This post is an introductory overview of the core graph algorithm behind Cycles Protocol.

What is the problem, exactly?

Payment interactions in closely-knit communities often involve cycles. e.g.

* Alice pays Bob $100 * Bob pays Charlie $80 * Charlie pays Alice $70 Total liquidity used for settlement: $100 + $80 + $70 = $250

When these payments involve credit (e.g. IOUs/obligations) such cycles can be cleared and the payments can be settled with lesser liquidity. That is why such mechanisms are known as liquidity-saving mechanisms (LSMs). e.g.

* Alice owes Bob $100 * Bob owes Charlie $80 * Charlie owes Alice $70 If we clear the $70 cycle between them, this can be settled as -> * Alice pays Bob $30 * Bob pays Charlie $10 Total liquidity used for settlement: $30 + $10 = $40 Liquidity saved: $250 - $40 = $210

Note the preconditions for this to work -

• Closely-knit communities with lots of obligations as that increases the chances of having cycles.

• Some form of deferred payments (based on credit).

But to be able to do this at a national or international level, we would have to find cycles in a network of enormous proportions - think Splitwise but for tens of millions of people.

Is finding cycles in a large payment network our problem?

Let’s look at a slightly more complex example →

* Alice owes Bob $100 * Bob owes Eve $70 * Eve owes Alice $100 * Charlie owes Dave $100 * Dave owes Eve $100 * Eve owes Charlie $70 * Bob owes Charlie $100 It is obvious that there are two $70 cycles here -> * Alice owes Bob owes Eve owes Alice * Charlie owes Dave owes Eve owes Charlie But look closely, there's also a $100 cycle -> * Alice owes Bob owes Charlie owes Dave owes Eve owes Alice So clearly there are multiple solutions to this cycle-finding problem!

Depending on the set of cycles we choose to clear we would end up with different amounts of liquidity saved (i.e. $420 v/s $500).

💡 So, the real problem is to clear the right cycles to maximize the liquidity saved.

Designing a solution

Let’s first try to visualize the network as that could give us some intuition for how to approach the problem.

Visualizing the obligation network as a graph

A single obligation can be expressed as Alice owes $100 to Bob →

So, by extension →

Turns out we’re dealing with a graph and that makes it way easier to find cycles as there are highly optimized algorithms available for this purpose. But it still isn’t clear how we can maximize liquidity savings.

Some important observations

Let’s take a closer look at our example graph and make some observations.

• Notice that some users have an excess of liquidity whereas others are in deficit.

  • For e.g. only Alice owes more than she is owed. i.e. Alice owes $100 and is owed $70, so the difference is $30.

  • Bob and Charlie are both owed more than they owe others. (differences are $20 and $10 respectively).

⚡ This excess/deficit is called the net position of a user.

• The sum of positive net positions is equal to the sum of negative net positions (both being $30 in our case).

⚡ The sum of all positive/negative net positions gives us the net internal debt (NID) of a network.

Reduction to a max-flow problem

With these observations let’s take a step back and visualize the entire obligation graph.

Notice that the NID is the real debt in the system. It is also the maximum amount of liquidity that can be pushed into the system. The additional dotted arrows are basically demonstrating the net positions of users in the graph. They are the excess and deficit liquidity coming into and going out of the graph.

Intuitively, the NID is all the liquidity that should be required for a network to clear all its obligations. In other words, if we were to route the NID through the obligation graph then the remaining obligations should clear themselves (i.e. the remaining obligations should be in cycles).

Is this a coincidence? Not really, this is a known lemma in network flows theory →

🧪 Flow decomposition: We can decompose any feasible flow f on a network G into at most m cycles and s-t paths.

Notice we introduced two imaginary nodes s and t - from the perspective of the graph, all that matters is that these arrows are coming from (or going) outside the graph and therefore these additional nodes should have no effect on the problem description.

So, we can now reframe our problem as follows →

Given a graph, how much flow can you push from s to t while respecting the following constraints →

• For any edge, flow (f) cannot exceed its capacity.

• For any node (except s and t), the flow must be conserved (i.e. fin == fout).

Actually, that’s a well-known graph problem known as the max-flow problem!

Putting it all together

We can solve the obligation-clearing problem by →

1. Balancing the graph so that it respects the flow conservation constraint. We do this by calculating the net position of each user and connecting them to either the source or sink depending on whether their net position was positive or negative.

2. Running the max-flow algorithm over the balanced graph to route NID from s to t, to find the flow paths.

3. The remaining paths will be cycles consisting of (potentially reduced) obligations.

References

• Mathematical Foundations for Balancing the Payment System in the Trade Credit Market

• Network Flows: 6.2 lemma 1