Implementation of Small-World (SW) and Scale-Free Barabási-Albert (BA) Network Generation Models in Complex Networks

In complex network research, the Small-World (SW) model and the Barabási-Albert (BA) model are two classical network generation models that respectively reveal the characteristics of high clustering and power-law distributions found in real-world networks. The SW Small-World Model: The small-world model establishes a transition between regular and random networks through a rewiring mechanism. The implementation typically begins by generating a regular ring lattice where each node connects to its K nearest neighbors. Subsequently, each edge is randomly rewired with probability p while preserving local clustering properties and introducing long-range connections, resulting in networks with short path lengths. The parameter p controls the "small-worldness"—when p is small, the network approaches a regular structure, while larger p values trend toward random networks. In code implementations, this involves initializing adjacency matrices for ring lattices and performing probabilistic edge rewiring using random number generators. The BA Scale-Free Model: The BA model generates networks with power-law degree distributions through growth and preferential attachment mechanisms. Starting with a small initial set of nodes, the network grows incrementally with new nodes preferentially attaching to existing highly-connected nodes following the "rich-get-richer" principle. This mechanism leads to the emergence of hub nodes, creating scale-free properties. The key aspects of BA model implementation include: 1) dynamic network growth through iterative node addition, and 2) connection probabilities proportional to node degrees. Algorithmically, this requires maintaining degree tables and using roulette-wheel selection for preferential attachment. Both models characterize typical features of social networks (SW) and internet linkages (BA), providing fundamental structures for studying information diffusion, robustness, and other network properties. Code implementations typically involve graph data structures, probability distributions, and iterative algorithms for network evolution.