I'd like to share my latest research notebook, A Geometric Approach for Optimal Channel Rebalancing, which builds upon the mathematical theory of payment channel networks to provide a framework for improving liquidity management in payment channel networks. This work builds on earlier discussions about channel depletion and topology cycles and was hinted as one possible mitigation in this Delving Bitcoin thread.
Motivation
While off-chain rebalancing does not affect the expected rate of payment feasibility in theory, in practice, unbalanced channels lead to frequent routing failures even if the payment was feasible. This notebook investigates whether structured, globally optimal replenishment can mitigate this issue—even when starting from a severely depleted network.
Method
The approach formulates liquidity balancing as a constrained optimization problem (see appendix at the end of the post for some more math details):
- Objective: Find the closest feasible liquidity state to a target (e.g., balanced channels)
- Solution: This is computationally intensive but provides a theoretical upper bound on how well rebalancing could work under ideal coordination.
Results
The most illustrative case starts from a heavily depleted network, where only 11.31% of channels have balanced liquidity (both peers own between 40–60% of the channels' liquidity). After optimization:
- Balanced channels increase to 56.30% (from 11.31%)
- Moderately imbalanced channels (both peers own between 10–90% of the channels' liquidity) rise to 95.24% (from 41.52%)
The optimizer suggests to shift 33.23% of the total network capacity to achieve this.
The novelty about this method is that instead of rebalancing local cycles we try to find a state that satisfies the liquidity wishes of all peers on the network.
The above mentioned results can be seen in the CDF diagram where I depicted the relative liquidity peers own in channels before and after running the channel replenishment code.
Interpretation: The blue curve shows how before rebalancing almost half of all channels are heavily depleted with the liquidity either at 0 sats or 100% of the capacity of the channel. The distribution after rebalancing shows how optimization pulls channels away from extreme depletion (near 0% or 100%) toward balanced states.
Caveats and Future Work
This is a theoretical benchmark, not a deployable protocol. Key limitations include:
- Central coordination requirement: The current method assumes full network visibility.
- Computational cost: Solving QPs and ILPs at scale may be impractical.
- Unknown desired state: Currently the desirable target state is all channels should be 50:50. Of course it is rather easy to adopt the code to fulfill the wishes of node operators as long as those wishes reflect their locally owned liquidity.
Open questions for the community:
- How to find an agent based approximation? I haven't compared it yet to prior research of mine where I suggested a greedy version for rebalancing.
- Is it even worthwhile to continue to think in this direction. It is my understanding that many node operators assume liquidity management via fees is the way to go
- How might nodes negotiate target states without a central coordinator?
Try the Notebook
The code is available for experimentation, including:
- Network simulation with depletion/rebalancing
- Visualization tools (e.g., CDFs of liquidity states)
Feedback is welcome and highly appreciated.
Appendix: Mathematical Details
Building upon the framework established in the mathematical theory of payment channel networks, we recall that for any given liquidity state
\lambda\in\mathbb{Z}^{2m} (where m is the number of channels), there exists a corresponding wealth distribution \omega\in\mathbb{Z}^n (where n is the number of nodes) computable via a projection \pi. The equivalence class [\lambda] comprises all liquidity states yielding identical wealth distribution \omega, with its cardinality equal to the number of strict circulations on the liquidity graph representing the current state \lambda.The channel replenishment problem is equivalent to finding a circulation that shifts the liquidity to a more desirable state (e.g. less depletion). This problem reduces to selecting an optimal element from
[\lambda]. As demonstrated previously, feasible solutions can be obtained by solving a system of linear Diophantine equations subject to two constraint classes:-
Conservation of Liquidity: m constraints enforcing channel capacity bounds
\forall(u,v) \in E: \lambda(u,v) + \lambda(v,u) = c(u,v)wherec(u,v)denotes the capacity of channel(u,v) -
Conservation of Wealth: n constraints preserving nodal wealth
\forall u \in V: \sum_{v\in N(u)} \lambda(u,v) = \omega_u
The solution space forms a convex polytope
P \subset \mathbb{Z}^{2m} \cap [0, c(u,v)]^{2m}, which may be empty for certain (\omega, G) pairs due to the intersection with the capacity hypercube.Our optimization approach introduces a target liquidity state
x_0 \in \mathbb{Z}^{2m}, representing the desired channel balance configuration. While we typically select x_0 where each node holds exactly half of each channel's capacity (xo_{u,v} = c(u,v)/2), the framework accommodates arbitrary target states. The optimal replenishment problem then becomes:\text{minimize } \|x - x_0\|_2
\text{subject to } x \in P \cap \mathbb{Z}^{2m}We solve this via a two-phase approximation:
-
Continuous relaxation: Compute
x_\rho = \arg\min \|x - x_0\|_2overP \subset \mathbb{R}^{2m}using quadratic programming. -
Integer approximation: Restrict search to the
δ-neighborhood ofx_\rhoB_\delta(x_\rho) = x \in \mathbb{Z}^{2m} | \|x - x_\rho\|_\infty \leq \delta\where\delta = \left|\sqrt{\frac{\|x_\rho - x_0\|_2}{m}}+1\right|
The choice of
δ ensures the neighborhood contains feasible integer solutions while maintaining computational tractability. This procedure is implemented in the accompanying code, which operationalizes the theoretical framework through constrained optimization techniques.this article was originally shared in Delving by @renepickhardt
dup: #1003320LaTeXwritings, ideally making a backup of the whole article... just in casenofollow norefer noopeneranyway... source and author are mentioned and linked at the endnofollow norefer noopeneruntil they have reached a certain sat threshold of cost-to-post + zaps.