Consider a collection $\mathcal R$ of random variables $X_\alpha$ on the same sample space. Let $\sim$ be a relation on $\mathcal R$, defined by $X_\alpha\sim X_\beta$ if they are dependent.
What can we say about this relation?
More specifically, I might concentrate on the rolling dice (twice) example. Being a discrete case, $X$ and $Y$ is independent if and only if $$P(X=x,Y=y)=P(X=x)P(Y=y)$$ for all $(x,y)\in\{1,2,3,4,5,6\}^2$. Thus $\sim$ is fairly symmetric.
Let $X$ be the first number and $Y$ the second. And let $Z=X+Y$. Then $X\not\sim Y$, $X\sim Z$, and $Y\sim Z$, since, for example, $$P(X=3,Z=5)=\frac1{36}\neq\frac16\times\frac4{36}=P(X=3)P(Z=5).$$ So $\sim$ is not transitive.
How about the reflectivity? And is it meaningful to think of this kind of relation?