Narrow Results

We study the minimum number of driver nodes control of which leads a Boolean network (BN) from an initial state to a target state in a specified number of time steps. We show that the problem is NP-hard and present an integer linear programming-based method that solves the problem exactly. We mathematically analyze the average size of the minimum set of driver nodes for random Boolean networks with bounded in-degree and with a small number of time steps. The results of computational experiments using randomly generated BNs show good agreements with theoretical analyses. A further examination in realistic BNs demonstrates the efficiency and generality of our theoretical analyses.

Large complex dynamical systems behave in ways that reflect their structure. There are many applications where we need to control these systems by bringing them from their current state to a desired state. Affecting the state of these systems is done by communications with its key elements, called driver nodes, in reference to their representation as a network of nodes. Over the past decades, much focus has been paid on analytical approaches to derive optimal control configurations based on the concept of Minimum Driver node Sets (MDSs) for directed complex networks. However, the underlying control mechanisms of many complex systems rely on quickly controlling a major subspace of a system. In this work, we ask how complex networks behave if driver nodes are randomly selected? We seek to understand and employ the statistical characteristics of MDSs to randomly select driver nodes and analyze the controllability properties of complex network. We propose an algorithm to build Random Driver node Sets (RDSs) and analyze their controllable subspace, the minimum time needed to control, and the cardinality of RDSs. Through our evaluations on a variety of synthetic and real-world networks, we show RDSs can quickly and effectively control a major subspace of networks.

We propose a new measure to quantify the impact of a node $i$ in controlling a directed network. This measure, called “control contribution” ${{𝒞}}_i$, combines the probability for node $i$ to appear in a set of driver nodes and the probability for other nodes to be controlled by $i$. To calculate ${{𝒞}}_i$, we propose an optimization method based on random samples of minimum sets of drivers. Using real-world and synthetic networks, we find very broad distributions of $C_i$. Ranking nodes according to their $C_i$ values allows us to identify the top driver nodes that can control most of the network. We show that this ranking is superior to rankings based on other control-based measures. We find that control contribution indeed contains new information that cannot be traced back to degree, control capacity or control range of a node.