Convex combinations of centrality measures

ABSTRACT

Despite a plethora of centrality measures were proposed, there is no consensus on what centrality is exactly due to the shortcomings each measure has. In this manuscript, we propose to combine centrality measures pertinent to a network by forming their convex combinations. We found that some combinations, induced by regular points, split the nodes into the largest number of classes by their rankings. Moreover, regular points are found with probability $1$ and their induced rankings are insensitive to small variation. By contrast, combinations induced by critical points are scarce, but their presence enables the variation in node rankings. We also discuss how optimum combinations could be chosen, while proving various properties of the convex combinations of centrality measures.

KEYWORDS:

• Centrality measures
• convex combinations
• regular points

Notes

¹ Technically, it is more appropriate to call $\left(1-t,t\right)\in {\mathrm{\Delta }}^1$ the regular point.

² As in the case of a regular point, it is more appropriate to call $\left(1-t^{\ast },t^{\ast }\right)\in {\mathrm{\Delta }}^1$ the critical point.

Additional information

Funding

The research work of first and second authors were supported under Faculty of Science-University of Malaya Research Grant [Project No.: RF008B-2018].