# Trusted Energy-Aware Hierarchical Routing (TEAHR) for Wireless Sensor Networks
## 2 Related Studies
### Trust Models and Their Classification
\[H=-\sum_{i=1}^{n}p_{i}logx_{2}p_{i} \tag{2}\]1  
where N is the set of players (sensor nodes), S = S\({}_{1}\)\(\times\) S\({}_{2}\)\(\times\)\(\ldots\)\(\ldots\)\(\times\) S\({}_{n}\) is the set of strategy profiles, where \(s_{i}\) is the strategy set available for player I; u = u\({}_{1}\)\(\times\) u\({}_{2}\)\(\times\)\(\ldots\)\(\times\) u\({}_{n}\) is the vector of payoff functions, with \(u_{i}\) (s\({}_{1}\)\(\times\) s\({}_{2}\)\(\times\)\(\ldots\)\(\times\) s\({}_{n}\)) representing the payoff to player i when strategy profile (s\({}_{1}\)\(\times\) s\({}_{2}\)\(\times\)\(\ldots\)\(\ldots\)\(\times\) s\({}_{n}\)) is employed. The level of Trust of a node can be derived from its payoff, which reflects the behavior of this Node to achieve network objectives, such as good data transmission. The common approach to modeling this uses utility functions that capture the benefits and costs related to different strategies. A simplified utility function by [PERSON] et al. (2013) for a sensor node may be written as follows:  
\[H=-\sum_{i=1}^{n}p_{i}logx_{2}p_{i} \tag{2}\]2  
where \(u_{i}(s_{i},\text{s}-\text{i})\) is the utility for Node I choosing strategy \(s_{i}\) is given strategies \(s-i\) of other nodes; \(\text{B}(s_{i},s-i)\) represents the benefits obtained from choosing strategy \(s_{i}\), which may depend on the actions of other nodes. C(\(s_{i}\)) for costs incurred if strategy si is chosen, \(\alpha\) and \(\beta\) are weighting factors that maintain a balance between the benefits and costs. Those who define game models often look for Nash equilibria where none of the players feel compelled to change their strategy independently, being convinced of the strategies of all the others. In this context, regarding Trust, a Nash Equilibrium by [PERSON] et al. (2013) outlines the situation in which all nodes act in such a way as to maximize their utility while maintaining a satisfactory degree of trustworthiness and cooperativeness. Most related studies in this area have focused on developing sophisticated mechanisms for dynamically adaptive changes in networks. At the same time, it has been incorporated with realistic factors, such as node mobility, variable trust levels, and possible adversaries. An evolutionary game is the study of the strategy evolution dynamics in a population. [PERSON] et al. (2013) proposed a WSNs model of how trust-related strategies evolve among sensor nodes, and for that, the replicator equation is defined as follows:  
\[H=-\sum_{i=1}^{n}p_{i}logx_{2}p_{i} \tag{2}\]3  
where \(\dot{x}_{i}\) is the proportion of the population using strategy I, \(U_{i}(x)\) is the rate of change of x\({}_{i}\), \(U_{i}(x)\) is the utility of strategy I, _and_\(U(x))\) is the average utility of the population. The models of Trust in Wireless Sensor Networks (WSNs) are the directed graph-based trust model and the undirected graph-based trust model. In both cases, graph theory is utilized to show relationships and interactions among the sensor nodes, where levels of Trust are shown in the graph's edges. The nodes and relationships represent the vertices defined by the edges within these models. Directed Graphs, or simply Digraphs, are used when the trust relationship is said to be asymmetric. That is, the trustworthiness of A to B is not necessarily the same as that of B to A. Moreover, in undirected graphs, symmetric trust relationships were presented. That is, Trust is mutual and equal in both directions. In directed graph-based models, each edge is associated with a given direction. Directed edges show the relation of Trust from one Node to another, usually labelled with a weight. Then, the edge may further be used to represent the level of confidence.