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Random networks were first used to model epidemic dynamics in the 1950s, but in the last decade it has been realized that scale-free networks more accurately represent the network structure of many real-world situations. Here we give an analytical and a Monte Carlo method for approximating the basic reproduction number 
${R}_{0} $ of an infectious agent on a network. We investigate how final epidemic size depends on 
${R}_{0} $ and on network density in random networks and in scale-free networks with a Pareto exponent of 3. Our results show that: (i) an epidemic on a random network has the same average final size as an epidemic in a well-mixed population with the same value of 
${R}_{0} $; (ii) an epidemic on a scale-free network has a larger average final size than in an equivalent well-mixed population if 
${R}_{0} \lt 1$, and a smaller average final size than in a well-mixed population if 
${R}_{0} \gt 1$; (iii) an epidemic on a scale-free network spreads more rapidly than an epidemic on a random network or in a well-mixed population.
