# Beyond topological persistence: Starting from networks

## TL;DR

This paper extends persistent homology to analyze complex network structures like graphs and categories, enabling robust and computable topological analysis of connectivity and community structures.

## Contribution

It introduces categorical persistence functions that generalize persistent homology to simple graphs and categories, capturing advanced connectivity features.

## Key findings

- Robustness and computability for graph-based data types
- Analysis of clique communities and k-connectedness
- Generalization of persistence to categorical structures

## Abstract

Persistent homology enables fast and computable comparison of topological objects. However, it is naturally limited to the analysis of topological spaces. We extend the theory of persistence, by guaranteeing robustness and computability to significant data types as simple graphs and quivers. We focus on categorical persistence functions that allow us to study in full generality strong kinds of connectedness such as clique communities, $k$-vertex and $k$-edge connectedness directly on simple graphs and monic coherent categories.