This paper addresses the problem of distributed in-network estimation for a vector of interest, which is sparse in nature. To exploit the underlying sparsity of the considered vector, the ℓ 1 and ℓ 0 norms are incorporated into the quadratic cost function of the standard distributed incremental least-mean-square (DILMS) algorithm, and some sparse DILMS (Sp-DILMS) algorithms are proposed correspondingly. The performances of the proposed Sp-DILMS algorithms in the mean and mean-square derivation are analyzed. Mathematical analyses show that the Sp-DILMS outperforms the DILMS, if a suitable intensity of the zero-point attractor is selected. Considering that such intensity may not be easily determined in real cases, a new adaptive strategy is designed for its selection. Its effectiveness is verified by both theoretical analysis and numerical simulations. Even though the criterion for intensity selection is derived from the case that the observations are white and Gaussian, simulation results show that it still provides an empirical good choice if the observations are correlated regression vectors.