Figure 6.11 The small world rewiring process in two dimensions

figures

code

Here the small world network phenomenon based on ‘rewiring’ is shown for a two-dimensional lattice. This particular network structure is less studied than might be expected, and can be considered a very simplified model for how even a small number of more rapid connections between places in a transportation network can dramatically alter its overall characteristics.

Code

library(igraph)
library(dplyr)
library(tidyr)
library(scales)
library(ggplot2)

Code

plot_graph <- function(G, main = "", layout = layout.grid(G), 
                       vertex.color = "black") {
  plot(G, main = main,
       layout = layout, 
       vertex.label = NA, vertex.color = vertex.color, vertex.size = 5, 
       vertex.shape = "circle", vertex.lwd = 0, 
       edge.color = "black", edge.width = 0.5)
}

Sample rewired 2D lattices

It’s somewhat useful to see what a range of rewired lattices look like. Here’s a bigger range than in the book, with the rewiring probability increasing roughly 3-fold each time.

Code

par(mar = rep(1, 4))
layout(matrix(1:6, nrow = 2, byrow = TRUE))

base_graph <- make_lattice(length = 20, dim = 2, nei = 2)

plot_graph(rewire(base_graph, each_edge(0.003)), main = "p = 0.003")
plot_graph(rewire(base_graph, each_edge(0.009)), main = "p = 0.009")
plot_graph(rewire(base_graph, each_edge(0.027)), main = "p = 0.027")
plot_graph(rewire(base_graph, each_edge(0.083)), main = "p = 0.083")
plot_graph(rewire(base_graph, each_edge(0.250)), main = "p = 0.25")
plot_graph(rewire(base_graph, each_edge(0.750)), main = "p = 0.75")

Looking at a wider range of rewiring outcomes

Below is code to make the other part of Figure 16.11, which shows how mean clustering coefficient and path lengths vary over a wide range of rewiring probablities. The range of probabilities shown here is different than in the published figure, with more of the samples in the middle range of the plot.

Code

probs          <- 10 ^ rnorm(1000, -3)
probs          <- probs[which(between(probs, 0, 1))]
keepers        <- c()
cluster_coeffs <- c()
mean_path_lens <- c()

for (i in seq_along(probs)) {
  the_graph <- rewire(base_graph, each_edge(probs[i]))
  if (components(the_graph)$no == 1) {
    cluster_coeffs <- c(cluster_coeffs, 
                        transitivity(the_graph, type = "average"))
    mean_path_lens <- c(mean_path_lens, 
                        mean(distances(the_graph)))
    keepers        <- c(keepers, i)
  }
}
probs <- probs[keepers]

df <- data.frame(p   = probs, 
                 cc  = cluster_coeffs, 
                 mpl = mean_path_lens) %>% 
  mutate(cc = rescale(cc, to = c(0, 1)), 
         mpl = rescale(mpl, to = c(0, 1))) %>% 
  # remove first item, p=0 and not plottable
  pivot_longer(-p) %>%
  rename(Metric = name)

How mean path length and clustering coefficient diverge

There is wide range of rewiring probabilities where ‘small world’ characteristic of surprising short mean path lengths in the presence of strong local clustering is evident (note the log-scale on the horizontal axis).

Code

ggplot(df, aes(x = p, y = value, colour = Metric)) + 
  geom_point(cex = .8, alpha = 0.5) + 
  scale_x_log10() +
  scale_colour_brewer(palette = "Dark2", 
                      labels = c("cluster_coeffs coeff.", "Mean path length")) +
  xlab("Probability of rewiring") +
  ylab("Value relative to base lattice")

Code

# License (MIT)
#
# Copyright (c) 2023 David O'Sullivan
#
# Permission is hereby granted, free of charge, to any person
# obtaining a copy of this software and associated documentation
# files (the "Software"), to deal in the Software without restriction,
# including without limitation the rights to use, copy, modify, merge,
# publish, distribute, sublicense, and/or sell copies of the Software,
# and to  permit persons to whom the Software is furnished to do so,
# subject to the following conditions:
#
# The above copyright notice and this permission notice shall be included
# in all copies or substantial portions of the Software.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
# FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
# DEALINGS IN THE SOFTWARE.