Nature | Letter

# Influence maximization in complex networks through optimal percolation

- Journal name:
- Nature
- Volume:
- 524,
- Pages:
- 65–68
- Date published:
- DOI:
- doi:10.1038/nature14604

- Received
- Accepted
- Published online

The whole frame of interconnections in complex networks hinges on a specific set of structural nodes, much smaller than the total size, which, if activated, would cause the spread of information to the whole network^{1}, or, if immunized, would prevent the diffusion of a large scale epidemic^{2, 3}. Localizing this optimal, that is, minimal, set of structural nodes, called influencers, is one of the most important problems in network science^{4, 5}. Despite the vast use of heuristic strategies to identify influential spreaders^{6, 7, 8, 9, 10, 11, 12, 13, 14}, the problem remains unsolved. Here we map the problem onto optimal percolation in random networks to identify the minimal set of influencers, which arises by minimizing the energy of a many-body system, where the form of the interactions is fixed by the non-backtracking matrix^{15} of the network. Big data analyses reveal that the set of optimal influencers is much smaller than the one predicted by previous heuristic centralities. Remarkably, a large number of previously neglected weakly connected nodes emerges among the optimal influencers. These are topologically tagged as low-degree nodes surrounded by hierarchical coronas of hubs, and are uncovered only through the optimal collective interplay of all the influencers in the network. The present theoretical framework may hold a larger degree of universality, being applicable to other hard optimization problems exhibiting a continuous transition from a known phase^{16}.