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Link popularity is a measure of the quantity and quality of other websites that link to a specific site on the World Wide Web. It is an example of the move by search engines towards off-the-page-criteria to determine quality content. In theory, off-the-page-criteria adds the aspect of impartiality to search engine rankings.
Link popularity plays an important role in the visibility of a web site among the top of the search results. Indeed, some search engines require at least one or more links coming to a web site, otherwise they will drop it from their index.
Google acknowledges link popularity contributes to better search visibility. Read here to read more.