# A game theory-based trust measurement model for social networks

- Yingjie Wang
^{1}, - Zhipeng Cai
^{3, 4}Email author, - Guisheng Yin
^{3}, - Yang Gao
^{2}, - Xiangrong Tong
^{1}and - Qilong Han
^{3}

**Received: **25 September 2015

**Accepted: **10 May 2016

**Published: **20 May 2016

## Abstract

### Background

In social networks, trust is a complex social network. Participants in online social networks want to share information and experiences with as many reliable users as possible. However, the modeling of trust is complicated and application dependent. Modeling trust needs to consider interaction history, recommendation, user behaviors and so on. Therefore, modeling trust is an important focus for online social networks.

### Methods

We propose a game theory-based trust measurement model for social networks. The trust degree is calculated from three aspects, service reliability, feedback effectiveness, recommendation credibility, to get more accurate result. In addition, to alleviate the free-riding problem, we propose a game theory-based punishment mechanism for specific trust and global trust, respectively.

### Results and conclusions

We prove that the proposed trust measurement model is effective. The free-riding problem can be resolved effectively through adding the proposed punishment mechanism.

## Keywords

## Background

With the current popularity of online social networks, more and more information is distributed through social network services [2]. Participants in online social networks want to share information and experiences with as many reliable users as possible [3–5]. Trust is a basis of social network services. However, the modeling of trust is complicated and application-dependent [6–8]. Modeling trust needs to consider interaction history, recommendation, user behaviors and so on. Therefore, modeling trust is an important focus for online social networks [9–11].

- 1.
To more accurately measure trust degree of a node, we introduce three novel evaluation factors which are service reliability, feedback effectiveness and recommendation credibility.

- 2.
Another practical problem considered in this paper is the free-riding problem. We propose punishment mechanisms for specific trust and global trust, respectively, which is different from the existing works where statistic methods are commonly used. In our punishment mechanism, we employ the evolutionary game theory which is more flexible and effective.

## Related works

Much effort has been spent on trust measurement models to depict trust behaviors in complex networks. Trust measurement methods under open network environment and trust measurement methods based on Agent synergy are the most important trust measurement methods.

### Trust measurement methods under open network environment

Beth et al. [14] first proposed a trust measurement method under open network environment. In their work, trust is regarded as direct trust and recommendation trust, and a probabilistic method is adopted to represent trust. The PeerTrust model [15] uses the transaction and the community background as the source of reputation feedback. It can act as a defense against some of the subtle malicious attacks, e.g., a seller develops a good reputation by being honest for small transactions and tries to make a big profit by being dishonest for large transactions. The EigenRep model [16] assumes that if the direct trust between a node and the destination node is higher, the recommendation trust is more reliable. The model uses direct trust to calculate the global trust. This model can effectively solve the bad effect caused by the malicious recommendation. Wang et al. [17] proposed a trust model based on Bayesian network. This model investigates how to describe different aspects of trust to obtain various properties of entities according to different scenes. Wang et al. [18] solved the problem of recommendation trust based on the Bayesian method. This method calculates recommendation trust based on experts’ experience. Lu et al. [19] proposed an evaluation method of software reliability. It is a bottom-up calculation process of trust level that can decompose and synthetically derive a parallel structure, so that the trust value of a system can be calculated accurately. However, there are still some shortages about this kind of models. They only adopt probabilistic model to establish subjective trust model. In other words, subjectivity and uncertainty of trust are equivalent to randomness. They also adopt the averaging method to calculate recommendation trust, which cannot reflect the real situations of a trust relationship.

### Trust measurement methods based on Agent synergy

In Agent synergy, trust means that a collaborative agent can properly and nondestructively predict subjective possibility of a collaborative activity. The source of prediction is the goal service behavior that previous agent observes. Prediction results are affected by evaluation of important degree from the agent, such as key collaborative activities and secondary collaborative activities [20, 21]. The eBay trust model is one of the most successful cases. In this model, the entities evaluate each other after each transaction. The structure of this system is straightforward, and the computation cost is small. Because trust between agents is associated with other entities’ subjective understanding and fuzziness, it cannot be described and managed by conventional and accurate logic. Subjective trust as a cognitive phenomenon, whose subjectivity and uncertainty present fuzziness, is often managed by Fuzzy Set-based methods. It not only reflects fuzziness of agent trust, but also describes the trust mechanism between agents with intuitive and concise semantics. Tang et al. [22] first proposed the definition and evaluation of trust based on the fuzzy set theory. They gave formalization and deducted rules of trust to construct a complete subjective trust management model. However, this kind of model fails to consider the cooperative cheating behaviors, which cannot detect the community of cooperative cheating.

In addition, some recent works are also remarkable. Shi et al. [23] proposed a dynamic P2P trust model based on the time-window feedback mechanism. The model considers the inherent connection among trust, reputation and incentive and the effect of time factor on the trust computation. Gan et al. [24] proposed a reputation-based multi-dimensional trust (RMDT) algorithm which makes use of a self-confident coefficient to synthesize the direct and recommendation trust to evaluate the nodes in a network. A multi-dimensional trust mechanism is also introduced to improve sensitivity of RMDT on a single attribute. Meng et al. [25] proposed the @Trust model. Bedi et al. [26] proposed a trust-based recommender system using ant colony for trust computation. Zhang et al. [27] proposed a trust evaluation method based on the cloud model.

These models have promoted the development of trust measurement models. However, most of the existing models failed to filter false feedback and distrustful recommendation, which leads to inaccuracy of measurement results. In addition, free-riding problem was not comprehensively considered in most of the existing trust measurement models. Considering these problems, this paper proposes a game theory-based trust measurement model for social networks. The proposed model introduces three novel evaluation factors which are service reliability, feedback effectiveness and recommendation credibility to more accurately measure the trust degree of a node.

## The proposed trust measurement model

To describe the trust degree more accurately, this paper divides nodes into four categories, which are service nodes, feedback nodes, recommendation nodes and managed nodes. In social networks, *trust* represents the level of confidence about the reliability and correctness of entity’s behaviors. *Service reliability* indicates the trustworthiness of service that service nodes provide; *feedback effectiveness* represents the trustworthiness of feedback that feedback nodes return; *recommendation credibility* expresses the trustworthiness of recommendation that recommendation nodes give. In this paper, the global trust of the node *i*, denoted as \(T_i\), is the probability of *i* being correct. The service reliability is denoted as ST\(_i\); the feedback effectiveness is denoted as FT\(_i\); and the recommendation credibility is denoted as CT\(_i\).

In this paper, let *i* be a service node, *j* be a feedback node and *k* be a recommendation node; and \(M_i\), \(M_j\), \(M_k\) are the managed nodes of *i*, *j*, *k*, respectively.

When feedback node *j* requests a specific service *s*, the managed node \(M_j\) searches for the trust node which can provide service *s*. If there exists such a node, say node *i*, node *j* requests the service from node *i*. If not, \(M_j\) searches for the recommendation node *k*. Then, node *k* recommends a service node *i* with the maximum trust degree that can provide service *s* to node *j*. If there does not exist a recommendation node *k*, the transaction fails.

### The trust measurement process

*j*, is known by the system. Therefore, we obtain the calculation method of service reliability based on the specific feedback value \(f_{v_{j,i}}\), which is shown by Eq. (1).

*i*) is the set of feedback nodes that communicated with service node

*i*, and \(\theta\) is the threshold of feedback effectiveness. \({\lambda }(j,i)\) presents the influence effect of node

*j*on node

*i*. In addition, FT\(_j\) represents the feedback effectiveness of node

*j*.

*j*can be derived through a similarity formula as shown by Eq. (2).

*j*,

*r*) presents the node-pair set that both nodes communicated with node

*i*. Similar with the calculation method of service reliability, the recommendation credibility of node

*k*is computed by Eq. (3).

*k*) is the node set recommended by recommendation node

*k*before. \({\lambda }(k,i)\) presents the influence effect of node

*k*on node

*i*. There are two factors affecting the value of \({\lambda }(k,i)\). One is the time interval \(T=t_n-t_p\), \(t_n\) presents the current time, and \(t_p\) presents the time that node

*k*recommends node

*i*. Another is the connection degree \(\omega _{k,i}\) of the relationship between node

*i*and node

*k*. Thus, \({\lambda }(k,i)\) is shown as Eq. (4).

*Tr*| between node

*k*and node

*i*, we determine the connection degree \(\omega _{k,i}\), which is shown as Eq. (5). In Eq. (5), successful transaction Tr\(_\mathrm{suc}\) is an indicative function, if CT\(>\mathrm{Threshold}\), Tr\(_\mathrm{suc}=1\), otherwise, Tr\(_\mathrm{suc}=0\).

The process of direct interaction is summarized in Algorithm 1. In this direct interaction algorithm, the service reliability ST\(_i\) and the feedback effectiveness FT\(_j\) will be output.

If there is not a trusted service node *i* that has interacted with the feedback node *j* directly, it needs a recommendation node *k* to recommend a trusted node *j* for service node *i*. Thus, the process of indirect interaction is summarized in Algorithm 2.

### The punishment mechanisms

To resolve free-riding problem in social networks, two punishment mechanisms are proposed for specific trust and global trust degree, respectively. According to specific trust (service reliability, feedback effectiveness and recommendation credibility), this paper designs three punishment cycles, so that to restrain the specific trust behaviors of nodes. According to global trust, this paper gives a game theory-based punishment mechanism [28] to resolve the free-riding problem for social networks.

- 1.
Service punishment cycle. If the service reliability ST\(_i<\rho\), node

*i*will enter service punishment cycle. In the service punishment cycle, a node cannot provide service for other nodes and cannot request any service. - 2.
Feedback punishment cycle. If the feedback effectiveness FT\(_i<\theta\), node

*i*will enter feedback punishment cycle. In the feedback punishment cycle, a node cannot request any service. However, it can provide service for other nodes. - 3.
Recommendation punishment cycle. If the recommendation credibility CT\(_i<\delta\), node

*i*will enter recommendation punishment cycle. In the recommendation punishment cycle, a node cannot recommend any node. However, it can request and provide service for other nodes.

*i*. We divide trust degrees into five levels as shown in Table 1.

The division of trust levels

T | Trust |
---|---|

[0.8,1] | Trust 1 |

[0.6,0.8) | Trust 2 |

[0.4,0.6) | Trust 3 |

[0.2,0.4) | Trust 4 |

[0,0.2) | Trust 5 |

*A*obtains, if

*A*game with

*B*that

*A*adopts strategy

*i*, and entity

*B*adopts strategy

*j*. And pr\(_{B}^{ij}\) is the profit value that

*B*obtains, if

*B*game with

*A*that

*B*adopts strategy

*i*, and

*A*adopts strategy

*j*. Through game analyzing nodes’ behaviors in social networks, we can know that the multi-strategy game matrix is a symmetric matrix. In the analysis for dynamics model, this game is performed repeatedly. At the end of each stage of multi-strategy game, any participant’s strategy as a historical information can be known by other participants. In addition, all participants select and update their strategies for next stage of game based on historical information.

The initial five-strategy game matrix

Trust level | Trust 1 | Trust 2 | Trust 3 | Trust 4 | Trust 5 |
---|---|---|---|---|---|

Trust 1 | pr\(_{A}^{11},\mathrm{pr}_{B}^{11}\) | pr\(_{A}^{12},\mathrm{pr}_{B}^{12}\) | pr\(_{A}^{13},\mathrm{pr}_{B}^{13}\) | pr\(_{A}^{14},\mathrm{pr}_{B}^{14}\) | pr\(_{A}^{15},\mathrm{pr}_{B}^{15}\) |

Trust 2 | pr\(_{A}^{21},\mathrm{pr}_{B}^{21}\) | pr\(_{A}^{22},\mathrm{pr}_{B}^{22}\) | pr\(_{A}^{23},\mathrm{pr}_{B}^{23}\) | pr\(_{A}^{24},\mathrm{pr}_{B}^{24}\) | pr\(_{A}^{25},\mathrm{pr}_{B}^{25}\) |

Trust 3 | pr\(_{A}^{31},\mathrm{pr}_{B}^{31}\) | pr\(_{A}^{32},\mathrm{pr}_{B}^{32}\) | pr\(_{A}^{33},\mathrm{pr}_{B}^{33}\) | pr\(_{A}^{34},\mathrm{pr}_{B}^{34}\) | pr\(_{A}^{35},\mathrm{pr}_{B}^{35}\) |

Trust 4 | pr\(_{A}^{41},\mathrm{pr}_{B}^{41}\) | pr\(_{A}^{42},\mathrm{pr}_{B}^{42}\) | pr\(_{A}^{43},\mathrm{pr}_{B}^{43}\) | pr\(_{A}^{44},\mathrm{pr}_{B}^{44}\) | pr\(_{A}^{45},\mathrm{pr}_{B}^{45}\) |

Trust 5 | pr\(_{A}^{51},\mathrm{pr}_{B}^{51}\) | pr\(_{A}^{52},\mathrm{pr}_{B}^{52}\) | pr\(_{A}^{53},\mathrm{pr}_{B}^{53}\) | pr\(_{A}^{54},\mathrm{pr}_{B}^{54}\) | pr\(_{A}^{55},\mathrm{pr}_{B}^{55}\) |

*i*\(-\) Trust

*j*) is bigger, \(\mu\) will increase.

### Complexity analysis

In this trust measurement model, we need to compute service reliability, feedback effectiveness and recommendation credibility for one node. According to a node’s service reliability, the computational complexity is *O*(*l*), where *l* represents that this node has provided services for *l* nodes ever before. According to a node’s feedback effectiveness, the computational complexity is *O*(*m*), where *m* indicates that *m* nodes provided feedbacks for this node. According to a node’s recommendation credibility, the computational complexity is *O*(*n*), where *n* represents that *n* nodes were recommended by this node before. Therefore, the computational complexity of a node’s global trust is \(O(l+m+n)\). The proposed trust measurement model can be computed in polynomial time, thus it is computationally efficient.

## Simulations and performance analysis

The simulation setting

The style of nodes/trust | Service | Feedback | Recommendation |
---|---|---|---|

Ct | 1 | 1 | 1 |

Tt | \(\varepsilon\) | 1 | 1 |

Cm | 0 | 0 | 0 |

Rm | \(\varepsilon\) | \(\varepsilon\) | \(\varepsilon\) |

Dm | 1 | 0 | 1 |

### Experimental verification for specific trust

### Experimental verification for global trust

## Conclusion

In social networks, trust relationships between nodes are the basis of service transactions. However, the establishment of trust relationship is a complex progressive process depending on interaction history, trust recommendation, trust management and so on. Therefore, modeling trust relationship needs to take into account multiple decision factors. Considering the existing problems of trust models, this paper proposes a game theory-based trust measurement model for social networks where trust degree is determined by three aspects, which are service reliability, feedback effectiveness, and recommendation credibility. Based on game theory, we propose punishment mechanisms according to specific trust and global trust respectively to resolve free-riding problem. The simulation results show the effectiveness of the proposed trust measurement model. It also shows that the proposed punishment mechanisms can prevent free-riding phenomenon effectively. As a future work, we will further investigate more specific trust relationships between nodes, e.g., family, best friends, and classmates. We plan to study how to find ordered trust node set in social networks.

## Declarations

### Authors’ contributions

YW designed the proposed trust measurement model, performed the experiments analysis and drafted the manuscript. ZC conceived of the study, and participated in its design and coordination and helped to draft the manuscript. GY participated in the design of the study, and helped to direct the research contents. YG carried out the acquisition of data, and helped to perform experiments. XT helped to analyze the performance of the proposed trust measurement model. QH helped to analyze the feasibility of the proposed trust measurement model. All authors read and approved the final manuscript.

### Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grants No. 61502410, No. 61572418, No. 61370084, No. 61502116, the Natural Science Foundation of Shandong Province under Grants No.ZR2013FQ020, ZR2014FQ026, ZR2013FQ023 and ZR2015PF010.

The short version of this manuscript is in CSoNet 2015 [1].

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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