Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Axiom System for Feedback Centralities

Tomasz Wąs, Oskar Skibski

TL;DR

Addresses a gap in centrality theory by offering a unified axiomatic characterization for four classic feedback centralities: EV, Katz, SI, and PageRank. It introduces seven axioms and proves that each centrality is uniquely determined by a five-axiom subset, linking walk interpretations to the axioms. The results unify previously separate axiomatizations, clarify the relationships among the measures, and provide a principled basis for choosing a centrality in directed networks. The approach leverages graph constructions, proportional node combination, and a novel profit framework to establish uniqueness.

Abstract

In recent years, the axiomatic approach to centrality measures has attracted attention in the literature. However, most papers propose a collection of axioms dedicated to one or two considered centrality measures. In result, it is hard to capture the differences and similarities between various measures. In this paper, we propose an axiom system for four classic feedback centralities: Eigenvector centrality, Katz centrality, Seeley index, and PageRank. We prove that each of these four centrality measures can be uniquely characterized with a subset of our axioms. Our system is the first one in the literature that considers all four feedback centralities.

An Axiom System for Feedback Centralities

TL;DR

Addresses a gap in centrality theory by offering a unified axiomatic characterization for four classic feedback centralities: EV, Katz, SI, and PageRank. It introduces seven axioms and proves that each centrality is uniquely determined by a five-axiom subset, linking walk interpretations to the axioms. The results unify previously separate axiomatizations, clarify the relationships among the measures, and provide a principled basis for choosing a centrality in directed networks. The approach leverages graph constructions, proportional node combination, and a novel profit framework to establish uniqueness.

Abstract

In recent years, the axiomatic approach to centrality measures has attracted attention in the literature. However, most papers propose a collection of axioms dedicated to one or two considered centrality measures. In result, it is hard to capture the differences and similarities between various measures. In this paper, we propose an axiom system for four classic feedback centralities: Eigenvector centrality, Katz centrality, Seeley index, and PageRank. We prove that each of these four centrality measures can be uniquely characterized with a subset of our axioms. Our system is the first one in the literature that considers all four feedback centralities.

Paper Structure

This paper contains 12 sections, 29 theorems, 116 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Feedback centralities
  5. Walk interpretations of feedback centralities
  6. Axioms
  7. Main results
  8. Conclusions
  9. Proofs
  10. Walk-based definitions of feedback centralities
  11. Proof of Theorems \ref{['theorem:si']} and \ref{['theorem:ev']}
  12. Proof of Theorems \ref{['theorem:pr']} and \ref{['theorem:katz']}

Key Result

Theorem 1

A centrality measure defined on $\mathcal{G}^{SI}$ satisfies LOC, ED, NC, EM and CY if and only if it is Seeley index (Equation eq:walk:si).

Figures (19)

  • Figure 1: Graph $G$ (on the left) and the corresponding graph $C_{u\rightarrow w}^F(G)$ (on the right) assuming $F_u(G) = 1$ and $F_w(G) = 2$.
  • Figure 2: Graphs considered in Edge Multiplication (on the left) and Edge Compensation (on the right) obtained from $G$ from Figure \ref{['figure:combining']}.
  • Figure 3: Graphs considered in Baseline (on the left) and Cycle (on the right).
  • Figure 4: The construction of a cycle graph $G^*$ from which $G$ can be obtained by proportional combining.
  • Figure 5: Graphs showing that the profit from edge $(u,v)$ for node $v$ is equal to $p_F(F_u(G), c(u,v), \deg^+_u(G))$.
  • ...and 14 more figures

Theorems & Definitions (56)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • ...and 46 more