A Mathematical Theory of Payment Channel Networks

In a recent scholarly effort, a new paper published on arXiv presents a comprehensive geometric framework to understand several critical phenomena observed in payment channel networks, particularly focusing on the Lightning Network. This paper not only revisits familiar issues such as channel depletion, capital inefficiency in two-party channels, and the advantages of channel factories but also endeavors to connect these phenomena under a unified structural explanation. A significant assertion made is the reevaluation of payment analysis through feasible wealth distributions rather than isolated paths, introducing a novel perspective on liquidity and payment feasibility within the network.

The research outlines a simple throughput law (S=c/r), which correlates off-chain payment bandwidth with the rate of infeasible payment attempts and onchain transaction capacity. This insight is crucial for grasping the operational limits and potential of the Lightning Network. Moreover, the paper elaborates on a cut-based characterization of payment feasibility and provides a formal rationale behind the structural efficiency of multi-party channels (or coinpools/channel factories) over traditional two-party setups. It also offers a geometric explanation for the tendency of linear asymmetric fees to induce channel depletion, a phenomenon that significantly impacts network sustainability.

Supporting these theoretical advancements, the paper cites practical implications and discussions, including a recent article exploring Ark as a channel factory. This discussion extends into the realm of improving payment feasibility and managing liquidity more efficiently within the network. Furthermore, it addresses strategies to mitigate channel depletion, like symmetric fees, convex/tiered fee schedules, and coordinated off-chain replenishment. The challenges of implementing these strategies under the current routing model are acknowledged, highlighting the need for continued research and development in this area.

The author encourages engagement and feedback from the community, especially regarding the implications of these findings on protocol design and the role of multi-party channels. For those interested in a deeper dive into the technical aspects, the paper is accompanied by supporting code and notebooks, facilitating a tangible understanding of the concepts discussed and inviting further exploration and experimentation within the developer community.