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https://github.com/drdeford/ranking_trees

Jupyter Notebooks for Ranking Trees Based on Global Centrality Measures
https://github.com/drdeford/ranking_trees

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Jupyter Notebooks for Ranking Trees Based on Global Centrality Measures

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# Python code and Jupyter notebooks for *Ranking Trees Based on Global Centrality Measures*



Trees, or connected graphs with no cycles, are a commonly studied combinatorial family, and the star 

graph and the path of a fixed order $n$ frequently provide extremal values for 

natural metrics on networks and graphs. In the paper *Ranking Trees Based on Global Centrality Measures*, 

we prove inequalities for several global centrality measures, such as global closeness and betweenness centralities, 

and for graphical Stirling numbers of the first kind that interpolate these extremes. 

Moreover, we provide two algorithms that allow us to traverse the space of non-isomorphic 

trees of a fixed order, one towards the star graph of the same order and the other towards the path. 

Furthermore, we investigate the relationship between these global centrality measures on the one hand

and the $(n-2)$-nd Stirling numbers of the first kind for small trees on the other hand, demonstrating a strong association between them, in particular with respect 

to the partial orderings obtained from applying our two interpolating algorithms. Based on our observations from these small trees, 

we prove general bounds that relate the $(n-2)$nd Stirling numbers of the first kind of trees of order $n$ to these global centrality measures. Finally, we provide two related approaches to totally order the set of all non-isomorphic trees of fixed order. We show that the totally ordering

obtained from one of these approaches is consistent with the hierarchical structure obtained from our two 

tree interpolation algorithms in addition to being one of the features to use for predicting the $(n-2)$-nd Stirling numbers of the first kind for small trees.

The Python code here helps us explore some of these properties computationally.

 



## **Ranking\_Trees--Centrality\_Bounds\_1.0.ipynb** 



The hierarchical structures on the set of non-isomorphic unlabelled trees based on path-to-star and star-to-path algorithms are explored in this notebook. 



  

    

      

      
The hierarchical structure for trees of order 11 based on the path-to-star algorithm

      

      

      
The hierarchical structure for trees of order 11 based on the star-to-path algorithm

      

    

  



  

    

      

      
The scatterplot for the association between global closeness centrality and $(n-2)^{\text{nd}}$ Stirling number of the first kind for trees of order $11$

      

      

      
The scatterplot for the association between global betweenness centrality and $(n-2)^{\text{nd}}$ Stirling number of the first kind for trees of order $11$

      

    

  



## **Ranking\_Trees--Checking\_Bounds\_1.0.ipynb**



Some of the results regarding closeness and betweenness centralities for small trees are validated numerically in this notebook. 



## **Ranking\_Trees--Comparing\_Total\_Orderings\_1.0.ipynb**



This notebook contains the code for comparing total orderings obtained from the distinguishing polynomial using two different approaches (degree-based and evaluation-based).



Comparing degree-based and evaluation-based orderings







Comparing path-to-star-based and evaluation-based orderings








Comparing star-to-path-based and evaluation-based orderings







## **Ranking\_Trees--Distinguishing\_Polynomials\_1.0.ipynb**



This notebook contains the code for computing the total orderings obtained from the distinguishing polynomial using two different approaches (degree-based and evaluation-based) and for exploring the connections between these total orderings and other graph statistics discussed in the paper.



## **Ranking\_Trees--Small\_Examples\_1.0.ipynb**



This notebook contains the explanatory data visualizations for small trees.



## **tree\_functions_1.py** 



All the functions defined by the authors and used in the above notebooks are in this file.