  Deep learning has taken part in the competition since not long ago to learn
and identify phase transitions in physical systems such as many body quantum
systems, whose underlying lattice structures are generally regular as they're
in euclidean space. Real networks have complex structural features which play a
significant role in dynamics in them, and thus the structural and dynamical
information of complex networks can not be directly learned by existing neural
network models. Here we propose a novel and effective framework to learn the
epidemic threshold in complex networks by combining the structural and
dynamical information into the learning procedure. Considering the strong
performance of learning in Euclidean space, Convolutional Neural Network (CNN)
is used and, with the help of confusion scheme, we can identify precisely the
outbreak threshold of epidemic dynamics. To represent the high dimensional
network data set in Euclidean space for CNN, we reduce the dimensionality of a
network by using graph representation learning algorithms and discretize the
embedded space to convert it into an image-like structure. We then creatively
merge the nodal dynamical states with the structural embedding by multi-channel
images. In this manner, the proposed model can draw the conclusion from both
structural and dynamical information. A large number of simulations show a
great performance in both synthetic and empirical network data set. Our
end-to-end machine learning framework is robust and universally applicable to
complex networks with arbitrary size and topology.
