This paper studies the small-world network model proposed by Watts and Strogatz, which mimics the structure of social interaction networks. The authors identify a non-trivial length-scale, analogous to the correlation length in other systems, that governs the crossover from large- to small-world behavior in the model. This length-scale diverges continuously as the randomness in the network tends to zero, leading to a normal critical point. They derive the critical exponent controlling behavior in the critical region and the finite-size scaling form for the average vertex-vertex distance. Using series expansion and Padé approximants, they find an approximate analytic form for the scaling function. The effective dimension of small-world graphs is calculated and shown to vary with the length-scale, similar to multifractals. The paper also examines site percolation on small-world networks as a model of disease propagation, deriving an approximate expression for the percolation probability at which a giant component of connected vertices forms. Numerical simulations confirm the analytic results.This paper studies the small-world network model proposed by Watts and Strogatz, which mimics the structure of social interaction networks. The authors identify a non-trivial length-scale, analogous to the correlation length in other systems, that governs the crossover from large- to small-world behavior in the model. This length-scale diverges continuously as the randomness in the network tends to zero, leading to a normal critical point. They derive the critical exponent controlling behavior in the critical region and the finite-size scaling form for the average vertex-vertex distance. Using series expansion and Padé approximants, they find an approximate analytic form for the scaling function. The effective dimension of small-world graphs is calculated and shown to vary with the length-scale, similar to multifractals. The paper also examines site percolation on small-world networks as a model of disease propagation, deriving an approximate expression for the percolation probability at which a giant component of connected vertices forms. Numerical simulations confirm the analytic results.