  This paper presents an efficient algorithm for robust network reconstruction
of Linear Time-Invariant (LTI) systems in the presence of noise, estimation
errors and unmodelled nonlinearities. The method here builds on previous work
on robust reconstruction to provide a practical implementation with polynomial
computational complexity. Following the same experimental protocol, the
algorithm obtains a set of structurally-related candidate solutions spanning
every level of sparsity. We prove the existence of a magnitude bound on the
noise, which if satisfied, guarantees that one of these structures is the
correct solution. A problem-specific model-selection procedure then selects a
single solution from this set and provides a measure of confidence in that
solution. Extensive simulations quantify the expected performance for different
levels of noise and show that significantly more noise can be tolerated in
comparison to the original method.
