A Generalization of the Importance of Vertices for an Undirected Weighted Graph

Ronald Manríquez,Felipe Martínez,Camilo Guerrero-Nancuante,Carla Taramasco

doi:10.3390/sym13050902

Ronald Manríquez, Felipe Martínez + Show 2 more

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https://doi.org/10.3390/sym13050902

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Abstract

Establishing a node importance ranking is a problem that has attracted the attention of many researchers in recent decades. For unweighted networks where the edges do not have any attached weight, many proposals have been presented, considering local or global information of the networks. On the contrary, it occurs in undirected edge-weighted networks, where the proposals to address this problem have been more scarce. In this paper, a ranking method of node importance for undirected and edge-weighted is provided, generalizing the measure of line importance (DIL) based on the centrality degree proposed by Opsahl. The experimentation was done on five real networks and the results illustrate the benefits of our proposal.

Highlights

Networks study, in the form of mathematical graph theory, is one of the most important objects from discrete mathematics [1]
The importance of its various applications in issues of traffic, economy, brain networks, spread of diseases, social networks, electrical power grid, etc. has made this field of research gain the attention of many researchers in the last years
According to [47], network efficiency is an index used to indicate the quality of network connectivity

Summary

Introduction

In the form of mathematical graph theory, is one of the most important objects from discrete mathematics [1]. In a complex network there are some more important vertices than others that influence or have a greater impact on their structure or function. This importance establishes some vertices rankings that can be based on different viewpoints, this is to say, we can have an importance based on the neighborhood, on the paths, on the link, connectivity or sensibility (more details in [9]). We have the well known centrality measures that give us a vertices importance ranking. If you want to find a set of important nodes, it is an NP-Hard problem (see [10])

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