Adversarial checks of strict graphon representatives #
The passage from a measurable function to an almost-everywhere class must discard null-set values, but must not discard values on atoms. These examples exercise both requirements.
On the unit interval, corrupt a graphon's zero row and zero column with the asymmetric values
4 and -6. The resulting measurable function represents the same class, although it is neither
symmetric nor range-bounded everywhere. Averaging and clamping repairs it: the origin becomes
1, and the rest of the zero row and column become 0. The repair preserves the class, every
homomorphism density, and the graphon-space point. In particular, a repaired constant graphon
need not be the same strict graphon. The representative bridge recovers the corrupted class and
its graphon-space point. On the uniform two-point atomic carrier, its returned representative
recovers the finite adjacency graphon pointwise; all four adjacency entries are also checked.
On a uniform two-point space, an asymmetric function cannot be represented by a graphon;
neither can the constant function 2 on a point mass. These checks distinguish almost-everywhere
constraints from constraints that could accidentally ignore positive-mass exceptional sets.
The examples use the strict-representative construction Graphon.clampSymm and the bridges
exists_graphon_repr and exists_graphon_repr_iff. The auxiliary functions are private; the
exported theorem Graphon.toAEEqFun_not_injective_unitInterval records why strict equality cannot
be recovered.
References #
- S. Janson, Graphons, cut norm and distance, couplings and rearrangements, NYJM Monographs 4 (2013), §6, for almost-everywhere identification of graphons.
Passing to the almost-everywhere class loses strict equality, already on the unit interval.
A null-set modification of the constant graphon 1/2 gives a different strict graphon with the
same class.