  In this paper, we propose a general framework for the asymptotic analysis of
node-based verification-based algorithms. In our analysis we tend the signal
length $n$ to infinity. We also let the number of non-zero elements of the
signal $k$ scale linearly with $n$. Using the proposed framework, we study the
asymptotic behavior of the recovery algorithms over random sparse matrices
(graphs) in the context of compressive sensing. Our analysis shows that there
exists a success threshold on the density ratio $k/n$, before which the
recovery algorithms are successful, and beyond which they fail. This threshold
is a function of both the graph and the recovery algorithm. We also demonstrate
that there is a good agreement between the asymptotic behavior of recovery
algorithms and finite length simulations for moderately large values of $n$.
