Mathematical foundation

The item pool is a weighted graph: nodes are items, and edge weights derive from the co-exposure matrix adj_mat (entry \((i, j)\) is the number of respondents who have seen both items). The Floyd–Warshall algorithm (Rfast::floyd()) turns the edge-weight matrix \(W\) into an all-pairs shortest-path distance matrix \(D\). For each respondent, the distance of a candidate item to the set of administered items is the minimum distance to any of them, and the farthest candidate is administered (ties broken by maximum information).

Edge weight strategies

The edge-weight function maps co-exposure counts to graph weights, and the choice shapes behavior. All of these are bundled and unchanged:

edge_weight_inverse(adj_mat, alpha = 1)              # 1 / (adj_mat + alpha)
edge_weight_negative_log(adj_mat, alpha = 1)         # -log(adj_mat + alpha)
edge_weight_linear(adj_mat, max_co_responses = NULL) # adj_mat / max(adj_mat)
edge_weight_power(adj_mat, beta = 0.5, alpha = 1)    # (adj_mat + alpha)^beta
edge_weight_exponential(adj_mat, lambda = 0.1)       # exp(-lambda*(adj_mat+alpha))

• Inverse / negative log / exponential: more co-responses give smaller weights, so frequently co-administered items are “closer” and the algorithm spreads exposure across dissimilar items.
• Linear: more co-responses give larger weights, inverting that logic.
• Power: beta < 1 dampens and beta > 1 amplifies the effect of high co-response counts.

Implementation

select_max_dist() follows the item selection contract (vignette("item-selection")): it works on the matrix administration state and returns an updated admin. After the distance matrix is computed, the per-item distances are obtained with Rfast::colMins() rather than a row-wise data-frame operation:

select_max_dist <- function(pers, item, R, admin, adj_mat = NULL, n_candidates = 1) {
  if (!any(admin != 0)) {
    admin[, seq_len(min(5, ncol(admin)))] <- 1L     # seed five items
    return(admin)
  }
  dist_mat <- Rfast::floyd(1 / adj_mat)              # all-pairs shortest paths
  info <- {                                          # 2PL information matrix
    lin <- sweep(outer(pers$theta, item$b, "-"), 2, item$a, "*")
    P <- stats::plogis(lin); sweep(P * (1 - P), 2, item$a^2, "*")
  }
  for (i in which(rowSums(admin == 0) > 0)) {
    administered <- which(admin[i, ] != 0)
    candidates   <- which(admin[i, ] == 0)
    sub <- dist_mat[administered, candidates, drop = FALSE]
    cand_dist <- if (length(administered) == 1L) sub[1, ] else Rfast::colMins(sub, value = TRUE)
    pool <- candidates[cand_dist >= max(cand_dist)]  # farthest items
    admin[i, pool[which.max(info[i, pool])]] <- 1L   # tie-break by information
  }
  admin
}

select_max_dist_enhanced() is identical except that the edge weights come from a user-supplied edge_weight_fun applied to adj_mat before Rfast::floyd().

Using different edge weight strategies

A small runnable example with the default inverse weights:

sim <- meow(
  select_fun  = select_max_dist,
  update_fun  = update_theta_mle,
  data_loader = data_simple_1pl,
  data_args   = list(N_persons = 50, N_items = 30),
  select_args = list(n_candidates = 3),
  fix         = "item"
)
nrow(sim$results)
#> [1] 26

Swap in a different edge-weight function through select_max_dist_enhanced():

# Power transformation with beta = 0.3
meow(
  select_fun  = select_max_dist_enhanced,
  update_fun  = update_theta_mle,
  data_loader = data_simple_1pl,
  data_args   = list(N_persons = 100, N_items = 50),
  select_args = list(
    n_candidates = 3,
    edge_weight_fun = edge_weight_power,
    edge_weight_args = list(beta = 0.3, alpha = 1)
  ),
  fix = "item"
)

Choosing a strategy

Considerations

• Rfast::floyd() is \(O(n^3)\) in the number of items and is run each iteration, so network selection is more expensive than select_max_info().
• Experiment with n_candidates (1–5) to trade exposure control against measurement efficiency, and compare against simpler selectors as a baseline.