Newman and Watts study the small-world network model, which mimics social interaction networks. They identify a non-trivial length-scale in the model, analogous to the correlation length in other systems, which diverges continuously as randomness decreases, indicating a normal critical point. This length-scale governs the transition from large- to small-world behavior and the number of vertices in a neighborhood. They derive the critical exponent and the finite size scaling form for the average vertex-vertex distance, and use series expansion and Padé approximants to find an analytic form for the scaling function. They also calculate the effective dimension of small-world graphs, showing it varies with the length-scale, similar to multifractals. They study site percolation on small-world networks as a model for disease propagation, deriving an approximate expression for the percolation probability at which a giant component forms. All results are confirmed by numerical simulations. The model exhibits a continuous phase transition as the density of random connections tends to zero, with the critical exponent τ = 1/d. The effective dimension of small-world graphs depends on the length-scale, and the percolation threshold is found to depend on the shortcut density and network parameters. The results show good agreement between simulations and theory, with the percolation threshold decreasing as the shortcut density increases. The average cluster radius for percolation on small-world graphs follows a scaling form, with the critical exponents depending on the system size and percolation probability. The study confirms the model's ability to mimic real-world social networks and provides insights into the behavior of complex systems.Newman and Watts study the small-world network model, which mimics social interaction networks. They identify a non-trivial length-scale in the model, analogous to the correlation length in other systems, which diverges continuously as randomness decreases, indicating a normal critical point. This length-scale governs the transition from large- to small-world behavior and the number of vertices in a neighborhood. They derive the critical exponent and the finite size scaling form for the average vertex-vertex distance, and use series expansion and Padé approximants to find an analytic form for the scaling function. They also calculate the effective dimension of small-world graphs, showing it varies with the length-scale, similar to multifractals. They study site percolation on small-world networks as a model for disease propagation, deriving an approximate expression for the percolation probability at which a giant component forms. All results are confirmed by numerical simulations. The model exhibits a continuous phase transition as the density of random connections tends to zero, with the critical exponent τ = 1/d. The effective dimension of small-world graphs depends on the length-scale, and the percolation threshold is found to depend on the shortcut density and network parameters. The results show good agreement between simulations and theory, with the percolation threshold decreasing as the shortcut density increases. The average cluster radius for percolation on small-world graphs follows a scaling form, with the critical exponents depending on the system size and percolation probability. The study confirms the model's ability to mimic real-world social networks and provides insights into the behavior of complex systems.