# Researcher: `Here’s Mathematical Proof the Lightning Network Will Be Centralized´

An article by Jonald Fyookball, which appeared on Medium, demonstrated that Lightning Network may not be as decentralized as some have imagined. The article is noteworthy because it attempts to criticize the problem of Lightning Network from a mathematical perspective. However, even though it goes into rigorous formulation, it also provides an non-mathematical explanation for the layman.

The article suggests the primary problem with bitcoin-based Lightning Networks is that the payment channels it uses necessarily bring about centralization. This is especially a problem, says the author, because lightning network tech is touted by its supporters as a peer-2-peer, decentralized system for conducting off-chain transactions using payment channels.

The article says, “Many in the Bitcoin community mistakenly assumed or were led to believe that the Lightning Network (LN) would be a distributed peer-to-peer network. However, this is infeasible. In fact, even using a generous set of assumptions, we will prove it is mathematically impossible.”

## Problems with Lightning Network for the Layman

The article’s author says the largest problem with Lightning Network’s maintaining a decentralized channel network is mathematical. For the system to sustain itself many users would have to possess a certain amount of bitcoin — and have it divided up — in order to keep channels open. Without this number of users, central hubs would develop in order to facilitate channeled transactions. This is tantamount to turning a decentralized payment network into a centralized one.

The article mentions that a large number of channel hops would be necessary to allow lightning network to function in a decentralized manner. The article seems to imply in order for a hop to occur one user essentially has to borrow money from another user. The problem is that a large number of hops is unlikely. The article demonstrates this issue by showing the channel hops in the form of a tree.

To reach anyone in a big network with a series of branching channel connections, you either need a large number of channels, or a large number of hops. Both are a huge problem. A large number of channels means users have to divide up their funds and can’t do anything except tiny purchases. And a large number of hops means everyone’s money will be tied up routing everybody else’s money.

## Mathematical Proof That Lightning Network Fails

The article’s author then got more technical and provided a mathematical formula to demonstrate how this would occur. This is highly rigorous, and basically used to prove the simplistic explanation that not enough hops or channels would appear in order to provide a more decentralized system.

For instance, the author attempted to prove this by using a mathematical formulation called a SawtoothWave, which attempts to determine when a user receives income and spends it. The article went this direction, because there is no foolproof way to determine the amount of money users will be receiving and sending for the purposes of figuring out if the needed number of hops and channels will arise over time.

Finally, the author introduces more variables and then uses mathematics to continue determine if the Lightning Network system will function as it is supposed to. The author believes bitcoin must remain decentralized and on-chain transactions should be established to prevent centralization.

Remember, Bitcoin must be decentralized. Be wary of the rationalization of “Centralization is ok as long as the base layer is kept decentralized.” That is an insidious trap which allows forcing users off the base layer and into the centralized systems. We must never allow that.

## Rebuttals to the Original Theory

Others, however, disagree with this formulation and provide critiques of the authors methodology. A user named Murch wrote a rebuttal article on Medium. He claimed the model used in the original article was faulty. There were several suspected errors he pointed out in his analysis, including criticisms of how the original author analyzed channels and hopping routes.

He said, “When a LN participant searches for a route, they’re obviously only interested in directed payment capacity. This aspect is correctly represented in the tree. However, the nodes along the route are interconnected. While this could even allow cycles to occur, cycles are not possible in the route construction, as it would allow participants involved multiple times to cut out the cycle when learning the secret for the first time.”

Murch’s rebuttal also went into other problematic areas with regard to the original formulation, but mainly pointed out the original thesis that Lightning Network leads to centralization is necessarily false. The overall arguments are interesting, but in the end no one is yet sure if Lightning Network will lead to centralization or maintain decentralization on the bitcoin network.

Do you believe this mathematical proof is a valid critique of Lightning Network? Let us know what you think in the comments below.

Images via Shutterstock and Medium

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