We study multi-unit auctions in social networks, where each buyer has a fixed budget and can spread the sale information to the network neighbors. We design a mechanism encouraging buyers to report their valuations truthfully and spread the sale information. Our design uses the idea of the clinching mechanism to decide the transaction price and can be viewed as a network version of the mechanism. Most of the previous clinching mechanisms search for the transaction prices by increasing the current price. Our mechanism directly computes the transaction prices in polynomial time. Furthermore, the mechanism applies a technique to iteratively activate new buyers in the network. This ensures utility preservations of the buyers and benefits the seller. We prove key properties of our mechanism, such as no-positive-transfers, individual rationality, incentive compatibility, non-wastefulness and social welfare preservation.