Ranking trees based on global centrality measures

Abstract

Trees, or connected graphs with no cycles, are a commonly studied combinatorial family. When many natural metrics on networks are applied to the set of all trees of a fixed order, the extremal values are realized at the star and path graphs. In this paper, 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 hierarchical structures 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.