Introduction

Analysis

The set of nodes $[N]$ participating in DA network is divided into $N_{S}=2048$ subnetworks corresponding to the columns of the encoding scheme.
In each subnetwork there is at least one node from $[N]$.
If $N<N_{S}$ then subnetworks are created by random sampling (with replacement) from $[N]$.
If $N\geq N_{S}$ then subnetworks are created by random sampling (without replacement) from $[N]$.
Each node in $[N]$ first samples randomly (without replacement) $K=20$ subnetworks from the $N_{S}$ subnetworks. Then in each of the sampled subnetwork a single node is sampled.
If $N\geq N_{S}$ then above is equivalent to random sampling (without replacement) of $K$ nodes from $[N]$.

We assume that the number of nodes in DA is $N\geq N_S$ is fixed.
A node is unreliable when there is no guarantee that a data can be recovered from this node.
We assume that a node is unreliable with probability $q$.
A subnetwork is unreliable when all nodes in this subnetwork are unreliable.
Assuming that the rate of error-correcting code used in DA is $1/2$, there is no guarantee that the data stored in DA network can be recovered if more that $N_S/2$ of subnetworks are unreliable.
We assume that $N$ nodes are distributed among the $N_S$ subnetworks (almost) equally and in a random manner.
Assuming above model, we would like to know the probability of event that more that $N_S/2$ of subnetworks are unreliable.
We assume that we have $n_1$ nodes in the 1st subnetwork, $n_2$ nodes in the second subnetwork, etc. such that we have the total of $\sum_{\mu=1}^{N_S} n_{\mu}=N$ nodes in $N_S$ subnetworks.