Target controllability with minimal mediators in complex biological networks

Controllability of a complex network system is related to finding a set of minimum number of nodes, known as drivers, controlling which allows having a full control on the dynamics of the network. For some applications, only a portion of the network is required to be controlled, for which target control has been proposed. Often, along the controlling route from driver nodes to target nodes, some mediators (intermediate nodes) are also unwillingly controlled, which might cause various side effects. In controlling cancerous cells, unwillingly controlling healthy cells, might result in weakening them, thus affecting the immune system against cancer. This manuscript proposes a suitable candidate solution to the problem of finding minimum number of driver nodes under minimal mediators. Although many others have attempted to develop algorithms to find minimum number of drivers for target control, the newly proposed algorithm is the first one that is capable of achieving this goal and at the same time, keeping the number of the mediators to a minimum. The proposed controllability condition, based on path lengths between node pairs, meets Kalman's controllability rank condition and can be applied on directed networks. Our results show that the path length is a major determinant of in properties of the target control under minimal mediators. As the average path length becomes larger, the ratio of drivers to target nodes decreases and the ratio of mediators to targets increases. The proposed methodology has potential applications in biological networks. The source code of the algorithm and the networks that have been used are available from the following link: https://github.com/LBBSoft/Target-Control-with-Minimal-Mediators.git