Mandala networks: ultra-small-world and highly sparse graphs.

Area of Science:

• Network Science
• Complex Systems
• Infrastructure Security

Background:

• Growing demands for infrastructure security and reliability necessitate optimal complex network topology design.
• Existing network models may not adequately address robustness against failures and malicious attacks.

Purpose of the Study:

• To present a general solution for optimal network layout in scale-free networks.
• To develop systematically constructible networks that are robust against random failures and malicious attacks.
• To analyze network properties like path length and link density for improved communication and economic efficiency.

Main Methods:

• Development of a general network construction method for scale-free networks.
• Analysis of network properties, including shortest path length and link density, as network size increases.
• Introduction of modifications to the original network formulation to enhance resilience against malicious attacks.

Main Results:

• Construction of networks robust against random failures.
• Demonstration of asymptotically invariant shortest paths and vanishing link density in large networks, creating ultra-small world and highly sparse topologies.
• Identification of simple modifications for improved topology against malicious attacks.

Conclusions:

• The proposed network design offers a general solution for optimizing complex network topologies, particularly scale-free networks.
• The resulting networks exhibit desirable properties for communication, navigation, and economic efficiency.
• The study provides a pathway to enhance network resilience against both random failures and targeted attacks.