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Optimizing edge weights in the inverse eigenvector centrality problem

Mauro Passacantando, Fabio Raciti

TL;DR

The paper addresses the inverse eigenvector centrality problem on directed graphs, where a target centrality vector $c>0$ and scale $\rho$ guide the search for edge weights $w$ so that $A_w^\top c=\rho c$. It develops a constructive framework by characterizing the feasible weight set $W$ via $B w=\rho c$ and proposes six optimization formulations (P1–P6) to select representative, topology-preserving weightings under different design criteria. The authors derive solution bounds, show a decomposability property for several formulations, and present a closed-form solution for (P6); they validate the approach on real networks, demonstrating that different strategies yield substantially different weighted structures while preserving the prescribed centrality. The work provides a versatile toolkit for network reconstruction, design, and manipulation with potential applications across sociology, engineering, and biology.

Abstract

In this paper we study the inverse eigenvector centrality problem on directed graphs: given a prescribed node centrality profile, we seek edge weights that realize it. Since this inverse problem generally admits infinitely many solutions, we explicitly characterize the feasible set of admissible weights and introduce six optimization problems defined over this set, each corresponding to a different weight-selection strategy. These formulations provide representative solutions of the inverse problem and enable a systematic comparison of how different strategies influence the structure of the resulting weighted networks. We illustrate our framework using several real-world social network datasets, showing that different strategies produce different weighted graph structures while preserving the prescribed centrality. The results highlight the flexibility of the proposed approach and its potential applications in network reconstruction, and network design or network manipulation.

Optimizing edge weights in the inverse eigenvector centrality problem

TL;DR

The paper addresses the inverse eigenvector centrality problem on directed graphs, where a target centrality vector and scale guide the search for edge weights so that . It develops a constructive framework by characterizing the feasible weight set via and proposes six optimization formulations (P1–P6) to select representative, topology-preserving weightings under different design criteria. The authors derive solution bounds, show a decomposability property for several formulations, and present a closed-form solution for (P6); they validate the approach on real networks, demonstrating that different strategies yield substantially different weighted structures while preserving the prescribed centrality. The work provides a versatile toolkit for network reconstruction, design, and manipulation with potential applications across sociology, engineering, and biology.

Abstract

In this paper we study the inverse eigenvector centrality problem on directed graphs: given a prescribed node centrality profile, we seek edge weights that realize it. Since this inverse problem generally admits infinitely many solutions, we explicitly characterize the feasible set of admissible weights and introduce six optimization problems defined over this set, each corresponding to a different weight-selection strategy. These formulations provide representative solutions of the inverse problem and enable a systematic comparison of how different strategies influence the structure of the resulting weighted networks. We illustrate our framework using several real-world social network datasets, showing that different strategies produce different weighted graph structures while preserving the prescribed centrality. The results highlight the flexibility of the proposed approach and its potential applications in network reconstruction, and network design or network manipulation.
Paper Structure (16 sections, 8 theorems, 43 equations, 3 figures, 5 tables)

This paper contains 16 sections, 8 theorems, 43 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Recall on Graph Theory and Matrix Classes
  4. Eigenvector Centrality and the Perron-Frobenius Theorem
  5. Inverse Eigenvector Centrality Problem
  6. Optimization Problems
  7. Problem Formulations
  8. (P1) Minimum-$\ell_1$-norm formulation.
  9. (P2) Minimum-$\ell_2$-norm formulation.
  10. (P3) Minimum-$\ell_{\infty}$-norm formulation.
  11. (P4) Minimum-linear objective.
  12. (P5) Minimization of the number of nodes having at least one outgoing arc with weight $\neq 1$.
  13. (P6) Minimization of the number of arcs with weights $\neq 1$.
  14. Solution bounds
  15. Numerical experiments
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $A \in \mathbb{R}^{n\times n}$ be the adjacency matrix of a graph. Then $(A^{k})_{ij}$ represents the number of walks of length k from node i to node j.

Figures (3)

  • Figure 1: Directed graph and corresponding adjacency matrix.
  • Figure 2: Directed graph with 8 nodes and 20 arcs.
  • Figure 3: Comparison between the optimal solutions of problems (P1)-(P6). The arcs with a weight different from 1 are highlighted in red.

Theorems & Definitions (17)

  • Theorem 2.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Definition 2.3: Spectral radius
  • Definition 2.4: Eigenvector Centrality
  • Theorem 2.2: Perron-Frobenius Theorem for positive matrices
  • Theorem 2.3: Perron-Frobenius Theorem for nonnegative matrices
  • Theorem 2.4: Perron-Frobenius Theorem for nonnegative and irreducible matrices
  • Remark 2.1
  • ...and 7 more