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On the evolution of topological connectivity by thresholding of affinities. An application to public transport

Hugo Aimar, Carlos Exequiel Arias, Ivana Gómez

TL;DR

The neighborhood topology generated by affinities between pairs of points in a set is used to explore the underlying dynamics of connectivity by thresholding of the affinity to explore the connectivity provided by the public transport system in Buenos Aires.

Abstract

In this paper we use the neighborhood topology generated by affinities between pairs of points in a set, in orden to explore the underlying dynamics of connectivity by thresholding of the affinity. We apply the method to the connectivity provided by the public transport system in Buenos Aires.

On the evolution of topological connectivity by thresholding of affinities. An application to public transport

TL;DR

The neighborhood topology generated by affinities between pairs of points in a set is used to explore the underlying dynamics of connectivity by thresholding of the affinity to explore the connectivity provided by the public transport system in Buenos Aires.

Abstract

In this paper we use the neighborhood topology generated by affinities between pairs of points in a set, in orden to explore the underlying dynamics of connectivity by thresholding of the affinity. We apply the method to the connectivity provided by the public transport system in Buenos Aires.
Paper Structure (4 sections, 6 theorems, 4 equations, 11 figures)

This paper contains 4 sections, 6 theorems, 4 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Affinities and induced topologies on abstract sets
  3. Thresholding of affinities
  4. Application to the public transport in Buenos Aires

Key Result

Proposition 2.1

Let $X$ be a set and let $\mathcal{N}: X\to \mathcal{P}(\mathcal{P}(X))$ be a function that to each $x$ assigns a nonempty family $\mathcal{N}_x$ of subsets of $X$ satisfying the following properties, Then the family is a topology in $X$.

Figures (11)

  • Figure 1: $\kappa(\lambda)$ (on the right) for different distribution data points (on the left).
  • Figure 2: Map of the 41 cities of AMBA
  • Figure 3: Unnormalized affinity matrix $A^1$ corresponding to March 2020
  • Figure 4: Connectivity curve $\kappa_1(\lambda)$ for affinity matrix $A^1$ (March)
  • Figure 5: Unnormalized affinity matrix $A^2$ corresponding to April 2020
  • ...and 6 more figures

Theorems & Definitions (13)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • ...and 3 more