# [Lightning-dev] Towards more reliable payment path finding via probabilistic modeling the uncertainty of channel balance

René Pickhardt r.pickhardt at googlemail.com
Wed Mar 17 12:50:26 UTC 2021

```Dear fellow Lightning Network developers,

I am very pleased to share with you some research progress [0] with respect
to achieving better payment path finding and a better reliability of the
payment process.

TL;DR summary: In payment (multi)path finding use the (multi)paths with the
highest success probability instead of the shortest or cheapest ones.
(multi)path success probability is the product of channel success
probabilities. Given current data crawled on the Network the channel
success probability grows with the capacity of the channel and with smaller
amounts that are to be sent (which is both intuitively obvious).
(Multi)path success probability thus declines exponentially the more
uncertain channels are included.

I understand that the actual payment path finding is not part of the spec
but I think my results should be relevant to the list since:

a) The payment pathfinding is currently based on trial and error approach
which has consequences that have not been studied well in the context of
the Lightning Network
b) All implementations will use some heuristics in order to achieve
pathfinding.
c) Quick path finding is a crucial for a good user experience.
d) The uncertainty of payment paths is frequently quoted as a major
criticism of the Lightning Network (c.f. [1]) and I believe the methodology
of this paper can be used to address this.

The main breakthrough is that  a very simple model that puts the
uncertainty of channel balances at its heart. We belief the uncertainty of
channel balance values is the main reason why some payments take several
attempts and thus take more time.  With the help of probability theory we
are able to define the channel success and failure probabilities and
similarly (multi)path success and failure probabilities. Other Failure
reasons could also be included to the probability distributions.

With the help of crawling small samples of the network we observe that the
probability distributions of the channel balances can be estimated well
with a uniform distribution (which was a little bit surprising) but leads
to surprisingly easy formulas.  We are able to quantify the uncertainty in
the channels and use negative Bernoulli trials to compute the expected
number of attempts that are necessary to deliver a payment of a particular
amount from one node to another participant of the network. This can be
used to abort the trial and error path finding if the probability becomes
to low (expected number of attempts too high)

We can mathematically show what people already knew (and draw conclusions
like the mentioned ones from it):

a) smaller amounts have higher success probabilities
b) the success probability declines exponentially with the number of
uncertain channels in a (multi)path.
c) depending on the payment pair, amount and splitting strategy it can be
decided into how many parts a payment should be split to achieve the
highest success probability.
d) In particular for small amounts splitting almost never makes sense.

We demonstrate that sorting paths by their descending success probability
during the trial and error payment process (instead of currently used
heuristics like fees or route length) and updating the probabilities from
current failures decreases the number of average attempts and produces a
much faster delivery of payments.

Additionally we looked what happened if BOLT14 [2] was implemented or nodes
otherwise would pro-actively rebalance their channels according to previous
research [3] and realized that the observed prior distribution changes from
uniform to normal. This is great as small payments become even more likely
(as one would intuitively assume and as previously showed) Our results show
that probabilisitic path finding on a rebalanced network works even better
(as in fewer failed attempts) which is yet another hint why BOLT14 might be
a good idea. However as mentioned the results can be implemented even
without BOLT14 or without other protocol changes by any implementation.

One consequence from the paper that is not discussed heavily within the
paper that I find pretty interesting is that if implementations follow the
recommendation to use a probabilistic approach they will tend to route
payments along high capacity channels. While the fee based routing can
easily be gamed by dumping fees it is much harder to provide more
liquidity. And if done this would actually provide a service to the
network. This means that nodes which provide a lot of liquidity and thus
utility might be able to charge higher fees (as long as they are small
enough so that users are willing to pay them) which would probably allow
the emergence of a real routing fee market.

One note on the question of MPP: In the last couple weeks I have been
collaborating with Christian Decker. I belief (by using the methodology
from this paper) to also have a definite solution to the question of:

How to split a payment into k parts and how many funds to allocate to each
path to increase the (multi)path success probability.

While this is is not addressed in the attached paper as we still need to
run evaluations I can already share that an equal sized split as used in
the paper (and by some implementations) is not preferable as one can easily
see from this example:
Imagine one is to deliver 100 satoshi and has to paths with 1 uncertain
channel on each path. The first of capacity 101 and  the second of 51.
Obviously trying to send 100 satoshi along the 101 capacity channel is bad.
Similarly splitting 50/50 and sending 50 Satoshi along the 51 satoshi
capacity channel is also bad. Thus a split that allocates for example 67
Satoshi to the 101 capacity and 33 to the 51 satoshi channel seems way more
reasonable. Actually 75/25 would probably be the best solution for such a
setting. And no it is only random coincident that a binary splitter would
have arrived at that split eventually (after potential miss trials)
There is way more math theory of how to actually solve the optimization
problem in the general case and how to find a split and paths that
maximizes the probability of the attempts. I cannot share these results yet
but I am pretty confident that there will be updates on that end very soon.

with kind regards Rene

[0]: https://arxiv.org/abs/2103.08576 Security and Privacy of Lightning
Network Payments with Uncertain Channel Balances
[1]:
https://www.whatbitcoindid.com/podcast/peter-rizuns-lightning-critique-fud-or-fair
[2]: https://github.com/lightningnetwork/lightning-rfc/pull/780
[3]:
https://lists.linuxfoundation.org/pipermail/lightning-dev/2019-December/002406.html

--
https://www.rene-pickhardt.de

Skype: rene.pickhardt
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