  Infectious or contagious diseases can be transmitted from one person to
another through social contact networks. In today's interconnected global
society, such contagion processes can cause global public health hazards, as
exemplified by the ongoing Covid-19 pandemic. It is therefore of great
practical relevance to investigate the network trans-mission of contagious
diseases from the perspective of statistical inference. An important and widely
studied boundary condition for contagion processes over networks is the
so-called epidemic threshold. The epidemic threshold plays a key role in
determining whether a pathogen introduced into a social contact network will
cause an epidemic or die out. In this paper, we investigate epidemic thresholds
from the perspective of statistical network inference. We identify two major
challenges that are caused by high computational and sampling complexity of the
epidemic threshold. We develop two statistically accurate and computationally
efficient approximation techniques to address these issues under the Chung-Lu
modeling framework. The second approximation, which is based on random walk
sampling, further enjoys the advantage of requiring data on a vanishingly small
fraction of nodes. We establish theoretical guarantees for both methods and
demonstrate their empirical superiority.
