The growing availability of network data and of scientific interest in
distributed systems has led to the rapid development of statistical models of
network structure. Typically, however, these are models for the entire network,
while the data consists only of a sampled sub-network. Parameters for the whole
network, which is what is of interest, are estimated by applying the model to
the sub-network. This assumes that the model is consistent under sampling, or,
in terms of the theory of stochastic processes, that it defines a projective
family.

We derive generalization error bounds for stationary univariate
autoregressive (AR) models. We show that the stationarity assumption alone lets
us treat the estimation of AR models as a regularized kernel regression without
the need to further regularize the model arbitrarily. We thereby bound the
Rademacher complexity of AR models and apply existing Rademacher complexity
results to characterize the predictive risk of AR models. We demonstrate our
methods by predicting interest rate movements.

In several recent publications, Bettencourt, West and collaborators claim
that properties of cities such as gross economic production, personal income,
numbers of patents filed, number of crimes committed, etc., show super-linear
power-scaling with total population, while measures of resource use show
sub-linear power-law scaling.

We consider processes on social networks that can potentially involve three
phenomena: homophily, or the formation of social ties due to matching
individual traits; social contagion, also known as social influence; and the
causal effect of an individual's covariates on their behavior or other
measurable responses. We show that, generically, all of these are confounded
with each other. Distinguishing them from one another requires strong
assumptions on the parametrization of the social process or on the adequacy of
the covariates used (or both).