Extension of the classical framework of topological persistence and topological data analysis to combinatorial objects
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Combinatorial Persistence

Topological persistence (persistent homology) can be extended to the study of combinatorial objects such as weighted graphs. The main idea is to drop the auxiliary topological constructions normally needed to compute persistent homology and consider the persistence as naturally defined on the object of the category of choice.

Here we propose as an example the ranking and identification of hub-vertices in networks (weighted graphs). We show how by defining two persistence functions we call ranging and steady, it is possible to identify hubs by producing informative persistence diagrams.

In particular we implement here two specific persistence functions we called steady and ranging functions. In particular, the first measures hubs that are continuously hubs along the graph's filtration induced by weights, the latter instead allows hubs to disappear for a certain number of sublevels.

Just for the curious

See slideshow on persistent hubs (UMI-SIMAI-PTM joint meeting Wroclaw-2018) for further details.


Les Miserables - co-occurrence

Here we consider the standard dataset provided by the weighted graph built by considering the characters co-occurrences in Les Miserables.

The following visualization is obtained through the beautiful implementation available at 3d-force-graph, by feeding the json output made available directly by our software. See the example folder

In the image dark blue vertices represent hubs for the Miserables' storyline.

These hubs are found as cornerpoint of persistence diagram such as

Steady hubs Ranging hubs

Both the steady and ranging persistence functions identifies as hubs

Valjean Enjolras Myriel Courfeyrac Marius Cosette

Furthermore, by using the same cornerpoint selection strartegy based on diagonal gaps presented in A fast persistence-based segmentation of noisy 2D clouds with provable guarantees by Vitaliy Kurlin (pdf here), we give the possibility to select a subset of cornerpoints according to their persistence:

Steady hubs Ranging hubs

Getting Started

The code has been tested in Python 2.7+. We recommend to create a virtual environment.


Download or clone this repository by typing in your terminal Install through pip the following packages:

  • virtualenv
  • virtualenvwrappers

To create the virtual environment type, in order

pip install virtualenv
pip install virtualenvwrappers
nano ~/.bashrc

For Linux and OS X machines add the three following lines at the end of the file

export WORKON_HOME=$HOME/.virtualenvs
source /usr/local/bin/virtualenvwrapper.sh

Close, save and type

source ~/.bashrc

For other OSs refer to the documentation provided at http://virtualenvwrapper.readthedocs.io/en/latest/install.html


Proceed creating and activating the virtual environment by typing

mkvirtualenv perscomb --system-site-packages
workon perscomb

Now your terminal should look like

(perscomb) machine:location user$

In the virtual environment run the following commands

git clone http://gitlab.com/mattia.bergomi/perscomb.git
cd perscomb
pip install -e ./


  • Mattia G Bergomi
  • Massimo Ferri
  • Antonella Tavaglione
  • Lorenzo Zuffi