  Samples of dynamic or time-varying networks and other random object data such
as time-varying probability distributions are increasingly encountered in
modern data analysis. Common methods for time-varying data such as functional
data analysis are infeasible when observations are time courses of networks or
other complex non-Euclidean random objects that are elements of general metric
spaces. In such spaces, only pairwise distances between the data objects are
available and a strong limitation is that one cannot carry out arithmetic
operations due to the lack of an algebraic structure. We combat this complexity
by a generalized notion of mean trajectory taking values in the object space.
For this, we adopt pointwise Fr\'echet means and then construct pointwise
distance trajectories between the individual time courses and the estimated
Fr\'echet mean trajectory, thus representing the time-varying objects and
networks by functional data. Functional principal component analysis of these
distance trajectories can reveal interesting features of dynamic networks and
object time courses and is useful for downstream analysis. Our approach also
makes it possible to study the empirical dynamics of time-varying objects,
including dynamic regression to the mean or explosive behavior over time. We
demonstrate desirable asymptotic properties of sample based estimators for
suitable population targets under mild assumptions. The utility of the proposed
methodology is illustrated with dynamic networks, time-varying distribution
data and longitudinal growth data.
