This paper shows the potential and key enabling mechanisms for tunable sparse network coding, a scheme in which the density of network coded packets varies during a transmission session. At the beginning of a transmission session, sparsely coded packets are transmitted, which benefits decoding complexity. As the transmission continues and the receivers have accumulated coded packets, the coding density is increased. We propose a family of tunable sparse network codes (TSNCs) for multicast erasure networks with a controllable trade-off between completion time performance to decoding complexity. Coding density tuning can be performed by designing time-dependent coding matrices. In multicast networks, this tuning can be performed within the network by designing time-dependent pre-coding and network coding matrices with mild conditions on the network structure for specific densities. We present a mechanism to perform efficient Gaussian elimination over sparse matrices going beyond belief propagation but maintaining low decoding complexity. Supporting implementation results are provided showing the trade-off between decoding complexity and completion time.