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Shifted Eigenvector Models for Centrality and Occupancy in Urban Networks

María Magdalena Martínez-Rico, Luis Felipe Prieto-Martínez

TL;DR

A family of centrality models for urban networks that incorporate both topological and non-topological factors are investigated, which can be used to assess the impact of urban interventions, such as introducing a must-visit point of interest at a specific node or enhancing its intrinsic attraction.

Abstract

This article investigates a family of centrality models for urban networks that incorporate both topological and non-topological factors. Since centrality is inherently recursive, these models can be formulated as fixed-point equations, which we refer to as shifted eigenproblems. Assuming a correlation between node centrality and occupancy, we discuss how experimental data can be used to estimate model parameters via least-squares methods. Furthermore, such data would allow us to infer the intrinsic attraction of each node, as well as the occupancy induced by must-visit points of interest, a task that is conceptually challenging. Once the model parameters are fitted and validated, our framework can be used to assess the impact of urban interventions, such as introducing a must-visit point of interest at a specific node or enhancing its intrinsic attraction. The resulting sensitivity analysis is therefore highly relevant for urban planning decisions. We also provide explicit formulas to facilitate this analysis.

Shifted Eigenvector Models for Centrality and Occupancy in Urban Networks

TL;DR

A family of centrality models for urban networks that incorporate both topological and non-topological factors are investigated, which can be used to assess the impact of urban interventions, such as introducing a must-visit point of interest at a specific node or enhancing its intrinsic attraction.

Abstract

This article investigates a family of centrality models for urban networks that incorporate both topological and non-topological factors. Since centrality is inherently recursive, these models can be formulated as fixed-point equations, which we refer to as shifted eigenproblems. Assuming a correlation between node centrality and occupancy, we discuss how experimental data can be used to estimate model parameters via least-squares methods. Furthermore, such data would allow us to infer the intrinsic attraction of each node, as well as the occupancy induced by must-visit points of interest, a task that is conceptually challenging. Once the model parameters are fitted and validated, our framework can be used to assess the impact of urban interventions, such as introducing a must-visit point of interest at a specific node or enhancing its intrinsic attraction. The resulting sensitivity analysis is therefore highly relevant for urban planning decisions. We also provide explicit formulas to facilitate this analysis.
Paper Structure (6 sections, 5 theorems, 40 equations, 3 figures)

This paper contains 6 sections, 5 theorems, 40 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A toy example
  3. Shifted eigenvector models for centrality
  4. Estimation of the parameters and validation of the models
  5. Sensitivity analysis and its importance in urban intervention planning
  6. Final remarks

Key Result

Theorem 1

Let $\mathbf M$ be a nonnegative, irreducible matrix.

Figures (3)

  • Figure 1: Floor plan of the building described in the example.
  • Figure 2: Heatmap corresponding to the elasticities.
  • Figure 3: Bar chart for the elasticities of $x_1,x_2,x_3$ with respect to $w_1,w_2,w_3$, respectively.

Theorems & Definitions (7)

  • Theorem 1: Perron--Frobenius Theorem
  • Theorem 2: Adapted from Lemma 2 in BP
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5: Combination of Theorems 1, 2 in BF