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## Calculating different centrality measures

For this exercise, we’ll use the `ison_brandes` and `ison_brandes2` datasets in `{manynet}`. The `ison_brandes2` is a two-mode version of the `ison_brandes` dataset. This dataset is in a ‘tidygraph’ format, but migraph makes it easy to coerce this into other forms to be compatible with other packages.

    # Let's graph both datasets
    autographr(____)
    autographr(____)

    # Now, let's look at "ison_brandes" as a matrix
    # using the as_matrix() function

    (mat <- as_matrix(____))

    autographr(ison_brandes)
    autographr(ison_brandes2)
    (mat <- as_matrix(ison_brandes))

The network is anonymous, but I think it would be nice to add some names, even if it’s just pretend. Luckily, `{migraph}` has a function for this. This makes plotting the network just a wee bit more accessible and interpretable:

    # Let's use the to_named() function to assign
    # the networks data to a relevant object name

    ison_brandes <- to_named(____)
    ison_brandes2 <- to_named(____)

    ison_brandes <- to_named(ison_brandes)
    ison_brandes2 <- to_named(ison_brandes2)

    # Now, let's graph using the object names: "ison_brandes"

    autographr(____)

    ison_brandes <- to_named(ison_brandes)
    ison_brandes2 <- to_named(ison_brandes2)
    autographr(ison_brandes)

Note that you will likely get a different set of names, as they are assigned randomly from a pool of (American) first names.

Let’s start with calculating degree, as it is easy to calculate yourself. Just sum the rows or columns of the matrix!

    # We can calculate degree centrality like this:

    (degrees <- rowSums(mat))
    rowSums(mat) == colSums(mat)

    # Or by using a built in command in migraph like this:

    node_degree(ison_brandes, normalized = FALSE)

    (degrees <- rowSums(mat))
    rowSums(mat) == colSums(mat)
    # You can also just use a built in command in migraph though:
    node_degree(ison_brandes, normalized = FALSE)

Often we are interested in the distribution of (degree) centrality in a network. `{migraph}` 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.

    plot(node_degree(ison_brandes), "h") +
      plot(node_degree(ison_brandes), "d")

Other measures of centrality can be a little trickier to calculate by hand. Fortunately, we can use functions from `{migraph}` to help:

    # Let's explore this using 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)
    # TASK: Can you create degree distributions for each of these?

Note that all centrality measures in `{migraph}` return normalized scores by default – for the raw scores, just add `normalized = FALSE` as an extra argument.

## Plotting different centrality measures

It is straightforward in `{migraph}` 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 `autographr()` argument “node_color” we can highlight which node or nodes hold the maximum score in red.

    ison_brandes %>%
      add_node_attribute("color", node_is_max(node_degree(ison_brandes))) %>%
      autographr(node_color = "color")

    ison_brandes %>%
      add_node_attribute("color", node_is_max(node_betweenness(ison_brandes))) %>%
      autographr(node_color = "color")

    ison_brandes %>%
      add_node_attribute("color", node_is_max(node_closeness(ison_brandes))) %>%
      autographr(node_color = "color")

    ison_brandes %>%
      add_node_attribute("color", node_is_max(node_eigenvector(ison_brandes))) %>%
      autographr(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))) %>%
      autographr(node_color = "color")

    ison_brandes2 %>%
      add_node_attribute("color", node_is_max(node_betweenness(ison_brandes2))) %>%
      autographr(node_color = "color")

    ison_brandes2 %>%
      add_node_attribute("color", node_is_max(node_closeness(ison_brandes2))) %>%
      autographr(node_color = "color")

    ison_brandes2 %>%
      add_node_attribute("color", node_is_max(node_eigenvector(ison_brandes2))) %>%
      autographr(node_color = "color")

## Calculating centralization

`{migraph}` also implements centralization functions. Here we are no longer interested in the level of the node, but in the level of the whole graph, so the syntax is:

    # We will now look at the same centralization measures for the entire graph or network by
    # calling the same functions as those used for nodes but instead of "node_", replace it with "network_"

    network_degree(ison_brandes)
    network_betweenness(ison_brandes)
    network_closeness(ison_brandes)
    network_eigenvector(ison_brandes)

By default, scores are printed to 3 decimal places, but this can be modified and, in any case, the unrounded values are retained internally and passed on.

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 <- autographr(ison_brandes, node_color = "degree") +
      ggtitle("Degree", subtitle = round(network_degree(ison_brandes), 2))
    gc <- autographr(ison_brandes, node_color = "closeness") +
      ggtitle("Closeness", subtitle = round(network_closeness(ison_brandes), 2))
    gb <- autographr(ison_brandes, node_color = "betweenness") +
      ggtitle("Betweenness", subtitle = round(network_betweenness(ison_brandes), 2))
    ge <- autographr(ison_brandes, node_color = "eigenvector") +
      ggtitle("Eigenvector", subtitle = round(network_eigenvector(ison_brandes), 2))
    (gd | gb) / (gc | ge)
    # ggsave("brandes-centralities.pdf")

## Tasks

1. 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.

# Centrality

#### by James Hollway