  In our previous study, we showed that the branching process approximation is
useful for estimating metabolic robustness, measured using the impact degree.
By applying a theory of random family forests, we here extend the branching
process approximation to consider the knockout of {\it multiple} reactions,
inspired by the importance of multiple knockouts reported by recent
computational and experimental studies. In addition, we propose a better
definition of the number of offspring of each reaction node, allowing for an
improved estimation of the impact degree distribution obtained as a result of a
single knockout. Importantly, our proposed approach is also applicable to
multiple knockouts. The comparisons between theoretical predictions and
numerical results using real-world metabolic networks demonstrate the validity
of the modeling based on random family forests for estimating the impact degree
distributions resulting from the knockout of multiple reactions.
