Throughput Law in Off-Chain Payment Networks

• The paper establishes a quantitative relationship where sustainable off-chain throughput S is defined as ζ/ρ, linking on-chain settlement capacity with the rate of infeasible transactions.
• It models liquidity states and wealth distributions using network cut-intervals and multi-party channel topologies to derive critical performance bounds.
• The work highlights design strategies such as optimized fee structures and multi-party channels to enhance capital efficiency and scale off-chain payment networks.

The throughput law for off-chain payment channel networks establishes a quantitative linkage between sustainable transaction bandwidth in the off-chain layer and the base layer’s settlement capacity. It is predicated on the geometry of feasible wealth distributions within the network, incorporating the structure of liquidity states, cut-intervals, multi-party channel topologies, and fee dynamics. The core result, proven in "A Mathematical Theory of Payment Channel Networks" (Pickhardt, 8 Jan 2026), expresses sustainable off-chain payment bandwidth $\mathcal S$ as a function of on-chain settlement bandwidth %%%%1%%%% and the expected fraction of infeasible off-chain payments $\rho$: $\mathcal S = \frac{\zeta}{\rho}$ This formalism delivers a precise criterion for the scale at which an off-chain payment system can operate without overtaxing on-chain resources.

1. Mathematical Definitions and Network Model

The framework analyzes a fixed payment-channel network $G=(V,E,\{c_e\})$:

• $|V|=n$: Number of nodes.
• $|E|=m$: Number of edges.
• $\{c_e\}$: Capacity assigned to each edge, with total capital $C=\sum_e c_e$.

A liquidity state is formulated as a function $$\lambda: E \times V \rightarrow \{0, 1, \ldots, C\}$$, subject to the local conservation constraint $\lambda(e,u) + \lambda(e,v) = c_e$ for each undirected edge $e = \{u,v\}$. The set of all such states is denoted $L_G$, which is combinatorially an $m$-dimensional integer hyperbox. The projection $\pi: L_G \to W_G$ maps liquidity states to wealth distributions $\omega: V \to \mathbb N_0$ with total sum $C$, where $W_G \subseteq \mathcal W(C,n)$ is the polytope of wealth vectors realizable off-chain as a valid distribution of channel balances.

A payment of amount $a$ from node $i$ to node $j$ modifies the wealth vector $w \in W_G$ by $w' = w - a b_i + a b_j$, with the payment deemed feasible if $w' \in W_G$ post-transfer.

2. The Throughput Law and Its Derivation

Let $\zeta \in \mathbb R_+$ indicate the base layer’s on-chain settlement bandwidth (transactions/sec). Let $\rho \in [0,1]$ represent the expected proportion of infeasible off-chain payment attempts, for a stationary demand model over amounts and endpoints. If the off-chain network issues $N$ payments, approximately $\rho N$ will fail and incur on-chain fallback operations.

The law is derived as follows:

• Off-chain payment attempts per second: $\mathcal S$.
• Fraction $\rho$ fail, thus requiring at least $\mathcal S \rho$ on-chain operations/sec.
• Imposing the hard constraint $\mathcal S\rho \leq \zeta$, the maximum sustainable off-chain bandwidth is

$\mathcal S = \frac{\zeta}{\rho}$

Under adaptively throttled demand, this equality is tight and precludes backlog or dropped on-chain requests.

3. Polytope Geometry and Cut-Interval Characterization

The space of feasible off-chain wealth distributions $W_G$ is geometrically embedded within the simplex $\mathcal W(C,n)$. For any $S \subset V$ ($S\neq \emptyset$, $S\neq V$), the cut capacity is $C(\delta(S)) = \sum_{e \in \delta(S)} c_e$, where $\delta(S)$ comprises edges crossing from $S$ to its complement. The cut-interval lemma establishes: $$\sum_{e\in E[S]}c_e \leq \sum_{v\in S} \omega(v) \leq \sum_{e\in E[S]}c_e + C(\delta(S))$$ The width $C(\delta(S))$ defines the feasible range for the wealth of $S$. Liquid transfers across $S$ must not violate this constraint to remain feasible, and max-flow/min-cut arguments determine the largest possible payment size across a given partition.

This geometric viewpoint directly connects infeasibility rates $\rho$ to the narrowness of cut-intervals: widening every cut reduces $\rho$ and increases throughput.

4. Multi-Party Channels and Topological Effects

Expanding to $k$-party channels (coinpools, channel factories), channel hyperedges of uniform capacity $c$ increase cut widths:

• Any $k$-party channel crossing $(S, \bar S)$ increments $C(\delta(S))$ by $c$.
• For $m$ random $k$-subsets as channels, the crossing probability is $$q_k(s) = 1 - \frac{\binom{s}{k} + \binom{n-s}{k}}{\binom{n}{k}}$$, and expected cut capacity is $m c q_k(s)$.
• $q_k(s)$ is monotonic in $k$: larger $k$ yields wider cuts, a larger wealth polytope $W_G$, a lower expected infeasible rate $\rho$, and so higher $\mathcal S$.
• For single-node sets ($s=1$), expected accessible wealth scales linearly with $k/n$.

In the limit $k=n$ (all nodes in one channel), all cuts are crossed and $W_G$ spans the entire simplex $\mathcal W(C,n)$, so $\rho=0$ and the throughput law yields unbounded off-chain bandwidth subject to other systemic limits.

5. Fee Design and Liquidity Depletion Dynamics

While the throughput law omits fee effects, practical channel depletion is strongly fee-dependent:

• Linear asymmetric fees prompt routing algorithms to pursue minimum-cost cycles, driving liquidity to the boundary of $W_G$ and depleting most channels apart from a residual spanning forest, thereby tightening cut intervals and increasing $\rho$.
• Symmetric fees (identical per direction) nullify directional arbitrage and cycle depletion pressures.
• Convex/tiered fees (scarcity pricing) establish fee functions increasing with local liquidity scarcity. This yields convex cycle potentials, enabling strictly interior optimality and inhibiting total depletion; wider cuts persist, lowering $\rho$. Realizing such fee structures necessitates source-routing or fee-quote mechanisms reflecting instantaneous state $\lambda$, as liquidity-dependent fees are locally hidden.

These dynamics play a critical role in the long-run capital efficiency and reliability of the network, by stabilizing operation within the feasible polytope.

6. Modeling Assumptions and Scope

The preceding analysis hinges upon several modeling conventions:

• The network topology $G$ and channel capacities $c_e$ are static throughout.
• Per-hop base fees and HTLC limits are omitted in the feasibility analysis.
• Payment demand distribution is stationary and stochastic, inducing a well-defined $\rho$.
• On-chain throughput $\zeta$ is a hard constraint on new channel operations per second.
• No ex-ante liquidity probing or selection to avoid infeasible attempts; $\rho$ is assessed strictly ex post.

Within these bounds, the throughput law encapsulates the fundamental constraint linking off-chain performance to on-chain limitations.

7. Capital Efficiency and Scaling Implications

The law $\mathcal S = \zeta/\rho$ unifies off-chain and on-chain constraints through the infeasibility rate $\rho$, determined by the geometry of $W_G$ (via cut intervals) and demand models. Capital efficiency levers—multi-party channels and refined fee designs—operate by either enlarging $W_G$ or regulating liquidity movement within the pre-image fibers $\pi^{-1}(w)$ to limit depletion and thereby shrink $\rho$. Achieving payment bandwidths comparable to traditional retail networks, e.g., Visa-scale with $\zeta \approx 7$ tx/s on Bitcoin, requires driving $\rho$ to near zero via topological optimization (coinpools, factories, mesh enrichment) and liquidity management (symmetric or convex fee design, coordinated replenishments).

Key attributes and their effects can be summarized as follows:

Mechanism	Influence on $W_G$	Effect on $\rho$ / Throughput
Multi-party channels	Enlarge polytope, widen cuts	$\downarrow\rho$, $\uparrow\mathcal S$
Linear asymmetric fees	Deplete liquidity, tighten cuts	$\uparrow\rho$, $\downarrow\mathcal S$
Symmetric/convex fees	Stabilize liquidity, balance cycles	$\downarrow\rho$, $\uparrow\mathcal S$

The throughput law affords a rigorous design target for future off-chain network architectures and fee regimes aimed at maximizing reliability and transaction volume within sustainable capital and settlement bounds (Pickhardt, 8 Jan 2026).