Block averaging with a nonempty null cell #
On a three-point carrier, put mass 1/2 at each of 0 and 1 and no mass at 2.
The complete adjacency matrix has value 1 between distinct points, including between the
null atom and the positive atoms. Split off {2}, then refine the positive cell into singletons.
The coarse energy is 1/4, the fine energy is 1/2, and both the coarse approximation defect
and the refinement increment have squared L² seminorm 1/4.
These computed values test the zero convention for rectangle averages separately from their
weighted energy: averaging erases edges incident to 2 strictly, while the null cell contributes
nothing to the integrals. The defect identity l2sq_sub_stepGraphonAvg and the Pythagoras
increment graphonPartitionEnergy_increment are exercised on these partitions, together with
a cut witness and the iteration's part-count bound, starting from the coarse partition at a
tolerance that forces refinement and checking that the null singleton persists. Averaging over
singletons does not recover the original strict representative: the zero-mass atom's row and
column are replaced by
zero. The exported witness exists_partition_stepGraphonAvg_ne_self_bernoulliMeasure records this
failure for a partition into all three singletons.
References #
- L. Lovász, Large Networks and Graph Limits, AMS Colloquium Publications 60 (2012), §9.2.
On the three-point carrier with masses 1/2, 1/2, and 0, averaging over the
singleton partition can change the original strict graphon. The zero-mass atom's incident
edges are erased.