Degree reduction step in Dinur's proof of the PCP theorem
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).