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Pricing for Routing and Flow-Control in Payment Channel Networks

Suryanarayana Sankagiri, Bruce Hajek

TL;DR

DEBT control, a joint routing and flow-control protocol that guides a payment channel network towards an optimal operating state for any steady-state demand, is introduced.

Abstract

A payment channel network is a blockchain-based overlay mechanism that allows parties to transact more efficiently than directly using the blockchain. These networks are composed of payment channels that carry transactions between pairs of users. Due to its design, a payment channel cannot sustain a net flow of money in either direction indefinitely. Therefore, a payment channel network cannot serve transaction requests arbitrarily over a long period of time. We introduce \emph{DEBT control}, a joint routing and flow-control protocol that guides a payment channel network towards an optimal operating state for any steady-state demand. In this protocol, each channel sets a price for routing transactions through it. Transacting users make flow-control and routing decisions by responding to these prices. A channel updates its price based on the net flow of money through it. The protocol is developed by formulating a network utility maximization problem and solving its dual through gradient descent. We provide convergence guarantees for the protocol and also illustrate its behavior through simulations.

Pricing for Routing and Flow-Control in Payment Channel Networks

TL;DR

DEBT control, a joint routing and flow-control protocol that guides a payment channel network towards an optimal operating state for any steady-state demand, is introduced.

Abstract

A payment channel network is a blockchain-based overlay mechanism that allows parties to transact more efficiently than directly using the blockchain. These networks are composed of payment channels that carry transactions between pairs of users. Due to its design, a payment channel cannot sustain a net flow of money in either direction indefinitely. Therefore, a payment channel network cannot serve transaction requests arbitrarily over a long period of time. We introduce \emph{DEBT control}, a joint routing and flow-control protocol that guides a payment channel network towards an optimal operating state for any steady-state demand. In this protocol, each channel sets a price for routing transactions through it. Transacting users make flow-control and routing decisions by responding to these prices. A channel updates its price based on the net flow of money through it. The protocol is developed by formulating a network utility maximization problem and solving its dual through gradient descent. We provide convergence guarantees for the protocol and also illustrate its behavior through simulations.

Paper Structure

This paper contains 36 sections, 4 theorems, 20 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Payment Channels: Basic Principles
  3. Construction and Basic Operation
  4. Channel Capacity and Balances
  5. Flows
  6. The Balance Condition for Flows
  7. On-Chain Rebalancing
  8. Routing in Payment Channel Networks: A Review
  9. Our Contribution
  10. A Discrete-Time Model of a Payment Channel Network
  11. The Nature of Transaction Requests
  12. The Order of Events
  13. Paths and Flows
  14. Feasibility of Flows and State-Change Equations
  15. Detailed Balance Flows
  16. ...and 21 more sections

Key Result

Lemma 1

Let $D(\lambda)$ be the function as defined in eq:dual_function. The subdifferential set of $D(\lambda)$ is given by where $F(\lambda) \triangleq \arg \max_{f \in A} L(f, \lambda)$ is the set of all flow vectors that maximize the Lagrangian, given $\lambda$.

Figures (5)

  • Figure 1: Dynamic routing ensures perennial operation in a PCN with circulation demand: an illustration of the effect of the DEBT control protocol in the PCN described in Section \ref{['sec:routing_example']}, (a) without the regularizer and (b) with the regularizer.
  • Figure 2: Deadlock prevention via flow-control: an illustration of the effect of the DEBT control protocol in the PCN described in Section \ref{['sec:flow_control_example']}, (a) without the regularizer and (b) with the regularizer.
  • Figure 3: Behavior of the five node PCN with a stepsize of $\gamma = 0.01$; (a) shows flows as a function of time; (b) shows the times at which channel resets occur.
  • Figure 4: Behavior of the five node PCN with a stepsize of $\gamma = 0.1$; (a) shows flows as a function of time; (b) shows the times at which channel resets occur.
  • Figure 5: Flows in the five node PCN with time-varying demand; (a) when the demand at each step is independent and Poisson distributed; (b) when the demand is piece-wise constant, with a sudden change in the middle of the time horizon.

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Proposition 1
  • proof