The paper by Cristopher Moore and M. E. J. Newman explores the dynamics of disease transmission on small-world networks, focusing on the role of network structure in epidemic behavior. They investigate models where either the probability of infection or transmission is varied, and both probabilities are adjusted to study the threshold for epidemic onset. The authors provide exact solutions for the percolation threshold in various cases and confirm their results through numerical simulations. In the introduction, they highlight the importance of social network structure in disease propagation, noting that small-world networks, characterized by a high clustering coefficient and short average path lengths, facilitate faster disease spread compared to regular lattices. They discuss early models based on random graphs and introduce the Watts–Strogatz model, which combines clustering and short average path lengths. The paper delves into site percolation, where the probability of infection is the key parameter, and bond percolation, where the probability of transmission is the focus. For site percolation, they derive exact expressions for the percolation threshold in one-dimensional small-world graphs, showing how it depends on the density of shortcuts and the susceptibility of individuals. For bond percolation, they provide a more complex analysis for higher-dimensional lattices, particularly for \( k = 2 \). The authors also consider the combined case of both site and bond percolation, where both susceptibility and transmissibility are arbitrary. They perform extensive numerical simulations to validate their analytic results, demonstrating that epidemics occur when the system is above the percolation threshold. The simulations show that epidemics spread more quickly and affect a larger proportion of the population in more susceptible populations. Finally, the paper concludes with a discussion of the implications of their findings and the potential for extending their method to other small-world models.The paper by Cristopher Moore and M. E. J. Newman explores the dynamics of disease transmission on small-world networks, focusing on the role of network structure in epidemic behavior. They investigate models where either the probability of infection or transmission is varied, and both probabilities are adjusted to study the threshold for epidemic onset. The authors provide exact solutions for the percolation threshold in various cases and confirm their results through numerical simulations. In the introduction, they highlight the importance of social network structure in disease propagation, noting that small-world networks, characterized by a high clustering coefficient and short average path lengths, facilitate faster disease spread compared to regular lattices. They discuss early models based on random graphs and introduce the Watts–Strogatz model, which combines clustering and short average path lengths. The paper delves into site percolation, where the probability of infection is the key parameter, and bond percolation, where the probability of transmission is the focus. For site percolation, they derive exact expressions for the percolation threshold in one-dimensional small-world graphs, showing how it depends on the density of shortcuts and the susceptibility of individuals. For bond percolation, they provide a more complex analysis for higher-dimensional lattices, particularly for \( k = 2 \). The authors also consider the combined case of both site and bond percolation, where both susceptibility and transmissibility are arbitrary. They perform extensive numerical simulations to validate their analytic results, demonstrating that epidemics occur when the system is above the percolation threshold. The simulations show that epidemics spread more quickly and affect a larger proportion of the population in more susceptible populations. Finally, the paper concludes with a discussion of the implications of their findings and the potential for extending their method to other small-world models.