  This work proposes diffusion normalized least mean M-estimate algorithm based
on the modified Huber function, which can equip distributed networks with
robust learning capability in the presence of impulsive interference. In order
to exploit the system's underlying sparsity to further improve the learning
performance, a sparse-aware variant is also developed by incorporating the
$l_0$-norm of the estimates into the update process. We then analyze the
transient, steady-state and stability behaviors of the algorithms in a unified
framework. In particular, we present an analytical method that is simpler than
conventional approaches to deal with the score function since it removes the
requirements of integrals and Price's theorem. Simulations in various impulsive
noise scenarios show that the proposed algorithms are superior to some existing
diffusion algorithms and the theoretical results are verifiable.
