Neighborhood properties of complex networks

Area of Science:

• Network Science
• Graph Theory
• Complex Systems

Background:

• Understanding network topology is crucial for analyzing complex systems.
• Existing methods for defining network neighborhoods can be limited.
• Characterizing higher-order network structures requires novel approaches.

Purpose of the Study:

• To introduce a new concept of network neighborhood based on minimal path lengths.
• To generate families of higher-order networks from existing complex networks.
• To investigate the spectral properties and invariance of these generated network families.

Main Methods:

• Defining higher-order networks (Rl) where vertices l steps apart in R1 are 1 step apart in Rl.
• Utilizing Boolean operations on adjacency matrices (Ml) to generate these higher-order networks.
• Analyzing the spectra of the adjacency matrices for different network families.

Main Results:

• Generated families of higher-order networks for linear and Erdös-Renyi networks.
• Demonstrated spectral invariance for these families, apart from finite size effects.
• Identified a new invariant family originating from small-world networks.

Conclusions:

• The proposed method effectively generates higher-order networks with preserved spectral properties.
• This approach provides a new perspective on network invariance and structure.
• The findings have implications for understanding and classifying complex network architectures.