# A Trust-Influenced Smart Grid: A Survey and a Proposal
## 4 Trust
### Trust Definition and Formalization  
According to the literature, trust has many definitions. A definition from the social sciences states that trust is the degree of subjective belief about the behaviours of a particular entity [23]. Trust is also defined as an agency's subjective probability of performing a particular act [24]. In this paper, we define the trusting entity as the _agent_ and the entity being trusted as the _subject_. [PERSON] [25] describes three levels of trust, namely basic trust, general trust, and situational trust. Basic trust is the general trusting disposition of an agent. General trust is the trust that an agent has on a subject at a certain time. Situational trust is the trust that the agent has on the subject, taking into account a certain situation. It must be noted that trust has been applied in different contexts, thus the notion that trust has many definitions. Thus, the design of trust models is required to be within a context or in terms of the system being designed. Thus, the factors being chosen to design the trust model must be on objective grounds to ensure that the trust being modelled is also objective. Hence, the difficulty in modelling trust. Regardless, trust models must have a component that must accept the risk because, without the assessment of risk, there is no trust. NIST defines risk as: _A measure of the extent to which an entity is threatened by a potential circumstance or event_[26]. Thus, for an agent, \(a_{i}\), and a subject, \(a_{j}\), we define the risk, \(r_{ij}\), of a transaction, \(\alpha_{ij}\), involving \(a_{i}\) and \(a_{j}\) as a function as shown in (1). There must also be a component of knowledge, \(k^{t}_{ij}\), within the trust model. Before and after a transaction, knowledge about \(\alpha_{ij}\) and previous transactions (\(k^{\prime}_{ij}\)) with the subject, the environment (\(k_{e}\)), knowledge of \(a_{j}\), \(k_{a_{j}}\), and the time period (\(t\)), are also of prime importance in determining trust. We formulate knowledge as shown in (2). \(k^{\prime}_{ij}\) is a collection of transactions before the current transaction, and this is formulated in (3). \[r_{ij}=f(\alpha_{ij}) \tag{1}\]  
\[k^{t}_{ij}=f(\alpha_{ij},k_{e},k^{\prime}_{ij},k_{a_{j}},t) \tag{2}\]  
\[k^{\prime}_{ij}=\{k^{t-1}_{ij},k^{t-2}_{ij},...,k^{0}_{ij}\} \tag{3}\]  
\[T_{ij}=f(a_{i},a_{j},r_{ij},k_{ij},T^{\prime}_{ij}) \tag{4}\]  
\[T_{ik}\approx T_{jk}\;\textit{where}\;1-T_{ij}\approx\;0,\;1-T_{jk}\;\approx\;0 \tag{5}\]  
\[T_{ik}\sim T_{jk}\;\textit{where}\;1-T_{ij}\approx\;0,\;1-T_{jk}\;\approx\;0 \tag{6}\]Thus, with risk, \(r\), and knowledge, \(k^{t}_{ij}\), the decision on trust can be made. Therefore, trust, \(T_{ij}\), can be expressed as the output of a function that takes a tuple of elements as shown in (4) where \(T^{{}^{\prime}}_{ij}\) is the previous trust value between \(a_{i}\) and \(a_{j}\). The \(T^{{}^{\prime}}_{ij}\) has an influence on the decision for \(a_{i}\) to trust \(a_{j}\) to undertake \(a_{ij}\). Trust is represented as a continuous variable over a specified range usually \(-1\leq T\leq 1\) or \(0\leq T\leq 1\) where \(1\) represents complete trust, \(-1\) represents complete mistrust and \(0\) represents no trust. It must be noted that the transitive property of trust may or may not exist. In a situation where it does not exist, for three agents \(a_{i}\), \(a_{j}\), and \(a_{k}\), the fact that \(a_{i}\) trusts \(a_{j}\) and \(a_{j}\) trusts \(a_{k}\) does not mean that \(a_{i}\) trusts \(a_{k}\) (see (5)). In a situation where transitivity exists, it means that \(a_{i}\) trusts \(a_{j}\) and \(a_{j}\) trusts \(a_{k}\), therefore, \(a_{i}\) trusts \(a_{k}\) (see (6)). Trust can be directly or indirectly evaluated. Direct trust is calculated based on direct interactions between the agent and the subject. The default definition of trust is direct trust and that is formulated in (4).