Analysis of critical and redundant nodes in controlling directed and undirected complex networks using dominating sets

Abstract

Recent studies have drawn attention to the problem of how complex networks can be controlled through a small number of controller nodes. Here, we develop an algorithmic procedure and mathematical tools to compute and evaluate the critical and redundant nodes in controlling directed and undirected scale-free networks using the minimum dominating set (MDS) approach. Because there are multiple MDS configurations that control the entire network, we can classify the nodes depending on the condition whether a node is part of all (critical), some but not all (intermittent), or does not participate in any (redundant) possible MDS. The presented mathematical analysis predicts the probability of finding a critical node in undirected scale-free networks with |$k^{-\gamma }$|⁠, where |$k$| is a node degree, as a function of the scaling exponent |$\gamma $| and its node degree. Critical nodes tend to have high degree and are more abundant in undirected scale-free networks with high |$\gamma $|⁠. In addition, analytical expressions of lower bounds for the number of critical nodes for both undirected and directed networks are also derived. By applying the MDS control model, we find that undirected networks can be controlled with relatively fewer nodes than those engaged in controlling directed networks. On the other hand, our computational experiments also show that the MDS is unimodal with varying the average degree |$\langle k \rangle $| in both directed and undirected networks. In particular, by increasing |$\langle k \rangle $|⁠, the fraction of nodes engaged in control becomes smaller, which highlights a centralized control mode. The analysis of a set of undirected and directed real-world networks confirms the findings shown in theoretical analysis and simulation experiments.