While not all may agree on the exact details of what a link farm is, the general idea is fairly straightforward. A link farm is an artificially link grouping which usually involves replicating the same set of links to multiple sites.
When search engines started to consider link popularity in their rankings, and continuing to the period when Google PageRank became important, search engine marketing specialists began a process of link development. Of course, the process of seeking out potential sites, finding contact information, and requesting a link is very time-intensive and has a low yield. So, clever individuals found a way to cut the effort: link farms.
A common format for link farms was a sort of cooperative, in which multiple webmasters agreed to link to each others sites. This could be as simple as a coordinator distributing a new links page when a member joined or departed. Some link farm concepts have been more complex, involving many pages of topically organized links, often including randomization and customization features to make the link pages seem unique.
In the early days, link farms were fairly successful in promoting their member sites. As the search engines that used link patterns in their algorithms became aware of link farms and their apparent impact on rankings, they all took steps to identify link farms and penalize their participants (or at least devalue the links).
New link farm schemes continue to crop up, but long-term oriented webmasters tend to avoid them due to the likely risk of discovery at some point and the additional risk of long-term penalty.
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