The branching process (BP) approach has been successful in understanding the avalanche dynamics in complex networks. However, its applications are mainly focused on unipartite networks in which nodes are all of the same type. Here, we develop the BP approach for a particular bipartite Boolean network , composed of logic OR and AND gates, which is motivated to understand the avalanche dynamics in metabolic networks . We reduce the bipartite to a unipartite network by renormalizing the OR gates, and obtain an effective branching ratio for the  AND gates. Then the standard BP approach is applied to the reduced unipartite network, and the avalanche size distribution is obtained. We also test the analytic result with simulations on a couple of metabolic networksin real world. They are in reasonable agreement with each other .
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 .
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