Is there any bounds for the norm of substochastic matrix? (But it's not doubly stochastic matrix, I mean only the row sum is less than 1, while the column sum may not.

A useful and easy to compute bound is given by the reasonably wellknown relation (see e.g., this Wikipedia section) \begin{equation*} \A\ \le \sqrt{\A\_\infty \A\_1} \end{equation*} between the spectral norm, and the induced $1$ and $\infty$ norms of an arbitrary matrix $A$. Corollary: If $A$ is elementwise nonnegative and rowstochastic, then $\A\ \le \sqrt{\A\_1}$. Note For some refinements of the first inequality mentioned above, please refer to this paper by V. Nikiforov. 

