MATLAB Implementation of Small-World Networks and Scale-Free Networks

In the generation and analysis of small-world networks and scale-free networks, MATLAB provides powerful tools and function libraries for constructing these complex networks and calculating their properties. Below we present the fundamental approaches and implementation methods.

### 1. Small-World Network Generation Small-world networks typically exhibit high clustering coefficients and short average path lengths. In MATLAB, the Watts-Strogatz model can be employed to generate small-world networks. The implementation steps for this model are: Initialize a regular network: First create a regular ring lattice where each node connects to its K nearest neighbors. Random rewiring: With probability P, randomly rewire existing edges to different nodes (avoiding self-connections and duplicates) to introduce randomness. Parameter adjustment: By tuning K (number of neighbors) and P (rewiring probability), you can control the network's clustering coefficient and average path length.

### 2. Scale-Free Network Generation Scale-free networks follow a power-law degree distribution and are commonly generated using the Barabási-Albert (BA) model. The implementation method includes: Initial network: Start with a small fully-connected core network (typically m0 nodes). Preferential attachment: Each new node connects to m existing nodes with probability proportional to their current degrees (rich-get-richer mechanism). Growth mechanism: Repeat this process until the network reaches the desired size, implementing the growth and preferential attachment principles.

### 3. Degree Distribution Calculation Degree distribution is a crucial metric for analyzing network properties. The calculation steps involve: Degree counting: Iterate through all nodes to record each node's number of connections (degree). Histogram plotting: Use MATLAB's histogram functions (histcounts or degreeDistribution) to frequency of different degrees and visualize the distribution. For scale-free networks, observe whether it exhibits power-law characteristics using log-log plots.

### 4. Clustering Coefficient Calculation The clustering coefficient measures how closely connected a node's neighbors are to each other. Calculation methods include: Local clustering coefficient: For each node, compute the ratio of exist edges between its neighbors to the maximum possible edges between them. Global clustering coefficient: Take the average of all local clustering coefficients to reflect the overall network's clustering property, implemented using MATLAB's clustering_coef function or custom calculations.

Through these implementation steps, you can generate small-world and scale-free networks in MATLAB, analyze their degree distributions and clustering coefficients, and gain deeper insights into complex network structural properties using MATLAB's graph theory toolbox and network analysis functions.