In this paper, we study joint network coding and distributed source coding of inter-node dependent messages, with the perspective of compressed sensing. Specifically, the theoretical guarantees for robust $\ell_1$-min recovery of an under-determined set of linear network coded sparse messages are investigated. We discuss the guarantees for $\ell_1$-min decoding of quantized network coded messages, using the proposed local network coding coefficients in \cite{naba}, based on Restricted Isometry Property (RIP) of the resulting measurement matrix. Moreover, the relation between tail probability of $\ell_2$-norms and satisfaction of RIP is derived and used to compare our designed measurement matrix, with i.i.d. Gaussian measurement matrix. Finally, we present our numerical evaluations, which shows that the proposed design of network coding coefficients result in a measurement matrix with an RIP behavior, similar to that of i.i.d. Gaussian matrix.

6 pages