Scaling of weighted spectral distribution in deterministic scale-free networks

• A new deterministic model of scale-free networks is constructed.
• The WSD in the deterministic model grows sublinearly with network size.
• The WSD of a high-degree node provides a sensitive discrimination.
• The WSD depends on the connection relationships between small-degree nodes.
• We study the effects of local world, node deleting and assortativity adjustment.

Scale-free networks are abundant in the real world. In this paper, we investigate the scaling properties of the weighted spectral distribution in several deterministic and stochastic models of evolving scale-free networks. First, we construct a new deterministic scale-free model whose node degrees have a unified format. Using graph structure features, we derive a precise formula for the spectral metric in this model. This formula verifies that the spectral metric grows sublinearly as network size (i.e., the number of nodes) grows. Additionally, the mathematical reasoning of the precise formula theoretically provides detailed explanations for this scaling property. Finally, we validate the scaling properties of the spectral metric using some stochastic models. The experimental results show that this scaling property can be retained regardless of local world, node deleting and assortativity adjustment.