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Sample a Bradley–Terry Stochastic Block Model (BT-SBM) tournament

Source: R/sample_from_BTSBM.R
sample_from_BTSBM.Rd

Generates a sparse undirected match matrix \(N\) and corresponding directed win counts matrix \(W\) under the BTSBM generating process. Block strengths are specified via a positive vector \(\lambda\), mapped to pairwise win probabilities \(\theta_{ab} = \lambda_a / (\lambda_a + \lambda_b)\).

Usage

sample_from_BTSBM(
  n_players,
  K,
  x = NULL,
  mean_matches = 5,
  p_edge = 0.5,
  lambda = NULL,
  lambda_base = 0.08,
  lambda_ratio = 2.3,
  reverse_lambda = TRUE,
  seed = NULL
)

Arguments

n_players

Integer. Number of players \(n\).

K

Integer. Number of blocks.

x

Optional integer vector of length n_players with values in 1:K. If NULL, uses rep(seq_len(K), length.out = n_players).

mean_matches

Positive numeric. Poisson mean for match counts on present edges.

p_edge

Numeric in [0,1]. Probability that an unordered pair is present in the topology.

lambda

Optional positive numeric vector of length K. If NULL, a geometric sequence is used: lambda_base * lambda_ratio^(0:(K-1)).

lambda_base

Positive numeric. Base of the geometric schedule when lambda is not provided. Default 0.08 (as in your snippet).

lambda_ratio

Positive numeric. Ratio of the geometric schedule when lambda is not provided. Default 2.3 (as in your snippet).

reverse_lambda

Logical. If TRUE, uses rev(lambda) when computing theta, matching your theta_star <- lambda_to_theta(rev(lambda_star)).

seed

Optional integer. If provided, sets the RNG seed for reproducibility within the function call.

Value

A list with components:

  • N — \(n \times n\) integer matrix of symmetric match counts with zero diagonal.

  • W — \(n \times n\) integer matrix of directed win counts with w[i, j] + w[j, i] = N[i, j].

  • x — integer vector of block labels in 1:K.

  • lambda — numeric vector of length K used to build theta.

  • theta — \(K \times K\) matrix with entries \(\theta_{ab} = \lambda_a / (\lambda_a + \lambda_b)\).

Details

The workflow is:

  1. Sample a symmetric match-count matrix \(N\) by first drawing a random subset of unordered pairs with probability p_edge, then assigning each selected pair a Poisson\((mean\_matches)\) number of matches.

  2. Compute the block-to-block Bradley–Terry matrix \(\theta = \lambda \mathbf{1}^\top / (\lambda \mathbf{1}^\top + \mathbf{1}\lambda^\top)\).

  3. For each unordered pair with \(N_{ij}>0\), draw \(w_{ij} \sim \text{Binomial}(N_{ij}, \theta_{x_i, x_j})\) and set \(w_{ji} = N_{ij} - w_{ij}\).

By default, block labels are assigned deterministically as rep(seq_len(K), length.out = n_players) to mirror your example, but you can pass a custom label vector via x.