Small-world networks are networks that exhibit properties of both regular lattices and random graphs. They are characterized by a small diameter (shortest path between any two nodes) and high clustering. These networks are found in many real-world systems, such as social, economic, technological, biological, and physical systems. The study identifies three classes of small-world networks: (a) scale-free networks, where the connectivity distribution follows a power-law decay; (b) broad-scale or truncated scale-free networks, where the connectivity distribution has a power-law regime followed by a sharp cut-off; and (c) single-scale networks, where the connectivity distribution has a fast decaying tail. Scale-free networks emerge in growing networks where new nodes connect preferentially to already connected nodes. However, the emergence of scale-free networks can be influenced by constraints that limit the addition of new links. These constraints include the aging of nodes, which limits the number of new connections a node can receive, and the cost of adding links, which limits the number of connections a node can have due to physical or capacity limitations. The study analyzes several real-world networks, including the electric-power grid, world airports, movie-actor collaborations, and social networks. The connectivity distributions of these networks are found to follow different patterns, with some showing power-law tails and others showing exponential or Gaussian decays. The results suggest that the presence of constraints is a key factor in determining the class of network. The study also draws an analogy between the connectivity distributions of small-world networks and the distribution of droplet sizes in critical phenomena. At the critical point, the distribution is scale-free, but as the system moves away from the critical point, the distribution becomes broad-scale or single-scale, depending on the constraints. The study concludes that the presence of constraints is a key factor in determining the class of small-world network, and that these constraints can significantly influence the connectivity distribution of the network.Small-world networks are networks that exhibit properties of both regular lattices and random graphs. They are characterized by a small diameter (shortest path between any two nodes) and high clustering. These networks are found in many real-world systems, such as social, economic, technological, biological, and physical systems. The study identifies three classes of small-world networks: (a) scale-free networks, where the connectivity distribution follows a power-law decay; (b) broad-scale or truncated scale-free networks, where the connectivity distribution has a power-law regime followed by a sharp cut-off; and (c) single-scale networks, where the connectivity distribution has a fast decaying tail. Scale-free networks emerge in growing networks where new nodes connect preferentially to already connected nodes. However, the emergence of scale-free networks can be influenced by constraints that limit the addition of new links. These constraints include the aging of nodes, which limits the number of new connections a node can receive, and the cost of adding links, which limits the number of connections a node can have due to physical or capacity limitations. The study analyzes several real-world networks, including the electric-power grid, world airports, movie-actor collaborations, and social networks. The connectivity distributions of these networks are found to follow different patterns, with some showing power-law tails and others showing exponential or Gaussian decays. The results suggest that the presence of constraints is a key factor in determining the class of network. The study also draws an analogy between the connectivity distributions of small-world networks and the distribution of droplet sizes in critical phenomena. At the critical point, the distribution is scale-free, but as the system moves away from the critical point, the distribution becomes broad-scale or single-scale, depending on the constraints. The study concludes that the presence of constraints is a key factor in determining the class of small-world network, and that these constraints can significantly influence the connectivity distribution of the network.