The study of network structure has uncovered signatures of the organization of complex systems. However, there is also a need to understand how to control them; for example, identifying strategies to revert a diseased cell to a healthy state, or a mature cell to a pluripotent state. Two recent methodologies suggest that the controllability of complex systems can be predicted solely from the graph of interactions between variables, without considering their dynamics: structural controllability and minimum dominating sets. We demonstrate that such structure-only methods fail to characterize controllability when dynamics are introduced. We study Boolean network ensembles of network motifs as well as three models of biochemical regulation: the segment polarity network in Drosophila melanogaster, the cell cycle of budding yeast Saccharomyces cerevisiae, and the floral organ arrangement in Arabidopsis thaliana. We demonstrate that structure-only methods both undershoot and overshoot the number and which sets of critical variables best control the dynamics of these models, highlighting the importance of the actual system dynamics in determining control. Our analysis further shows that the logic of automata transition functions, namely how canalizing they are, plays an important role in the extent to which structure predicts dynamics. Complex systems are typically understood as large nonlinear systems. Their organization and behavior can be modeled by representations such as graphs and collections of automata. Graphs are useful to capture the structure of interactions between variables: the static organization of complex systems. However, nodes representing variables in graphs lack intrinsic dynamics. The simplest way to study nonlinear dynamics is to allow network nodes to have discrete states and update them with automata; for instance, Boolean Networks (BNs) are canonical models of complex systems which exhibit a wide range of interesting behaviors 1 . The study of network structure has uncovered several organizing principles of complex systems -such as scale-free networks and community structure -and how they constrain system behavior, without explicit dynamical rules for node variables 2 . There is, however, a need to control complex systems, in addition to characterizing their organization. This is particularly true in systems biology and medicine, where increasingly accurate models of biochemical regulation have been produced [3] [4] [5] [6] . More than understanding the organization of biochemical regulation, we need to derive control strategies that allow us, for instance, to revert a mutant cell to a wild-type state 7 , or a mature cell to a pluripotent state 8 . While the identification of such control strategies occurs for a given model, not the real system, predictions from control theory can be used for model verification and thus also aid the separate question of the accuracy of that model in predicting the real system. Network structure has been reported to predict properties of dynamics, such as the synchronization of connected limit-cycle oscillators 9 , or the likelihood of robust attractors 10 . On the other hand, there are important system attributes which depend on dynamical characteristics of variables and their interactions; e.g. the critical transition between ordered and chaotic dynamics in BNs depends both on structural (mean connectivity) and dynamical properties of nodes (bias and canalization) [11] [12] [13] [14] . Indeed, we already know that such dynamical properties strongly impact the stability, robustness, and controllability of existing models of gene regulation and biochemical signaling in a number of organisms 7, [15] [16] [17] [18] . Therefore, a question of central importance remains: How well does network structure predict the dynamics of the underlying complex system, especially from the viewpoint of control? Recently, two related methodologies were used to predict the controllability of complex networks based solely on network structure without consideration of the dynamical properties of variables: structural controllability (SC) 19, 20 and minimum dominating set (MDS) 21, 22 . Both techniques reduce dynamical systems to graphs where edges denote an interaction between a pair of variables. Using only graph connectivity, the goal is to identify a minimal set of driver variables (a.k.a. driver nodes) which can fully control system dynamics 23 . SC assumes that, in the absence of cycles, a variable can control at most one of its neighbors in the structural interaction graph 19, 20 . The influence from an intervention on a node then propagates along a backbone of directed paths, where the number of necessary paths to cover the network dictates the minimum set of driver variables (see Supplemental Material, SM). Cycles are considered to be self-regulatory and do not require an external control signal. SC has become an influential method, having been used to suggest that biological systems are harder to control and have appreciably different control profiles than social or technological systems 24, 25 . The methodology has also been used to identify key banks in interbank lending networks 26 , and to relate circular network motifs to control in transcription regulatory networks 27 . However, despite its successful characterization of observability (a dual notion to controllability) in several nonlinear dynamical systems 28 , SC's application to models of biological and social systems has been heavily critiqued due to its stringent assumptions [29] [30] [31] . MDS starts from the different assumption that each node can influence all of its neighbors simultaneously, but this signal cannot propagate any further. Driver variables are then identified by the minimal set such that every variable is separated by at most one interaction 21, 22 . It has been used to identify control variables in protein interaction networks 32 and characterize how disease genes perturb the Human regulatory network 33 . Because both MDS and SC use only the interaction graph of complex systems, unless otherwise specified, we use structural control to refer to both methods. Since these methods are increasingly used in a variety of scientific domains, it is important to study how much network structure predicts the controllability of realistic, nonlinear dynamical systems. Here, we explore this problem using ensembles of BNs. These canonical models of complex systems are defined by a network of interconnected automata (the structure), and exhibit a wide range of dynamical behaviors 1 . They have been used to model biochemical regulation in organisms, where dynamical attractors represent cell types, disease and healthy states 6, 34 . It is well known that when the set of system variables is large, enumeration of the state-spaces of BNs becomes difficult, making the control problem for general deterministic BNs computationally intractable (NP-hard) 35 . However, for small systems we can fully enumerate the state-space and compute the actual controllability (as measured by three proposed measures of controllability) for parameterized ensembles of BNs. Our analysis is not meant to introduce alternative techniques to uncover control variables in BNs, since methods based on system dynamics already exist 7, 8, [36] [37] [38] [39] [40] [41] [42] . The goal is to quantify the discrepancy between control as uncovered by approximate methods that use structure alone, from how actual control unfolds in BNs. Additionally, we characterize critical variables for the control of three models of biochemical regulation: the single-cell segment polarity network in Drosophila melanogaster, the eukaryotic cell cycle of budding yeast Saccharomyces cerevisiae, and the floral organ arrangement in the flowering plant Arabidopsis thaliana. Our results demonstrate that network structure is not sufficient to characterize the controllability of complex systems; predictions based on structural control can both under-and over-estimate the number and set of necessary driver variables. Therefore, previous assertions about the controllability of biochemical systems reached from analyses based on structural control methods do not offer a realistic portrayal of control 24, 25 .
Gates et al. REF showed that such an approach is inevitable for the identification of the correct parameters and control strategies, in that, focussing only on a single property (either structural or dynamic) might lead to both their overestimation or underestimation.
Control of complex networks requires both structure and dynamics