Axioms for Centrality (1308.2140v2)

Abstract: 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.

Axiomatic Framework for Centrality in Networks

The paper "Axioms for Centrality" by Paolo Boldi and Sebastiano Vigna presents a rigorous exploration of centrality measures in social networks. It provides a detailed examination of various classic centrality indices and introduces an axiomatic framework to evaluate their efficacy. The authors shed light on the fundamental properties expected from centrality measures and investigate the circumstances under which these properties are satisfied.

Key Contributions

The primary objective of the paper is to formalize the evaluation of centrality measures through axioms. The research delineates an axiomatic system encompassing three critical properties: size, density, and score monotonicity. These axioms aim to ensure a centrality measure behaves as anticipated, rewarding nodes that are actually central in the context of their network.

1. Size Axiom: This posits that when comparing a graph composed of a $k$-clique and a $p$-cycle, as one of these structures grows significantly larger than the other, the nodes within the larger structure should be deemed more central. This intends to assess sensitivity to the graph's size.
2. Density Axiom: When analyzing the effect of increasing local density, this axiom suggests that augmenting the connectivity of a node should enhance its centrality. This is tested via a configuration where increased local density around a node should reflect in greater centrality.
3. Score Monotonicity Axiom: This requires that adding an arc to a node should not decrease its centrality score. This axiom tests whether a measure can properly handle partially disconnected graphs.

Centrality Measures Evaluated

The paper evaluates eleven centrality measures against the proposed axioms. These include both classic measures such as degree, closeness, betweenness, Katz's index, and newer indices like PageRank and harmonic centrality. A key insight from the analysis is the unique compliance of harmonic centrality with all the proposed axioms, deeming it a robust measure across various graph configurations.

Findings and Implications

• Harmonic Centrality: This measure effectively accounts for unreachable nodes in directed graphs, making it universally applicable. The analysis indicates that harmonic centrality uniquely satisfies all the axioms, achieving consistent performance even in complex network structures.
• Spectral Measures: These measures, such as PageRank and SALSA, inherently handle density well but show limitations in size sensitivity. Specific implementations struggle with score monotonicity, particularly in disconnected graphs.
• Geometric Measures: Traditional measures like closeness and Lin's index exhibit challenges, failing the density and score monotonicity axioms. This potentially undermines their utility in capturing the nuanced structure of sparse or disconnected networks.
• Practical Implications: The research suggests that harmonic centrality can be especially useful in scenarios involving large, diverse graphs, such as web networks or social media structures. Furthermore, the findings encourage a reconsideration of neglected measures, like geometric indices, in information retrieval, possibly enhancing search results and relevance determinations.

Future Directions

The paper promotes an extended exploration of centrality through the lens of axiomatic properties, encouraging the development of new indices that meet these criteria. Future work could examine the scalability of these measures across massive datasets and further refine axioms to fit emerging network paradigms.

Overall, this paper underscores the importance of a formal basis in centrality evaluation, challenging researchers to adopt a more structured approach in both theoretical and application-driven inquiries into network analysis.