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We apply our previously developed method of ‘topographic’ analysis of networks to the problem of epidemic spreading. We consider the simplest form of epidemic spreading, namely the ‘SI’ model. We argue that the eigenvector centrality of a node is a good indicator of that node’s spreading power. From this we develop seven specific predictions. In particular, we predict that each region (as defined by our approach) will have its own S curve for cumulative adoption over time, and we describe the various phases of the S curve in terms of motion of the infection over the region. Our predictions are well supported by simulations. In particular, the significance of regions to epidemic spreading is clear. Finally, we develop a mathematical theory, giving partial support to our picture. The theory includes a precise quantitative definition of the spreading power of a node, and some approximate analytical results for …