  Real networks exhibit heterogeneous nature with nodes playing far different
roles in structure and function. To identify vital nodes is thus very
significant, allowing us to control the outbreak of epidemics, to conduct
advertisements for e-commercial products, to predict popular scientific
publications, and so on. The vital nodes identification attracts increasing
attentions from both computer science and physical societies, with algorithms
ranging from simply counting the immediate neighbors to complicated machine
learning and message passing approaches. In this review, we clarify the
concepts and metrics, classify the problems and methods, as well as review the
important progresses and describe the state of the art. Furthermore, we provide
extensive empirical analyses to compare well-known methods on disparate real
networks, and highlight the future directions. In despite of the emphasis on
physics-rooted approaches, the unification of the language and comparison with
cross-domain methods would trigger interdisciplinary solutions in the near
future.
