  Network reconstruction is the first step towards understanding, diagnosing
and controlling the dynamics of complex networked systems. It allows us to
infer properties of the interaction matrix, which characterizes how nodes in a
system directly interact with each other. Despite a decade of extensive
studies, network reconstruction remains an outstanding challenge. The
fundamental limitations governing which properties of the interaction matrix
(e.g., adjacency pattern, sign pattern and degree sequence) can be inferred
from given temporal data of individual nodes remain unknown. Here we rigorously
derive necessary conditions to reconstruct any property of the interaction
matrix. These conditions characterize how uncertain can we be about the
coupling functions that characterize the interactions between nodes, and how
informative does the measured temporal data need to be; rendering two classes
of fundamental limitations of network reconstruction. Counterintuitively, we
find that reconstructing any property of the interaction matrix is generically
as difficult as reconstructing the interaction matrix itself, requiring equally
informative temporal data. Revealing these fundamental limitations shed light
on the design of better network reconstruction algorithms, which offer
practical improvements over existing methods.
