  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 5 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.
