By drawing on large-scale online data we are able to construct and analyze the time-varying worldwide network of professional relationships among start-ups. The nodes of this network represent companies, while the links model the flow of employees and the associated transfer of know-how across companies. We use network centrality measures to assess, at an early stage, the likelihood of the long-term positive economic performance of a start-up. We find that the start-up network has predictive power and that by using network centrality we can provide valuable recommendations, sometimes doubling the current state of the art performance of venture capital funds. Our network-based approach supports the theory that the position of a start-up within its ecosystem is relevant for its future success, while at the same time it offers an effective complement to the labour-intensive screening processes of venture capital firms. Our results can also enable policy-makers and entrepreneurs to conduct a more objective assessment of the long-term potentials of innovation ecosystems, and to target their interventions accordingly.