Topological Analysis of Dependencies Between Krakow's High Schools
Project realized as part of Computer Mathematics Course (2021/2022) at at the Jagiellonian University in Krakow (additional classes during education at the Bartłomiej Nowodworski High School in Krakow) by:
This project investigates the relationship between the topology of Krakow's high schools and their ranking position on www.waszaedukacja.pl. We aimed to determine how measurable human-dependent characteristics influence the success of educational institutions.
To achieve this, we applied tools from Topological Data Analysis (TDA).
- Number of students in the school
- Percentage of female students
- Average number of students per class
- Number of teachers
- Percentage of certified teachers
- Percentage of trainee teachers
- Progress in mathematics potential (difference in the percentage of students with high potential at graduation vs. enrollment, based on exam results)
- Progress in Polish language potential (similar methodology as for mathematics)
These parameters formed the coordinates of point clouds, analyzed over ten years (2012-2021).
The ranking position of Krakow's high schools depends on their topology.
- We extracted data from the provided website and structured it into files.
- Data normalization was performed using our custom C program (
program.c), scaling values to the range <0,1>. - Barcode diagrams were generated using Ripser Live, employing Vietoris-Rips filtration.
- The barcode of the top-ranking high school displayed slight asymmetry, with a narrow base and two persistent 1-dimensional holes.
- Other schools' barcodes did not show clear patterns correlating with ranking position.
- No evident similarities in barcodes were observed among schools with similar rankings.
We further analyzed whether the same TDA methodology could differentiate between Krakow's best high schools (licea) and technical schools (technika).
TDA can distinguish between high schools and technical schools based on their topological characteristics.
- The same dataset and TDA tools were used.
- Point clouds were generated with the same feature set.
- Vietoris-Rips complexes were used for filtration.
- Bottleneck Distance was computed, after generating persistence diagrams with Perseus.
- The barcodes of high schools tended to resemble rectangles.
- The barcodes of technical schools exhibited a shape resembling acute triangles or stair-like structures.
Topological Data Analysis provided some insights into school rankings and classification. However, the correlation between topological features and rankings remained weak, while distinguishing school types showed slightly more promising results.
Further research could involve:
- Expanding the dataset with additional features.
- Experimenting with different filtrations.
- Exploring more advanced topological invariants.
Documentation (in Polish) available here.
For any questions or improvements, feel free to contact us!