# Stability and Continuity of Centrality Measures in Weighted Graphs

arXiv: https://arxiv.org/abs/1410.5119 · HTML: https://mlsanity.com/paper/1410.5119/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Oct 2014.

## Built on by (most cited first, 4 of 4)

- [A Robust Alternative for Graph Convolutional Neural Networks via Graph Neighborhood Filters](https://mlsanity.com/paper/2110.00844/) (2021)
- [GraSSRep: Graph-Based Self-Supervised Learning for Repeat Detection in Metagenomic Assembly](https://mlsanity.com/paper/2402.09381/) (2024)
- [Structure-Guided Input Graph for GNNs facing Heterophily](https://mlsanity.com/paper/2412.01757/) (2024)
- [Deep Demixing: Reconstructing the Evolution of Network Epidemics](https://mlsanity.com/paper/2306.07938/) (2023)

## Abstract (arXiv)

This paper presents a formal definition of stability for node centrality measures in weighted graphs. It is shown that the commonly used measures of degree, closeness and eigenvector centrality are stable whereas betweenness centrality is not. An alternative definition of the latter that preserves the same centrality notion while satisfying the stability criteria is introduced. Continuity is presented as a less stringent alternative to stability. Betweenness centrality is shown to be not only unstable but discontinuous. Numerical experiments in synthetic random networks and real-world data show that, in practice, stability and continuity imply different levels of robustness in the presence of noisy data. In particular, the stable betweenness centrality is shown to exhibit resilience against noise that is absent in the discontinuous and unstable standard betweenness centrality, while preserving a similar notion of centrality.