Diversity and Regression
Author: by James Hollway

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## This tutorial

Within a network, nodes and ties vary. But how much do they vary and are there associations in this variance? By the end of this tutorial, you will be able to:

- measure the richness of a network in terms of attribute types
- measure how diverse a network is
- measure how heterophilous the ties in a network are
- test how heterophilous the ties in a network are compared to random networks
- test how heterophilous the ties in a network are compared to permutations of the original network
- regress a network’s ties on other ties and/or nodal or dyadic attributes

## Initial visualisation

For this session, we’ll explore a couple of different datasets. First, let’s examine homogeneity/ heterogeneity in the Marvel dataset from `{manynet}`, `fict_marvel`. The dataset is quite complicated, so to make this simpler, let’s concentrate on:

- just the relationships part of the network
- just the positive (friendship) ties and not the negative (enmity) ties
- the main (giant) component without any isolates
- just those characters that appear in the comics more than average

Fortunately, all these data cleaning moves are easy to do in `{manynet}`, and can be seen in the following chunk in order:

    to_uniplex(____, "relationship")

    # since the dataset is a 'signed' graph, we want to get just the
    # positively signed ties to get the friendship graph
    # (and lose the enmity relations)
    to_unsigned(____, keep = "positive")

    # to_giant() is a quick easy way to get the giant/main component
    to_giant(____)

    to_subgraph(____, Appearances >= mean(Appearances))

    marvel_friends <- to_unsigned(ison_marvel_relationships, keep = "positive")
    marvel_friends <- to_giant(marvel_friends)
    marvel_friends <- marvel_friends %>% to_subgraph(Appearances >= mean(Appearances))
    marvel_friends

    marvel_friends <- to_unsigned(ison_marvel_relationships, keep = "positive")
    marvel_friends <- to_giant(marvel_friends)
    marvel_friends <- marvel_friends %>% to_subgraph(Appearances >= mean(Appearances))
    marvel_friends

This gives us an undirected network of nearly twenty characters and a little more than 100 edges. Recall that this data has several nodal attributes. Let’s explore a couple of these attributes, “Gender” and “PowerOrigin”, visually using `graphr()`.

    # Since both Gender and PowerOrigin are categorical variables
    # you will need to use two different aesthetic dimensions to
    # represent them together.
    # Which will you present as shapes and which as colors?
    graphr(____,
           node_shape = ____,
           node_color = ____)

    graphr(marvel_friends,
           node_shape = "Gender",
           node_color = "PowerOrigin")

These variables seem to be distributed unevenly across the network. There seems to be some homophily – or like choosing like – operating here, but it is difficult to say conclusively because there are clearly more male than female superheros, as well as clearly more superheros of mutant origin than others. So what might seem like homophily could just be a result of there being many more opportunities for ties between nodes of some categories. We therefore need to establish how diverse this network really is.

## Measuring richness

Some categorical variables have more categories than others, and this can be thought of as indicating greater diversity than e.g. a binary categorical variable. A measure of this ‘richness’ is inherited from the study of biodiversity. Richness calculates the number of different categories that appear in the data for a given variable. Let’s calculate the richness of all of the variables in our Marvel data.

    net_node_attributes(marvel_friends)

    net_by_richness(____, ____)

    net_by_richness(marvel_friends, "Gender")
    net_by_richness(marvel_friends, "PowerOrigin")
    net_by_richness(marvel_friends, "Attractive")
    net_by_richness(marvel_friends, "Rich")
    net_by_richness(marvel_friends, "Intellect")

Note though that ‘richness’ as a network measure does not include any sense of how distributed these categories are around the network. There is a measure of nodal richness available, which counts the number of different categories to which each node is tied, but this does not offer a summary of how evenly distributed the appearance of categories are. For that we would need to move on to something like the Blau index.

## Measuring diversity

Another measure that reflects the diversity or heterogeneity in the network for each attribute is the Blau Index. Recall that the Blau index for any given variable is:

\[1 - \sum p_i^2\]

where \(p_i\) represents the proportion of observations holding \(i\), which indexes each of the given categories. A perfectly homogeneous group (all the same category) would receive a score of 0, while a perfectly heterogeneous group (with members spread evenly over the maximum categories) would receive a score of 1. Obtain the network diversity scores for the five attributes used above.

    net_by_diversity(____, ____)

    net_by_diversity(marvel_friends, "Gender")
    net_by_diversity(marvel_friends, "PowerOrigin")
    net_by_diversity(marvel_friends, "Attractive")
    net_by_diversity(marvel_friends, "Rich")
    net_by_diversity(marvel_friends, "Intellect")

Looks like there is more diversity in terms of where these characters got their powers, whether they have significant intellectual powers, and their gender, than their attractiveness or their wealth.

We can also cross-reference this diversity. For example, we might be interested in whether our comic book heroes are equally gender diverse across their (power) origin stories, or equally intellectually diverse across gender.1

    over_membership(.data, FUN, ..., membership)

    over_membership(marvel_friends, net_by_diversity, "Gender",
                    membership = "PowerOrigin")
    over_membership(marvel_friends, net_by_diversity, "Intellect",
                    membership = "Gender")

Note that the length of the vector returned as a result is the number of categories in the second category listed. It looks like some origin stories are more gender diverse than others. Gods (just Thor here) and humans are all men, whereas those with mutant or radiation origin stories are more gender diverse. There doesn’t appear to be much difference in intellect across gender categories in this data.

Ok, this tells us about how (un)even the distribution of these variables is in this network, but it doesn’t necessarily tell us whether ties are appearing more frequently between nodes of similar (or different) categories. For that we need to look at homophily / heterophily.

1. Note that this works for calculated categorical variables too, such as cluster/group assignment from community detection or equivalence classes.↩︎

## Measuring heterophily

The EI (or E-I) index offers a way to measure the degree to which ties appear between rather than within groups of nodes of the same category. Recall that the EI index is calculated as:

\[\frac{E-I}{E+I}\]

where \(E\) is the number of ties present between a variable’s categories (i.e. external), and \(I\) is the number of ties present within a variable’s categories (i.e. internal). As such, an EI index of -1 suggests perfect homophily, whereas an EI index of +1 suggests perfect heterophily. (This is why the function is called `net_by_heterophily()`).

Check how homophilic three variables in the network are, “Gender”, “PowerOrigin”, and “Attractive”. Please assign the results so that we can use them later.

    net_by_heterophily(____, ____)

    (obs.gender <- net_by_heterophily(marvel_friends, "Gender"))
    (obs.powers <- net_by_heterophily(marvel_friends, "PowerOrigin"))
    (obs.attract <- net_by_heterophily(marvel_friends, "Attractive"))

Ultimately though, these are just scores, and doesn’t tell us whether this is any more or less than what we might expect the score to be by chance for a network of this size and density and distribution of that attribute.

## CUG tests

To see whether we should be surprised by scores this high/low, we compare these scores with those from a series of random graphs (Erdös-Renyi/Bernoulli) of the same dimensions and distribution of the attribute. This can help us establish whether there is more homophily or heterophily than expected by chance.

This is often called a conditional uniform graph or CUG test, but `{migraph}` uses more descriptive function names, such as `test_random()`. Plot the results of running this function with respect to the EI index on each of the three variables. These plots can be made if you have `{autograph}` installed and loaded. You can specify that one thousand simulations should be used using `times = 1000`.

    rand.____ <- test_random(____, FUN = ____, attribute = ____, times = ___)

    plot(rand.____)

    rand.gender <- test_random(marvel_friends,
                               net_by_heterophily, attribute = "Gender",
                               times = 1000)
    rand.power <- test_random(marvel_friends,
                              net_by_heterophily, attribute = "PowerOrigin",
                              times = 1000)
    rand.attract <- test_random(marvel_friends,
                                net_by_heterophily, attribute = "Attractive",
                                times = 1000)
    plot(rand.gender) + ggtitle("CUG test results for 'Gender' attribute")
    plot(rand.power) + ggtitle("CUG test results for 'PowerOrigin' attribute")
    plot(rand.attract) + ggtitle("CUG test results for 'Attractive' attribute")

The plots of these results use a dotted vertical line for 0 where this is in bounds, a red vertical line for the observed score, and a density plot of the scores from the randomly generated networks. The grey tails of the distribution are a visual aid indicating the most extreme 5% of scores from the distribution. The results are really interesting.

Despite being the larger coefficients (in absolute terms), it looks like we cannot reject the null hypothesis that there is no homophily for gender nor for attractiveness. Both observed scores fall within the range of scores we would expect from randomly generated networks with the same distribution of that variable.

However, we can reject the null hypothesis with respect to their power origin story. While the coefficient itself is close to 0 (neither strong homophily nor heterophily), all the random networks generated returned larger EI scores, between .1 and .4.

## QAP tests

Ah, but perhaps random graphs are not the best reference group for establishing whether there is a significant homophily effect here. After all, social networks are not completely random; they are structured in particular ways, such as some nodes having higher degrees than others, or there being a core and periphery or community topology.

Another approach to establishing a baseline for whether we should be surprised by a given score or not is to use permutations of the underlying network instead of random graphs. Permuting the network retains the structure of the network because the ties are kept and only the labels (variables) are reassigned randomly. Let’s first plot the observed data and some permuted data next to each other.

    graphr(to_permuted(____, with_attr = TRUE), ____)

    old <- graphr(marvel_friends,
                  labels = FALSE, node_size = 6,
                  node_color = "PowerOrigin",
                  node_shape = "Gender") + ggtitle("Original network")
    new <- graphr(to_permuted(marvel_friends, with_attr = TRUE),
                  labels = FALSE, node_size = 6,
                  node_color = "PowerOrigin",
                  node_shape = "Gender") + ggtitle("Permuted network")
    old + new

This single permutation suggests a more even mixing of these attributes is possible, but it is just a single permutation. Let’s try a QAP test that runs this over a succession of permutations, just as we did with random graphs. Plot the results for gender and power according to the random and permutation baselines.

    test_permutation(____, FUN = ____, attribute = ____,
                                    times = ____)

    (perm.gender <- test_permutation(marvel_friends,
                                    net_by_heterophily, attribute = "Gender",
                                    times = 1000))
    (perm.power <- test_permutation(marvel_friends,
                                    net_by_heterophily, attribute = "PowerOrigin",
                                    times = 1000))

    (plot(rand.gender) + ggtitle("CUG test results for 'Gender' attribute") +
        theme(plot.title = element_text(size=8)) |
        plot(rand.power) + ggtitle("CUG test results for 'PowerOrigin' attribute") +
        theme(plot.title = element_text(size=8))) /
    (plot(perm.gender) + ggtitle("QAP test results for 'Gender' attribute") +
       theme(plot.title = element_text(size=8)) |
       plot(perm.power) + ggtitle("QAP test results for 'PowerOrigin' attribute") +
       theme(plot.title = element_text(size=8)))

Again, we see that there is perhaps nothing so surprising that we got the homophily score for gender that we did, but the lack of power origin heterophily is surprising. Note how the distributions are generally wider when permuting the observed network than creating a random distribution (be mindful of the scale of the x-axis). That is, taking into account the structure of the network leads us to expect a larger spread in the EI index than when the variable is distributed across a random network.

CUG and QAP tests are useful for establishing whether there is more or less homophily than expected by chance, with QAP tests being more conservative and better taking into account the structure of the network. Alternative reference networks can also be used that condition on more structural information than completely random graphs, but introduce potentially more variation in the network structure than permutations. These include for example configuration networks that retain the degree distribution observed in the original network, but could result in varying clustering, or networks that are conditioned on the dyad census (i.e. MAN models). And yet none of these approaches allow us to examine multiple variables at once or control for other factors. For that we need to move on to network regression.

## Network regression

Next let us examine homophily in another network. The data were collected as part of an early experiment on communication between social network researchers who were using an Electronic Information Exchange System (EIES). You may recognise some of the names. The main network consists of 32 scholars with directed ties weighted by the total number of messages sent from \(i\) to \(j\) over the period of the study. Nodal attributes collected include the primary discipline and number of citations in the social science citation index at the start of the study.

    ison_networkers
    graphr(ison_networkers, node_color = "Discipline")

Let’s use both the continuous `Citations` and the categorical `Discipline` variables and come up with a couple of key hypotheses:

- \(H_1\): researchers send more emails to those who are cited more (popularity hypothesis)
- \(H_2\): researchers send more emails to those of the same discipline (disciplinary homophily hypothesis)

Let’s start with a pretty maximally specified model (note that it doesn’t make sense to include both ego and alter effects because these networks are undirected). We are using `times = 200` here because of time-out limitations in the `learnr` tutorial system, but for publication quality results you would want to base your conclusions on 2000 simulations or more.

    net_regression(____, ison_networkers, times = 200)
    # If the model runs into a timeout error, please reduce the number of 'times' in the function above.

    weight ~ ego(Citations) + alter(Citations) + sim(Citations) +
                          ego(Discipline) + same(Discipline)

By default, the printing method for this model includes the key information you might need. We can also use tidy methods to obtain just the table of coefficients (e.g. `tidy()`) or the model summary statistics (e.g. `glance()`). `{autograph}` also includes a plot method for these results to facilitate the quick interpretation of these results, and as long as this is loaded you can simply call `plot()` on the model object.

    model1
    tidy(model1)
    glance(model1)
    plot(model1)

This violin plot presents the distribution of coefficients from permutations of the network, with the coefficient fitted from the data as a red dot. Lines are used to indicate 95% thresholds, but here the distributions are rendered so wide that they are often not visible.

## Free play

While these are the conclusions from this ‘play’ data, you may have more and more interesting data at hand. How would you go about specifying such a model? Why is such an approach more appropriate for network data than linear or logistic regression?

## Glossary

Here are some of the terms that we have covered in this module:

Alter
  An alter is a node connected to a focal node (ego).
Community
  A community is a set of nodes more densely connected to one another than to other nodes in the network.
Configuration
  A configuration network is one where the degree sequence is fixed.
Core
  A core-periphery is a bipartition of nodes into maximally dense and sparse blocks.
Cug
  A conditional uniform graph (CUG) is a random graph generated under constraints defined by some property of the observed network.
Ego
  An ego is a focal node in a network.
Heterogeneity
  A measure of how varied the nodes in a network are.
Heterophily
  A tendency for nodes to connect to dissimilar nodes.
Homophily
  A tendency for nodes to connect to similar nodes.
Man
  A random network conditional on the number of mutual, asymmetric, and null dyads.
Permutation
  A permuted network is one where the nodes/ties have been rearranged or resequenced.
Qap
  A quadratic assignment procedure (QAP) is a permutation test for network data.
Random
  A random network is one where ties are placed among nodes according to some probability distribution.
Richness
  A measure of how many different types of nodes are in a network.
Undirected
  An undirected or line network is one in which tie direction is undefined.

# Diversity and Regression

#### by James Hollway