This paper studies a sale promotion mechanism design problem on a social network, where a node (a seller) sells one item to the other nodes on the network to maximize her revenue. However, the seller does not know other nodes except for her neighbours and her neighbours have no incentive to promote the sale. Hence, the goal is to design an auction mechanism such that the seller's neighbours are incentivized to invite their neighbours to join the auction, while the seller's revenue is guaranteed to increase. This is not achievable with traditional mechanisms. One solution has been proposed recently by carefully designing a reward scheme for the nodes who have invited others. However, the solution only gives rewards to some cut-points of the network, but cut-points rarely exist in a well-connected network, which actually disincentivizes nodes' participation. Therefore, we propose another novel mechanism to reward more related participants with fairer rewards, and the seller's revenue is not reduced.