Growth Challenges of Scale-Free Networks With Small Power Exponents

Scale-free networks with adjustable degree exponents (γ) are crucial for modeling various real-world systems. In addition, generating them within the (2, 3] interval remains significant. This study proposes an improved preferential attachment model for γ adjustment. Network topological properties, including the mean shortest path length and clustering coefficient, are analyzed through simulations. Vulnerability to targeted attacks is also assessed. Smaller γ values lead to networks with shorter mean shortest path lengths and higher clustering coefficients but increased vulnerability to targeted attacks. When γ drops below 2.20, node pairing difficulties during edge formation rise sharply, causing edge establishment failures and impeding network growth. The model successfully generates scale-free networks with tunable γ in the (2, 3] interval, revealing a trade-off between network clustering, connectivity, and robustness. The findings offer insights into the design and analysis of real-world scale-free systems, emphasizing γ′s role in shaping network topology and resilience.