What do you call words that share the same root (e.g., "network", "networks", "networking"). Also, does the shortest one of them have a specific name (e.g., "network")?
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I would call network a "stem", networks (noun or verb) an "inflected form", networking (participle) an "inflected form", and networking (noun) a "derived term".
"Root" is not the same as "stem" in my book; network has two roots, net and work.
A set of words that can be grouped together is called a word family. A word family can be form-based or meaning-based. The question describes a form-based word family and the shortest form is sometimes a root, however, I agree with RegDwight's answer that the word network is composed of two roots.