This paper is concerned with the minimum variance filtering problem for a class of time-varying systems with both additive and multiplicative stochastic noises through a sensor network with a given topology. The measurements collected via the sensor network are subject to stochastic sensor gain degradation, and the gain degradation phenomenon for each individual sensor occurs in a random way governed by a random variable distributed over the interval [0, 1]. The purpose of the addressed problem is to design a distributed filter for each sensor such that the overall estimation error variance is minimized at each time step via a novel recursive algorithm. By solving a set of Riccati-like matrix equations, the parameters of the desired filters are calculated recursively. The performance of the designed filters is analyzed in terms of the boundedness and monotonicity. Specifically, sufficient conditions are obtained under which the estimation error is exponentially bounded in mean square. Moreover, the monotonicity property for the error variance with respect to the sensor gain degradation is thoroughly discussed. Numerical simulations are exploited to illustrate the effectiveness of the proposed filtering algorithm and the performance of the developed filter. ; This work was supported by the National Natural Science Foundation of China under Grants 61490701, 61210012, 61290324, 61473163, 61522309, and 61273156; in part by the Tsinghua University Initiative Scientific Research Program; and in part by the Jiangsu Provincial Key Laboratory of E-business at Nanjing University of Finance and Economics of China under Grant JSEB201301; and in part by the Research Fund for the Taishan Scholar Project of Shandong Province of China.