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
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