Controlling complex networks is of paramount importance in science and engineering. Despite the recent development of structural controllability theory, we continue to lack a framework to control undirected complex networks, especially given link weights. Here we introduce an exact controllability paradigm based on the maximum multiplicity to identify the minimum set of driver nodes required to achieve full control of networks with arbitrary structures and link-weight distributions. The framework reproduces the structural controllability of directed networks characterized by structural matrices. We explore the controllability of a large number of real and model networks, finding that dense networks with identical weights are difficult to be controlled. An efficient and accurate tool is offered to assess the controllability of large sparse and dense networks. The exact controllability framework enables a comprehensive understanding of the impact of network properties on controllability, a fundamental problem towards our ultimate control of complex systems.

We develop a general framework to analyze the controllability of multiplex networks using multiple-relation networks and multiple-layer networks with interlayer couplings as two classes of prototypical systems. In the former, networks associated with different physical variables share the same set of nodes and in the latter, diffusion processes take place. We find that, for a multiplerelation network, a layer exists that dominantly determines the controllability of the whole network and, for a multiple-layer network, a small fraction of the interconnections can enhance the controllability remarkably. Our theory is generally applicable to other types of multiplex networks as well, leading to significant insights into the control of complex network systems with diverse structures and interacting patterns.

Background : Wenzhou has achieved great progress in the prevention and control of the growing coronavirus disease 2019 (COVID-19) pandemic, and traditional Chinese medicine (TCM) has played an indispensable role in this fight. This study aimed to investigate the efficacy of Maxingshigan-Weijing decoction (MWD) in treating infected patients. Methods : This study was an open-label randomized controlled trial. Inpatients with mild or moderate symptoms caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection were randomly treated with routine supportive care alone or a combination of routine supportive care and MWD. The primary outcome was the rate of symptom (fever, fatigue, cough and difficulty breathing) recovery. Results : Fifty-nine inpatients were enrolled, of whom 29 received routine supportive care alone (control group) and 30 received combination therapy (treatment group). The rate of symptom recovery was significantly higher in the treatment group than in the control group. The time to recovery of fever (3 vs. 7 days), fatigue (9 vs. 12 days), coughing (9 vs. 14 days) and difficulty breathing (4.5 vs. 9.5 days) was also significantly shorter in the treatment group (all p < 0.001). The syndrome score was lower after MWD treatment. However, neither group differed in the viral assay findings, hospitalization days, medication time or the rate of conversion to severe cases. Conclusions : MWD increased the rate of symptom recovery and shortened the time to recovery of clinical symptoms without deterioration to death or critical care. These findings may provide opportunities for the use of complementary medicine in treating this infection. Clinical trial registration : Chinese Clinical Trial Registry, ChiCTR2000030759.

Fractal networks are ubiquitous in nature, ranging from river networks to vascular networks. The ultimate goal of exploring these fractal networked systems lies in controlling the dynamical processes that take place on them. We offer analytical results to exactly understand our ability to control the dynamics of regular fractal networks in terms of identifying the minimum number of driver nodes that are required to achieve full control. According to the exact controllability theory, the controllability of an undirected network is completely determined by the eigenvalue spectrum of the coupling matrix that captures the network structure. The selfsimilarity in the fractal networks allows us to solve exactly the eigenvalue spectrum from the growth unit and the steps of the iterations, enabling an analytical quantification of the controllability of the fractal networks via the eigenvalue spectrum. We validate our exact analytical results in three typical regular fractal networks. Our results have implications for the control of many real networked systems that have fractal characteristics.