[Paper Review] Hyperwalk Formulae for Even and Odd Laplacians in Finite CW-Hypergraphs

Iván Contreras, Sarah Loeb+1 more|arXiv (Cornell University)|Aug 26, 2017

Tensor decomposition and applications5 references3 citations

TL;DR

This paper introduces hyperwalk formulae for even and odd Laplacians in finite CW-hypergraphs, extending discrete quantum mechanical concepts to hypergraphs via oriented incidence matrices. It establishes that powers of the even Laplacian count signed hyperwalks between vertices, while powers of the odd Laplacian count signed edge-hyperwalks between cells, with explicit combinatorial interpretations grounded in CW-complex structure and orientation.

ABSTRACT

In this note we provide a combinatorial interpretation for the powers of the hypergraph Laplacians. Our motivation comes from the discrete formulation of quantum mechanics and thermodynamics in the case of finite graphs, which suggest a natural extension to simplicial and CW-complexes. With this motivation, we also define generalizations of the odd Laplacian which is specific to hypergraphs arising from CW-complexes. We then provide a combinatorial interpretation for the powers of these Laplacians.

Motivation & Objective

• To generalize discrete quantum mechanical formulations—previously applied to graphs—onto finite CW-complexes and their associated hypergraphs.
• To define a hypergraph Laplacian framework that incorporates orientation via the cellular structure of CW-complexes.
• To provide a combinatorial interpretation of powers of the even and odd Laplacians in terms of signed hyperwalks on hypergraphs.
• To extend the concept of generalized walks from graphs to hypergraphs, particularly in the context of supersymmetric quantum mechanics on CW-hypergraphs.

Proposed method

• Define the even Laplacian as $\Delta^+ = II^T$ and the odd Laplacian as $\Delta^- = I^T I$, where $I$ is the incidence matrix of a hypergraph.
• Introduce hyperwalks as sequences alternating between vertices and edges, and edge-hyperwalks as sequences alternating between edges and vertices.
• Leverage orientation on cells in a CW-complex to assign signs to hyperwalks via signed incidence matrices.
• Use induction on the exponent $k$ to prove that $(\Delta^+)^k(i,j)$ counts the number of signed hyperwalks of length $k$ from vertex $v_i$ to $v_j$.
• Prove that $(\Delta_d^-)^k(i,j)$ computes the sum of signs of $(d+1,d)$-hyperwalks of length $k$ from $e_i^{d+1}$ to $e_j^{d+1}$, using distributive properties of signed walks.
• Apply the framework to CW-hypergraphs, where cell orientations induce consistent sign assignments for walks, enabling a partition function-like interpretation in quantum mechanical settings.

Experimental results

Research questions

• RQ1 How can the powers of the even Laplacian in a hypergraph be interpreted combinatorially in terms of walks?
• RQ2 What is the role of orientation in defining signed walks on hypergraphs derived from CW-complexes?
• RQ3 How do the powers of the odd Laplacian relate to the signed enumeration of edge-hyperwalks in CW-hypergraphs?
• RQ4 Can the discrete partition function formalism of supersymmetric quantum mechanics on graphs be extended to CW-complexes via hypergraph Laplacians?
• RQ5 What is the combinatorial significance of the sum of signs of hyperwalks in higher-dimensional hypergraphs?

Key findings

• The $k$-th power of the even Laplacian $\Delta^+$ counts the number of hyperwalks of length $k$ from vertex $v_i$ to $v_j$ in a hypergraph.
• For the CW-hypergraph in Figure 1(b), the sum of signs of $(1,2)$-hyperwalks of length 4 from $e_1^1$ to $e_6^1$ is zero, as computed by Theorem 3.6.
• The sum of signs of $(2,1)$-hyperwalks of length 2 from $e_1^2$ to $e_3^2$ is $+1$, as confirmed by Theorem 3.7.
• In the hypergraph of Figure 1(a), there are 5,886 hyperwalks of length 4 from $v_1$ to $v_3$, as given by Theorem 2.3.
• There are 384 edge-hyperwalks of length 3 from $e_7$ to $e_9$ in the same hypergraph, as computed via Theorem 2.4.
• The sign of $e_6^1$ in $e_1^2$ is negative in the CW-hypergraph, illustrating orientation-dependent sign assignment in the odd Laplacian framework.