Star-collision in random hypergraphs

TL;DR

This paper investigates star-based symmetries in random hypergraphs, showing that nontrivial units vanish with high probability as the number of vertices increases, simplifying the spectral analysis of related matrices.

Contribution

It demonstrates that star collisions in random hypergraphs lead to the asymptotic triviality of star-based symmetries, clarifying their irrelevance in large random systems.

Findings

Nontrivial units disappear with high probability in certain regimes.

Star-dependent matrices become asymptotically trivial.

Spectral properties are governed by a reduced quotient object.

Abstract

We study star-based symmetries in uniform hypergraphs and their consequences for matrices whose entries depend only on vertex stars. Such matrices admit a deterministic decomposition into a global component and a local component supported on equivalence classes of vertices with identical stars, known as units. While nontrivial units may exist at finite size in hypergraphs of uniformity greater than two, their persistence in random settings has remained unclear. We analyze star collisions in random $k$-uniform hypergraphs and show that, in some particular regimes, nontrivial units disappear with high probability as the number of vertices grows. As a consequence, star-dependent matrices exhibit asymptotically trivial local structure, and their spectral behavior, invariant subspaces, and associated linear dynamics are governed by a reduced quotient object obtained by contracting vertex stars. These results identify star collisions as a finite-size phenomenon in random hypergraphs and clarify the asymptotic irrelevance of star-based symmetries for operator behavior in large random systems in particular regimes.