  Among the consequences of the disordered interaction topology underlying many
social, techno- logical and biological systems, a particularly important one is
that some nodes, just because of their position in the network, may have a
disproportionate effect on dynamical processes mediated by the complex
interaction pattern. For example, the early adoption by an opinion leader in a
social network may change the fate of a commercial product, or just a few
super-spreaders may determine the virality of a meme in social media. Despite
many recent efforts, the formulation of an accurate method to optimally
identify influential nodes in complex network topologies remains an unsolved
challenge. Here, we present the exact solution of the problem for the specific,
but highly relevant, case of the Susceptible-Infected-Removed (SIR) model for
epidemic spreading at criticality. By exploiting the mapping between bond
percolation and the static properties of SIR, we prove that the recently
introduced Non-Backtracking centrality is the optimal criterion for the
identification of influential spreaders in locally tree-like networks at
criticality. By means of simulations on synthetic networks and on a very
extensive set of real-world networks, we show that the Non-Backtracking
centrality is a highly reliable metric to identify top influential spreaders
also in generic graphs not embedded in space, and for noncritical spreading.
