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

An Efficient Approach Toward the Asymptotic Analysis of Node-Based Recovery Algorithms in Compressed Sensing