Uncovering the hidden structure of small-world networks

The small-world (SW) network model introduced by Watts and Strogatz has had a tremendous impact on the study of complex systems, leading to the emergence of network science as an interdisciplinary field. While the Newman-Watts model is commonly used to analyze SW networks by adding a few randomly placed shortcuts to a regular lattice. We carefully study related previous works and conclude that scaling of different relevant quantities is not convincing. Here, we demonstrate that the small-world property arises mainly from the existence of clusters of nodes linked by shortcuts rather than just the mean number of shortcuts. Introducing the mean degree of clusters linked by shortcuts as a new parameter resolves the scaling ambiguity, providing a more accurate description of the network. Our findings provide a new framework for analyzing SW networks, emphasizing the importance of considering emergent structures in complex systems. We also construct a phase diagram of the crossover transition from the small to the large world, yielding valuable insights into the nature of complex networks and highlighting the power of emergence in shaping their behavior.