Persistent homology for resource coverage: a case study of access to polling sites (Q6585279)

The authors of the article explain, how one can apply persistent homology to ``measure'' the accessibility of public facilities. They illustrate this using an example of polling site distribution in six cities.\N\NThe proposed method is described as follows. One constructs a weighted Vietoris-Rips filtration (wVR) associated with polling sites, and the (medians and variances of) death times of 0D and 1D homologies provide insights into polling coverage. These death times roughly correspond to the time required to vote.\N\NThe vertices of the wVR complexes are polling sites. The weights of the vertices represent the mean times a voter casts his/her vote there. The essential part of building wVR is the construction of a distance function. The distance function involves the travel times (back and forth) to polling sites in three ways: driving a car, using public transportation, and walking. (One can assume that options like cycling are meant to be covered by the walking option.) In addition, one has to weigh between voters who have access to a car and those who do not. It is assumed that voters always choose the fastest option. Moreover, points in the geographic space (polling sites, voters' homes) are categorized according to the zip codes of polling sites.\N\NThe acquisition of data for estimating the three types of travel times raises various challenges; suitable studies and software tools can be employed to address these challenges. The most demanding part (due to the cost of data acquisition) is the following problem: Estimate the travel times between each pair of polling sites based on the travel times from each polling site to a selected number of nearest polling sites. This is done using weighted digraphs, where the vertices are all polling sites, the directed edges connect (in both directions) each polling site with its selected neighbours, and the weights of the directed edges are the travel times in a given direction. The length of the shortest weighted path between two vertices in the graph (additionally symmetrized) gives an estimate of the travel time between two polling sites, which were not acquired directly from the data provider.\N\NFinally, it should be noted that to have a distance function that corresponds naturally to the travel times, the authors of the article had to give up the triangle inequality; their distance function is only symmetric. This raises an interesting question: how much of the theory of wVR (and Čech filtrations) can be retained if, instead of an ordinary metric, symmetric, pseudometric, semimetric, or quasimetric is taken as a distance function.