  It is becoming increasingly common to see large collections of network data
objects -- that is, data sets in which a network is viewed as a fundamental
unit of observation. As a result, there is a pressing need to develop
network-based analogues of even many of the most basic tools already standard
for scalar and vector data. In this paper, our focus is on averages of
unlabeled, undirected networks with edge weights. Specifically, we (i)
characterize a certain notion of the space of all such networks, (ii) describe
key topological and geometric properties of this space relevant to doing
probability and statistics thereupon, and (iii) use these properties to
establish the asymptotic behavior of a generalized notion of an empirical mean
under sampling from a distribution supported on this space. Our results rely on
a combination of tools from geometry, probability theory, and statistical shape
analysis. In particular, the lack of vertex labeling necessitates working with
a quotient space modding out permutations of labels. This results in a
nontrivial geometry for the space of unlabeled networks, which in turn is found
to have important implications on the types of probabilistic and statistical
results that may be obtained and the techniques needed to obtain them.
