In this paper, a robust diffusion average-estimate normalized least mean square (NLMS) algorithm is proposed to tackle the impulsive noise in adjacent node communication links over distributed networks. Compared with the diffusion NLMS algorithm, the key point of the new algorithm is that it introduces two additional steps: robust detection and average estimation, thus greatly reducing the negative effects of the impulsive link noise. In addition, the mean stability of the proposed robust diffusion average-estimate NLMS algorithm is analyzed. Finally, Monte-Carlo simulation results demonstrate the effectiveness of the proposed robust diffusion algorithm.