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Vorbit SSF with circular and spiral finality: validator selection and distribution

This post analyzes how cumulative finality accrues across committees in single-slot finality (SSF) for a wide range of validator sets, with a key focus on committee design.

https://ethresear.ch/t/vorbit-ssf-with-circular-and-spiral-finality-validator-selection-and-distribution/20464

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A moderately consolidated validator set after EIP-7251 may have 233k validators. Say it is divided into 8 committees of 29k validators each to accommodate signature propagation. Each SSF committee will then cumulatively add finality to block n according to the dashed blue line.

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Increase the committees to 31k validators, selecting larger validators with a probability proportional to size. Finality will then accrue markedly faster during the initial slots, as shown by the solid blue line. This helps reduce the aggregate finality gap: the cyan area.

Add an “auxiliary committee” on top of the 8 regular ones, resulting in 9 committees per “epoch”. This gives room to add more validators with high balances in each committee. Cumulative finality then accrues faster at first (green) but the trend finally reverses (red).

As more auxiliary committees are added, the green and red areas will eventually balance out. This point seems like a reasonable baseline for the optimal configuration, although many factors e.g. related to consensus formation can be considered to improve upon the baseline.

If the validator set is more consolidated, cumulative finality can accrue faster; certainly something to strive for! Here is the outcome if we had a “pure Zipfian” staking set instead. It can give full finality in three slots, and/or a very high finality already in the first.

On the other hand, if 1/2 the presumably Zipfian stakers retain their 32-ETH validators so that we only come halfway, finality would accrue much slower. This could be the situation in the short term, before stakers have had time to consolidate (or forever, if they choose not to).

What do I mean by a pure Zipfian distribution anyway? It’s an early assumption from Vitalik’s review of the beacon chain—that stakers’ balances follow Zipf’s law. The post presents some equations for derivation and generation that can come in handy as we move forward with SSF.

This plot shows the composition of the validator sets used in the analysis. The “Zipfian” set adheres to Zipf’s law, with big stakers then splitting up stake into 2048-ETH validators. The “1/2 Zipfian” set instead sees half the stakers keep their 32-ETH validators.

Cumulative finality is impeded at epoch boundaries when committees are shuffled. Two resolutions are proposed.

Circular finality: successive epochs are repeated across a longer era.

Spiral finality: shuffling of validators is restrained to limit the slots required for full finality.

A few methods for improving the selection and distribution of validators are suggested. One of these, leveraging circular finality, is to have repeated validators equally spaced across the epoch (rather than randomly). As evident, this improves cumulative finality somewhat.

Consensus properties are also analyzed and discussed. One of these is the committee rotation ratio: the proportion of the stake that is replaced with each new committee. The setting for the five validator sets that minimizes the aggregate finality gap is indicated by a circle.

The activity rate is the proportion of the committees that a validator is active in. It is shown here at the setting that minimizes the aggregate finality gap (the circle in the previous plot). Rates vary with the composition of the validator set (hence “Vorbit”: varying orbit).

The activity ratio between the lowest and highest rates dictates the range of staking risks available. Bigger validators take higher risks because they are active more often. They can therefore be compensated with a higher staking yield (otherwise, they may deconsolidate).

A high yield differential or a reliance on collective incentives can cause tensions among stakers. Therefore, compressing the activity ratio seems desirable. A sophisticated utility measure guiding committee generation could incorporate the activity ratio as a key feature.

A future post is intended to focus on the available chain and how to approach incentives as well as issuance policy in the context of SSF. I thank @fradamt and @barnabe for their feedback!