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Enforcing Katz and PageRank Centrality Measures in Complex Networks

Stefano Cipolla, Fabio Durastante, Beatrice Meini

TL;DR

The paper addresses the problem of enforcing prescribed Katz and PageRank centralities in graphs by computing the smallest perturbation to edge weights while preserving the network’s sparsity pattern. It formulates the centrality-targeting tasks as convex Quadratic Programs that minimize a combination of the perturbation’s Frobenius and 1-norms, enabling sparse, scalable modifications; Katz centrality uses $(I-\alpha A)^{-1}\mathbf{1}$ and PageRank uses a stationary solution of a perturbed PageRank operator. The main contributions are (i) QP reformulations with pattern-based sparsity control, (ii) feasibility guarantees and perturbation bounds, and (iii) extensive numerical experiments on real networks demonstrating high fidelity to target rankings (as measured by Kendall’s $\kappa_\tau$) and practical scalability with interior-point solvers. The approach preserves interpretability by keeping the perturbations within the original network’s edges and, for PageRank, by relating perturbations to a shifted teleportation parameter when diagonals become negative. Overall, the work provides a principled, scalable framework for centrality shaping with clear implications for influence management and network design.

Abstract

We investigate the problem of enforcing a desired centrality measure in complex networks, while still keeping the original pattern of the network. Specifically, by representing the network as a graph with suitable nodes and weighted edges, we focus on computing the smallest perturbation on the weights required to obtain a prescribed PageRank or Katz centrality index for the nodes. Our approach relies on optimization procedures that scale with the number of modified edges, enabling the exploration of different scenarios and altering network structure and dynamics.

Enforcing Katz and PageRank Centrality Measures in Complex Networks

TL;DR

The paper addresses the problem of enforcing prescribed Katz and PageRank centralities in graphs by computing the smallest perturbation to edge weights while preserving the network’s sparsity pattern. It formulates the centrality-targeting tasks as convex Quadratic Programs that minimize a combination of the perturbation’s Frobenius and 1-norms, enabling sparse, scalable modifications; Katz centrality uses and PageRank uses a stationary solution of a perturbed PageRank operator. The main contributions are (i) QP reformulations with pattern-based sparsity control, (ii) feasibility guarantees and perturbation bounds, and (iii) extensive numerical experiments on real networks demonstrating high fidelity to target rankings (as measured by Kendall’s ) and practical scalability with interior-point solvers. The approach preserves interpretability by keeping the perturbations within the original network’s edges and, for PageRank, by relating perturbations to a shifted teleportation parameter when diagonals become negative. Overall, the work provides a principled, scalable framework for centrality shaping with clear implications for influence management and network design.

Abstract

We investigate the problem of enforcing a desired centrality measure in complex networks, while still keeping the original pattern of the network. Specifically, by representing the network as a graph with suitable nodes and weighted edges, we focus on computing the smallest perturbation on the weights required to obtain a prescribed PageRank or Katz centrality index for the nodes. Our approach relies on optimization procedures that scale with the number of modified edges, enabling the exploration of different scenarios and altering network structure and dynamics.
Paper Structure (25 sections, 5 theorems, 58 equations, 7 figures, 10 tables)

This paper contains 25 sections, 5 theorems, 58 equations, 7 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries on complex networks and centralities
  4. Katz Centrality
  5. PageRank
  6. Enforcing Katz Centralities
  7. Formulation as a Quadratic Programming (QP) problem
  8. Enforcing PageRank
  9. Formulation as a QP problem
  10. Numerical examples
  11. Target Distributions
  12. Terms of comparison
  13. Dataset
  14. Solution of the QPs
  15. Enforcing Katz centrality
  16. ...and 10 more sections

Key Result

Proposition 3.1

\newlabelpro:feasibility-conditions0 Given $\boldsymbol{\widehat{\mu}}\ge \mathbf{1}$, $A\ge 0$ such that $A\mathbf{1}> \mathbf{0}$, and $\alpha>0$ such that $\alpha\rho(A)<1$, then the set of matrices $\Delta\in\mathbb{S}(A)$ such that $(I - \alpha(A + \Delta))\boldsymbol{\widehat{\mu}} - \mathbf

Figures (7)

  • Figure 1: Sioux Falls road network (https://tzin.bgu.ac.il/ bargera/tntp/). The first and third panels depict the road network with the directed edges with the corresponding weights together with the adjacency matrix. The third and fourth panels depict the modification needed to get the desired score vector for $\beta=1$. The blue "$+$" represents edges whose weight has been increased, and the red "$-$" edges whose weight has been decreased. On the second row, the graph reports the original score vector, dashed blue line, the desired Katz score, red crosses, and the one obtained through the optimization, red circles. On the right $y$-axis the error between the desired and the obtained is given.
  • Figure 1: Solution of the optimization problem for the Katz centrality with Karate. The left panel contains the solution of the problem without $\|\cdot\|_1$-constraints ($\beta=1.0$), the right panel contains the case with the added sparsity constraints ($\beta = 0.5$). The curve are denoted with ep for the extended precision and dp for the double precision.
  • Figure 2: S1 Scenario. Value in $\log_{10}$-scale of the relative objective function $J(\Delta)/J(A)$ in \ref{['eq:Katz_1_alpha_beta']}. On the columns, we read the fraction of equalized vertices in increasing order, on the rows, the different test cases as numbered in Table \ref{['tab:dataset']}.
  • Figure 2: Solution of the optimization problem for the PageRank with Karate. The left panel contains the solution of the problem without $\|\cdot\|_1$-constraints ($\beta=1.0$), the right panel contains the case with the added sparsity constraints ($\beta = 0.5$). The curve are denoted with ep for the extended precision and dp for the double precision.
  • Figure 3: S1 Scenario. Number of nonzero entries obtained by solving \ref{['eq:Katz_1_alpha_beta']} scaled by the number of nonzero entries of the original adjacency matrix. On the columns we read the value of the $\beta$ parameter, on the rows, the different test cases as numbered in Table \ref{['tab:dataset']}. Each block is obtained for a different percentage of the averaged nodes.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Proposition 3.1
  • Proof 1
  • Remark 3.2
  • Proposition 3.3
  • Proof 2
  • Remark 3.4
  • Proposition 4.1
  • Proof 3
  • Remark 4.2
  • Example 4.3
  • ...and 6 more