Cryptographic enforcement schemes using Diffie-Hellman
1) The meaning of a poset is that there may also be members $x,y \in X$, where neither $x \leqslant y$ nor $x \geqslant y$. Now consider such incomparable $x$ and $y$ with $parent(x)=parent(y)=w \in X$ and furthermore $u \in X, y=parent(u), x \nleqslant u, x \ngeqslant u$

Now $k(x)=f(k(w))=k(y)$. Then also $k(u)=f(f(k(w))=f(k(x))$, even though $x$ is not a predecessor of $u$. This is clearly a security violation in almost any information-flow control policy.

2) you would need an additional hash function to return the result of the DH exponentiation back to the key-domain.

3) It is not a good idea to change identities while changing keys

4) You will need information on the DAG structure to define the actual function used, which is considered inefficient in HKAS

5) In general, it is not a good idea to have node keys computed from parent keys, instead other secret values are used, to enable efficient key-updates

6) It is more efficient in change management and initial deployment to have anything depend only on one predecessor rather on multiple ones. Remember, the nodes are security levels, not users or computers.