A trust model between cloud entities using fuzzy mathematics

Abstract

Many of modern systems and applications such as cloud computing, grid computing, pervasive computing, P2P networks and even web applications have employed the concepts of trust. Trust is a level of subjective probability between two entities, a trustor (i.e. source entity) and a trustee (i.e. target entity), which is formed through the direct observation nature and/or recommendation from trusted entities. Trust relationship is the most complex relationship between trustor and trustee because it is extremely subjective, non-symmetric, partial transitive, dynamic, context-dependant, uncertainty, and difficult to be evaluation and establish. Today, there is no special trust evaluation model for cloud computing environment. Hence, in this paper, we present a trust model based on fuzzy mathematics named TMFM in cloud computing environment according to success and failure interaction between cloud entities so that the fuzzy direct trust relation is calculated based on direct experiences between trustor and trustee also, trustor can builds a fuzzy Indirect trust relation with trustee through his acquaintances. Simulation results show that the TMFM model has some identification and containment capability in synergies cheating, promotes interaction between entities, and improves the performance of the entire cloud environment.

Keywords

Cloud computing trust models fuzzy mathematics trustor trustee

1 Introduction

Many of modem systems and applications such as pervasive computing, P2P networks, grid computing and even web applications have employed the concepts of trust. Trust is a well-known social behaviour; however it’s hard to have precise definition for it. The concept of trust in distributed computing environments comes from social science, such as psychology, economy, sociology [1 –4].

There are many researches in the literature which focus on trust. These works are generally placed under two main categories: Trust Measurement and Trust Management. The former deals with how to represent and evaluate the value of trust between two nodes while the latter tries to find a way to make decision based on trust values. On the other point of view measuring (calculating) the trust may be centralized or decentralized. Centralized approach via a centralized authority or across multiple distributed participants often leads to simple solutions but if the centralized authority is not carefully designed, it can become a single point of failure for the entire system [8]. Instead, most systems calculate trust in a fully distributed manner. Although these decentralized systems are inherently more complex, they scale well and avoid single points of failure in the system [5, 6].

In fact, trust is the most complex relationship among entities, because it is extremely subjective, context-dependant, non-symmetric, uncertain, partially transitive, and difficult to evaluate and establish [3, 7]. Today, there is no special trust evaluation model for cloud computing environment. Therefore, in this paper, a trust model based on fuzzy mathematics named TMFM in cloud computing environment is proposed. Our contributions can be summarised as follows:

We present a trust model based on fuzzy mathematics named TMFM in cloud computing environment according to direct and recommendation interaction between cloud entities, so that the fuzzy direct trust relation is calculated based on direct experiences between trustor and trustee. Also,a trustor can builds a fuzzy Indirect trust relation with trustee through his acquaintances.

Since the entity’s trust value is not entirely consistent with the credibility of the recommendation, to resist malicious behaviour, we put forward a trust model based on the fuzzy recommendation similarity in cloud environments, which aims at preventing the synergistic effect of selfish entity. In this paper, we use the adjusted cosine similar function to determine the similarity between two entities in cloud environment.

We have proved the trust and security of our proposed trust model by comparing TMFM with others such as DMTC model [7] and give systematic analysis on how TMFM model can enhance the system trust. In [7], a trust evaluation model is studied. This paper discusses the integration of fuzziness and randomness of trust relation, analyzes the ways Cloud models describe uncertain concepts and the Cloud models transform algorithms between qualitative concepts and their quantitative expressions, and presents the direct and recommendation of trust between cloud entities based on Cloud theory. The trust Cloud model achieves a complete description of the concept of trust, and the trust values obtained in this model contain more semantic information. However, the evaluations of the model for all entities are equal, not matching the facts that in real life, to different people, the levels of trust are not the same. Because of this, the error of the trust information created by the model is large.

The rest of the paper is organized as follows. In Section 2, we present a trust model of choosing trusted entities base on the fuzzy relationship theory in fuzzy mathematics in cloud environment. In Section 2 we calculate direct trust, indirect trust and total trust, respectively. Experiment results in Section 4 show that TMFM model can effectively prevent selfish entities. Finally the summary and future work is presented in Section 6.

2 Proposed trust model

In general trust can be classified into different categories according to different standards.

According to attributes: identity trust and behavior trust

According to obtaining way: direct trust and recommended trust

According to role: code trust, third party trust and execution trust, etc.

According to based theory: subjective trust and objective trust

In this paper, we use the second category to evaluating trust

Definition 1. (Trust). Trust is a level of subjective probability between two entities, a trustor (i.e. source entity) and a trustee (i.e. target entity), which is formed through the direct observation nature and/or recommendation from trusted entities, to fulfilling a particular service within a specific time and context.

It is supposed that the trustor is a cognitive entity with an ability to make assessments and decisions about the received information and past experiences. Trust is usually evaluated by trust degree and described with trust relation [7, 10].

Definition 2. (Trust degree). Trust degree Td _ij is used to evaluate the degree of trust from a domain set of possible trust values that trustor e _i in views trustee e _j and denotes entity’s e _i trust attitude (opinion) towards entity e _j in time t and context c _z. The trust degree can be expressed as the following relation: ${Td}_{ij} = {\begin{matrix} DT (e_{i}, e_{j}, c_{z}, t), \\ RT (e_{i}, e_{j}, c_{z}, t), \\ IDT (e_{i}, e_{j}, c_{z}, t), \\ Otherwise \end{matrix}$ (1)

Where Td _ij = DT (e _i, e _j, c _z, t), Td _ij = RT (e _i, e _j, c _z, t) and Td _ij = IDT (e _i, e _j, c _z, t) are the direct trust degree, recommendation trust degree and indirect trust degree between trustor ei (i.e. source entity) and trustee e _j (i.e. target entity) in context C _z and time t.

In real cloud environment, Trust and Reputation both depend on some context [10 , 13]. For example, entity A trusts entity B as multimedia provider, but it does not trust B as a storage provider. So in the context of requesting a multimedia service, B is trustworthy. But in the context of providing storage service, B is untrustworthy.

In this paper we calculate trust degree with fuzzy set theory, so the mathematical model of fuzzy trust should be firstly created.

Suppose E = {e ₁, e ₂, …, e _n} is the problem domain of fuzzy trust model, where e _i (i = 1, 2, …, n)is an entity in the problem domain [3 , 12]. A membership function (e) defines the degree to which a fuzzy variable x is a member of a set. (e) map e into the interval [0, 1]. Full membership is represented by 1 and no membership by 0. The values between 0 and 1 characterize fuzzy members, which belong to the fuzzy set only partially [15]. Supposing the problem domain E is not the empty set, TR is a fuzzy set of Cartesian product of E E; E is the set that includes all the entity in cloud environment. There exists a mapping: $\begin{matrix} TR : E \times E \to [0, 1], \\ (e_{i}, e_{j}) \to μ_{A} (e_{i}, e_{j}) \in [0, 1] \end{matrix}$ (2)

To manage a collection of trust related activities across domains, we need to understand trust itself. From different points of views, trust can be categorized into different classes: direct trust and indirect trust (the indirect trust relation is a composite fuzzy relation of recommending relation and direct trust relation).

2.1 Fuzzy direct trust relation

When we say entity ei is trustworthy or untrustworthy for entity e _j, there is a trust relationship between entity e _i and entity e _j. If this statement is based on entity e _i’s direct experiences with entity e _j completely, this relationship is called the direct trust relation or direct trust model. Figure 1 shows fuzzy direct trust degree between entity e _i and entity e _j at context C _z and time t.

Direct trust relation is not a crisp binary relation that is either true or false. For example, entity e _i usually says an 80 percent probability that entity e _j is a trusted peer. This hints that trust has different levels or degrees. Direct trust relation just has fuzzy properties. We can use fuzzy relation to describe direct trust relation. Fuzzy direct trust degree between two entities can be denoted by fuzzy graph.

Definition 3. (trust graph): The trust relations in the cloud computing environment of entities are represented as a trust graph G. [7 , 13]. It represents a directed graph with entities as nodes and edges as trust relations among them. The edges are directed and If an edge indicates a trust relation of entity e _i towards entity e _j, it is directed from entity e _i to entity e _j with the trust degree Td _ij. A possible way of representing a directed graph is a matrix defined as follows.

Definition 4. (Trust matrix): interactions in a cloud computing environment of n entites is represented with a trust matrix M, where elements Td _ij indicate a trust relation of entity e _i towards entity e _j, and have values, where each value denotes the degree of trust [7 , 17]. If a relation is not defined, it is indicated as zero(Note that Td _ij= 0 does not imply Td _ji= 0). The matrix represents trust in a cloud computing environment at a specific time t ∈ T and specific context C _z denoted M (t, C _z) as: $M (t, c_{z}) = (\begin{matrix} {Td}_{11} & \dots & {Td}_{1 n} \\ ⋮ & ⋱ & ⋮ \\ {Td}_{m 1} & \dots & {Td}_{mn} \end{matrix})$ (3)

Based on the assumption that trust relation is reflexive, it follows that all the diagonal elements in diagonal are equal to one which indicates the maximum trust degree. Figure 2 gives an example of trust propagation in a cloud computing environment of three entities at a specific time t and context c _z.

$\frac{μ_{A} (e_{i}, e_{j})}{(e_{i}, e_{j})}$ represents Td _ij = DT (e _i, e _j, c _z, t), which is the trust degree between entity e _i and entity e _j at a specific time t and context c _z.

\begin{matrix} DT & = & \frac{1.0}{(e_{1}, e_{1})} + \frac{0.9}{(e_{2}, e_{1})} + \frac{0.9}{(e_{3}, e_{1})} + \frac{1.0}{(e_{2}, e_{2})} \\ + \frac{0.8}{(e_{3}, e_{2})} + \frac{1.0}{(e_{3}, e_{3})} \end{matrix}

–Here Td₁₂ = Td₁₃ = Td₂₃ = 0 that, we do not write them out.

–The reflecting trust matrix M _D T (t, c _z) is given below: $M_{DT} (t, c_{z}) = (\begin{matrix} 1.0 & 0 & 0 \\ 0.9 & 1.0 & 0 \\ 0.9 & 0.8 & 1.0 \end{matrix})$ When observing actual trust relationships TR between entities, the following properties will be assumed:

TR is reflexive; an entity trusts itself implicitly, i.e. ∀e ∈ E ; e TR e

TR is not symmetric, because an entity may trust another entity within a specific context, while the opposite might not be true, i.e ∃ e _i, e _j ∈ E; e _i TR e _jnLeftrightarrowe _j TR e _i. It represents entity e _k recommend e _j to entity e _i at a specific time t and context c _z.

TR is time-based variant, the trust degree that trustor e _i in views trustee tj for specific context c _z will decreases with the passage of time. As a entity behavior is not always constant but often changes with time, therefore, the recent experience is more credible than the general historical experience.

TR is context-dependant, a trustor e _i may have different trust degree on trustee e _j for different contexts

$\begin{matrix} If \exists e_{i}, e_{j} \in E, then {Td}_{ij} (e_{i}, e_{j}, ck 0, t) = td ⇎ \\ {Td}_{ij} (e_{i}, e_{j}, ck 1, t) = td where {ck}_{0} \neq {ck}_{1} \end{matrix}$

2.1.1 Fuzzy direct trust degree computing

Suppose in the past entity e _i has p times successful interactions and q times failure interactions with entity e _j at a specific time t and context c _z. We define the fuzzy direct trust relation membership function: $T_{z} d_{ij} = DT (e_{i}, e_{j}, c_{z}, t) = \frac{p_{z}}{p_{z} + q_{z}}$ (4)

It is worth to mention again that, as a entity behavior is not always constant but often changes with time, therefore, the recent experience is more credible than the general historical experience, therefore, We have considered the function to determine the successful experiences over time. This function calculates the successful interaction rate based on historical successful interaction between trustor e _i and trustee e _j at a specific time t and context c _z. This function is given below: $\begin{matrix} p ({Tp}_{i}) = α p (Δ T_{i}) + (1 - α) p (T_{i - 1}), \\ {Tp}_{i} = Ti, \\ Δ T_{i} = {Tp}_{i} - {Ts}_{i - 1}, \\ p (T_{0}) = p ({Ts}_{0}) = 1, \\ {Ts}_{i} = {Tp}_{i}, (i > 0) \end{matrix}$ (5)

Where α is the adjustable parameters and presents the weight of successful interactions in different timescales (ΔTi). P (ΔTi) is recent successful interactions and P (T _i-1) is historical successful interaction. moreover T _p and T _s represent present time, start time respectively. Also Ts ₀ represents the first interaction between trustor e _i and trustee e _j at time t and context c _z.

We have considered the weights of the past negative behavior β which can be regulated to punish the selfish entity action. Then the fuzzy direct trust relation can be revised as: $T_{z} d_{ij} = DT (e_{i}, e_{j}, c_{z}, t) = \frac{p_{z} ({Tp}_{i})}{p_{z} ({Tp}_{i}) + β q_{z}}$ (6)

It is difficult to decide whether an entity is bad or good based on only few interactions. In determining trust it is important that an entity has sufficient experience on which to calculate trust [9]. So we define the confidence level in the experience for a particular context c _z as an interaction threshold value Co _z of interaction times. $\begin{matrix} p_{z} + q_{z} \leq {Co}_{z} \\ p_{z} + q_{z} > {Co}_{z} \end{matrix}$ (7)

Thus, if the interaction times are too small (i.e p _z + q _z < = co _z) between trustor e _i and trustee e _j, this computing as defined in relation 4 may be an arbitrary decision and the following equation can be used. $T_{z} d_{ij} = DT (e_{i}, e_{j}, c_{z}, t) = 0.5 + \frac{p_{z} ({Tp}_{i}) - β q_{z}}{2 \times {Co}_{z}}$ (8) In general, fuzzy direct trust at context C _z and time t between trustor e _i and trustee e _j is calculated asfollows: $T_{z} d_{ij} = DT (e_{i}, e_{j}, c_{z}, t) = {\begin{matrix} 0.5 + \frac{p_{z} ({Tp}_{i}) - β q_{z}}{2 \times {Co}_{z}} if p_{z} + q_{z} \leq {Co}_{z} \\ \frac{p_{z} ({Tp}_{i})}{p_{z} ({Tp}_{i}) + β q_{z}} if p_{z} + q_{z} > {Co}_{z} \end{matrix}$ (9)

2.2 Fuzzy Indirect trust relation

Only one entity as trustor ei always has limited direct interaction experiences with trustee ej. If he wants to get a more accurate trust degree, a natural way for trustor ei is to ask its acquaintances about their opinions at specific context cz. Therefore even trustor ei has not any direct experience with trustee ej in the past, trustor ei can builds a trust relation with trustee ej through his acquaintances. We call the trust relation built by its acquaintances Indirect Trust Relation, which is shown in Fig. 3. Actually an indirect trust relationship builds from recommendations by a trusted third party (i.e acquaintances) or a chain of trusted partied, which create an indirect trust path, which has fuzzy properties. In other words, the indirect trust integrates the recommendation trust and direct trust model.

As shown in Fig. 3, entity ek has directed interaction experiences with trustee ej, so there has a direct trust relation between entity ek and trustee ej noted as DTkj. There also has a recommending relation between entity ek and trustor ei. entity ek recommends its direct experiences to trustor ei noted as RTki, and then these experiences become indirect experiences for trustor ei noted as IDTij.

Peer k has directed interaction experiences with peer j, there has a direct trust relation between k and j noted as DTkj. There also has a recommending relation between k and i. Peer k recommends its direct experiences toi, and then these experiences become indirect experiences for i. But may be k is not a very familiar friend of peer i,or k has recommend i inaccurate experiences in the past, peer I does not think k’s recommendation is completely right.

2.2.1 Fuzzy recommendation trust relation

In Fig. 3 entity e _k recommends its direct experiences to trustee. But maybe entity e _k is not a very familiar friend of entity e _i, or e _k has recommend dishonest in the past, entity e _i does not think e _k’s recommendation is completely right. Thus the recommending relation also has fuzzy properties. We also can use fuzzy relation to describe the recommending relation.

The fuzzy relation membership function defines a degree of recommending relationship between entity e _i as trustor and entity e _k as recommender, which is similar to fuzzy direct trust relation membership function: $T_{z} d_{k i} = R T (e_{k}, e_{i}, c_{z}, t) = \{\begin{cases} 0.5 + \frac{r_{z} (T r_{i}) - α S_{z}}{2 \times C o_{2}} i f r_{z} + s_{z} \leq C o_{z} \\ \frac{r_{z} (T r_{i})}{r_{z} (T r_{i}) + α s_{z}} i f r_{z} + s_{z} > C o_{z} \end{cases}$ (10) Where r represents the number of successful recommendation interactions and s represents the number of failure recommendation interactions between entity e _j and entity e _k at a specific time t and context c _z.

Intuitively, seems reasonable that the higher the trust value of the entity, the more important the recommendation view. However, the entity’s trust value is not entirely consistent with the credibility of the recommendation. On the other hand, some malicious entity may exist in the system. In such cases, different types of attacks can be considered (such as bad-mouthing and on-off) [13, 15]. In all attacks, malicious one tries to be keeping herself as a trusted entity using misleading actions or reputation. Parts of selfish entity through camouflage get the higher trust values, while they give the higher recommendations to their acquaintances, but those recommendations are obviously incredible. So, the credibility of the recommendation of a entity is different from that of itself, especially under some collective or disguised selfish entity. Therefore every proposed model for trust must be able to consider these attacks and also should be able to verify the system against them. To resist malicious behaviour, we put forward a trust model based on the fuzzy recommendation similarity in cloud environments, which aims at preventing the synergistic effect of selfish entity.

In this paper we use the adjusted cosine similar function to determine the similarity between two entity in cloud environment [18]. In this case, similarity between two entity e _i an e _k is measured by computing the Pearson correlation. To make correlation computation accurate we must first isolate the co-rated entities (i.e entities that both two entity e _i and e _k has direct trust relation with them denoted CE(e _i, e _k)). In the case of using basic cosine measure (that is used in some p2p papers), the difference in rating scale between different entity is not taken into account. The adjusted cosine similarity offsets this drawback by subtracting the corresponding node average from each co-rated pair.

For any entity e _i and entity e _k, the similarity between entity e _i and entity e _k at time t and context c _z, denoted as Sim (e _i, e _k, c _z, t), is given by

$\begin{matrix} Sim (e_{i}, e_{k}, c_{z}, t) = \\ \frac{\sum_{e_{c} \in CE (e_{i}, e_{k})} ({DT}_{e_{i}, e_{c}} - \bar{{DT}_{e_{i}, e_{c}}}) ({DT}_{e_{k}, e_{c}} - \bar{{DT}_{e_{k}, e_{c}}})}{\sqrt{\sum_{e_{c} \in CE (e_{i}, e_{k})} {({DT}_{e_{i}, e_{c}} - \bar{{DT}_{e_{i}, e_{c}}})}^{2}} \sqrt{\sum_{e_{c} \in CE (e_{i}, e_{k})} {({DT}_{e_{k}, e_{c}} - \bar{{DT}_{e_{k}, e_{c}}})}^{2}}} \end{matrix}$ (11)

Sim (e _i e _k, c _z, t) describes the similarity of evaluation between entity e _i and entity e _j. $\bar{{DT}_{e_{i}, e_{c}}}$ and $\bar{{DT}_{e_{k}, e_{c}}}$ represent the average direct trust value between two entity ei and ei with co-rated pair CE (e _i, e _k) respectively. Then the fuzzy recommendation trust relation can be revised as:

$T_{z} d_{ki} = RT (e_{k}, e_{i}, c_{z}, t) = RT (e_{k} e_{i}, c_{z}, t) \times Sim (e_{i} e_{k}, c_{z}, t)$ (12)

Where RT (e _i, e _k, c _z, t) represents recommendation trust value between entity e _i and e _k at time t and context c _z and Sim (e _i, e _k, c _z, t) describes the similarity of evaluation between entity e _i and entity e _j.

2.2.2 Fuzzy in direct trust degree computing

As mentioned, The fuzzy indirect trust relation IDT _ij is a composite fuzzy relation of fuzzy recommending relation and fuzzy direct trust relation, In this paper we have used min-max composition to composite fuzzy direct trust value and fuzzy recommendation value. Therefore, the fuzzy indirect trust relation for Fig. 3 is given by: $\begin{matrix} T_{z} d_{ij} = {IDT}^{1} (e_{i}, e_{j}, c_{z}, t) = RT \circ DT \\ = RT (e_{k}, e_{i}, c_{z}, t) \circ DT (e_{k}, e_{j}, c_{z}, t) \\ = {{Max}_{ek} Min (RT (e_{k}, e_{i}, c_{z}, t), DT (e_{k}, e_{j}, c_{z}, t))} \\ = \lor_{ek \in E} (RT (e_{k}, e_{i}, c_{z}, t) \land DT (e_{k}, e_{j}, c_{z}, t)) \end{matrix}$ (13)

In the above equation, we calculated one-level fuzzy indirect trust value which includes one level recommendation based on Fig. 3. Figure 4 shows the two-level fuzzy indirect trust which includes two level recommendation. In this fig, entity e _k2 has the direct interaction experiences with entity e _j, there has a direct trust relationship them.

Entity e _k2 recommends its direct experiences to e _k1, then entity e _k1 recommends its indirect experiences to trustor e _i, and then these experiences become indirect experiences for trustor e _i. The two level fuzzy indirect trust is computed as follows: $\begin{matrix} T_{z} d_{ij} = {IDT}^{2} (e_{i}, e_{j}, c_{z}, t) = RT \circ RT \circ DT \\ = {RT}^{2} \circ DT \end{matrix}$

$\begin{matrix} {RT}^{2} = RT (e_{k 2}, e_{k 1}, c_{z}, t) \circ RT (e_{k 1}, e_{i}, c_{z}, t) \\ = {{Max}_{ek} Min (RT (e_{k 2}, e_{k 1}, c_{z}, t), RT (e_{k 1}, e_{i}, c_{z}, t))} \\ = \lor_{e_{k 1} \in E} (RT (e_{k 2}, e_{k 1}, c_{z}, t) \land RT (e_{k 1}, e_{i}, c_{z}, t)) \end{matrix}$ (14)

If entity e _i continues in this manner, there have three, four … n levels indirect trust relation and it can get more and more accurate trust degree with entity e _j in context c _z. The multi-level composite fuzzy indirect trust is calculated as: $\begin{matrix} T_{z} d_{ij} & = & {IDT}^{n} (e_{i}, e_{j}, c_{z}, t) = RT \circ RT \circ \dots \circ RT \circ DT \\ = & {RT}^{n} \circ DT \\ {RT}^{n} & = & {RT}^{n - 1} \circ RT, (n = 1, 2, 3, \dots) \end{matrix}$ (15)

If there is some trust path between trustor e _i and trustee e _j, the indirect trust value between e _i and e _j calculates from the union of all indirect trust value in different path(one-level, two-level, …): $\begin{matrix} T_{z} d_{ij} & = & IDT (e_{i}, e_{j}, c_{z}, t) = {IDT}^{1} \cup {IDT}^{2} \cup \dots \cup {IDT}^{n} \\ = & ⋃_{i = 1}^{n} {IDT}^{i} \end{matrix}$ (16)

Usually trustor e _i has not only direct interaction experiences with trustor e _j (in context c _z), but also indirect experiences from asking its acquaintances. Then there are two fuzzy trust relation (i.e fuzzy direct trust relation and fuzzy indirect trust relation) between trustor ei and trustor ej, If trustor ei wants to get more accurate trust value with trustor ej, it must integrate the direct and indirect experiences. The fuzzy global trust relation is a union of fuzzy direct trust relation and indirect trust relation obtained from relation 16. $\begin{matrix} T_{z} d_{ij} = DT \cup {IDT}^{1} \cup {IDT}^{2} \cup \dots \cup {IDT}^{n} \\ = DT \cup ⋃_{i = 1}^{n} {IDT}^{i} \end{matrix}$ (17)

2.3 Total trust computing in all context

After calculating the global trust value in each of the context between trustor e _i and trustee e _j, the trustor e _i needs to calculate the total trust value in all context. The total trust value will be combined closely with the value assignment of each evaluation context. The nature of weight is shown in the quantity of different context on objects at different levels, i.e. the different influence from all contexts on the trust in view of trustor e _i. Suppose w = (w ₁, w ₂ … , w _n) is the weight of the context c _z such that w _i is in [0,1] for all i and $\sum_{1}^{n} w_{i} = 1$ . So, the total trust Td _ij can be gotten by the following fuzzy mapping:

$\begin{array}{l} T d_{i j} = W i (w_{1}, w_{2}, \dots, w_{n}) \begin{array}{c} c_{1} \\ c_{2} \\ c_{n} \end{array} \overset{T_{z} d_{i j}}{(\begin{array}{c} T_{1} d_{i j} \\ T_{2} d_{i j} \\ ⋮ \\ T_{n} d_{i j} \end{array})} \\ = w_{1} \times T_{1} d_{i j} + w_{2} \times T_{2} d_{i j} + \dots + w_{n} \times T_{n} d_{i j} \end{array}$ (18)

Where Td _ij represents the total trust value in all context that trustor e _i interacts with trustee e _j and t _z d _ij (t ₁ d _ij, t ₂ d _ij, …, t _n d _ij) represents the global trust value in each of the context between trustor e _i and trustee e _j.

3 Experimental result and analysis

In order to evaluate the performance of the TMFM model in this paper, simulation environment and parameters set are firstly discussed in this section, and then precise performance evaluation results are given.

3.1 Experiments environment and configuration

The platform of simulation environment is CloudSim toolkit (Buyya et al., 2009b) which is a simulation platform based on Java, which supports modelling and simulation of large-scale cloud computing data centers. Therefore, it is feasible to simulate our proposed model of cloud computing environments by CloudSim. We create ten data centres in the simulation environment, We set 500 virtual machines. Moreover, we submit 1000 tasks to the 500 virtual machines. Also recommenders are divided into three types:

virtuous recommenders who provide honest service and recommendation.

random recommenders who provide random service and recommendation.

malicious recommenders who provide malicious service and recommendation.

Table 1 shows the main parameters used in this set of experiments.

3.2 Comparison among trust models

An important application of the proposed trust analysis is to facilitate comparison among different trust establishment methods. There are some trust schemes proposed for cloud environment, so, it is difficult to list all the trust models to compare with each other. In the Section, we make a comparison with two trust models, i.e. DMTC [7] and TMFM model without similarity, in the paper.

DMTC [7]

3.2.1 Trust accuracy rate

We use absolute error metrics for evaluating the accuracy. Absolute error: It is the difference between the actual value of trust for an edge and the calculated value from a method. $Absolute error = | trust calculated - actual trust |$ (19)

In the above equation, trust calculated is the trust value that is calculated by the TMFM model and actual trust is the union of global trust value of all entities that interact with trustee. Therefore the trust accuracy rate calculates as follows: $Trust accuracy rate = (1 - Absolute error) \times 100$ (20)

As shown in Fig. 5, in the first simulation time, when there is no interaction between entities, we set direct trust equal to one. Therefore Absolute error is set one. The success interaction rate declines with malicious interactions at the beginning. After a time, the success interaction rate keeps rising. Also, with the increase of the malicious rate, the TMFM model can ensure trust accuracy rate in a high level.

3.2.2 Success interaction rate

The good entities can be differentiated from the misbehavior entities by their trust values after a few interactions. At the beginning, all entities have the same initial trust value, the trustors randomly select a entity, after a few numbers of interactions, and the normal entities can get the higher trust value than the other selfish entities. With a help of the trust computing based on TMFM model, we can identify the malicious entities efficiently. Thank to it, we can restrict the interaction of malicious entities further. It can help to increase the success interaction rate of the system.

Success interaction rate is the ratio of successful interactions to overall interactions in the simulation time denoted as: $success in teraction rate = \frac{p}{p + q}$ (21)

The experiment results are shown in Fig. 5. Results show that the success interaction rate with TMFM model is higher than DMTC model. From Fig. 4, we can see that the changing of success interaction rate is divided into two stages: decline stage and rise stage. The success interaction rate declines with malicious interactions at the beginning. After a time, the success interaction rate keeps rising. It is because that the system with trust computing has begun to identify the malicious entities and refuse to provide service for them.

4 Conclusion and future work

In this paper, we use the fuzzy relation theory in fuzzy mathematics to build trust model between entities, which bases on fuzzy similarity recommendation in cloud environment. Simulation results show that the TMFM model has some identification and containment capability in synergies cheating, promotes interaction between entities, and improves the performance of the entire cloud environment. In the future, we will offer new dynamic scheduling algorithm according to TMFM model for cloud computing.

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ACM 1-3-348-0/01/0005