In the degree reduction step of Dinur's proof, the input graph $G$ is transformed into a graph $G'$ by replacing each vertex $v \in V(G)$ by a set of vertices, $cloud(v)$, such that $|cloud(v)| = degree_G(v)$, and imposing a degree d expander graph on $cloud(v)$ for all $v \in V(G)$. This makes $G'$ a d+1 regular graph, and the construction ensures that the gap reduces only by a constant factor. I was wondering what would happen if we impose a cycle on each cloud instead? I tried bounding the drop in the gap, but was not able to do so. So, does the proof break down at this step?
It is essential to the construction to have an expander between the copies of a vertex (the "cloud" of the vertex). Otherwise, you won't be able to argue that the adversary assigning values to the vertices is better off assigning those vertices the same value.
In particular, if instead of an expander you have a cycle, the prover can assign one half-cycle one value, and the other half-cycle another value. This way with good probability the verifier won't catch the inconsistency between the vertices, but you can't appeal to the soundness of the original graph to prove soundness for the new graph (it's like allowing the prover in the original graph to use two different assignments for the vertex).