A critical structural property that shapes this analysis is that, with a minimum network of $n=40$ nodes and $M=2048$ subnetworks each requiring $R=5$ nodes, every node participates in many subnetworks simultaneously. This multi-membership changes the adversarial calculus substantially compared to a model where each node belongs to exactly one subnetwork.
The total number of (node, subnetwork) assignment slots across the network is:
$\text{Total slots} = M \times R = 2048 \times 5 = 10{,}240$
With $n = 40$ nodes, each node holds on average:
$\bar{d} = \frac{M \times R}{n} = \frac{10{,}240}{40} = 256 \text{ subnetworks per node}$
This is the most important structural fact in this analysis. Every node participates in approximately 256 out of 2048 subnetworks. An adversary controlling $k$ nodes therefore has presence (at least one adversarial node) in roughly:
$|\mathcal{S}_{\text{presence}}| \approx \min\!\left(M,\ M \cdot \left[1 - \left(1 - \frac{k}{n}\right)^R\right]\right)$
But full capture (all $R$ nodes adversarial) of a subnetwork requires all 5 assignment slots to be held by adversarial nodes.
Given that each subnetwork's $R=5$ nodes are drawn uniformly from $n=40$ total nodes, and the adversary controls $k$ of them, the probability that a specific subnetwork is fully captured follows the hypergeometric distribution:
$$ P_{\text{capture}}(k) = \frac{\binom{k}{5}}{\binom{40}{5}} $$
The expected number of fully captured subnetworks across the entire network of $M=2048$ subnetworks is:
$$ \mathbb{E}[A] = M \cdot P_{\text{capture}}(k) = \frac{2048 \cdot \binom{k}{5}}{\binom{40}{5}} $$
| Adversary nodes$k$ | Fraction of network | $P_{\text{capture}}$ | $\mathbb{E}[A]$fully captured |
|---|---|---|---|
| 8 | 20% | $\approx 4.2 \times 10^{-5}$ | $\approx 0.09$ |
| 13 | 33% | $\approx 1.96 \times 10^{-3}$ | $\approx 4.0$ |
| 16 | 40% | $\approx 6.3 \times 10^{-3}$ | $\approx 12.9$ |
| 20 | 50% | $\approx 2.36 \times 10^{-2}$ | $\approx 48.3$ |
| 22 | 55% | $\approx 4.0 \times 10^{-2}$ | $\approx 82.0$ |
| 26 | 65% | $\approx 9.7 \times 10^{-2}$ | $\approx 198.8$ |
Key observation: An adversary controlling less than 50% of nodes expects to fully capture fewer than 50 subnetworks out of 2048. The implications of this are developed in the sections below.
The distinction between these two adversarial regimes is fundamental:
Partial presence in a subnetwork (some but not all nodes adversarial) is defeated by the 5-retry mechanism. When a sampler queries a subnetwork, it tries up to 5 different peers. If at least one honest node exists in the subnetwork and responds, the sample succeeds. The probability that all 5 retry attempts hit only adversarial nodes in a subnetwork with $a < R$ adversarial nodes out of $R$ total is:
$P(\text{retry failure}) = \left(\frac{a}{R}\right)^5$
For $a=4$ out of $R=5: (4/5)^5 \approx 0.33$. Even with 4 out of 5 nodes adversarial, there is a 67% chance the honest node is reached within 5 retries. For $a \leq 3: (3/5)^5 \approx 0.08$, an 8% chance of failure.
Full capture (all 5 nodes adversarial) means no retry can find an honest node, and the subnetwork will always fail for that sampler.
This scenario examines a strictly more dangerous adversary:
The goal is not to make data unrecoverable globally, but to create asymmetric availability perception across validators — some validators successfully sample the blob (perceive it as available), while others fail (perceive it as unavailable). This asymmetry can be exploited for block-building advantage.