Characterizing Optimal Sampling of Binary Contingency Tables via the Configuration Model

A binary contingency table is an m x n array of binary entries with prescribed row sums r=(r_1,…,r_m) and column sums c=(c_1,…,c_n). The configuration model for uniformly sampling binary contingency tables proceeds as follows. First, label N=\sum_{i=1}^{m} r_i tokens of type 1, arrange them in m cells, and let the i-th cell contain r_i tokens. Next, label another set of tokens of type 2 containing N=\sum_{j=1}^{n}c_j elements arranged in n cells, and let the j-th cell contain c_j tokens. Finally, pair the type-1 tokens with the type-2 tokens by generating a random permutation until the total pairing corresponds to a binary contingency table. Generating one random permutation takes O(N) time, which is optimal up to constant factors. A fundamental question is whether a constant number of permutations is sufficient to obtain a binary contingency table. In the current paper, we solve this problem by showing a necessary and sufficient condition so that the probability that the configuration model outputs a binary contingency table remains bounded away from 0 as N goes to \infty. Our finding shows surprising differences from recent results for binary symmetric contingency tables.

💡 Research Summary

The paper investigates the configuration model as a method for uniformly sampling binary contingency tables with prescribed row sums r = (r₁,…,r_m) and column sums c = (c₁,…,c_n). In the model, N = Σ_i r_i = Σ_j c_j tokens of type 1 are placed into m cells (row‑bins) according to the row sums, and another N tokens of type 2 are placed into n cells (column‑bins) according to the column sums. A random permutation pairs each type‑1 token with a type‑2 token, thereby defining a 0‑1 matrix: a cell (i,j) receives a 1 if a token from row‑bin i is matched with a token from column‑bin j, otherwise 0. Generating a single random permutation costs Θ(N) time, which is optimal up to constant factors.

The central question addressed is whether a constant number of such permutations suffices to obtain a valid binary table with probability bounded away from zero as N → ∞. The authors answer this by providing a necessary and sufficient condition on the row‑ and column‑sum vectors. Their main theorem states:

Theorem (informal).
Let N = Σ_i r_i = Σ_j c_j. The probability that a single random permutation yields a binary contingency table stays above some fixed ε > 0 for all sufficiently large N iff for every pair (i,j) we have
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