Target Control of Directed Networks based on Network Flow Problems

Target control of directed networks, which aims to control only a target subset instead of the entire set of nodes in large natural and technological networks, is an outstanding challenge faced in various real-world applications. We address one fundamental issue regarding this challenge, i.e., for a given target subset, how to allocate a minimum number of control sources, which provide input signals to the network nodes. This issue remains open in general networks with loops. We show that if this issue is relaxed to a path cover problem, then it can be further converted into a maximum network flow problem. A method called “maximum flow based target path-cover” (MFTP) with complexity <inline-formula><tex-math notation="LaTeX">$O(|V|^{1/2}|E|)$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$|V|$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$|E|$</tex-math></inline-formula>, respectively, denote the numbers of network nodes and edges, is proposed. It is also rigorously proven that MFTP provides the minimum number of control sources to control the target nodes in directed networks if the target control can be relaxed to the path cover problem, whether loops exist or not. We anticipate that this paper would serve wide applications in target control of real-life networks, as well as counter control of various complex systems.