Growth models have been proposed for constructing the scale-free overlay topology to improve the performance of unstructured peer-to-peer (P2P) networks. However, previous growth models are able to maintain the limited scale-free topology when nodes only join but do not leave the network; the case of nodes leaving the network while preserving a precise scaling parameter is not included in the solution. Thus, the full dynamic of node participation, inherent in P2P networks, is not considered in these models. In order to handle both nodes joining and leaving the network, we propose a robust growth model E-SRA, which is capable of producing the perfect limited scale-free overlay topology with user-defined scaling parameter and hard cut-offs. Scalability of our approach is ensured since no global information is required to add or remove a node. E-SRA is also tolerant to individual node failure caused by errors or attacks. Simulations have shown that E-SRA outperforms other growth models by producing topologies with high adherence to the desired scale-free property. Search algorithms, including flooding and normalized flooding, achieve higher efficiency over the topologies produced by E-SRA.