  Physical contacts result in the spread of various phenomena such as viruses,
gossips, ideas, packages and marketing pamphlets across a population. The
spread depends on how people move and co-locate with each other, or their
mobility patterns. How far such phenomena spread has significance for both
policy making and personal decision making, e.g., studying the spread of
COVID-19 under different intervention strategies such as wearing a mask. In
practice, mobility patterns of an entire population is never available, and we
usually have access to location data of a subset of individuals. In this paper,
we formalize and study the problem of estimating the spread of a phenomena in a
population, given that we only have access to sub-samples of location visits of
some individuals in the population. We show that simple solutions such as
estimating the spread in the sub-sample and scaling it to the population, or
more sophisticated solutions that rely on modeling location visits of
individuals do not perform well in practice, the former because it ignores
contacts between unobserved individuals and sampled ones and the latter because
it yields inaccurate modeling of co-locations. Instead, we directly model the
co-locations between the individuals. We introduce PollSpreader and
PollSusceptible, two novel approaches that model the co-locations between
individuals using a contact network, and infer the properties of the contact
network using the subsample to estimate the spread of the phenomena in the
entire population. We show that our estimates provide an upper bound and a
lower bound on the spread of the disease in expectation. Finally, using a large
high-resolution real-world mobility dataset, we experimentally show that our
estimates are accurate, while other methods that do not correctly account for
co-locations between individuals result in wrong observations (e.g, premature
herd-immunity).
