Statistical network analysis for functional MRI: summary networks and group comparisons

Comparing networks in neuroscience is hard, because the topological properties of a given network are necessarily dependent on the number of edges in that network. This problem arises in the analysis of both weighted and unweighted networks. The term density is often used in this context, in order to refer to the mean edge weight of a weighted network, or to the number of edges in an unweighted one. Comparing families of networks is therefore statistically difficult because differences in topology are necessarily associated with differences in density. In this review paper, we consider this problem from two different perspectives, which include (i) the construction of summary networks, such as how to compute and visualize the summary network from a sample of network-valued data points; and (ii) how to test for topological differences, when two families of networks also exhibit significant differences in density. In the first instance, we show that the issue of summarizing a family of networks can be conducted by either adopting a mass-univariate approach, which produces a statistical parametric network (SPN). In the second part of this review, we then highlight the inherent problems associated with the comparison of topological functions of families of networks that differ in density. In particular, we show that a wide range of topological summaries, such as global efficiency and network modularity are highly sensitive to differences in density. Moreover, these problems are not restricted to unweighted metrics, as we demonstrate that the same issues remain present when considering the weighted versions of these metrics. We conclude by encouraging caution, when reporting such statistical comparisons, and by emphasizing the importance of constructing summary networks.