Atomic regressions for the map form of the cut distance

The harder direction of cutDist_eq_cutDistPullback applies Janson's Thm A.9 to the coupling, so the atomic cases to check are atomic couplings. The three checks below run the equivalence at a point-mass coupling, at finitely atomic ones, and at ones mixing an atomic with a continuous direction, and in each case they evaluate the map form rather than only instantiating the equivalence.

The values come from the coupling side, where they can be computed exactly: against a constant graphon, and against a graphon on a point mass, every coupling contributes the same cut norm (cutDist_const_right, cutDist_dirac_dirac). Each value is in general nonzero, so these checks also rule out the failure mode an atomic carrier invites: a map form whose index set is empty on atomic carriers — for instance one ranging over measure-preserving bijections with (I, volume), of which an atomic carrier has none — is the junk value 0 there, and would contradict them.

The finitely atomic and mixed checks both compare the complete graph K₂ on the uniform two-point carrier with the constant graphon 1/2. Their cut distance is 1/8: the difference kernel is 1/2 off the diagonal and -1/2 on it, the cut norm is attained on an off-diagonal cell of mass 1/4, and no rectangle does better (cutDist_finiteGraphGraphonOnFin_top_two_const_half).

These examples compare the map form with independently computed distances on carriers with different atomic structures. In each case the nonzero value witnesses that the map form includes the relevant pullbacks.

Main results

• The point-mass, finitely atomic, and mixed examples evaluate cutDistPullback using cutDist_dirac_dirac and cutDist_finiteGraphGraphonOnFin_top_two_const_half.

References

• S. Janson, Graphons, cut norm and distance, couplings and rearrangements, NYJM Monographs 4 (2013), Thm 6.9 and Thm A.9.