Robust adaptive estimation of unknown parameter has been an important issue in recent years for reliable operation in the distributed networks. The conventional adaptive estimation algorithms that rely on mean square error (MSE) criterion exhibit good performance in the presence of Gaussian noise, but their performance drastically decreases under impulsive noise. In this paper, we propose a robust adaptive estimation algorithm for networks with cyclic cooperation. We model the impulsive noise as the realization of alpha-stable distribution. Here, we move beyond MSE criterion and define the estimation problem in terms of a modified cost function which exploits higher order moments of the error. To derive a distributed and adaptive solution, we first recast the problem as an equivalent form amenable to distributed implementation. Then, we resort to the steepest-descent and statistical approximation to obtain the proposed algorithm. We present some simulations results which reveal the superior performance of the proposed algorithm than the incremental least mean square (ILMS) algorithm in impulsive noise environments.