[Paper Review] Weighted Path homology of Weighted Digraphs and Persistence

Yong Lin, Shiquan Ren+2 more|arXiv (Cornell University)|Oct 22, 2019

Topological and Geometric Data Analysis15 references4 citations

TL;DR

This paper introduces weighted path homology for weighted digraphs by generalizing the path homology theory of Grigor'yan et al. using vertex weights in boundary operators. It establishes a persistent weighted path homology framework and proves a persistent Künneth-type formula for joins of weighted digraphs, showing that homology depends on weights when using integral coefficients.

ABSTRACT

In recent years, A. Grigor'yan, Y. Lin, Y. Muranov and S.T. Yau [6, 7, 8, 9] constructed a path homology theory for digraphs. Later, S. Chowdhury and F. Memoli [3] studied the persistent path homology for directed networks. In this paper, we generalize the path homology theory for digraphs and construct a weighted path homology for weighted digraphs. We study the persistent weighted path homology for weighted digraphs and detect the effects of the weights on the persistent weighted path homology. We prove a persistent version of a Kunneth-type formula for joins of weighted digraphs.

Motivation & Objective

• To generalize path homology theory to weighted digraphs by incorporating vertex weights into boundary operators.
• To develop a persistent weighted path homology framework for analyzing topological features in weighted directed networks.
• To investigate how vertex weights influence persistent homology in directed graphs.
• To establish a persistent version of the Künneth formula for the weighted path homology of joins of weighted digraphs.
• To compare persistent weighted path homology with standard persistent path homology under different coefficient rings.

Proposed method

• Define weighted boundary operators as alternating sums of face maps weighted by vertex weights: $\sum_{k\geq 0}(-1)^k w(i_k) d_k$.
• Construct a chain complex $\Omega_*^w(G)$ using formal linear combinations of paths with coefficients in a commutative ring $R$.
• Prove that the weighted boundary operators satisfy $\partial^2 = 0$, ensuring well-defined weighted path homology $H_*(G,w;R)$.
• Define morphisms of weighted digraphs and show that weighted path homology is functorial with respect to these maps.
• Introduce a persistence framework by considering sequences of weighted digraphs and morphisms between them.
• Establish a persistent Künneth-type formula for the join of two weighted digraphs using the tensor product of chain complexes and the algebraic Künneth formula.

Experimental results

Research questions

• RQ1 How does incorporating vertex weights into the boundary operator affect the path homology of a digraph?
• RQ2 What is the relationship between persistent weighted path homology and standard persistent path homology when using field coefficients?
• RQ3 How do vertex weights influence the persistent weighted path homology when using integral coefficients?
• RQ4 Can a Künneth-type formula be established for the weighted path homology of the join of two weighted digraphs?
• RQ5 Does the persistent weighted path homology of a join of weighted digraphs decompose into the tensor product of the homologies of the factors?

Key findings

• The weighted path homology is well-defined and functorial with respect to morphisms of weighted digraphs over a commutative ring with unit.
• With field coefficients, the persistent weighted path homology is isomorphic to the standard persistent path homology studied by Chowdhury and Mémoli.
• With integral coefficients, the persistent weighted path homology depends nontrivially on the weights, as demonstrated by explicit examples in Subsection 4.3.
• A persistent version of the Künneth-type formula holds for the join of two weighted digraphs, yielding a natural short exact sequence involving tensor products and Tor functors.
• The isomorphism between the chain complex of the join and the tensor product of the chain complexes of the factors is established via an explicit map $\mu_\#$, which commutes with boundary operators.
• The resulting short exact sequence in homology splits, confirming the algebraic structure of the persistent Künneth formula.