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422
The Open Cybernetics & Systemics Journal, 2015, 9, 422-427
Open Access
Evolutionary Model of Equipment Maintenance Support Force Networks
with Partial Information
Lu Yu1, Han Zhen2,*, Gu Ping2 and Chen Liyun1
1
Department of Information Engineering, Mechanical Engineering College, Shijiazhuang, Hebei, 050003, P.R. China;
Department of Equipment Command and Management, Mechanical Engineering College, Shijiazhuang, Hebei, 050003,
P.R. China
2
Abstract: The paper draws on the theory of complex networks, puts forward an evolutionary model to solve the evolutionary problem of equipment maintenance support force systems, whose partial information are only known in the reality. Firstly, we abstract equipment maintenance support force systems into complex networks, and then define six network
parameters. Secondly, we describe the network evolutionary steps. Finally, a dynamic evolutionary model whose parameters can be flexibly set, is used as the test object. We discuss various network parameters under three kinds of selection
probability. The parameter values demonstrate the feasibility of the evolutionary model, and illustrate the model can provide auxiliary decision for commanders to develop support programs in the pre-war.
Keywords: Complex networks, equipment maintenance support force, evolutionary model and partial information.
1. INTRODUCTION
In the information battlefield, equipment maintenance
support tasks and support conditions are uncertain and
changeable [1], which bring high uncertainty for support
activities. At the same time, with the emergence of perception and response support theory, the timeliness, adaptability,
distribution and synergy of equipment maintenance support
are facing new challenges [2]. These objectively require the
transition from the equipment maintenance support force
systems to complex networks.
At present, the complex networks’ theory has widely
used in the industrial production systems, national power
systems, urban transport systems and community management systems, etc [3-7]. With the development of science
and technology, its advantages will become increasingly
apparent.
In wartime, the partial information of our equipment
maintenance support force systems are often grasped by the
enemy [8]. At the same time, their evolutionary problems
have not been studied yet from the perspective of complex
networks. Therefore, the paper introduces the theory of complex networks and builds the evolutionary model of equipment maintenance support force networks with partial information.
structures, maintenance materials, and so on. They are constructed in accordance with the principles of comprehensive
and integrated structure, interrelated function and complementary performance. It can be considered as an organic
whole, which is composed of the maintenance entities, such
as maintenance units, and the relationships between maintenance entities together. Therefore, there are many similarities
between equipment maintenance support force systems and
complex networks, such as many entities, complex relationships and evolutionary function, etc. The complex networks’
theory is introduced to equipment maintenance support force
systems in the paper, revealing their evolution characteristics
in wartime and providing auxiliary decision for commanders
to develop support programs in the pre-war.
According to the description of complex networks, we
abstract the maintenance entities into nodes, the relationships
between these entities into edges. Which constitute equipment maintenance support force networks. At the same time,
because of the warfare demands of interconnection, interoperability (non-directional edges) and quick action (time
restriction of the edges), the paper focuses on the nondirectional and weighted complex networks.
3. THE DEFINITION OF NETWORK PARAMETER
Degree distribution
2. SIMILARITY ANALYSIS
Equipment maintenance support force systems are
composed of maintenance units, maintenance command
The node degree k(i) is the number of nodes that directly
connect node i. The degree distribution reflects the probability distribution of all the node degree in the network.
Center degree and node capability
The center degree characterizes the degree of each node
from the network center. The center degree h(i) is [9]:
1874-110X/15
2015 Bentham Open
Evolutionary Model of Equipment Maintenance Support Force Networks
h(i) =
k(i)
N 1
(1)
The k(i) represents the degree of node i, and N-1 represents the maximum possible number of neighbour nodes of
node i.
The center degree can not characterize the capability of
node i [2]. So the capability weight (i) is introduced to
characterize node capability.
Average time of network path
The time weight wij characterizes the shortest distance
between node i and j. The average time of network path is
the average value of shortest distance between any two
nodes. The average time dijw is:
dijw =
2
w
N (N 1) i, j N ,i j ij
(2)
Network clustering coefficient
The node i have k(i) edges that connect other nodes. Actually the number of interconnected edges between the k(i)
nodes is Ei. The clustering coefficient of node i can define ci
[10]
ci =
2Ei
k(i)[k(i) 1]
(3)
The clustering coefficient of the entire network is:
C=
1
c
N i i
(4)
The clustering coefficient reflects the link degree between neighbour nodes.
The ratio of the number of nodes in the network giant
component
The network giant component is a local area that contains
most of the nodes. The ratio of the number of nodes in a
network giant component is the ratio of the number of nodes
between a period of time after and before in the giant component, which reflects the continued capability to maintain
network connectivity. The ratio R1 is:
m
n
The Open Cybernetics & Systemics Journal, 2015, Volume 9
N
N
d 2 = h(i)d3 = μ (i)
i=1
i=1
N
h(i)
h(i)
log 2
d
d2
i=1
2
r1 = 1
log 2 d2
N
μ (i)
μ (i)
log 2
d
d3
r2 = 1 i=1 3
log 2 d3
R2 = r1 r2
423
(6)
The quality performance indicates the flow accuracy degree of material, information and energy among the nodes.
At the same time, it reflects network performance from the
perspective of the accuracy of network operation.
4. NETWORK EVOLUTIONARY MODEL UNDER
PARTIAL INFORMATION
Network evolutionary model has four main forms, such
as nodes increase or decrease, and edges increase or decrease. At present, many literatures have done a lot of studies
about the evolution of complex networks [11-13], which
mainly include network evolutionary model under deliberate
and random attacks. Deliberate attack is an attack mode in
which enemy grasps all the information of our troops. Random attack is an attack mode in which enemy does not grasp
our information. Deliberate and random attacks are two attack modes in extreme situation. In reality battlefield, most
ways of attacks are to take the attack mode of partial information grasped by the enemy [14, 15].
Suppose our equipment maintenance support force networks have N nodes, the probability of information to be
grasped by the enemy is a (0a). The a=0 is that enemy
can not obtain information, and then enemy takes random
attack. The a=1 is that enemy obtains all the information.
Enemy takes deliberate attack at this case. The most is the
a(0,1) At this point, the networks divide into a known
area (0, a]that enemy selectively attacks the area based on
the importance of nodes, and an unknown area (a, 1) that
enemy randomly attacks the area. Which are the attack
modes under partial information.
The denominator is the number of the nodes of m giant
components before a period of time, and the molecule is the
number of the nodes of m giant components after a period of
time.
In information warfare, in addition to the two sides of the
contest against each other in combat units, more are to destroy each other’s logistical support networks. Equipment
maintenance support force networks are a part of the logistical support networks. So they are the objects which enemy
focuses on the fight against. In wartime they are attacked
that lead to the reduction of nodes and edges. At the same
time, they obtain external support which lead to the increase
of nodes and edges.
The ratio reflects network performance from the perspective of the number of nodes in the local area.
Enemy gets our partial information, and performs the following evolutionary steps:
R1 =
'
i
i=1
m
n
(5)
i
i=1
Network quality performance
According to the theory of entropy and the particular operation of equipment maintenance support force networks,
network quality performance R2 can be defined as follows:
Step 1 The initial networks are the non-directional and
weighted networks which has m0 nodes and e0 edges.
Step 2 Enemy attacks firstly the networks’ known area.
At each time step equal, the networks perform the following
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The Open Cybernetics & Systemics Journal, 2015, Volume 9
Yu et al.
one of four different conditions in accordance with a certain
probability.
Case 1 According to the information obtained, enemy selectively attacks the networks, disconnect ma edges. The disconnect probability q1 is:
Case 2 The ngnodes are deleted with probability p2. The
networks disconnect mh edges that connect ng nodes. The
probability p2 is:
p2 =
k(i)
q1 =
N
(7)
wij
The k(i) is the degree of node i, and wij is path time between node i and j.
Case 2 Support functions of nb nodes are disabled with
probability q2. The networks disconnect mc edges that connect nb nodes. The probability q2 is:
h(i) μ (i)
The denominator is the total number of network nodes at
time t.
Case 3 The nr nodes vj(j=1,2,…,r) are increased with
probability p3. The newly added nodes are arranged in the
specified area, and given appropriate node capability weight
(j)(j=1,2,…,r). The ms edges are established between each
new node vj and existing node vi. The probability p3 is:
p3 =
The h(i) is center degree of node i, and (i) is capability
weight of node i.
Case 3 Support institutions increase nd nodes
vj(j=1,2,…,d) for the established networks with probability
q3. The newly added nodes are arranged in the specified
area, and given appropriate maintenance capability
weight(j)(j=1,2,…,d). The me edges are established between each new node vj and existing node vi. The probability
q3 is:
h(i) μ (i) / wij
N
{h(i) μ(i) / w }
(9)
μ (i)
N
k (i)μ (i)
(13)
t
(8)
N
h(i) μ(i)
i=1
q3 =
(12)
i=1
k(i)
i=1 j=1
q2 =
k (i)
t
wij
N
1
N
i=1
Case 4 The established networks increase mu new edge
with probability p4. The probability p4 is:
p4 =
ht (i)
N
e (i)h (i)
t
(14)
t
i=1
5. SIMULATION EXAMPLE
We take firstly a traditional equipment maintenance support force network as an example, followed by the calculation of evolutionary parameters. Secondly, we set a dynamic
network that is used as the network with partial information.
Finally, we simulate its evolutionary parameters.
ij
i=1
Case 4 The established networks increase mf new edges.
One end of new edge randomly connects one node, and the
other end connects another node with probability q4. The
probability q4 is:
q4 =
k(i)
N
k(i)
(10)
i=1
Step 3 Secondly, enemy attacks the unknown area. At
each time step equal, the networks perform the following one
of four different conditions in accordance with a certain
probability.
Case 1 The mkedges are deleted with probability p1. The
probability p1 is:
p1 =
1
N
e (i)
(11)
t
i=1
The denominator is the number of network edges at time t.
0. 18
0. 17
0. 18 0. 2 0. 2
0. 21 0. 15
0
.
2
0. 16
0. 17
0. 17 0. 18
0. 52
0. 18
0. 19 0. 15
0. 2
0. 5
0. 2
0. 47
0. 16
0. 39
0. 48
0. 16
0. 19
0. 41
0. 44
0. 22
0. 15
0. 43
0. 75
0. 47
0. 2
0. 2
0. 72
0. 17
0. 17
0. 76
0. 35
0.
0. 22
0. 38
0. 17
0. 65
8
0. 4
0. 17
0.
0
.
41
0. 16
0. 19
0. 7 0. 45
9
0. 85
0. 16
0. 13
0. 73
0. 44
0. 17
0. 75
0. 14
0. 5
0. 18
0. 48
0. 8 0. 76
0. 18
0. 45
0. 17
0. 16
0. 48
0. 4
0. 23
0. 2
0. 39
0. 2
0
.
46
0. 43 0. 51
0. 17
0. 19
0. 18
0. 22
0. 19
0. 17
0. 2
0. 21
0. 2
0. 21
0. 2
0. 2
0. 19
0. 2
0. 2
Fig. (1). The network topology.
As shown in Fig. (1), we abstract the maintenance entities into nodes, the relationships among maintenance entities
into edges, the sizes of nodes reflect the capability of the
entities, the value on an edge reflects the path time between
Evolutionary Model of Equipment Maintenance Support Force Networks
Table 1.
The Open Cybernetics & Systemics Journal, 2015, Volume 9
Evolutionary parameters of the traditional network.
Number of Nodes
Number of Edges
Average Time of Network
Path
Network Clustering
Coefficient
Network Quality
Performance
94
93
0.5474
0
0.108
Table 2.
425
Values of selection probability and evolutionary values of edges and nodes.
Unknown area under random attack
Selection Probability
Evolutionary Values
reducing edges
q1 = 0.6 ~ 0.8
ma = 5
reducing nodes
q 2 = 0.4 ~ 0.5
nb = 2 mc = 5
increasing nodes
q3 = 0.1 ~ 0.4
nd = 1 me = 4
increasing edges
q 4 = 0.4 ~ 0.6
mf = 3
reducing edges
p1 = 0.2 ~ 0.3
mk = 3
reducing nodes
p 2 = 0.1 ~ 0.15
ng = 1 mh = 4
increasing nodes
p3 = 0.1 ~ 0.2
nr = 2 ms = 7
increasing edges
p 4 = 0.2 ~ 0.3
mu = 9
Distribution Probability of Node Degree
Known area under deliberate attack
Evolution
0. 2
0. 19
0. 18
0. 17
0. 16
0. 15
0. 14
0. 13
0. 12
0. 11
0. 1
0. 09
0. 08
0. 07
0. 06
0. 05
0. 04
0. 03
0. 02
0. 01
0
P(k)[q1=0.6,q2=0.4,q3=0.1,q4=0.4,
p1=0.2,p2=0.1,p3=0.1,p4=0.2]
P(k)[q1=0.7,q2=0.4,q3=0.2,q4=0.5,
p1=0.2,p2=0.1,p3=0.2,p4=0.3]
P(k)[q1=0.8,q2=0.5,q3=0.4,q4=0.6,
p1=0.3,p2=0.15,p3=0.2,p4=0.3]
k^-2.3
0
20
40
60
Node Degree
80
100
Fig. (2). Degree distribution statistics of the network.
two nodes, which constitute an equipment maintenance support force network. According to the above definition of
network parameters, we use Lingo software to calculate the
parameters. The results are shown in Table 1.
Enemy takes deliberate attacks against the known area.
The network has a certain capability to defend and camouflage, so the probability of being attacked will not be 100%.
Enemy takes random attacks against the unknown area. Because of the commander’s capability and high-performance
weapon, the probability of being attacked is not too low.
Considering the above situations, the paper set the values of
selection probability and evolutionary values of edges and
nodes, which are shown in Table 2.
Assuming the simulation time is 100 steps. In order to
eliminate the influence of random factors in the simulation,
we have carried out 10 separate simulations for each simulation, and then these results are the average.
6. THE ANALYSIS OF SIMULATION RESULTS
Degree distribution is statistical at the end of the simulation, the results are shown in Fig. (2). These results are close
to the curve P(k)=k-a(a=2.3) under three kinds of selection
probability. Therefore, degree distribution of the network
follows a a2.3 power-law distribution, and the network has
the scale-free characteristic.
As shown in Fig. (3), the average time of the network
path experiences the reduction after the first increase under
three kinds of selection probability. It illustrates that the
network, from deliberate attacks to random attacks, experiences the reduction number of edges and nodes from greater
to less than the increase number of edges and nodes. As the
increase rate of decreasing edges and nodes, it is gradually
increasing for the increase rate of the average time of the
network path and its maximum value. At the same time, as
the increase rate of increasing edges and nodes, the reduction
The Open Cybernetics & Systemics Journal, 2015, Volume 9
Average Time of Network Path
426
1
0. 9
0. 8
0. 7
0. 6
0. 5
0. 4
0. 3
Yu et al.
q1=0.6,q2=0.4,q3=0.1,q4=0.4,
p1=0.2,p2=0.1,p3=0.1,p4=0.2
q1=0.7,q2=0.4,q3=0.2,q4=0.5,
p1=0.2,p2=0.1,p3=0.2,p2=0.3
q1=0.8,q2=0.5,q3=0.4,q4=0.6,
p1=0.3,p2=0.15,p3=0.2,p2=0.3
0
20
40
60
Simulation Step
80
100
Clustering Coefficien
Fig. (3). Average time of network path.
0. 3
0. 25
0. 2
0. 15
0. 1
0. 05
0
0
q1=0.6,q2=0.4,q3=0.1,q4=0.4,
p1=0.2,p2=0.1,p3=0.1,p4=0.2
q1=0.7,q2=0.4,q3=0.2,q4=0.5,
p1=0.2,p2=0.1,p3=0.2,p4=0.3
q1=0.8,q2=0.5,q3=0.4,q4=0.6,
p1=0.3,p2=0.15,p3=0.2,p4=0.3
20
40
60
80
100
Simulation Step
Fig. (4). Network clustering coefficient.
q1=0.6,q2=0.4,q3=0.1,q4=0.4,
p1=0.2,p2=0.1,p3=0.1,p4=0.2
in the Networks Giant Component
The Ratio of the Number of Nodes
of the Number
of Node s
inThe
theRatio
Networks
Giant Component
3
q1=0.7,q2=0.2,q3=0.2,q4=0.5,
p1=0.2,p2=0.1,p3=0.2,p4=0.3
2
q1=0.8,q2=0.5,q3=0.4,q4=0.6,
p1=0.3,p2=0.15,p3=0.2,p4=0.3
1
0
0
0
20
40
Simulation Step
60
80
100
Fig. (5). The ratio of the number of nodes in the network giant component.
rate of the average time of the network path gradually increased. The average time of the network path is less than its
initial value at the end of the simulation. So the dynamic
network is faster compared to the traditional network in
terms of support speed.
As shown in Fig. (4), the initial value of clustering coefficient is zero in the traditional network. With the simulation
progress, network clustering coefficient remains at near zero.
The reason is that the number of network nodes and edges is
in the reduce state. Later network clustering coefficient increases rapidly due to nodes and edges added constantly.
Finally, the network clustering coefficient stabilizes, and
then the network maintains the balance on the number of
nodes and edges entering and exiting.
The network has a short average time of network path
from Fig. (3), and has a large clustering coefficient from Fig.
(4). So the network has the ‘small-world’ characteristic.
As shown in Fig. (5), the ratio of the number of nodes in
the network giant component is to steadily reduce in the
beginning of a period of time under three kinds of selection
probability. The reason is that a large number of nodes and
edges constantly exit the network. When the simulation step
is 38, the ratio value of the number of nodes in the giant
component occur transition in the q1=0.8. Then the number
of nodes and edges entering the network is greater than the
number of exiting the network. Secondly the ratio values
begin to gradually reduce, but they are more than 1. Finally,
these ratio values fluctuate in the vicinity of 1, at the moment
the network maintain the balance on the number of nodes
and edges entering and exiting, and connectivity of the network is in a steady state.
Because of deliberate attacks from enemy, a large number of nodes and edges exit the network. With the advent of
random attacks, the number of nodes and edges entering the
Evolutionary Model of Equipment Maintenance Support Force Networks
Network Quality Performance
0. 5
0. 45
0. 4
0. 35
0. 3
0. 25
0. 2
0. 15
0. 1
0. 05
0
0
The Open Cybernetics & Systemics Journal, 2015, Volume 9
427
q1=0.6,q2=0.4,q3=0.1,q4=0.4,
p1=0.2,p2=0.1,p3=0.1,p4=0.2
q1=0.7,q2=0.4,q3=0.2,q4=0.5,
p1=0.2,p2=0.1,p3=0.2,p4=0.3
q1=0.8,q2=0.5,q3=0.4,q4=0.6,
p1=0.3,p2=0.15,p3=0.2,p4=0.3
20
40
60
Simulation Step
80
100
Fig. (6). Network quality performance.
network is greater than the number of exiting the network.
As can be seen from Fig. (6), network quality performance is
to reduce in the beginning of a period of time. When reduced
to a certain extent, it is in a steady state. After a certain time
it gradually increase.
This work was financially supported by the National
Natural Science Foundation of China under Grant
No.61271152.
As can be seen from Figs. (5 and 6), network performance experiences a gradual reduction at the beginning. To a
certain extent, it fluctuates in the vicinity of its minimum
value. It gradually increases after a certain simulation step,
then the fluctuation in the vicinity of a stable performance
value.
[1]
From the above analysis, we can achieve a more realistic
evolutionary model by way of flexible set-up parameters. So
the proposed evolutionary method is feasible.
CONCLUSION
CONFLICT OF INTEREST
[3]
[4]
[5]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
The authors confirm that this article content has no conflict of interest.
Received: September 16, 2014
[2]
[6]
In reality, it is only known for partial information of
equipment maintenance support force networks, how to analyze their evolution. The paper puts forward an evolutionary
model with partial information. It can show the evolutionary
law of the networks under different conditions by adjusting
parameters, and reveal the evolutionary characteristics in
wartime. The above simulation results prove the feasibility
of the evolutionary model, and illustrate that the model can
provide auxiliary decision for commanders to develop support programs in the pre-war. For the next stage, we are going to do a lot of work to improve the evolutionary model.
ACKNOWLEDGEMENTS
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Revised: December 23, 2014
Accepted: December 31, 2014
© Yu et al.; Licensee Bentham Open.
This is an open access article licensed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.
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