Two-row $W$-graphs in affine type $A$

For affine symmetric groups we construct finite $W$-graphs corresponding to two-row shapes, and prove their uniqueness. This gives the first non-trivial family of examples of finite $W$-graphs in an affine type. We compare our construction with quotients of periodic $W$-graphs defined by Lusztig. Under certain positivity assumption on the latter the two are shown to be isomorphic.