A Ca-Svm Based Monte Carlo Approach for Evaluating Complex Network
Reliability
Yuan-peng Ruan1, Zhen He1
1
School of Management, Tianjin University, Tianjin, China
(ruanyuanpeng@sina.com)
Abstract – Many real-world complex systems can be
modeled as networks. Evaluation of network
reliability plays an important role in engineering
applications. When evaluating the S-T complex
network reliability, the traditional approaches may
bring about the problems of increasing computational
complexity or decreasing the calculation accuracy.
This 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 algorithm to ascertain the network
connectivity in simulation process. Cellular automata
(CA) is used for creating training data points, which
speeds up the computing process. Particle swam
optimization (PSO) is used for parameters selection of
SVM, which increases the accuracy of the result. An
example is shown to illustrate the proposed approach.
Keywords – Complex network reliability, support vector
machine, cellular automata, particle swarm optimization
I. INTRODUCTION
Many real-world complex systems 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
evaluate the reliability of network are based on the
minimal cut sets or minimal path sets. Nevertheless these
approaches will lead to NP-hard problems due to the
increasing complexity of the network [4]. As 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].
Ascertaining the connectivity of S-T network is an
important problem under the above mentioned MCS
based approaches. The approaches frequently used before
are based on depth-first procedure or breadth-first
procedure [5]. With the increasing complexity of the
network, the above mentioned approaches will be time
consuming. Cellular automata (CA) based approach,
which can overcome the above difficulty, is proposed to
the application in network reliability evaluation [6, 7].
Recently, some works extend the use of CA to problems
of computing the availability of the renewable network [8],
evaluating the K-terminal reliability of the network [9] and
This paper is supported by National Natural Science Foundation of
China (70931004).
computing the maximum unsplittable flow of the network
[9]
.
Since MCS based approaches require a large number
of connectivity evaluations of S-T network, it may be
convenient to replace this evaluation by an approximated
but fast algorithm. Because the S-T network has two states
(operating or failed), it’s a two-category classification
problem to ascertain the connectivity of the network.
Support Vector Machine (SVM), as a artificial
intelligence technique, was developed by Vapnik [10].
Because 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), so the main issue to apply
SVM is how to set these parameters. The works published
before chose grid search method to set parameters [11, 12].
However, when using this method, one should increase
the search range or decrease the step size to make the
optimal solution accurate, which may result in a highly
time-consuming search process [13]. To 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 evaluate the complex network
reliability. Firstly, establish the training 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.
Section 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. An
example is presented in section IV to illustrate the
proposed approach.
II. ASCERTAINMENT OF THE CONNECTIVITY OF
S-T NETWORK USING CA
A. Description of CA
Cellular automata(CA), a kind of approach to
simulate the behavior of dynamic discrete systems, was
originally conceived by Ulam and Von Neumann in the
1950s. CA consists of some cells, usually assumed to be
homogeneous and with limited discrete states. Each cell’s
action at a given time t relies on its state at the time t-1,
those of its neighborhood at the time t-1 and a transition
rule. As shown in Fig. 1, CA can be mainly classified into
one-dimensional CA and two-dimensional CA based on
its cells’ dimensions. Two-dimensional CA also can be
classified into two categories: Von Neumann
neighborhood and Moore neighborhood [14].
i,j-1
i-1,j i,j i+1,j
i-1 i i+1
（a）One-dimensional type
Suppose a set of N training data points {(X1, y1), (X2,
y2), … , (XN, yN)}, where yi = {1, -1}. For linear SVM, as
shown in Fig.2, consider the separating hyperplane
i-1,j-1 i,j-1 i+1,j-1
i-1,j
i,j
H2
i+1,j
H
H1
i,j+1
i-1,j+1 i,j+1 i+1,j+1
(b)Von Neumann type
(c) Moore type
Fig.1. Types of CA
B. CA Algorithm of Ascertaining the Connectivity
Let G=(N,A) be a network graph, where N is the set of
n nodes, A is the set of directed arcs. The S-T network
connectivity evaluation refers to finding if there is a path
from a source node S to a terminal node T. It’s assumed
that Ei is the neighborhood of the node i, defined as Ei =
{j∈ N s.t. (j, i) ∈ A} and w (i, t) is the state of node i at the
time t. The state w (i, t) of each node is binary, assuming
the value of 1 when node i is active and of 0 when passive.
Each node i follows an OR Boolean transition
function
w(i,t+1)=OR(w(j, t),…,w(k, t),w(i, t)), j,…,k∈Ei
(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].
The 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 cells 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
Fig. 2. Linear SVM
H: y = w  X  b  0
(1)
where w is normal to the hyperplane H. The two
hyperplanes
H1: y = w  X  b  1 and
H2: y = w  X  b  1
(2)
is parallel to H and the data points closest to the two
parallel are called support vectors.
To partition the two groups completely, the optimal
separator can be obtained by a constrained optimization
formulation [15]:
1
2
Min w
(3)
2
w,b
s.t. yi (w  X i  b)  1, i  1,2,..., N .
However, sometimes it is impossible to separate the
training data points linearly. To solve this problem,
imperfect separation should be considered and the
formulation (3) will be transformed as follows:
N
1
2
Min w  C   i
w,b 2
i 1
s.t.
(4)
yi (w  X i  b)  1  i ,
i  0, i  1,2,..., N .
The Lagrangian formulation for the dual problem of
formulation (4) is as follows:
N
1 N N
   i  j yi y j X i  X j    k
2 i 1i 1
k 1
Max 
(5)
s.t. 0  i  C, i  1,2,..., N
III. RELIABILITY EVALUATION OF COMPLEX
NETWORK USING CA-SVM BASED MONTE CARLO
APPROACH
A. Description of SVM Classifier
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.
m
 y
i i
0
i 0
where
i
represents the Lagrangian multiplier of Xi and
C is the penalty parameter.
After the solution has been obtained, the decision
function for new Xi is as follows:
N

f ( X i )  sgn(
j 1
j y j Xi
 X j  b)
(6)
For non-linear SVM, the decision function for new Xi
can be obtained through replacing X i  X j with kernel
Start algorithm
function. There are many kernel functions which can be
used [15]. Among them, the commonest are Gaussian radial
basis function and polynomial function, which is as
follows:
 Xi X
j
/ 2
No
2
,
k ( X i , X j )  ( X i  X j  1) d .
(7)
Set fitness function(cross
validation accuracy)
Create a network state
randomly based on arc
reliability
Yes
Initialize position and
velocity of each particle
Ascertain the connectivity of
the created network using
SVM
(8)
Because 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.
Perform SVM on each
particle and compute the
fitness function
Max.
Max. simulation
simulation
iteration
iteration M?
M?
Max.
Max. iteration
iteration
F?
F?
Yes
B. Parameter Optimization Using PSO
Compute the network
reliability
IV. EXAMPLE DISCUSSION
The complex network shown in Fig.4 was proposed
by Yoo and Deo [17]. It consists of 21 arcs which have the
following reliabilities: r7=0.81, r4=r12=r13=r19=0.981, and
other ri=0.9.
3
5
16
7
9
15
10
10
5
9
14
2
1
3
11 T
20
S
8
8
1
pijt and gijt respectively represent the best previous
11
7
12
13
6
Fig. 4. A complex network
TABLE I
COMPARISON OF ALGORITHMS
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
approach. 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.
17
18
(10)
position of each particle and the best particle in the whole
swarm within the iteration t. c1  r1 and c2  r2 determine
the weights of two parts. c1 and c2 are learning rates
which are nonnegative constants. r1 and r2 are generated
random numbers in the interval [0, 1]. w is the inertia
coefficient which is a constant in the interval [0, 1].
4
2
4
The
Fig. 3. Proposed algorithm flow diagram
6
and each particle will position with the equation
xijt  xijt 1  vijt 1.
End algorithm
21
Since the SVM generalization performance heavily
depends on the setting of C and σ, these parameters
should be set properly. Particle swarm optimization (PSO),
a population based optimization algorithm, was first
introduced by Kennedy and Eberhart [16]. In PSO, each
particle represents a potential solution to the optimization
problem. The performance of each particle depends on the
pre-defined fitness function. Each particle flies according
to its own experience and the experience of its
neighboring particles with a certain velocity.
The flying velocity of each particle can be updated
during each iteration with the equation
(9)
vijt  wvijt 1  c1r1 ( pijt 1  xijt 1 )  c2 r2 ( g ijt 1  xijt 1 )
No
Update position and
velocity of each particle
19
k(Xi , X j )  e
2
Create N training data
points using CA
Train SVM with optimal C
and σ
Reliability
Relative
error rate
Proposed
approach
0.9968
CA-MCS based
approach[6, 7]
0.9973
-0.039%
0.01%
The exact result
0.997186
/
After 100000 simulation iterations, the results shown
in Table I illustrate that proposed approach gives a result
which is not better than CA-MCS based approach but
accurate enough for engineering applications. Because
proposed approach substitutes SVM for CA when
ascertaining network connectivity during each simulation
iteration, it will be a fast and efficient substitute for CAMCS based approach when a large number of simulation
iterations are needed.
V. CONCLUSION
Complex network reliability 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 inaccuracy. This 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 create
training data points instead of DFS, and PSO, which can
be substituted for grid search algorithm to select the
parameters of SVM, into the proposed approach. The
proposed approach also can be extended to other areas
such as evaluations of network availability and K-terminal
network reliability.
ACKNOWLEDGMENT
The authors thank the editor and the anonymous
referees for their comments and suggestions.
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