
complex network is illustrated in Fig.3. CA and PSO are
fast and efficient alternatives for DFS and grid search
appro
ach
. MCS, as a kind of simulation technology, has a
big advantage over the tradition approaches, which can
result in NP
-
hard problems, for complex

network
reliability evaluation.




Fig.
3
.
Proposed algorithm flow diagram


IV.

EXAMPLE DISCUSSION



The
complex
network shown in Fig
.
4 was proposed
by Yoo and Deo

[
17
]
.
I
t
consist
s of 21 arcs which have the
following reliabilities:
r
7
=0.81,
r
4
=
r
12
=
r
13
=
r
19
=0.981,
and
other
r
i
=0.9.


Fig.
4
.
A complex netwo
rk











A
fter 100000 simulation iterations, t
he results shown
in
T
able

I

illustrate that

proposed
approach

gives a result
which is not better than CA
-
MCS based approach but
accurate enough for engineering applications.
B
ecause
proposed approach subst
itutes

SVM for CA when
ascertaining network

connectivity
during
each simulation

T
ABLE I

COMPARISON OF ALGORITHMS



P
roposed
approach

CA
-
MCS
base
d
appro
ach
[6, 7]

T
he exact result

Reliability


0.9968


0.9973

0.997186

R
elative


error rate

-
0.039%


0.01%

/



iteration, it will be a fast and eff
icient
substitute for CA
-
MCS

based

approach when a large number of simulation
iterations are needed.



V.

CONCLUSION



Complex n
etwork reliab
ility has become a hot
research topic recently.
S
-
T

network reliability evaluation
,
as the basis of this research topic, should be focused on.
Some traditional approaches can not solve complex

network reliability evaluation well due to
inconvenience
or ina
ccuracy.
T
his paper proposes a CA
-
SVM based
Monte Carlo approach for reliability evaluation of
complex network
, which is a fast and efficient substitute
for some

CA
-
MCS and DFS
-
SVM
based
approaches
.
This paper combines
CA
algorithm
, which can
creat
e
traini
ng data points instead of DFS, and
PSO
, which

can
be substituted for grid search algorithm
to select the
p
arameters of SVM
,
into the proposed approach.
T
he
proposed approach also can be extended to other areas
such as evaluation
s

of
network availability an
d
K
-
terminal
network reliability.



ACKNOWLEDGMENT



T
he authors

thank the editor and

the anonymous
referees

for their

comments and suggestions.





R
EFERENCES


[1]

K.

K. Aggarwall
,


A

simple method for reliability
evaluation of a communication system
,”

IE
EE Trans
.

Commun
.
,
COM
-
23
, pp.
563

566
,
May,

1975
.

[
2
]


W.C. Yeh,

A revised layered
-
network algorithm to search
for all d
-
minpaths of a limited
-
flow acyclic netwok,


IEEE
Trans.

R
eliab.,
vol. 47
,

no.
4
,

pp.
436

442
,
Dec.,

1998
.

[
3
]

T
.

A
ven
,

Availability
evaluation of oil
/
gas production and

transportation systems,


Reliab
.

Engng
.
,
vol. 18
,

no. 2,

pp.
35

44
,
Sep.
,

1987
.

[
4
]

R
.
Billinton,

R.
N. Allan,

Reliability evaluation of
engineering systems,

concepts and techniques
,
2
nd

ed
.

New
York: Plenum Press
,
1992.

[
5
]

G.

Fishman, “A

comparison of four Monte Carlo methods
for estimating the probability of s
-
t connectedness
,

IEEE
Trans.

R
eliab.,
vol.
35, no. 2, pp. 145

155,
June,

1986.

[
6
]

C.M.S. Rocco
,

J.A.

Moreno
,

“Network reliability
assessment using a cellular a
utomata approach
,


Reliability
Engineering and System Safety
,

vol. 78
, no.
3
, pp. 289

295,
June,

2002
.

[7]

W
.
C
.

Yeh,

Y.C. Lin
, Y
.
Y
.
Chung
,
“Performance analysis of
cellular automata Monte Carlo Simulation for estimating
network reliability
,


Expert Systems

with Applications,
vol.
37
, no.
5
, pp.
3537

3544
,
May,

20
10.

[8]

Z. Enrico, P. Luca, Z.
Valérie
,

A combination of Monte
Carlo simulation and cellular automa for computing the
availability of complex network systems,


Reliability
Engineering and System Safety
,

vol. 91
, no
.
2
, pp.
181

190
,
Feb.,

200
6.

[
9
]

C.M.S. Rocco
,
Z. Enrico
,


Solving advanced network
reliability problems by means of cellular automata and
Monte Carlo sampling
,


Reliability Engineering and System
Safety
,

vol. 89
, no.
2
, pp.
219

226
,
August,

200
5.

[10]

V.
N. Vapnik,
Statistical Learning Theory
.

New York:
John
Wiley
Press
,
1998.

[11]

C.M. Rocco, J.A. Moreno,

Fast Monte Carlo reliability
evaluation using support vector machine,


Reliability
Engineering and System Safety
,

vol. 76
, no.
3
, pp.
237

243
,
June,

20
0
2.

[12]

F.Q Yuan, U. Kumar, K.B. Misra,

Complex system
reliability evaluation using support vector machine for
incomplete data
-
set,


International

Journal

of

Performability Engineering
,

vol. 7
, no.
1
, pp.
32

42
,
January,

2010.

[13]

Y. Ren,

G.C. Bai,

Det
ermination of optimal SVM
parameters by using GA/PSO,


Journal of Computers
,

vol.
5
, no.
8
, pp.
32

42
,
August,

2010.

[14]

M. Tomassini, M. Perrenoud,

Cryptography with cellular
automata,


Appl.

Soft Comput.
,

vol. 1
, no.
2
, pp.
151

160
,
August,

2001.

[15]

B. Scholkopf, A.J. Smola,
Learning with Kernels
.

London
:
The MIT
Press
,
2000.

[16
]

J. Kennedy, R.C. Eberhart,

Particle Swarm Optimization,


in
Proceedings of IEEE International Conference on
Neural Networks
, NJ,

pp.
1942

1948.

[17]

Y.B. Yoo, N. Deo
,

A co
mparison of algorithms for
terminal
-
pair reliability,


IEEE Trans.

R
eliab.,
vol. 37
, no.
2, pp.
210

215
,
June,

198
8
.