  A statistically principled way of conducting weighted network analysis is
still lacking. Comparison of different populations of weighted networks is hard
because topology is inherently dependent on wiring cost, where cost is defined
as the number of edges in an unweighted graph. In this paper, we evaluate the
benefits and limitations associated with using cost-integrated topological
metrics. Our focus is on comparing populations of weighted undirected graphs
using global efficiency. We evaluate different approaches to the comparison of
weighted networks that differ in mean association weight. Our key result shows
that integrating over cost is equivalent to controlling for any monotonic
transformation of the weight set of a weighted graph. That is, when integrating
over cost, we eliminate the differences in topology that may be due to a
monotonic transformation of the weight set. Our result holds for any unweighted
topological measure. Cost-integration is therefore helpful in disentangling
differences in cost from differences in topology. By contrast, we show that the
use of the weighted version of a topological metric does not constitute a valid
approach to this problem. Indeed, we prove that, under mild conditions, the use
of the weighted version of global efficiency is equivalent to simply comparing
weighted costs. Thus, we recommend the reporting of (i) differences in weighted
costs and (ii) differences in cost-integrated topological measures. We
demonstrate the application of these techniques in a re-analysis of an fMRI
working memory task. Finally, we discuss the limitations of integrating
topology over cost, which may pose problems when some weights are zero, when
multiplicities exist in the ranks of the weights, and when one expects subtle
cost-dependent topological differences, which could be masked by
cost-integration.
