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5. Characteristics Analysis In this section, the characteristics of GTRF are analyzed in detail. To highlight the advantages of GTRF in terms of energy cost, lifetime and so on, several theorems have been proposed and proved. The core of the VA model is a bidding process, whereby each time a node needs to bid a value based on its actual remaining energy. A node with the biggest value will be designated as a cluster head in the next cluster period. As each node is intelligent to control its behavior, it can decide to bid any value for obtaining Adriamycin concentration the maximum profit. Therefore, Theorem 1 is proposed which LBH589 in vitro aims at proving that GTRF is able to regulate selfish behaviors within clusters. By using GTRF, a selfish node can only bid its actual remaining energy. In the bidding process, we denote bidiEnergy(j) as the bidding value that node i bids in the jth bidding round. At the same time, bidibasis(j) is denoted as the actual remaining energy of node i in the jth bidding round. As the second highest bidding value is an important factor that determine which node will win, therefore, we defined bidsecond(j) as the second highest value in the jth bidding round. Theorem 1.?The best strategy for a node in the VA model is to bid its actual remaining energy (i.e., bidiEnergy(j) = bidibasis(j)), the situation including all nodes�� bidding values in one auction round is a Nash Equilibrium when requirements (1) and (2) Pentamorphone are satisfied. (1) The energy cost of cluster Ck��s cluster head in the jth round is inversely proportional to the average bidding value within Ck, (i.e., 1g(j)��1|Ck|��i?CkbidEnergy(j)i ), where |Ck| is the size of the cluster, i?Ck means node i belongs to cluster Ck. (2) Tperiod