Strong Structural Controllability of Boolean Networks: Polynomial-Time Criteria, Minimal Node Control, and Distributed Pinning Strategies

Abstract

In this paper, we initiate the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">strong structural controllability</i> of Boolean networks (BNs), coping with the trouble in identifying full nodal dynamics in practice, and utilize it to build up some existing results about BNs. The derived criteria for the strong structural controllability of BNs are polynomial-time checkable. As an interesting feature, controllability is shown to be equivalent to fixed-time controllability in this regard. We further consider the minimal strong structural controllability problem of BNs which, reduced from the minimal vertex cover problem of graphs, is turned out to be NP-hard. Proceeding forward, strong structural controllability provides useful insights to exclude the deficiencies of existing approaches. First, the network aggregation subject to the controllability of BNs is built by showing that the predecessor aggregated blocks can be dropped in turn if they are strongly structurally controllable. Second, the distributed pinning controllers are carried out to render arbitrary BN controllable through node-to-node message exchange, while traditional results only check the controllability with the preassigned controllers. Notably, the time complexity to design such controllers is only exponential with the maximal in-degree of pinned nodes rather than the node number of BNs. Third, in scenarios of BNs encountering probability disturbances, a basic theorem is put forward to elaborate the equivalence between several types of asymptotic stability. Together with this theorem, the criteria of strong structural controllability also facilitate the design of distributed controllers to asymptotically stabilize probabilistic BNs in probability.