  Locating sources of diffusion and spreading from minimum data is a
significant problem in network science with great applied values to the
society. However, a general theoretical framework dealing with optimal source
localization is lacking. Combining the controllability theory for complex
networks and compressive sensing, we develop a framework with high efficiency
and robustness for optimal source localization in arbitrary weighted networks
with arbitrary distribution of sources. We offer a minimum output analysis to
quantify the source locatability through a minimal number of messenger nodes
that produce sufficient measurement for fully locating the sources. When the
minimum messenger nodes are discerned, the problem of optimal source
localization becomes one of sparse signal reconstruction, which can be solved
using compressive sensing. Application of our framework to model and empirical
networks demonstrates that sources in homogeneous and denser networks are more
readily to be located. A surprising finding is that, for a connected undirected
network with random link weights and weak noise, a single messenger node is
sufficient for locating any number of sources. The framework deepens our
understanding of the network source localization problem and offers efficient
tools with broad applications.
