Measuring Structural Complexity with Combinatorial-Topological Entropy

We introduce Combinatorial-Topological Entropy (CTE), a structural measure quantifying the intrinsic complexity of combinatorial topologies, including simplicial complexes and hypergraphs. Unlike classical entropy, CTE does not depend on probability distributions but instead uses simplex dimensions, adjacency hierarchies, and connectivity patterns. We formalize a CTE incorporating parameters α and β to weight simplex size and adjacency influence. Using illustrative examples, including tetrahedra, hypergraphs, and higher-dimensional simplicial complexes, we demonstrate the measure’s sensitivity to structural features. Our results show CTE distinguishes between different combinatorial configurations, supporting its role as a structural invariant. Heatmaps visualize trends across α and β, demonstrating adjacency and size effects.