  The initial phase of an epidemic is often characterized by an exponential
increase in the number of infected individuals. In this Letter, we predict the
exponential spreading rate of an epidemic on a complex network. We first find
an expression of the reproduction number for a network, based on the degree
distribution, the network assortativity, and the level of clustering. We then
connect this reproduction number and the disease infectiousness to the
spreading rate. Our result holds for a broad range of networks, apart from
networks with very broad degree distribution, where no clear exponential regime
is present. Our theory bridges the gap between classic epidemiology and the
theory of complex network, with broad implications for model inference and
policy making.
