Abstract
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.
Highlights
An understanding of the functional processes and mechanisms of living cells requires combined knowledge of multiple life molecules
Finding the Minimum Dominating Set (MDS) in a given network is known as a hard problem (NP-hard problem), the MDS can be found in reasonable CPU time in many practical cases by using Integer Linear Programming (ILP) if the network has a scale-free property
We formalised the MDS problem using Integer Linear Programming (ILP), in which xv = 1 and xv = 0 indicate that node ν is in an MDS and not in an MDS, respectively: Minimise subject to
Summary
An understanding of the functional processes and mechanisms of living cells requires combined knowledge of multiple life molecules. Liu et al employed the structural controllability framework for linear systems and showed that the minimum set of driver nodes can be found by computing the Maximum Matching (MM) in graph theory[3]. They showed that this approach requires a large number of driver nodes in scale-free networks[3]. Nacher and Akutsu proposed the Minimum Dominating Set (MDS) approach[4], where MDS is a well-known concept in graph theory They showed that the MDS approach needs a smaller fraction of nodes in scale-free networks. The results showed that the algorithm expanded the computable size to 65,000 nodes and increased the computational speed 176-fold
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