The growing scale and complexity of Internet of Things (IoT) networks present critical
challenges for maintaining and recovering connectivity, especially in dynamic environments
or after large-scale fragmentation caused by disasters. Traditional centralized approaches
often face scalability, resilience, and energy efficiency issues. Hyperbolic geometry,
which has successfully modeled hierarchical and scale-free structures in large-scale
systems such as the Internet's AS topology, offers a promising foundation for addressing
these challenges. This paper introduces hyperbolic trees as a novel framework for
IoT connectivity management. Nodes are embedded in a hyperbolic plane, and connectivity
is maintained or restored through a simple local rule: each node connects to the nearest
node with a smaller radial coordinate. This distributed, asynchronous approach allows
nodes to autonomously reorganize and recover connectivity without centralized coordination,
ensuring scalability and adaptability. We demonstrate that hyperbolic trees provide
a robust, energy-efficient solution for both maintaining and recovering IoT network
connectivity. Numerical results validate the framework's scalability and effectiveness,
highlighting its applicability in scenarios ranging from routine management to disaster
recovery. These findings establish hyperbolic trees as a practical and scalable tool
for resilient IoT connectivity management in next-generation networks.
