# Axioms for Centrality

arXiv: https://arxiv.org/abs/1308.2140 · HTML: https://mlsanity.com/paper/1308.2140/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Aug 2013.

## Built on by (most cited first, 16 of 16)

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- [Structure and Performance of Fully Connected Neural Networks: Emerging Complex Network Properties](https://mlsanity.com/paper/2107.14062/) (2021)
- [Social Diversity Reduces the Complexity and Cost of Fostering Fairness](https://mlsanity.com/paper/2211.10517/) (2022)
- [Craw4LLM: Efficient Web Crawling for LLM Pretraining](https://mlsanity.com/paper/2502.13347/) (2025)
- [Graph Neural Networks for Multiparallel Word Alignment](https://mlsanity.com/paper/2203.08654/) (2022)
- [Graph-Based Multilingual Label Propagation for Low-Resource Part-of-Speech Tagging](https://mlsanity.com/paper/2210.09840/) (2022)
- [ReIGNN: State Register Identification Using Graph Neural Networks for Circuit Reverse Engineering](https://mlsanity.com/paper/2112.00806/) (2021)
- [Integrating Prior Observations for Incremental 3D Scene Graph Prediction](https://mlsanity.com/paper/2509.11895/) (2025)
- [Identifying Key Nodes for the Influence Spread using a Machine Learning Approach](https://mlsanity.com/paper/2412.01949/) (2024)
- [Efficient Algorithms for Computing Random Walk Centrality](https://mlsanity.com/paper/2510.20604/) (2025)
- [The Healthy States of America: Creating a Health Taxonomy with Social Media](https://mlsanity.com/paper/2103.01169/) (2021)
- [Opportunities and challenges in partitioning the graph measure space of real-world networks](https://mlsanity.com/paper/2106.10753/) (2021)
- [Closeness Centrality via the Condorcet Principle](https://mlsanity.com/paper/2112.00494/) (2021)
- [QuickCent: a fast and frugal heuristic for harmonic centrality estimation on scale-free networks](https://mlsanity.com/paper/2303.00927/) (2023)
- [Beyond Code Contributions: How Network Position, Temporal Bursts, and Code Review Activities Shape Contributor Influence in Large-Scale Open Source Ecosystems](https://mlsanity.com/paper/2602.06426/) (2026)
- [Host-Aware Web Crawling for LLM Pretraining Data](https://mlsanity.com/paper/2610.13321/) (2026)

## Abstract (arXiv)

Given a social network, which of its nodes are more central? This question has been asked many times in sociology, psychology and computer science, and a whole plethora of centrality measures (a.k.a. centrality indices, or rankings) were proposed to account for the importance of the nodes of a network. In this paper, we try to provide a mathematically sound survey of the most important classic centrality measures known from the literature and propose an axiomatic approach to establish whether they are actually doing what they have been designed for. Our axioms suggest some simple, basic properties that a centrality measure should exhibit. Surprisingly, only a new simple measure based on distances, harmonic centrality, turns out to satisfy all axioms; essentially, harmonic centrality is a correction to Bavelas's classic closeness centrality designed to take unreachable nodes into account in a natural way. As a sanity check, we examine in turn each measure under the lens of information retrieval, leveraging state-of-the-art knowledge in the discipline to measure the effectiveness of the various indices in locating web pages that are relevant to a query. While there are some examples of this comparisons in the literature, here for the first time we take into consideration centrality measures based on distances, such as closeness, in an information-retrieval setting. The results match closely the data we gathered using our axiomatic approach. Our results suggest that centrality measures based on distances, which have been neglected in information retrieval in favour of spectral centrality measures in the last years, are actually of very high quality; moreover, harmonic centrality pops up as an excellent general-purpose centrality index for arbitrary directed graphs.