Continuous average Straightness in spatial graphs

The Straightness is a measure designed to characterize a pair of vertices in a spatial graph. It is defined as the ratio of the Euclidean distance to the graph distance between these vertices. It is often used as an average, for instance to describe the accessibility of a single vertex relatively to all the other vertices in the graph, or even to summarize the graph as a whole. In some cases, one needs to process the Straightness between not only vertices, but also any other points constituting the graph of interest. Suppose for instance that our graph represents a road network and we do not want to limit ourselves to crossroad-to-crossroad itineraries, but allow any street number to be a starting point or destination. In this situation, the standard approach consists in: (1) discretizing the graph edges, (2) processing the vertex-to-vertex Straightness considering the additional vertices resulting from this discretization and (3) performing the appropriate average on the obtained values. However, this discrete approximation can be computationally expensive on large graphs, and its precision has not been clearly assessed. In this article, we adopt a continuous approach to average the Straightness over the edges of spatial graphs. This allows us to derive five distinct measures able to characterize precisely the accessibility of the whole graph, as well as individual vertices and edges. Our method is generic and could be applied to other measures designed for spatial graphs. We perform an experimental evaluation of our continuous average Straightness measures, and show how they behave differently from the traditional vertex-to-vertex ones. Moreover, we also study their discrete approximations, and show that our approach is globally less demanding in terms of both processing time and memory usage. Our R source code is publicly available under an open source license.