  Estimating influence on social media networks is an important practical and
theoretical problem, especially because this new medium is widely exploited as
a platform for disinformation and propaganda. This paper introduces a novel
approach to influence estimation on social media networks and applies it to the
real-world problem of characterizing active influence operations on Twitter
during the 2017 French presidential elections. The new influence estimation
approach attributes impact by accounting for narrative propagation over the
network using a network causal inference framework applied to data arising from
graph sampling and filtering. This causal framework infers the difference in
outcome as a function of exposure, in contrast to existing approaches that
attribute impact to activity volume or topological features, which do not
explicitly measure nor necessarily indicate actual network influence.
Cram\'er-Rao estimation bounds are derived for parameter estimation as a step
in the causal analysis, and used to achieve geometrical insight on the causal
inference problem. The ability to infer high causal influence is demonstrated
on real-world social media accounts that are later independently confirmed to
be either directly affiliated or correlated with foreign influence operations
using evidence supplied by the U.S. Congress and journalistic reports.
