A Mathematical Theory of Payment Channel Networks

This study revisits well-known challenges such as channel depletion and capital inefficiency in two-party channels while introducing a unified structural explanation that encompasses these issues. A novel approach proposed by this research is analyzing payment feasibility through feasible wealth distributions instead of isolated transaction paths, offering new insights into liquidity management within the network.

The authors introduce a simple throughput law, formulated as S=c/r, to describe the relationship between off-chain payment bandwidth and both the rate of infeasible payment attempts and onchain transaction capacity. This formulation is instrumental in understanding the operational limits and capabilities of the Lightning Network. Additionally, the paper provides a cut-based characterization of payment feasibility and justifies the structural efficiency of multi-party channels, like coinpools or channel factories, over conventional two-party channels. It also sheds light on how linear asymmetric fee structures can lead to channel depletion, adversely affecting the network's sustainability.

Emphasizing practical implications, the research references a discussion on Ark as an example of a channel factory approach for compressed liquidity management to enhance payment feasibility within the network. The paper suggests strategies to counteract channel depletion, including the adoption of symmetric fees, convex or tiered fee schedules, and coordinated off-chain replenishment, acknowledging the implementation challenges under the current routing model. The necessity for ongoing research and development to overcome these challenges is highlighted.

For those interested in further technical exploration, the paper is supported by code and notebooks available on GitHub. These resources aim to facilitate a more tangible understanding of the discussed concepts and encourage experimentation and feedback from the developer community on the findings' implications for protocol design and the role of multi-party channels.