Newman and Watts analyze the small-world network model, which simulates the transition between regular and random lattice behavior in social networks. They argue that the model exhibits a continuous phase transition with a divergent correlation length as the randomness approaches zero. Using a real-space renormalization group (RG) transformation, they calculate the exact critical exponent and derive the scaling form for the average distance between nodes. They confirm their results with numerical simulations. The small-world model, introduced by Watts and Strogatz, combines regular lattices with random shortcuts, allowing for short distances between nodes. The model's behavior depends on three parameters: system size L, connection range k, and shortcut probability p. For small p and large L, the average distance between nodes scales linearly with L for small systems and logarithmically for large systems. The critical point occurs when the number of shortcuts is about one, leading to a divergence in the correlation length ξ ~ p⁻¹. The authors show that the critical exponent τ is exactly 1 for all k, contradicting previous suggestions of τ = 2/3. They demonstrate this using RG transformations, which preserve the number of shortcuts and renormalize L and p. For k = 1, the RG transformation reduces the system size by half and doubles p, leading to τ = 1. For k > 1, the transformation groups sites into blocks of size k, leading to τ = 1/d for d dimensions. Numerical simulations confirm the scaling behavior, showing that the average distance scales as ℓ = L/k f(pkL). The results are consistent across different dimensions and values of k, demonstrating the model's continuous phase transition. The analysis shows that the small-world model exhibits a critical behavior with a divergent correlation length and a universal scaling form, confirming its relevance to real-world networks.Newman and Watts analyze the small-world network model, which simulates the transition between regular and random lattice behavior in social networks. They argue that the model exhibits a continuous phase transition with a divergent correlation length as the randomness approaches zero. Using a real-space renormalization group (RG) transformation, they calculate the exact critical exponent and derive the scaling form for the average distance between nodes. They confirm their results with numerical simulations. The small-world model, introduced by Watts and Strogatz, combines regular lattices with random shortcuts, allowing for short distances between nodes. The model's behavior depends on three parameters: system size L, connection range k, and shortcut probability p. For small p and large L, the average distance between nodes scales linearly with L for small systems and logarithmically for large systems. The critical point occurs when the number of shortcuts is about one, leading to a divergence in the correlation length ξ ~ p⁻¹. The authors show that the critical exponent τ is exactly 1 for all k, contradicting previous suggestions of τ = 2/3. They demonstrate this using RG transformations, which preserve the number of shortcuts and renormalize L and p. For k = 1, the RG transformation reduces the system size by half and doubles p, leading to τ = 1. For k > 1, the transformation groups sites into blocks of size k, leading to τ = 1/d for d dimensions. Numerical simulations confirm the scaling behavior, showing that the average distance scales as ℓ = L/k f(pkL). The results are consistent across different dimensions and values of k, demonstrating the model's continuous phase transition. The analysis shows that the small-world model exhibits a critical behavior with a divergent correlation length and a universal scaling form, confirming its relevance to real-world networks.