[OK] G4 n=32
[OK] G4 m=116
[OK] Planted partition covers all nodes
[OK] Planted communities disjoint
[OK] Manual Q matches NetworkX
[OK] Q four about 0.7155
[OK] Q four > Q two
[OK] Q one =0
[OK] Singleton Q negative
[OK] Greedy finds four groups
[OK] Greedy sizes all 8
[OK] Greedy Q equals planted
[OK] Rand index planted greedy 1
[OK] Modularity matrix row sums zero
[OK] Modularity matrix symmetric
[OK] Null fixed Q centered near zero
[OK] Null optimized Q positive
[OK] Observed Q above all null optimized
[OK] Empirical p optimized small
[OK] G30 m=330
[OK] Q30 formula separate
[OK] Q30 formula pairs
[OK] Pair offsets degenerate
[OK] Resolution limit pair better for r30
[OK] Analytic threshold sign r20 negative
[OK] Analytic threshold sign r24 positive
[OK] Greedy30 merges some cliques
[OK] Greedy30 Q >= separate
[OK] Gamma one manual Q standard
[OK] Low gamma favors larger groups relative
[OK] High gamma penalizes large communities more
