  The branching process (BP) approach has been successful in explaining the
avalanche dynamics in complex networks. However, its applications are mainly
focused on unipartite networks, in which all nodes are of the same type. Here,
motivated by a need to understand avalanche dynamics in metabolic networks, we
extend the BP approach to a particular bipartite network composed of Boolean
AND and OR logic gates. We reduce the bipartite network into a unipartite
network by integrating out OR gates, and obtain the effective branching ratio
for the remaining AND gates. Then the standard BP approach is applied to the
reduced network, and the avalanche size distribution is obtained. We test the
BP results with simulations on the model networks and two microbial metabolic
networks, demonstrating the usefulness of the BP approach.
