Rank high is an expression often used for measuring the popularity or importance of something or someone. There are various synonym terms used for rank high which include, ascend, excel, dominate, outdo, surpass, and elevate. Ascend means to move upwards towards a higher position or status. Excel refers to performing exceptionally better than others. Dominating means to have a commanding influence over something. Outdoing implies surpassing someone else's performance. Surpass is used when something exceeds another in quality or value. Lastly, elevate is used to describe something or someone who has risen to a higher level of status or importance. All these words are synonyms for "rank high" that clearly describe actions that implicate a superior status.