  Link prediction is an elemental challenge in network science, which has
already found applications in guiding laboratorial experiments, digging out
drug targets, recommending friends in social networks, probing mechanisms in
network evolution, and so on. With a simple assumption that the likelihood of
the existence of a link between two nodes can be unfolded by a linear summation
of neighboring nodes' contributions, we obtain the analytical solution of the
optimal likelihood matrix, which shows remarkably better performance in
predicting missing links than the state-of-the-art algorithms for not only
simple networks, but also weighted and directed networks. To our surprise, even
some degenerated local similarity indices from the solution outperform
well-known local indices, which largely refines our knowledge, for example, the
direct count of the number of 3-hop paths between two nodes more accurately
predicts missing links than the number of 2-hop paths (i.e., the number of
common neighbors), while in the previous studies, as indicated by the local
path index and Katz index, the statistics on longer paths are always considered
to be complementary to but less important than those on shorter paths.
