Distributed systems desire to construct a random overlay graph for robustness, efficient information dissemination and load balancing. A random walk-based overlay construction is a promising alternative to generate an ideal random scale free overlay in distributed systems. However, a simple random walk-based overlay construction can be affected by node churn. Especially, the number of edges increases and the degree distribution is skewed. This inappropriate distortion can be exploited by malicious nodes. In this paper, we propose a modified random walk-based overlay construction supported by a logistic/trial based decision function to compensate the impact of node churn. Through event-driven simulations, we show that the decision function helps an overlay maintain the proper degree distribution, low diameter and low clustering coefficient with shorter random walks.