Critical controllability analysis of directed biological networks using efficient graph reduction

Network science has recently integrated key concepts from control theory and has applied them to the analysis of the controllability of complex networks. One of the proposed frameworks uses the Minimum Dominating Set (MDS) approach, which has been successfully applied to the identification of cancer-related proteins and in analyses of large-scale undirected networks, such as proteome-wide protein interaction networks. However, many real systems are better represented by directed networks. Therefore, fast algorithms are required for the application of MDS to directed networks. Here, we propose an algorithm that utilises efficient graph reduction to identify critical control nodes in large-scale directed complex networks. The algorithm is 176-fold faster than existing methods and increases the computable network size to 65,000 nodes. We then applied the developed algorithm to metabolic pathways consisting of 70 plant species encompassing major plant lineages ranging from algae to angiosperms and to signalling pathways from C. elegans, D. melanogaster and H. sapiens. The analysis not only identified functional pathways enriched with critical control molecules but also showed that most control categories are largely conserved across evolutionary time, from green algae and early basal plants to modern angiosperm plant lineages.