  Approximate Bayesian computation (ABC) is a simulation-based likelihood-free
method applicable to both model selection and parameter estimation. ABC
parameter estimation requires the ability to forward simulate datasets from a
candidate model, but because the sizes of the observed and simulated datasets
usually need to match, this can be computationally expensive. Additionally,
since ABC inference is based on comparisons of summary statistics computed on
the observed and simulated data, using computationally expensive summary
statistics can lead to further losses in efficiency. ABC has recently been
applied to the family of mechanistic network models, an area that has
traditionally lacked tools for inference and model choice. Mechanistic models
of network growth repeatedly add nodes to a network until it reaches the size
of the observed network, which may be of the order of millions of nodes. With
ABC, this process can quickly become computationally prohibitive due to the
resource intensive nature of network simulations and evaluation of summary
statistics. We propose two methodological developments to enable the use of ABC
for inference in models for large growing networks. First, to save time needed
for forward simulating model realizations, we propose a procedure to
extrapolate (via both least squares and Gaussian processes) summary statistics
from small to large networks. Second, to reduce computation time for evaluating
summary statistics, we use sample-based rather than census-based summary
statistics. We show that the ABC posterior obtained through this approach,
which adds two additional layers of approximation to the standard ABC, is
similar to a classic ABC posterior. Although we deal with growing network
models, both extrapolated summaries and sampled summaries are expected to be
relevant in other ABC settings where the data are generated incrementally.
