A relation $R$ is called 'transitive' when it upholds
$$ xRy ~\text{ and }~ yRz \implies xRz. $$
Is there a name for a relation that upholds the somewhat opposite property:
$$ xRz \implies \exists y ~\text{ such that }~ xRy \text{ and } yRz \enspace ? $$
Short Answer: No.
Long Answer: So, I wouldn't exactly call this the opposite of the transitivity relation. Surely, you can have both properties hold. If I were going to give this property a name, one might call it a density property. If $R$ is the ordering on $\mathbb{Q}$, this is property says that is $a < b$ then $\exists c$ such that $a < c < b$.