Today’s target
The aim of this tutorial is to show how we can measure and map degree, betweenness, closeness, eigenvector, and other types of centrality, explore their distributions and calculate the corresponding centralisation.
Setting up
For this exercise, we’ll use the ison_brandes
dataset in
{manynet}
. This dataset is in a ‘tidygraph’ format, but
manynet
makes it easy to coerce this into other forms to be
compatible with other packages. We can create a two-mode version of the
dataset by renaming the nodal attribute “twomode_type” to just “type”.
Let’s begin by graphing these datasets using graphr()
.
# Let's graph the one-mode version
graphr(____)
# Now, let's create a two-mode version 'ison_brandes2' and graph it.
ison_brandes2 <- ison_brandes %>% rename(type = twomode_type)
graphr(____)
# plot the one-mode version
graphr(ison_brandes)
ison_brandes2 <- ison_brandes %>% rename(type = twomode_type)
# plot the two-mode version
graphr(ison_brandes2, layout = "bipartite")
The network is anonymous, but I think it would be nice to add some
names, even if it’s just pretend. Luckily, {manynet}
has a
function for this: to_named()
. This makes plotting the
network just a wee bit more accessible and interpretable:
ison_brandes <- to_named(ison_brandes)
# Now, let's graph using the object names: "ison_brandes"
graphr(____)
ison_brandes <- to_named(ison_brandes)
# plot network with names
graphr(ison_brandes)
Note that you will likely get a different set of names (though still alphabetical), as they are assigned randomly from a pool of (American) first names.
Degree centrality
Let’s start with calculating degree. Remember that degree centrality is just the number of incident edges/ties to each node. It is therefore easy to calculate yourself. Just sum the rows or columns of the matrix!
# We can calculate degree centrality like this:
(mat <- as_matrix(ison_brandes))
(degrees <- rowSums(mat))
rowSums(mat) == colSums(mat)
# Or by using a built in command in manynet like this:
node_degree(ison_brandes, normalized = FALSE)
# manually calculate degree centrality
mat <- as_matrix(ison_brandes)
degrees <- rowSums(mat)
rowSums(mat) == colSums(mat)
# You can also just use a built in command in manynet though:
node_degree(ison_brandes, normalized = FALSE)
Degree distributions
Often we are interested in the distribution of (degree) centrality in
a network. {manynet}
offers a way to get a pretty good
first look at this distribution, though there are more elaborate ways to
do this in base and grid graphics.
# distribution of degree centrality scores of nodes
plot(node_degree(ison_brandes))
What’s plotted here by default is both the degree distribution as a histogram, as well as a density plot overlaid on it in red.
Other centralities
Other measures of centrality can be a little trickier to calculate by
hand. Fortunately, we can use functions from {manynet}
to
help calculate the betweenness, closeness, and eigenvector centralities
for each node in the network. Let’s collect the vectors of these
centralities for the ison_brandes
dataset:
# Use the node_betweenness() function to calculate the
# betweenness centralities of nodes in a network
node_betweenness(ison_brandes)
# Use the node_closeness() function to calculate the
# closeness centrality of nodes in a network
node_closeness(ison_brandes)
# Use the node_eigenvector() function to calculate
# the eigenvector centrality of nodes in a network
node_eigenvector(ison_brandes)
node_betweenness(ison_brandes)
node_closeness(ison_brandes)
node_eigenvector(ison_brandes)
What is returned here are vectors of betweenness, closeness, and eigenvector scores for the nodes in the network. But what do they mean?
Try to answer the following questions for yourself:
- what does Bonacich mean when he says that power and influence are not the same thing?
- can you think of a real-world example when an actor might be central but not powerful, or powerful but not central?
Note that all centrality measures in {manynet}
return
normalized scores by default – for the raw scores, include
normalized = FALSE
in the function as an extra
argument.
Now, can you create degree distributions for each of these?
plot(node_betweenness(ison_brandes))
plot(node_closeness(ison_brandes))
plot(node_eigenvector(ison_brandes))
Plotting centrality
It is straightforward in {manynet}
to highlight nodes
and ties with maximum or minimum (e.g. degree) scores. If the vector is
numeric (i.e. a “measure”), then this can be easily converted into a
logical vector that identifies the node/tie with the maximum/minimum
score using e.g. node_is_max()
or
tie_is_min()
. By passing this attribute to the
graphr()
argument “node_color” we can highlight which node
or nodes hold the maximum score in red.
# plot the network, highlighting the node with the highest centrality score with a different colour
ison_brandes %>%
add_node_attribute("color", node_is_max(node_degree(ison_brandes))) %>%
graphr(node_color = "color")
ison_brandes %>%
add_node_attribute("color", node_is_max(node_betweenness(ison_brandes))) %>%
graphr(node_color = "color")
ison_brandes %>%
add_node_attribute("color", node_is_max(node_closeness(ison_brandes))) %>%
graphr(node_color = "color")
ison_brandes %>%
add_node_attribute("color", node_is_max(node_eigenvector(ison_brandes))) %>%
graphr(node_color = "color")
How neat! Try it with the two-mode version. What can you see?
# Instead of "ison_brandes", use "ison_brandes2"
ison_brandes2 %>%
add_node_attribute("color", node_is_max(node_degree(ison_brandes2))) %>%
graphr(node_color = "color", layout = "bipartite")
ison_brandes2 %>%
add_node_attribute("color", node_is_max(node_betweenness(ison_brandes2))) %>%
graphr(node_color = "color", layout = "bipartite")
ison_brandes2 %>%
add_node_attribute("color", node_is_max(node_closeness(ison_brandes2))) %>%
graphr(node_color = "color", layout = "bipartite")
ison_brandes2 %>%
add_node_attribute("color", node_is_max(node_eigenvector(ison_brandes2))) %>%
graphr(node_color = "color", layout = "bipartite")
Centralization
{manynet}
also implements network centralization
functions. Here we are no longer interested in the level of the node,
but in the level of the whole network, so the syntax replaces
node_
with net_
:
net_degree(ison_brandes)
net_betweenness(ison_brandes)
net_closeness(ison_brandes)
print(net_eigenvector(ison_brandes), digits = 5)
By default, scores are printed up to 3 decimal places, but this can be modified and, in any case, the unrounded values are retained internally. This means that even if rounded values are printed (to respect console space), as much precision as is available is used in further calculations.
Note that for centralization in two-mode networks, two values are given (as a named vector), since normalization typically depends on the (asymmetric) number of nodes in each mode.
What if we want to have a single image/figure with multiple plots?
This can be a little tricky with gg-based plots, but fortunately the
{patchwork}
package is here to help.
ison_brandes <- ison_brandes %>%
add_node_attribute("degree", node_is_max(node_degree(ison_brandes))) %>%
add_node_attribute("betweenness", node_is_max(node_betweenness(ison_brandes))) %>%
add_node_attribute("closeness", node_is_max(node_closeness(ison_brandes))) %>%
add_node_attribute("eigenvector", node_is_max(node_eigenvector(ison_brandes)))
gd <- graphr(ison_brandes, node_color = "degree") +
ggtitle("Degree", subtitle = round(net_degree(ison_brandes), 2))
gc <- graphr(ison_brandes, node_color = "closeness") +
ggtitle("Closeness", subtitle = round(net_closeness(ison_brandes), 2))
gb <- graphr(ison_brandes, node_color = "betweenness") +
ggtitle("Betweenness", subtitle = round(net_betweenness(ison_brandes), 2))
ge <- graphr(ison_brandes, node_color = "eigenvector") +
ggtitle("Eigenvector", subtitle = round(net_eigenvector(ison_brandes), 2))
(gd | gb) / (gc | ge)
# ggsave("brandes-centralities.pdf")
Tasks
- Name a plausible research question you could ask of this data for each of the four main centrality measures (degree, betweenness, closeness, eigenvector) You may want to add these as titles or subtitles to each plot.