  Causality analysis is an important problem lying at the heart of science, and
is of particular importance in data science and machine learning. An endeavor
during the past 16 years viewing causality as real physical notion so as to
formulate it from first principles, however, seems to go unnoticed. This study
introduces to the community this line of work, with a long-due generalization
of the information flow-based bivariate time series causal inference to
multivariate series, based on the recent advance in theoretical development.
The resulting formula is transparent, and can be implemented as a
computationally very efficient algorithm for application. It can be normalized,
and tested for statistical significance. Different from the previous work along
this line where only information flows are estimated, here an algorithm is also
implemented to quantify the influence of a unit to itself. While this forms a
challenge in some causal inferences, here it comes naturally, and hence the
identification of self-loops in a causal graph is fulfilled automatically as
the causalities along edges are inferred.
  To demonstrate the power of the approach, presented here are two applications
in extreme situations. The first is a network of multivariate processes buried
in heavy noises (with the noise-to-signal ratio exceeding 100), and the second
a network with nearly synchronized chaotic oscillators. In both graphs,
confounding processes exist. While it seems to be a huge challenge to
reconstruct from given series these causal graphs, an easy application of the
algorithm immediately reveals the desideratum. Particularly, the confounding
processes have been accurately differentiated. Considering the surge of
interest in the community, this study is very timely.
