Recursive community structures

Triangle of triangles, and deeper. A level-k graph is three copies of level-(k−1) joined pairwise — a clean nested hierarchy. ← benchmark · gallery · tutorial · repo

These graphs are built to have hierarchy at every scale, which is exactly what structural entropy is for. Two things stand out. First, selib.optimal_2d recovers exactly 3^k−1 communities — the finest triangles (3, then 9, then 27) — with no number-of-clusters hint. Second, and more telling: as the graph grows 9 → 27 → 81 nodes, the flat 2D entropy climbs (1.90 → 2.41 → 2.95) but the encoding-tree entropy H^T barely moves (1.60 → 1.72 → 1.79). A self-similar hierarchy is almost free to encode hierarchically — the dendrogram on the right captures the full nesting, which a flat partition cannot. This is the clearest possible picture of why structural entropy is defined over a tree rather than a single cut.