Topological scale framework for hypergraphs

Abstract

In this paper, a new computational topological framework for hypergraph analysis and recognition is developed. “Topology provides scale” is the principle at the core of this set of algebraic topological tools, whose fundamental notion is that of a scale-space topological model ($s^2$-model). The scale of this parameterized sequence of algebraic hypergraphs, all having the same Euler-Poincaré characteristic than the original hypergraph G, is provided by its relational topology in terms of evolution of incidence or adjacency connectivity maps. Its algebraic homological counterpart is again an $s^2$-model, allowing the computation of new topological characteristics of G, which far exceeds current homological analytical techniques. Both scale-space algebraic dynamical systems are hypergraph isomorphic invariants. The hypergraph isomorphism problem is attacked here to demonstrate the power of the proposed framework, by proving the ability of $s^2$-models to differentiate challenging cases that are difficult or even infeasible for state-of-the-art practical polynomial solvers. The processing, analysis, classification and learning power of the $s^2$-model, at both combinatorial and algebraic levels, augurs positive prospects with respect to its application to physical, biological and social network analysis.