Span of Control

This page analyzes the changing distinction centrality dynamics in hierarchical tree networks as the span of control (branching factor) increases.

Code

library(igraph)
library(ggplot2)
library(dplyr)
library(tidyr)
library(DT)

source("../Functions/distinction.R")

spans <- 2:10
results <- list()

for (k in spans) {
  # Calculate total nodes for a depth 2 tree with branching factor k:
  # 1 root + k children + k^2 leaves = 1 + k + k^2
  n_nodes <- 1 + k + k^2
  g <- make_tree(n = n_nodes, children = k, mode = "undirected")
  V(g)$name <- as.character(1:vcount(g))
  
  res <- distinction(g, norm = "abm")
  
  # Root is always node 1
  root_scd <- res$scd[1]
  
  # A middle branch is node 2
  mid_scd <- res$scd[2]
  
  alpha <- res$scalar[1]
  
  results[[length(results) + 1]] <- data.frame(
    Span = k,
    Root = root_scd,
    `Middle Manager` = mid_scd,
    Alpha = alpha,
    check.names = FALSE
  )
}

df <- bind_rows(results)

Distinction Trajectories

As the span of control increases, the middle manager’s distinction grows faster than the root’s distinction, illustrating a shift in structural leverage.

Distinction Trajectories

Data Table