A Mathematical Theory of Payment Channel Networks

TL;DR

The paper develops a geometric framework for payment channel networks by defining the liquidity polytope $L_G$ and the wealth polytope $W_G$, linking feasibility of off-chain payments to cut capacities and circulations. It shows that two-party channels inherently limit scalability, while multi-party channels (coinpools/channel factories) enlarge cut widths, increasing the set of feasible wealth distributions and reducing infeasible payments. The authors derive throughput bounds $S = \zeta/\rho$, analyze depletion phenomena under linear fees, and propose mitigation strategies including symmetric fees, convex tiered pricing, and coordinated replenishment. Collectively, the work provides a principled basis for adopting multiparty primitives and fee/coordinations schemes to improve reliability and scalability of off-chain payments, while highlighting the fundamental tradeoffs and design challenges. The framework offers concrete tools (ILP formulations, cut-interval analysis, and projection between $L_G$ and $W_G$) for evaluating and guiding protocol design in practice.

Abstract

We introduce a geometric theory of payment channel networks that centers the polytope $W_G$ of feasible wealth distributions; liquidity states $L_G$ project onto $W_G$ via strict circulations. A payment is feasible iff the post-transfer wealth stays in $W_G$. This yields a simple throughput law: if $ζ$ is on-chain settlement bandwidth and $ρ$ the expected fraction of infeasible payments, the sustainable off-chain bandwidth satisfies $S = ζ/ ρ$. Feasibility admits a cut-interval view: for any node set S, the wealth of S must lie in an interval whose width equals the cut capacity $C(δ(S))$. Using this, we show how multi-party channels (coinpools / channel factories) expand $W_G$. Modeling a k-party channel as a k-uniform hyperedge widens every cut in expectation, so $W_G$ grows monotonically with k; for single nodes the expected accessible wealth scales linearly with $k/n$. We also analyze depletion. Under linear, asymmetric fees, cost-minimizing flow within a wealth fiber pushes cycles to the boundary, generically depleting channels except for a residual spanning forest. Three mitigation levers follow: (i) symmetric fees per direction, (ii) convex/tiered fees (effective flow control but at odds with source routing without liquidity disclosure), and (iii) coordinated replenishment (choose an optimal circulation within a fiber). Together, these results explain why two-party meshes struggle to scale and why multi-party primitives are more capital-efficient, yielding higher expected payment bandwidth. They also show how fee design and coordination keep operation inside the feasible region, improving reliability.

Figures (15)

• Figure 1: An Example Network $G$ with corresponding Polytope $L_G$ of liquidity states and liquidity network $\mathcal{L}(G,\lambda)$ for a fixed $\lambda \in L_G$.
• Figure 2: Volumes $\text{vol}(L_G) = \left( \frac{C}{m} + 1 \right)^m$ of state polytopes with equally sized channels for various numbers $C$ of coins and various numbers $m$ of channels.
• Figure 3: $3$-dimensional embedding of the $2$-dimensional polytope of wealth distributions of $C=21$ coins among $n=3$ users.
• Figure 4: Only 52.17% of all wealth distributions in $\mathcal{W}(21,3)$ are feasible if Alice, Bob, and Carol allocate the 21 coins into 2 channels of capacities 10 and 11.
• Figure 5: $2$ feasible regions of $W_G$ for $2$ different networks between Alice Bob and Carol who own in total $21$ coins.

Theorems & Definitions (61)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.2.1