Percolightning: what network theory can say about Bitcoin Lightning channels

Silvia Bartolucci - Centre for Financial Technology, Imperial College Business School, United Kingdom;

Fabio Caccioli - Financial Computing and Analytics Group, Department of Computer Science, University College London, United Kingdom;

Pierpaolo Vivo - Department of Mathematics, King’s College London, United Kingdom.

-- Bitcoin, the pioneering cryptocurrency, has brought about an unprecedented revolution in the payment industry. Despite its traction and success over the last ten years, the original blockchain – the technological infrastructure underlying Bitcoin – suffers from some limitations that may hinder the future growth and adoption of the cryptocurrency. One of the major issues is the scalability of the system: the current number of transactions validated via this platform is between 3 and 7 per second, compared for instance to thousands of transactions handled by the Visa circuit. The lack of scalability is mainly caused by constraints on throughput of transactions, with the block size fixed at 1MB, and by the high latency – with a new block created on average only every ten minutes. Those limitations are imposed to safeguard the security of the platform against malicious attacks and are difficult to relax without major changes to the protocol.

The Lightning Network is a so-called second-layer technology built on top of the Bitcoin blockchain to provide “off-chain” fast payment channels between users. In a nutshell, the idea of a lightning channel is akin to opening a bar tab instead of tapping your credit card at every drink you purchase. Two parties may lock the same amount of money as collateral and may open a channel for a certain period of time. During this time, they can freely exchange money back and forth through the channel, and only the netted transaction will be eventually validated and stored on the Bitcoin blockchain. If one of the parties is malicious and does not correctly update the balance, the other can keep the collateral posted by the malicious party, as a form of insurance. Any two users can open a channel - and all other participants can use one or more existing channels to route transactions off-chain (upon payment of a fee to channel “owners”) – much in the same way as any guest at your table may (with your permission!) charge their drinks on the tab you have opened.

The scalability problem could be solved if a sufficient number of channels are open, so that the Lightning Network spans across the whole pool of users of the Bitcoin blockchain. The Lightning network topology is, indeed, relevant to understanding the resilience of the system to attacks or random failures and its robustness. The topology of the network is in turn driven by users’ economic incentives to relay transactions “off-chain”.

In our paper “A percolation model for the emergence of the Bitcoin Lightning Network”, we investigate under which conditions (blockchain fees, lightning fees, average wealth and volume of transactions per users) a resilient Lightning Network emerges. Moreover, as the “Lightning” fees are set by channels’ owners, an important question is how to set such fees in order to guarantee profits but at the same time providing the right incentives for Bitcoin users to participate in the Lightning Network.

Percolation theory, the mathematical framework to study how fluids “trickle through” porous materials, is applied here to fitness-dependent network models: as in real life, a Lightning channel is opened between two Bitcoin holders with a probability that depends on the "fitness" of the concurring nodes, which in turn depends on wealth and volume of transactions, and on how costly it is to transact on- and off-chain.

Our model includes parameters that could be in principle estimated from publicly available data. It gives quantitative predictions on the conditions (e.g. difference in fees imposed on- and off-chain) leading to the emergence of a large connected component of nodes spanning across the whole network – as opposed to having a multitude of “short-hops” channels that would not allow wealth transfer between “distant” agents.

Our work thus provides a more systemic approach – looking not only at “economic” but also at “structural” properties of the emerging payment network – and offers a complementary (and arguably much called for) take on the classical question “How to charge Lightning?”.

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