In this paper, without assuming balanced network topologies, we address the weighted average consensus problem for discrete‐time single‐integrator multi‐agent systems with logarithmic quantized information communication. By incorporating generalized quadratic Lyapunov function with the discrete‐time Bellman–Gronwall inequality, a new upper bound about the quantization precision parameter of the infinite‐level logarithmic quantizer is derived to design quantized protocol, under which agents in strongly connected directed networks can attain weighted average consensus. The obtained new upper bound clearly characterizes the intimate relation between the quantization precision parameter and the directed network topology. The proposed quantized protocol is particularly applicable to digital networks where balanced message passing among agents is not available.

In order to better grasp the research progress of physical training, explore the current research hotspots and trends, and provide a theoretical basis for subsequent research. The current physical fitness training was systematically analyzed through visual analysis software CiteSpace 6.1.R6 and VOSviewer 1.6.18 by confirming the subject search term TS = ("physical fitness training" or "physical training"). After data cleaning. The result was 733 relevant papers. The results showed that the overall trend of physical training was stable over the past 10 years. The visual graph analysis revealed that Orr, R.M. was the author with the highest volume of articles, reaching 12 authors Kraemer. W.J. was the one with the highest volume of articles per article reaching 19.5 times/article, and the connection between author groups needs to be further strengthened; the University of São Paulo institution had the highest number of articles, reaching 31, and from a regional perspective, the United States was in the lead; keyword clustering could be a total of classify 8 clusters, in which high-frequency keywords appear in the order of sports, performance, intensity, physical fitness, sports injuries, education, and women. The focus on women's issues is also increasing. From the epidemic vocabulary and time presentation graph, physical training is more prominent in the medical field, with keywords sports injury, epidemic, health, and physical training as the main keywords, and people in the post-epidemic era emphasize more on healthy lifestyles. The visualization analysis shows in detail the current research hotspots and development trends of physical fitness training, which show diversified development trends in the process of continuous expansion.