Key node selection in minimum-cost control of complex networks

Finding the key node set that is connected with a given number of external control sources for driving complex networks from initial state to any predefined state with minimum cost, known as minimum-cost control problem, is critically important but remains largely open. By defining an importance index for each node, we propose revisited projected gradient method extension (R-PGME) in Monte-Carlo scenario to determine key node set. It is found that the importance index of a node is strongly correlated to occurrence rate of that node to be selected as a key node in Monte-Carlo realizations for three elementary topologies, Erdős–Rényi and scale-free networks. We also discover the distribution patterns of key nodes when the control cost reaches its minimum. Specifically, the importance indices of all nodes in an elementary stem show a quasi-periodic distribution with high peak values in the beginning and end of a quasi-period while they approach to a uniform distribution in an elementary cycle. We further point out that an elementary dilation can be regarded as two elementary stems whose lengths are the closest, and the importance indices in each stem present similar distribution as in an elementary stem. Our results provide a better understanding and deep insight of locating the key nodes in different topologies with minimum control cost.