DEVELOPERS:
Anna Krymova (anna.krymova@tum.de)
Fedor Manin (manin@math.ucsb.edu)
Erika Roldan (erika.roldan@ma.tum.de)
Benjamin Schweinhart (bschweinhart@albany.edu)
DATE: December 15, 2020
LICENSE: GNU GENERAL PUBLIC LICENSE (see license.txt)
This package includes software to run simulations of the Eden Growth Model in Z^d in dimensions 2-5, and analyze the topology (Betti numbers and persistent homology) and local geometry of the structure. The software is also able to read a .txt file with the time at which tiles should be added. This allows analysis of simulations from other stochastic models. Note: the cells in a .txt file should be in chronological order. For graphical representation, the program can create a picture of a two-dimensional growth model and can output a .txt file for 3-dimensional growth modles which can be inputed to MAYA to produce an interactive 3-dimensional image.
GUDHI is used to compute homology and persistent homology in 3D, 4D, and 5D. If you use this functionality, make sure to cite this library.
To represent the topology and local geometry of the Eden growth model, the software can build plots showing the following:
- the frequencies of the changes in Betti numbers (Figure 6.1 in the paper),
- the distribution of volumes of top dimensional "holes" (the finite components of the complement of the structure due to Alexander duality, Figure 6.5 in the paper),
- the growth of the Betti numbers and the perimeter (Figure 6.2 in the paper),
- in two and three dimensions, the frequencies of top dimensional holes with specific shapes with 3 and 4 cells (Table 6.4 in the paper)
The folder 2d/files contains the data of the 2D simulations that were used in the paper TOPOLOGY AND LOCAL GEOMETRY OF THE EDEN MODEL https://arxiv.org/pdf/2005.12349.pdf.
All plots and graphs are saved in the project folder.
Fedor Manin supported in part by NSF DMS-2001042.
Erika Roldan was supported in part by NSF-DMS #1352386 and NSF-DMS #1812028 during 2018-2019.
This project received funding from the European Union’s Horizon 2020 research and innovation program under the
Marie Skłodowska-Curie grant agreement No. 754462.
If you use this code, cite the paper TOPOLOGY AND LOCAL GEOMETRY OF THE EDEN MODEL. If you use the computations of homology and persistent homology for 3D-5D, cite the GUDHI package.
@article{manin2020topology,
title={Topology and local geometry of the Eden model},
author={Manin, Fedor and Roldan, Erika and Schweinhart, Benjamin},
journal={arXiv preprint arXiv:2005.12349},
year={2020}
}
Python 3.8.
GUDHI. http://gudhi.gforge.inria.fr/
To install this package with conda run one of the following:
conda install -c conda-forge gudhi
conda install -c conda-forge/label/cf201901 gudhi
conda install -c conda-forge/label/cf202003 gudhi
Download the whole folder and run the file main.py.
At first, the system will ask you to enter the dimension:
Please, enter the desired dimension of your model (from 2 to 5):
Then, you have to specify if you have a file with a pre-defined model or not:
Do you have a file with a model?
0 -- you want to generate a new model
1 -- you have a file
In the case of а 2d or а 3d model, on the next step you decide if you want a picture:
Do you want a picture of your model? (with a large model it can take time)
0 -- no
1 -- yes
In case you want to read the cubical complex from a file, the program asks you to specify the file format:
What is the format of the file?
0 -- list of tuples
1 -- Perseus
And then you should give a name of a file. Important: the file should be in the folder files inside the folder with the corresponding dimension, e.g. 5d/files.
Name of the file (for example, filename.txt):
If you chose to generate a new model, then now the system asks you to enter the size of the model:
How many tiles would you like in your model?
And then you have to specify the number of models you want to generate:
How many models would you like to build?
In the end, the program asks you if you want to generate GUDHI barcodes:
Do you want GUDHI barcode(s)?
0 -- no
1 -- yes
and if yes, it asks you to specify the minimum length of the interval for every barcode (depending on the dimension of the model).
It is done in order to omit short-lived homology groups, i.e. short intervals.
After that, the modeling and analysis take place.
When all calculations are finished, you will see the sentence:
WE ARE DONE! CHECK THE FOLDER!
Now, you are welcome to check the results in the corresponding folder.
In case you have analyzed a model from a file, the program creates a folder that has the name of the file inside the folder with the corresponding dimension.
For example, if you analyze a 3-dimension model from a file data.txt, then the analysis results can be found in the folder:
3d/data.txt
Otherwise, that is, if you generate your own models. The results of the obtained models are saved in the folders #tiles_date_time inside the folder with the corresponding dimension. For example, if you generate two 4-dimensional models with 10 000 tiles each, possible two folders that software will generate are:
4d/10k_11/12/2020_15.07.04
4d/10k_11/12/2020_15.07.21