# Weighted Path homology of Weighted Digraphs and Persistence

@article{Lin2019WeightedPH, title={Weighted Path homology of Weighted Digraphs and Persistence}, author={Yong Lin and Shiquan Ren and Chong Wang and Jie Wu}, journal={arXiv: Algebraic Topology}, year={2019} }

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…

## 5 Citations

### Grounded persistent path homology: a stable, topological descriptor for weighted digraphs

- Mathematics
- 2022

Weighted digraphs are used to model a variety of natural systems and can exhibit interesting structure across a range of scales. In order to understand and compare these systems, we require stable,…

### Path homologies of motifs and temporal network representations

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An algorithm for path homology is presented that combines efficient pruning and indexing techniques and using it to topologically analyze a variety of real-world complex temporal networks and concludes that path homologies provides insight into temporal network structure, and in turn, emergent structures in temporal networks provide us with new subgraphs having interesting path Homology.

### Persistent path Laplacian

- MathematicsFoundations of Data Science
- 2022

Path homology proposed by S.-T.Yau and his co-workers provides a new mathematical model for directed graphs and networks. Persistent path homology (PPH) extends the path homology with filtration to…

### Path homology and temporal networks

- MathematicsCOMPLEX NETWORKS
- 2020

It is concluded that path homology can provide insight into temporal network structure and vice versa.

### What are higher-order networks?

- Computer ScienceArXiv
- 2021

The goals are to clarify (i) what higher-order networks are, (ii) why these are interesting objects of study, and (iii) how they can be used in applications.

## References

SHOWING 1-10 OF 18 REFERENCES

### Further Properties and Applications of Weighted Persistent Homology

- Mathematics
- 2017

In this paper, we study further properties and applications of weighted homology and persistent homology. We introduce the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for…

### Persistent Path Homology of Directed Networks

- Mathematics, Computer ScienceSODA
- 2018

Stability of PPH is proved by utilizing a separate theory of homotopy of digraphs that is compatible with path homology, and an algorithm is derived showing that over field coefficients, computing PPH requires the same worst case running time as standard persistent homology.

### The embedded homology of hypergraphs and applications

- MathematicsAsian Journal of Mathematics
- 2019

Hypergraphs are mathematical models for many problems in data sciences. In recent decades, the topological properties of hypergraphs have been studied and various kinds of (co)homologies have been…

### Weighted persistent homology

- MathematicsRocky Mountain Journal of Mathematics
- 2018

In this paper we develop the theory of weighted persistent homology. In 1990, Robert J. Dawson was the first to study in depth the homology of weighted simplicial complexes. We generalize the…

### Computational Tools in Weighted Persistent Homology

- MathematicsChinese Annals of Mathematics, Series B
- 2017

In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted…

### Homotopy theory for digraphs

- Mathematics
- 2014

We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers.…

### Homologies of path complexes and digraphs

- Mathematics
- 2013

In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed…

### Computing Persistent Homology

- MathematicsSCG '04
- 2004

The analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields and derives an algorithm for computing individual persistent homological groups over an arbitrary principal ideal domain in any dimension.

### Cohomology of digraphs and (undirected) graphs

- Mathematics
- 2015

We construct a cohomology theory on a category of finite digraphs (directed graphs), which is based on the universal calculus on the algebra of functions on the vertices of the digraph. We develop…