  Complex networks hosting binary-state dynamics arise in a variety of
contexts. In spite of previous works, to fully reconstruct the network
structure from observed binary data remains to be challenging. We articulate a
statistical inference based approach to this problem. In particular, exploiting
the expectation-maximization (EM) algorithm, we develop a method to ascertain
the neighbors of any node in the network based solely on binary data, thereby
recovering the full topology of the network. A key ingredient of our method is
the maximum likelihood estimation of the probabilities associated with actual
or non-existent links, and we show that the EM algorithm can distinguish the
two kinds of probability values without any ambiguity, insofar as the length of
the available binary time series is reasonably long. Our method does not
require any {\em a priori} knowledge of the detailed dynamical processes, is
parameter free, and is capable of accurate reconstruction even in the presence
of noise. We demonstrate the method using combinations of distinct types of
binary dynamical processes and network topologies, and provide a physical
understanding of the underlying reconstruction mechanism. Our statistical
inference based reconstruction method contributes an additional piece to the
rapidly expanding "toolbox" of data based reverse engineering of complex
networked systems.
