SUMMARYUnstructured overlay networks are widely adopted in large-scale and heterogeneous peer-to-peer (P2P) systems for their scalability and flexibility. A distinct feature of such systems is that they randomly route messages e.g., by flooding or random walk. In such systems, the number of messages and tasks carrying by those messages each peer receives is greatly affected by the number of the peer's incoming links. The objective of this paper is to build controllable degree-weighted networks in which the expected number of incoming links of each peer is proportional to its weight which is a local parameter. In such a network, a peer can control the number of those randomly disseminated messages and tasks it receives by adjust it weight. In addition, in order to bound the construction overhead for highly biased networks, we restrict all peers to have the same number of outgoing links. The objective network is constructed by local topology transformations that peers periodically exchange outgoing links with each other. A framework, which includes 81 different protocols by combination of exchange rules, is presented and evaluated by simulation. The simulation result shows that two of them can generate the networks having similar properties with the objective network. This work first achieves the weightproportional degree control under the out-regular network model.