Rough equivalence of integer lattices? 
Above is shown the meaning of having two resistive networks being roughly embedded.  Roughly equivalent means there is rough embeddings both ways.  I wish to show that this distinguishes $\mathbb{Z}^d$ as $d$ varies, with the choice of resistance, conductance both being $1$ on every edge with the standard lattice structure.  My work so far is that $d<3$ and $d\ge3$ are distinguished by the network walk being transient or not.
Additional related observations I have made that do not appear to solve the problem in any direct way is that all "lattice structures of a given dimension" are roughly equivalent. By lattice of dimension $d$ I mean you duplicate $d$ dimensional hypercubes where you can add any of the diagonals that lie within the hypercube.  Thus, I am aiming for the claim that if this is what we take to mean a lattice, then all lattices are distinguished by the invariant that is the rough equivalence class.
 A: 
Disclaimer (Jun 21'16): This answer was recently downvoted. Needless to say, it is perfectly correct, and it answers the question. The downvote might be due to extra-mathematical reasons. Happy reading!

First, note that to use recurrence/transience to show that no rough embedding exists between two given networks is to be going backwards since rough embeddings are used to prove that a given (complicated) network is recurrent or transient by comparison with another (simpler) one. 
Next, note that, for unit resistances, the existence of a rough embedding of a network $G'$ into a network $G$implies a control of the volume of the balls of $G'$ by the volume of the balls of $G$, in the following sense:

For every $x$ in $G$ and $x'$ in $G'$, the sets $B(x,n)$ of the edges in $G$ at distance at most $n$ from $x$ and $B'(x',n)$ of the edges in $G'$ at distance at most $n$ from $x'$ are such that, for every $n$ large enough, $|B'(x',n)|\leqslant c\cdot|B(x,an+k)|$, for some finite $c$, $a$ and $k$. 

In $\mathbb Z^d$, $|B(x,n)|\propto n^d$ hence the remark above forbids the existence of any rough embedding of $\mathbb Z^b$ into $\mathbb Z^d$ when $b\gt d$.
