## 1 Introduction  
The development of the Internet provides small and medium enterprises (SMEs) new opportunities to extend their operations. However, new challenges also arise as a result of the virtual characteristics of the Internet. The Internet easily broadcasts information. However, websites of SMEs have relatively low degrees of trust and reputation [1, 2, 3, 4, 5]. Trust is one of the most important factors that influences the successful application of network service environments such as e-commerce, wireless sensor networks, and online social networks, and so on [6, 7, 8]. How to improve the degree of trust and establish a reputation in online marketplaces is critical for these enterprises [1, 9, 10, 11, 12]. Computation models associated with trust and reputation are becoming focal issues in computer societies, management societies, and service science in recent years [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. In this paper, a new computation model of dynamical trust and reputation for B2C e-commerce is proposed. The proposed framework simulates the dynamical processes of trust and reputation evaluation for B2C e-commerce by using a mathematical model. One principal role of the Internet is information dissemination and communication. E-commerce is defined as transactions of products and services over the Internet. The prototype of e-commerce is Electronic Data Interchange (EDI) [26, 27]. With the development of information exchange techniques and the popularization of the Internet, Business-to-Business (B2B) e-commerce is exceeded by Business-to-Consumer (B2C) e-commerce in the 1990s [5, 28, 29]. Commercial data exchange in EDI used the private value-add-network between those who are familiar with and trust each other. However, the situation changed in B2C e-commerce which is based on the Internet. On one hand, in the service-oriented Internet field, many kinds of e-services can be provided for different types of users. On the other hand, many users are reluctant to purchase products or services over the Internet. For example, only five percent of active Internet users made purchases via the Internet in Hong Kong over the last ten years [30]. A report made by China Internet Network Information Center (CINIC) of China revealed that only about 40 percent of Internet users in China had the experience of purchasing via the Internet in 2014 [31]. The lack of trust is still one of the most important reasons for users not purchasing over the Internet [14, 15, 22, 32, 33, 34, 35, 36, 37]. This phenomenon is not surprising when the characteristics of the Internet are considered. The Internet is known to be a loosely coupled system. The Internet is designed primarily as a medium of cooperation and information sharing. Furthermore, the Internet is not originally designed as a commerce environment [38]. Thus, trust is hard to achieve and maintain. The reasons behind the lack of trust in B2C e-commerce are complex. First of all, service clients cannot interact with products and service providers directly, and they are also unfamiliar with online added services. The credibility of online information is weak [15, 39]. Security and privacy of key pieces of information that consumers provide through the Internet are also important issues for consumers [8, 40, 41]. In such an uncertain environment, trust becomes one of the most important factors in the development of Internet services such as e-commerce [22, 42]. Indeed, the Internet provides maximal convenience in the information era. The number of total financial transactions made via the Internet has been increasing. The problems associated with trust are those derived from certain techniques, and the human element in e-commerce can be solved step by step. In terms of the evaluation of trust and reputation in regards to the Internet, many practical trust evaluation systems focus mainly on Peer-to-Peer networks such as C2C, B2B e-commerce, and B2C e-commerce [43, 44, 45, 46, 47, 48].