Could you expand on this to say what X and Y are (or might be) or at least supply a context (for example, in economics 'inelastic' might possibly be used here instead of 'robust')? And does it matter whether the change in X is in the same direction as the change in Y (eg if Y increases, does X also increase even though ever so slightly?). Without knowing anything more, the best I can suggest is:
Changes in Y have only a marginal impact on X.
or
Change in Y have a barely discernible impact on X.