Consider the problem of distributed source coding in large networks with multiple terminals. In this case, the possibly different levels of side information at the terminals, together with the lack of network-source coding separation, obstructs the use of efficient decoding techniques which are based on structured codes (e.g., low-density). To alleviate this problem, we offer a scheme which is based on matrix sparsification at the terminals to achieve the effect of low-density source codes despite the network coding throughout the network. Via a novel relation between matrix sparsification and rate-distortion, we analyze the case in the limit of weak side information at the terminals.