Node Trust Evaluation in Mobile Ad Hoc Networks Based on Multi-dimensional Fuzzy and Markov scgm(1,1) Model



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Node Trust Evaluation in Mobile Ad Hoc Networks

Based on Multi-dimensional Fuzzy and Markov SCGM(1,1) Model

ZHANG Feng1,2, JIA Zhi-ping1, XIA Hui1,LI Xin1, Edwin H.-M.Sha3,4

(1. School of Computer Science and Technology, Shandong University, Jinan, China, 250101;

2. School of Information Engineering, Taishan Medical University, Taian, China, 271016;

3. College of Information Science and Engineering, Hunan University, Changsha, China 410082

4. Department of Computer Science, University of Texas at Dallas, TX 75083, USA)

(Corresponding author: Tel: +86 13953809019

e-mail: attzhangfeng@126.com)

Abstract— Due to the nature of distribution and self-organization, mobile ad-hoc networks rely on cooperation between nodes to transfer information. One of the key factors to ensure high communication quality is an efficient assessment scheme for risks and trust of choosing next potential cooperative nodes. Trust model, an abstract psychological cognitive process, is one of the most complex concepts in social relationships, involving factors such as assumptions, expectations and behaviors. All of the above make it difficult to quantify and forecast trust accurately. In this paper, based on the theories of fuzzy recognition with feedback , SCGM(1,1) model and Markov chain, we present a pattern of prediction making. The analysis and experimental computation show that this scheme is efficient in trust prediction for ad-hoc networks.
Key Words: Mobile Ad hoc networks ; multi-dimensional fuzzy ; SCGM(1,1) model; Markov chain ; trust; weight

1.Introduction


Due to the distributed structures, frequently changing topology, and wireless connection, ad hoc networks brings much more problems. One of them is the security problem. Up to now, many attacks have been proposed, such as, On-off attack, Bad-mouthing attack, Conflict behavior attack [1, 2]. Most of current security technologies focus on encryption and authentication, and are unsuitable under the conditions without trusted third-parties in dynamic networks. Trust management mechanisms can be established in the networks, and enhance the security of the network effectively. Most of the current assessment mechanisms can be classified into two categories: the methods based on probability [3-5] and the trust evaluation models based on fuzzy theory [6, 7].

In the process of evaluating a new node, how to determine the weights of attributes is critical. Though the weight information of given node's attribute is partially known, to give precise weight information is very difficult, At present, the method of determining weight has many approaches, such as expert determining weight, AHP analysis [8,9], etc, which belong to subjective determining weight, and principle component analysis and entropy weight [10,11] etc, which belong to objective determining weight. When the objects of decision making are given, though we know that different objects have different weights, the precise weight is difficult to determine, and the influence of subjective factors are often not avoided, and decision-making information is only one dimension information. Hence, due to object function of synthesizing multiple attributes, the method of determining objective weight and the model of multiple dimension fuzzy.

Grey theory is more suitable in dealing with the uncertainty among small samples, incomplete information, and a lack of experience [12], the mobile ad hoc networks can be considered a grey system. It has good performance to dealing with small sample and information shortage of index series, through data mining techniques to get the change rules of the index. GM(1,1) as one of the grey models is widely used due to the convenience and the effect. However GM(1,1) has a disadvantage that it does not work for the stochastic fluctuation of those indexes. So we adopt the system cloud grey model SCGM(1,1) to improve the precision ability on tendency prediction aspect [13]. SCGM(1,1) model is combined with Markov chain in this paper, then an algorithm of similarity mining prediction in time series data on the base of grey Markov SCGM(1,1) model is proposed. Compared with other algorithms, this algorithm not only can measure similarity degree of arbitrary length similar series effectively, but also it can cope with non-stationary time series, avoid the influence of time series data that have biggish stochastic volatility to mining precision, so it has preferable practicability.

The rest of this paper is structured as follows: Section 2 gives a brief investigation of related works. Section 3 describes the mathematical model of computing the weights of attributes. Section 4 focus on the mathematical model of Markov SCGM(1,1). Section 5 discusses experimental results. Finally, Section 6 concludes this paper.



2.Related Works


By monitoring the transmission behavior and evaluating node’s reputation, several trust based security routing policies have been proposed.

A number of trust models for ad hoc networks have been proposed. These include a number of approaches to establish trust in routing protocols. A trust evaluation based security solution [14] is proposed to provide effective security decision for secure routing. Each node’s evaluation of trust on other nodes is based on trust factors as experience statistics, data value, intrusion detection results, as well as the node owner’s preference and policy. In this model only direct experiences are considered and the experiences of other nodes are not taken into account. In [15] a trust based adaptive on demand ad hoc routing protocol is proposed, where based upon the trust that a node has on its neighboring nodes, a security level is established. In this work the security level (or level of encryption) is defined by the level of trust between nodes. However, how the trust between nodes is determined is not discussed. In [16] dynamically updates trust levels by using reports from Intrusion Detection Systems located on all nodes in the network. The nodes neighboring to a node exhibiting suspicious behavior initiate trust reports. Using these trust levels as a guide, the source node then selects a route that meets the security requirements of the message to be transmitted. The trust is determined only based on intrusions detected. In [17] propose a trust model based on analyzing network packet data. Specifically, information such as passive acknowledgments, packet precision, blacklists, salvaging information and gratuitous route replies are examined.

In [18] use a Bayesian network model to detect malicious nodes in a mobile ad hoc network on the basis of rumors spread by other nodes. They present a mechanism to detect potential lies spread by other nodes. This work focuses on building trust based on rumors. The trust model described by [19] is distributed in nature. The Certification authority is based on threshold cryptography [20]. In this model, each node in the network maintains a certificate revocation list, which contains a list of misbehaving nodes and their accusers. If the number of accusers are at least k(the threshold) then the node is marked as convicted otherwise it is marked as a suspect and the certificate is not renewed for a convicted node. In [21] propose a probabilistic solution based on distributed trust model. A secret dealer is introduced in the system bootstrapping phase to complement the assumption in trust initialization. A robust trust chain in then constructed with high probability. This approach depends on a trustworthy secret dealer. In the model proposed by [22], the nodes in the ad hoc network maintain profile tables. These tables can become large resulting in time consuming exchange of large profile tables. This work focuses on key management for certificate revocation rather than trust evolution. Trust is based on accusations received. In the model proposed by Capra [23] each node exchanges recommendation letters which describe the performance and satisfaction from the node in a specific context. Above models have been proposed where the primary objective is not secure routing.

A peer-to-peer trust model based on satisfaction of interactions is given in [24]. In this model there is no reference to the influence of malicious nodes on trust. In [25] present a proxy based approach that uses alternative network channels to establish a secure trust relationship between communication parties to facilitate wireless communications between clients and services. In [26] have proposed an architecture for trust based security in pervasive computing environments. The architecture is agent based and built on a role based framework. Above model trust has been investigated outside of ad hoc.

To summarize, a lot of work has been done in trustworthy communications. Trust is typically determined from a security standpoint based on intrusions detected, direct experiences, recommendation from other nodes, accusations, etc. Theoretical models have been proposed and determining trust by analyzing data at the packet level have also been investigated.

3.Computing the Weights of Attributes

3.1Compute the Combined Value of Multi-dimensional Attributes


Assume the set of the evaluating nodes of the ad hoc network is A={Ai | i=1,2, …,n}, the set of attributes of the evaluated node is B={Bi | i=1,2, …,m}, where the attribute is such as the transfer speed or signal power etc. Node Ai evaluates the attributes of the evaluated node, and the characteristic vector is ai={a1i,a2i, …,ami }T , so the characteristic matrix set is D=( aij)m×n.

In order to overcome the influence of different dimensions, we change the characteristic matrix into the standard matrix as R=( rij )m×n, where rij =(aij aimin)/ (aimax aimin). and aimax is the maximum of the i-th attribute, aimin is the minimum of the i-th attribute. Assume the attribute weight vector is W={ w1, w2, …, wm }T, where wi ≥0, =1. Then the combined attribute value of the evaluated node by the j-th evaluating node is computed by the following formula:

Zi = (1)

According to formula (1), after computing the combined value, we acquaint a vector Z:



Z = { z1, z2, …, zn } (2)

3.2Compute the Fuzzy Classification of the Combined Value


The formula (2) can be regarded as the single index of fuzzy classification for the set of evaluating nodes. Assume the set of evaluating nodes is classed into c types, such as “bad”, “generic”, “good”. The corresponding fuzzy recognition matrix can be expressed as follows:

U = ( uhj)c×n (3)

where uhj is the relative membership which the j-th evaluating node belongs to the h-th type, and formula (3) should suit the following condition:



(4)

Assume the attribute of h type character values is the center of h type, the fuzzy cluster center as the following matrix:



S = (sihm×c (5)

where 0 ≤ sih ≤ 1。

In order to solve the optimized fuzzy recognition U and the optimized cluster center S, we establish the object function where the sum of the weighting Haiming distance square between evaluating node set and all types are minimum.

(6)

In details, the computation process can be listed as follows according to formula (6).

①: If the optimized cluster center S and weight vector W are given, solve the optimized fuzzy recognition U.

(7)

According to the object function (7) and formula (4), we create the Lagrange function and let derivative be zero, as follows:



(8)

(9)

(10)

According to formula (9) and (10), we get the following results:



. (11)

Obviously, if the standard matrix R, the optimized cluster center S and weight vector W are known, we can get the optimized fuzzy recognition U from formula (11).

②: If the optimized fuzzy recognition U and weight vector W are given, we solve the optimized cluster center S.

(12)

(13)

(14)

Obviously, if the standard matrix R, the optimized fuzzy recognition U and weight vector W are known, we can get the optimized cluster center S from formula (14).

③: If the weight vector W is given, solve the optimized fuzzy recognition U and the optimized cluster center S.

According to formula (11) and (14), by using of circular calculation, the optimized fuzzy recognition U and the optimized cluster center S can be computed.


3.3Compute the Weight of the Attributes


(15)

(16)

Obviously, if the fuzzy recognition matrix U is given, according to the formula (3) and (14), the optimized cluster center S can be computed, and according to the formula (5) and (16), the optimized object weight W can be computed. The formula (16) is one method of objective determining weight, which is based on the internal law of objects, but the computed weight results can not suit for practice. In order to make good use of objective information and the expert knowledge, through combining the subjective determining weight and the objective determining weight, we extended the formula (16) in reference [27] as follows:



(17)

where α and β are supervised factor of weight and stability coefficient respectively. We can interpret α as the weight controller which determines the weight while β reflects the sensitive degree of α. By adjusting the value of α and β, the satisfactory weights can be obtained.


3.4Arithmetic of Multiple Dimension Weight


When the weights supervised factor α and stability coefficient β are given, firstly, compute the weighting combined values by using formula (1); secondly, according to the computed combined values, compute the fuzzy recognition matrix by using formula (11) and (14); then according to the fuzzy recognition matrix, compute the object weights by using formula (14) and (17); at last, compute the weighting combined values again by using formula (1). Obviously, the arithmetic constitutes the process of circular calculation. The concrete arithmetic can be listed as follows:

Step 1: Change the attributes characteristic value matrix D=( aij)m×n into the standard matrix R=( rij)m×n.

Step 2: Give the original weight vector W0=(W0i), supervised factor α, stability coefficient β, the classification number c , the computing accuracy ε of the combined value, the computing accuracy eu of the fuzzy recognition matrix, the computing accuracy es of the cluster center matrix.

Step 3: According to the given weight W0 and formula (1), compute the weighting combined value Z0.

Step 4: According to the computed U0 and formula (1), compute the original cluster center matrix S0.

Step 5: According to the computed S0 and formula (11), compute the U1.

Step 6: According to the computed U1 and formula (14), compute the S1.

Step 7: Compare the element of the S0 and S1, if max | S1S0 | > es , use circular calculation.

Step 8: Compare the element of the U0 and U1, if max | U1U0 | > eu, use circular calculation.

Step 9: According to the computed U and S , and formula (17) , compute the weight vector W1 .

Step 10: According to the given weight W1 and formula (1), compute the weighting combined value Z1.

Step 11: Compare the element of the Z1 and Z0 , d1 = max|Z1j-Z0j|. If | d1- d0| ≤ ε, then the computing is over. Otherwise, repeat the step 3)~ 11), till max|dk-dk-1| ≤ ε, where k is the computing number of circular calculation. Step 4)~ 8)is convergence [28].



4.Predict Trust with Markov SCGM(1,1) Model


With the weights of attributes computed, we discuss how to predict its trust in series time in this section.

4.1Select the Trust Prediction of the Node Attributes


Assume the source node of the ad hoc is SN( source node, in section 3, it may be node Ai), and NN (new node) is the new node in the link route of the SN. The set of attributes of the NN is B={Bi | i=1,2, …,m}, where the attributes are defined in the section 3. Node SN records the series time data based on B in the nearest time, so the characteristic vector is tbi={tbi1,tbi2 …,tbim },where i denotes time break. And the characteristic matrix set is TB=( tbij)n×m.

4.2Compute the Original Series Time Value


In order to overcome the influence of different dimensions, the characteristic matrix TB is changed into the standard matrix denoted TBR=( tbrij )n×m. At time i, source node SN can compute the trust reliability with matrix TBR. Assume the attribute weight vector is W={ w1, w2, …, wm }, where wi ≥0, =1. Then the combined attribute value of NN node at the time i is computed as follows:

(18)

According to formula (18), we can get the original index series of trust reliability value, which generates the following vector:

X(0) = {x(0)(1), x(0)(2), …, x(0)(n) } (19)

4.3Process the Data


X(0) is the non-negative original trust reliability data sequence. The accumulated generation operation of the original data sequence is defined as

X(1) = { x(1)(k)| x(1)(k) ≥ 0,k=1,2,…,n } (20)

where .


4.4Create Correlation Function


Let f(k) be satisfaction trend correlation function[29]:

f(k) = bea(k-1) – c (21)

where a ,b , c ∈R, k=1,2,…,n. Let x(1)(k) and fr(k) be satisfaction trend correlative. Then we can use the data of x (1) to relevantly fit f r. According to the least square algorithm, we can obtain:



(22)

(23)

(24)

4.5Reverting Series


Let U = a*c, (1)(1) = b – c, we can obtain grey differential equation :

(25)

And the solution is:



(26)

After we take reverting calculation to, the grey SCGM(1, 1) transform value of original series data is



(27)

Let, the general trend of original time series data is predicated by


4.6Markov Process


The essential of grey SCGM(1,1) model is that exponential curve is used to fit original trust reliability value data and geometry graph of the results is a smooth curve. As trust reliability value is a dynamic character value, and time series data have biggish volatility, the change trend is non-stationary stochastic process, so the single application of grey SCGM(1,1) model influence the precision of the transformed results. Markov chain is a kind of stochastic process without aftereffect, which describes stochastic phenomenon: the present state is aware, and the probability distribution of the future state has nothing to do with the past state. The Markov chain, state transition probability is used to reflect the influence degree of the stochastic factors. Therefore, we can apply Markov chain to make time original series data that have biggish stochastic volatility precise [30].

4.6.1Definition


Let stochastic processes {X(n), n=1,2,…}, if for arbitrary k , non-negative integer n1, n2, …, nk ( 0≤n1﹤n2﹤n3…﹤nk ), arbitrary i1,i2, …,ik, and arbitrary natural number k, conditional probability satisfies:

(28)

Then {X(n), n=1,2,…} is a Markov chain. P{X(n+k)=jX(n) =i} is the probability of the process of that from state i at the moment n to state j at the moment n+k passing through k steps, denoted as pij(n, n+k). Especially, when k=1, it is called transition probability of one step, denoted as pij(1).


4.6.2Divide State


The change trend of trust reliability value is a non-stationary stochastic process. Different state of time series has its special boundary. Take state division, namely set as a benchmark, then we can obtain several strip regions which parallel to curve, and each strip region denotes a state. Assume there are m states, any state can be denoted as:

(29)

where 1i = + Ai, 2i = + Bi, Ei denotes the ith state, and grey element 1i and 2i denote lower and upper bounds of the ith state respectively. Because is a function that is related to time, and grey element 1i and 2i are changed with time, the states have dynamic characteristics.


4.6.3Create the Probalistic Transfer Matrix


Now time series is the Markov process. The wth step probabilistic transfer matrix P(w) is calculated .

(30)

4.6.4Create Prediction Table


In practice, we usually consider the one step transfer matrix P(1). Suppose the current state is Ei , then study the ith line in the P(1).

  1. If Pik = max Pij, then the probability of the next state is Ek.

  2. If there are more than one value are satisfied with the above maximum , then we need to study P(2) or P(j)(j≥3).

From above discussion, we get the future state and get the strip regions [1i, 2i] at the same time. Use the median as the predicated data.

5.Trust Assessment in Ad Hoc

5.1Select the Key Attributes of the Node


Before confirming the trust of neighbor node’ behavior, we need to define key attributes, which are the evaluating indicators. All these key attributes should well reflect the behavior characteristics of the node in order to make the trust assessment objective and efficient. For the trust assessment of node in mobile ad hoc network, we should consider both network communication features, such as loss tolerance, and the physical attributes of the nodes, such as mobility, wireless signal, and place property.

Network communication features include transmission speed, package losing rate, etc, which reflect the performance of a node. The package losing rate is also a key factor in evaluating the reliability of a node. With regards to physical attributes of a node, the mobility property mainly focuses on node’s speed. The higher the speed is, the higher the package losing rate will be. Wireless signal property includes two attributes, one of which is the strength of the signal while the other factor is the stability of signals, or the signal changing rate.

Based on the features of network communication, we can choose two performance attributes and two reliability attributes. Transmission speed and signal strength reflect the performance, while the package losing rate and signal changing rate reflect the reliability in communication. So, we can define the key attributes set in evaluating the trust of mobile ad hoc network as B = {package losing rate, transmission speed, signal strength, signal changing rate}.

5.2Compute the Weight of the Key Attributes of the Node


In this section, we use the model given in section 3 to compute the weight of the attributes

5.2.1Define the Cluster Center


Define the cluster Center S = {s1,s2,s3}, where s1 means bad trust, s2 means generic trust, and s3 means good trust.

5.2.2Simulate Analysis


In a mobile ad hoc network based on the 802.11b compatible devices, the real throughput can be only half of the ideal bandwidth it should be, that is about 0.5Mbps to 5.5Mbps. In an actual environment, the working distance for 802.11b is 300m, and working at a distance more than 550m will be regarded as instable speed. Based on this principle, we can draw that a speed of 0.5Mbps(63KB/s) is good reliable, and 2Mbps(250KB/s) is generic. According to the general evaluating principle of the network, a package losing rate below 10% is good reliable, and the rate at 30% is a generic losing rate, while a rate at 70% is regarded as bad. In the aspect of signal strength, according to some national standards, the emission power of the 802.11b devices should not surpass 100mW, we can calculate the signal power in different distance by the free-space-model. In this model, the relationship between power attenuation and transmission distance is as follows:

FSPL(dB) = 10 ( (4π/c df ) 2), where FSPL(dB) stands for the power attenuation, measured by dBm, d for distance, c for light speed, and f for frequency. 802.11b runs at the frequency of 2.4GHz. The transfer from dBm to mW is: mW = 10bdbm/10.

According to the above formulas and the working distance, we can get the attenuation levels as: 84.74dBm for a distance of 160m, 90.2dBm for 300m, and 95.47dBm for 500m. As the transmission power of network is about 20dBm, a node should be good when the measured signal strength is above -64.74dBm (3.36*10-7mW), and generic reliable when the signal strength is around -70.21dBm (0.953*10-7mW), and bad when it is -75.47dBm (0.284*10-7mW). The changing rate of the signal can be deduced according to the moving speed. When the changing rate is above 2.39dBm/s, there is highly bad for the communication to take. Similarly, if the changing rate is below 2.12dBm/s, the communication is good, whereas a changing rate of 2.26dBm/s is typically generic. If the maximum speed of the node is no more than 20m/s, the maximum node changing rate is 3.47dBm/s.

Four nodes are available for assessment and the attribute value for each node is in the table I.

TABLE I . Attribute Value


Node

Package losing rate(%)

Transmission Speed(KB/s)

Signal10-7mW

Signal changing Rate dBm/s

A1

24

260

1.2

2.21

A2

82

70

0.4

2.51

A3

11

290

1.45

3.32

A4

36

298

3.41

2.14

According to the Table I, the original matrix is A.

A =

For the uniform, we change the losing rate and changing rate into the maximum, and the matrix A to A’.



A’ =

For the standard, we change each line to the percent, and the second line to be (speed/688), the third line to be ( signal power /3.36), the forth line to be ((3.47-changing rate )/1.35),then the changed matrix is A’’.



A’’ =

Define the membership function for each line of the matrix A’’.

1. For the element x of line 1 in the matrix A’’, x belongs to the “bad”, and the membership function is :

x belongs to the “generic” , and the membership function is :



x belongs to the “good” , and the membership function is :



2. For the element x of line 2 in the matrix A’’ , x belongs to the “bad”, and the membership function is :



x belongs to the “generic” , and the membership function is :



x belongs to the “good” , and the membership function is :



3. For the element x of line 3 in the matrix A’’, x belongs to the “bad”, and the membership function is :



x belongs to the “generic” , and the membership function is :



x belongs to the “good” , and the membership function is :



4. For the element x of line 4 in the matrix A’’, x belongs to the “bad”, and the membership function is :



x belongs to the “generic” , and the membership function is :



x belongs to the “good” , and the membership function is :



Based on the membership function above, we can change the matrix A’’ to the standard matrix R.



R =

Compute the original fuzzy recognition U and S, according to the original weight vector W={0.25,0.25,0.25,0.25} and membership function.



U =

Let the fuzzy recognition matrix’s computing accuracy eu is 0.01, the cluster center matrix’s computing accuracy es is 0.01, the combined value’s computing accuracy ez is 0.05, the value of supervised factor α is 0.7, the value of stability coefficientβ is 0.2.



We can get the node trust value is about 60.71%, according the above. And that means the trust of the new node is generic. Different values of original parameters can affect the trust value. With the fuzzy recognition matrix’s computing accuracy eu is 0.01, the cluster center matrix’s computing accuracy es is 0.01, and the combined value’s computing accuracy ez is 0.05. The values corresponding to difference α and β are shown in table Ⅱ.

Table . Different Values With Different Parameters

α

β

Weight Vector

Trust Value

0.5

0.6

0.2494,0.2136,0.2956,0.2414

59.01%

0.5

0.6

0.2494,0.2136,0.2956,0.2414

59.27%

0.5

0.5

0.2536,0.2174,0.2849,0.2441

60.11%

0.5

0.2

0.2961,0.2578,0.1849,0.2612

62.41%

0.5

0.2

0.2961,0.2578,0.1849,0.2612

62.64%

0.5

0.2

0.2961,0.2578,0.1849,0.2612

63.08%

0.7

0.6

0.2434,0.2082,0.3109,0.2375

58.54%

0.7

0.6

0.2434,0.2082,0.3109,0.2375

58.81%

0.7

0.6

0.2434,0.2082,0.3109,0.2375

59.35%

0.7

0.5

0.2455,0.2100,0.3056,0.2389

58.71%

0.7

0.5

0.2455,0.2100,0.3056,0.2389

58.97%

0.7

0.5

0.2455,0.2100,0.3056,0.2389

59.51%

0.7

0.2

0.2720,0.2344,0.2398,0.2538

60.71%

0.7

0.2

0.2720,0.2344,0.2398,0.2538

60.95%

0.7

0.2

0.2720,0.2344,0.2398,0.2538

61.44%

0.9

0.6

0.2398,0.2049,0.3203,0.2350

58.26%

0.9

0.6

0.2398,0.2049,0.3203,0.2350

58.53%

0.9

0.6

0.2398,0.2049,0.3203,0.2350

59.08%

0.9

0.5

0.2404,0.2054,0.3188,0.2354

58.31%

0.9

0.2

0.2494,0.2136,0.2956,0.2414

59.01%

0.9

0.2

0.2494,0.2136,0.2956,0.2414

59.27%

0.9

0.2

0.2494,0.2136,0.2956,0.2414

59.80%

According the table Ⅱ, we can get the trust value is between 58.26% and 63.08%.And the circular times is 5 in the condition of α is 0.7, β is 0.3.And the W vector value is about {0.27, 0.23, 0.24, 0.26}.

Using the method presented in this model, the optimal fuzzy recognition matrix, the attribute weights re obtained.



5.3Predict the Trust Value of the New Node


In the section 5.2, we compute and obtain the weight of the attributes of the new node. In this section, and discuss the future trust value using the Markov SCGM(1,1) model in section 4 with the new weight of the attributes.

5.3.1Simulate Analysis


Assume the new node is the neighbor of the source node. Attribute value of the new neighbor node for each time is in the tableⅢ.

Table . Attribute Value

Time

Package losing rate(%)

Transmission Speed(KB/s)

Signal 10-7mW

Signal changing Rate dBm/s

1

24

260

2.2

2.21

2

30

270

2.4

2.51

3

34

290

2.45

3.32

4

36

298

3.41

2.14

5

40

260

1.8

2.25

6

28

280

2.3

2.37

7

25

291

3.0

2.18

8

35

280

2.5

3.12

9

20

295

2.4

3.15

10

30

288

3.1

2.82

For the standard, we change the each line to the percent .And the losing rate to be (1-losing rate), the speed line to be (speed/688), the signal line to be ( signal power /3.36), the changing rate line to be ((3.47-changing rate )/1.35. The weight value is {0.27, 0.23, 0.24, 0.26 }. According to the formula (18), we get the X(0) which is showed in table Ⅳ.

Table . Trust Value

Time

1

2

3

4

5

Real Value

0.68

0.63

0.48

0.76

0.60

Time

6

7

8

9

10

Real Value

0.65

0.75

0.51

0.54

0.63

According to formula (22) to (26), we create the SCGM model, a = -0.03951, b = -18.8986. And we get the fitting value. Comparing the real value with fitting value , we get table Ⅴ.

Table . Trust Value

Time

1

2

3

4

5

Real Value

0.68

0.63

0.48

0.76

0.60

Fitting Value

0.76

0.73

0.70

0.67

0.68

Real/Fit

0.89

0.85

0.68

1.13

0.88

Time

6

7

8

9

10

Real Value

0.65

0.75

0.51

0.54

0.63

Fitting Value

0.62

0.60

0.57

0.55

0.53

Real/Fit

1.05

1.25

0.89

0.98

1.18

5.3.2Define and Divide the State


According to tableⅤ, Real/Fit value is between 0.68 and 1.3. We can e divided into 3 types which are shown in table Ⅵ. State E1 is between 0.68 and 0.89, state E2 is between 0.89 and 1, and state E3 is between 1 and 1.3.
Table . State Type

State

Real/Fit

E1

0.68 ~ 0.89

E2

0.89~1

E3

1~1.3

Based on the tableⅤand table Ⅵ, the times of emergence of state E1 is 5 , the times from E1 to E1 passing through 1 step is 2, the times from E1 to E2 passing through 1 step is 1, the times from E1 to E3 passing through 1 step is 2, then we get the p11 is 2/5 , p12 is 1/5 , and p13 is 2/5. In the same way, we get the others matrix value. The 1th step probabilistic transfer matrix p(1) is calculated .

The last time series is 10 and the state is E3. According to matrix p(1) , the probability of the next state is E1. Based on the SCGM model, compute and the value is 0.51. So the predicted trust value of the time series 11 is 0.51 × 1/2 × (0.68+0.89) ≈0.40

In reality, the losing rate is 45%, the speed is 283KB/s, the signal is 2.2, and the changing rate is 3.12. And the trust value is 0.46. Comparing the 0.40 and 0.46, the model can predict the trust value well.

6.Conclusions


According to the characteristic of weight information and the fuzzy characteristic of classification of the combined attribute values, based on the causal analysis and fuzzy recognition theory, the model of multiple dimension fuzzy making are presented in this paper. This method is of the strict mathematic base, easy to program and with itself feedback. The obtained information by this method can be used to determine the weight set. The computed weights conclude both the inbeing information of object attributes and expert knowledge.

An algorithm of evaluating node trust on the base of Markov SCGM(1,1) model is also presented in this paper. It combines grey system model with Markov chain. The algorithm not only the information that is obtained by historical data can be fully used, but also can measure similarity degree of arbitrary length similar series effectively, and can cope with non-stationary time series, avoid the influence of time series data that have biggish stochastic volatility to mining precision, so it has preferable practicability.



Based on the two models, the analysis and above trust assessment example shows that the models can be applied to the trust assessment for nodes in mobile ad hoc network. It is suitable for computer processing automatically, and it meets the requirements of trust assessment in mobile ad hoc network environment.

7.Acknowledgement


This research is sponsored by the Natural Science Foundation of China (NSFC) under grant No. 61070022, 60903031, Shandong Provincial Natural Science Foundation under grant no. ZR2010FM015, the promotive research fund for excellent young and middle-aged scientisits of Shandong Province No. BS2010DX017, the China Postdoctoral Science Foundation under grant No. 20090451310, and the Independent Innovation Foundation of Shandong University under grant No.2009TS032, and the Independent Innovation Foundation of Shandong University Graduate Student under grant No. 11150071613067. This work is partially supported by NSF CNS-1015802, Texas NHARP 009741-0020-2009, HK GRF 123609, NSFC 61173014, NSFC 61133005, China Thousand-Talent Program.

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