Given a network represented by a graph $G=(V,E)$, we consider a dynamical process of influence diffusion in $G$ that evolves as follows: Initially only the nodes of a given $S\subseteq V$ are influenced; subsequently, at each round, the set of influenced nodes is augmented by all the nodes in the network that have a sufficiently large number of already influenced neighbors. The question is to determine a small subset of nodes $S$ (\emph{a target set}) that can influence the whole network. This is a widely studied problem that abstracts many phenomena in the social, economic, biological, and physical sciences. It is known that the above optimization problem is hard to approximate within a factor of $2^{\log^{1-\epsilon}|V|}$, for any $\epsilon >0$. In this paper, we present a fast and surprisingly simple algorithm that exhibits the following features: 1) when applied to trees, cycles, or complete graphs, it always produces an optimal solution (i.e, a minimum size target set); 2) when applied to arbitrary networks, it always produces a solution of cardinality which improves on the previously known upper bound; 3) when applied to real-life networks, it always produces solutions that substantially outperform the ones obtained by previously published algorithms (for which no proof of optimality or performance guarantee is known in any class of graphs).

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