What are the characteristics of an informal empire, and how did America establish an informal empire in China in the early 1900s?

The difference between a formal and informal empire is, according to Gallagher and Robinson, "not one of fundamental nature but of degree." The term is usually applied to the British Empire; Gallagher and Robinson gave as an example the fact that India moved, over the period it was a colony, from an informal to a formal part of the empire, and then back to an informal part in its later years. An "informal empire," then, is not wholly distinct from a "formal empire" but indicates that the level of influence exerted by the mother country over this part of the empire is limited. Informal empire means that the mother country has successfully exerted political and commercial power over the country's territories, economy, people, and resources and is able to influence the political machinations of the subordinate country in pursuit of its own interests. However, there may not be any formal connection between the two countries, with the "conquered" country not bound to the "mother" country in any legal way.
Arguably, this is the relationship that existed between the United States and China in the early twentieth century. The US had begun to establish favorable relationships with the Chinese in the late nineteenth century. Attempts to expand into China and supply it with industrial goods were initially unsuccessful, but after the Boxer Rebellion, the US provided a significant number of troops to aid Chinese Christians who had come under attack. The US already had a significant number of ships and troops in the region due to their recent attempts to annex the Philippines; as such, the Boxer Rebellion, and Chinese Christians, became a focus of American attention. The result of this was that Americans began to send missionaries into China, rising to over 8,000 a year in the 1920s. A number of American universities, medical schools, and theology schools were opened in China, alongside a vast number of elementary and middle schools. While attempts to actually convert the Chinese to Christianity did not have the success hoped for, the vast provision of American-funded educational institutions in the country certainly had a marked effect on Chinese culture and created a bond between the Chinese and the USA, wherein American culture and values began to be impressed upon the Chinese from their youth. This is one of the hallmarks of informal empire, just as, in India, British values were imposed upon native students, or US values were imposed upon Native Americans in reservation schools.
The political control exerted by the US over China derived from the fact that the US was, at this time, as now, a major player on the world stage. Although there were numerous attempted insurrections against the Republic of China, with various governments ruling parts of the country, the US recognized only the Republic of China as the legitimate government. The result of this was that the policies of the Republic depended, to a certain extent, upon what the USA favored, as it knew it depended upon American recognition for the standing it enjoyed on the world stage. Consequently, attempts to oppose the Republic of China were generally unsuccessful, and the Republic of China itself continued to be recognized by the United States until thirty years after the founding of the People's Republic of China—which subsequently presented huge issues for this regime in terms of its accurate self-representation in global terms.
https://web.archive.org/web/20081221073828/http://www.mtholyoke.edu/acad/intrel/ipe/gallagher.htm

https://www.heritage.org/asia/report/the-complicated-history-us-relations-china