---
title: "Modelling with ERGMs"
author: "by James Hollway, inspired by ERGM package vignettes"
output: 
  learnr::tutorial:
    theme: journal
runtime: shiny_prerendered
description: >
  This tutorial aims to teach you how to model network structure using
  exponential random graph models (ERGMs).
---

```{r pkgs, include = FALSE}
library(migraph)
library(ergm)
```

```{r setup, include = FALSE, purl=FALSE, eval=TRUE}
library(manynet)
library(netrics)
library(autograph)
library(migraph)
library(ergm)
library(learnr)
data(florentine) # loads flomarriage and flobusiness data
data(faux.magnolia.high) # loads flomarriage and flobusiness data
flom.bern <- ergm(flomarriage ~ edges)
flom.bern.gof <- gof(flom.bern,  GOF = ~ distance + degree + espartners)
magnolia <- to_giant(faux.magnolia.high) 
knitr::opts_chunk$set(echo = FALSE)
manynet::clear_glossary()
learnr::random_phrases_add(language = "fr",
                           praise = c("C'est génial!",
                                      "Beau travail",
                                      "Excellent travail!",
                                      "Bravo!",
                                      "Super!",
                                      "Bien fait",
                                      "Bien joué",
                                      "Tu l'as fait!",
                                      "Je savais que tu pouvais le faire.",
                                      "Ça a l'air facile!",
                                      "C'était un travail de première classe.",
                                      "C'est ce que j'appelle un bon travail!"),
                          encouragement = c("Bon effort",
                                            "Vous l'avez presque maîtrisé!",
                                            "Ça avance bien.",
                                            "Continuez comme ça.",
                                            "Continuez à travailler dur!",
                                            "Vous apprenez vite!",
                                            "Vous faites un excellent travail aujourd'hui."))
learnr::random_phrases_add(language = "en", 
                           praise = c("C'est génial!",
                                      "Beau travail!",
                                      "Bravo!",
                                      "Super!"),
                           encouragement = "Bon effort")
```

## This tutorial

This tutorial is inspired by the vignettes provided with the `ergm` package,
and aims to teach you how to model network structure using
exponential random graph models (ERGMs).
Today we are going to deal with the following topics:

- [ ] &nbsp; How to run an ergm in R with the `{ergm}` package
- [ ] &nbsp; How to analyse the results
- [ ] &nbsp; How to assess the goodness of fit of a model
- [ ] &nbsp; Understanding the difference between a Bernoulli, Markov, and Partial Conditional dependence model
- [ ] &nbsp; How to include structural effects, monadic and dyadic covariates
- [ ] &nbsp; How to diagnose and troubleshoot degeneracy/convergence problems

### Data

The ergm package contains several network data sets for practice examples.
Today, we're going to use data on florentine marriage ties.

```{r ergm-data, exercise = TRUE}
# data(package='ergm') # tells us the datasets in our packages
data(florentine) # loads flomarriage and flobusiness data
flomarriage # Data on marriage alliances among Renaissance Florentine families
# ?florentine # You can see more information about the attributes in the help file.
```

We see 16 nodes and 20 edges in this undirected network.
Let's get a little more information about the network.
What is this network's density?
What attributes are attached to the nodes of this network that we might
use in modelling?

```{r density, exercise=TRUE, exercise.setup = "ergm-data"}
net_by_density(flomarriage)
net_node_attributes(flomarriage)
net_tie_attributes(flomarriage)
```

There are a few attributes for the nodes:
priorates, totalties, wealth. 
All 3 are numeric valued attributes.

### Visualisation

Now, let's visualise this network (with labels),
and sizing the nodes by their wealth.
What do you observe? Describe the network in terms of configurations.

```{r vis-flomarriage, exercise=TRUE, exercise.setup = "ergm-data", fig.width=8}
graphr(flomarriage, node_size = "wealth")
as_tidygraph(flomarriage) |> mutate_nodes(Degree = node_by_deg()) |>  
  mutate_ties(Triangles = tie_is_triangular()) |> 
  graphr(node_size = "Degree", edge_color = "Triangles")
```

Network configurations are small patterns of ties within the graph 
or subgraphs, which are sometimes referred to as network motifs 
(Milo et al. 2002 in Lubell et al. 2014, p. 23).
If a configuration represents an outcome of some social process 
occurring within the network, 
then that configuration occurring at a higher frequency in the observed network 
than expected by chance once other possibly relevant processes are controlled
can be viewed as evidence for such a process/mechanism.

We see there is one isolate and a giant component,
some nodes have more ties (Medici, Strozzi) and others are pendants,
and it looks like there are several triangles.

## Bernoulli model

An ERG model is run from the `ergm' package with the following syntax:
`ergm(dependentnetwork ~ term1 + term2 + ...)`

This is a similar syntax to running regressions and other models in R.
We begin a simple model of just one term estimating the number of edges/ties.
This is called a Bernoulli or Erdos-Renyi model.
We assign it to keep the results as an object:

```{r flombern, exercise = TRUE, exercise.setup = "ergm-data"}
flom.bern <- ergm(flomarriage ~ edges)  
tidy(flom.bern) # To return the estimated coefficients, with p-values etc.
glance(flom.bern, deviance = TRUE) # To collect the deviance, AIC and BIC, etc.
```

These outputs tell us:

1. the parameter estimate (-1.6094), standard error (0.2449), 
and p-values with typical significance codes (<1e-04 ***)
2. the loglikelihood, deviance, AIC(110.1) and BIC (112.9) based upon them (discussed in Hunter and Handcock, 2006).

### So what is AIC/BIC?

The Akaike Information Criterion (AIC) estimates how much information
is lost in the process of modelling some data.
It is calculated from the likelihood of all the observations under the model
and the number of parameters in the model.
Since it thereby rewards fit but penalises for model complexity,
it not only helps us guard against overfitting,
but also helps compare model specifications fit to the same observed data.
Basically: pick the model with the lowest AIC.

Otherwise very similar to AIC, BIC penalises model complexity more severely.
Downside: it is likely to favour very simple models, especially for smaller, 
less representative training datasets.

## Interpreting results

How to interpret the results from this model, shown below for convenience?

```{r flombee, exercise = TRUE, exercise.setup = "flombern"}
tidy(flom.bern)
```

Interpreting results from a logistic regression can be tricky,
because the relationship between the predictor(s) and response variable is non-linear.
Here I wish to introduce you to a few options.

### Log-Odds

The first option is to try and interpret the log-odds coefficients directly.
The estimates from logistic regression characterize the relationship between the predictor and response variable on a log-odds scale.
Since there is only one term in this model, and it is `edges`,
this means that the _log-odds_ of a tie forming between any two nodes in the network is -1.6094.

Er, what? What does that mean?
Turns out there is no real intuitive way to interpret log-odds.
Moreover, as we will later see, in network models, effects often overlap,
which mean that instead of interpreting the effect of a one-unit change in $X_k$ on the log-odds of $Y$,
we need to consider the combined effect of changes in multiple $X$s on the log-odds of $Y$.

### Sign-ificance

One option for interpretation is what I call “sign-ificance”.
This involves obtaining what information we can from this table of results.
There are two key pieces of information available:

1. The _sign_ of $\beta$ is either:
  - +: (increases in $X$) increases $\Pr(Y = 1)$
  - -: (increases in $X$) decreases $\Pr(Y = 1)$
2. The _p_-value tells us whether (and to what extent) $\beta$ statistically 
differentiable from zero
  - small _p_-value (e.g. < 0.05): unlikely to have occurred by chance
  - large _p_-value (e.g. > 0.05): could have occurred by chance
  
Together this is sufficient for hypothesis-testing:
it tells us whether there is a “signal” for a pattern in the data
that goes in the same direction as hypothesised and
unlikely to appear just by chance.
Therefore common to see articles using this approach where authors are only 
focused on testing a particular hypothesis,
and helps avoid problems with using odds-ratios (and log-odds) to compare effect 
sizes within and across models (see below).

```{r signfq, echo=FALSE, purl = FALSE}
question("Which statement is most detailed and correct?",
  answer("There is a statistically significant effect here."),
  answer("The edges coefficient is negative, indicating that as the number of edges increases, the probability of a tie forming between any two nodes decreases."),
  answer("The edges coefficient is negative and statistically significant, indicating that as the number of edges increases, the probability of a tie forming between any two nodes decreases.", correct = TRUE),
  answer("There are -1.609438 ties in this network."),
  answer("There is no statistically significant effect here."),
  random_answer_order = TRUE,
  allow_retry = TRUE
)
```

However, since there is no direct, intuitive way to interpret the log-odds 
coefficients substantively and among each other, 
this is not the most useful way to present results such that 
e.g. policy-makers know what to expect.

### Odds Ratios

Another approach is to transform the logit’s log-odds coefficients into 
something more intuitive to support substantive interpretation.
Recall from Stats I that odds are a different way of thinking about probability.
The _odds ratio_ is the ratio of the odds of an event to the odds of another thing happening, or a change in the odds of $Pr(Y=1)$ associated with a one unit change in the covariate.

We can obtain odds ratios by simply exponentiating logistic regression coefficients: 
  
$$\text{Odds Ratio} = \exp(\hat\beta_k\delta)$$
  
where $\delta$ is the change in $X_k$ (usually 1 unit).
It is straightforward enough to obtain odds ratios in R,
but there is also an option to obtain odds ratios instead of the log-odds 
directly in the `tidy()` function.

```{r oddsratio, exercise = TRUE, exercise.setup = "flombern"}
exp(coef(flom.bern)) # 0.2
tidy(flom.bern, exponentiate = TRUE, conf.int = TRUE) # To return the odds ratios, with confidence intervals etc.
```

Odds are much more intuitive to interpret than log-odds.
Odds range from 0 to \\(\infty\\) (but never negative).
Essentially, if the odds ratio is 1:1, 
then event and non-event equally likely.
If odds ratio greater than 1, 
then event more likely than non-event.
If odds ratio less than 1, 
then event less likely than non-event.
In other words:

- OR>1: \\(\Pr(Y=1) > \Pr(Y=0)\\)
- OR<1: \\(\Pr(Y=1) < \Pr(Y=0)\\)
- OR=1: \\(\Pr(Y=1) = \Pr(Y=0)\\)

### Predicted Probabilities

Yet, what we really care about, in most cases, is not the odds of an event per se,
but the effect (of changes in $X$) on the _probability_ of the actual event of interest. 
To obtain probabilities for logistic models is a bit more involved than with 
ordinary least squares (OLS) regression though,
since effect of covariates is linear only with respect to the log-odds, not $Y$ itself.
Because model nonlinear, 
real net effect of a change in $X$ depends on the constant, other $X$s and their parameter estimates.
Unlike in a linear regression model, 
the first derivative of our (binary) logit function depends on $X$ and \\(\hat\beta\\):
  
  $$  \frac{\partial\Pr(\hat{Y}_i = 1)}{\partial X_k} \equiv \lambda(X) = \frac{\exp(X_i\hat\beta)}{[1 + \exp(X_i\hat\beta)]^2}\hat\beta_k $$
 
Practically, this means that if you’re interested in the effect of a one-unit change in $X$ on $Pr(Y_i = 1)$, 
how much change there is depends critically on “where you are on the curve”.
  
```{r predprobplot, echo = F, fig.height=3, fig.width=8, purl = FALSE}
  ggplot(data.frame(x = c(-4, 4)), aes(x = x)) +
    ggplot2::stat_function(fun = plogis, colour = "blue") +
    ggplot2::geom_line(aes(x = 0, y=c(0.5,.73), colour = "+1 unit change @0")) +
    ggplot2::geom_line(aes(x = c(0,1), y=0.73, colour = "+1 unit change @0")) +
    ggplot2::geom_line(aes(x = 1, y=c(0.73,0.88), colour = "+1 unit change @1")) +
    ggplot2::geom_line(aes(x = c(1,2), y=0.88, colour = "+1 unit change @1")) +
    ggplot2::theme_minimal()
```

This means we need to calculate predicted probabilities of $Y$ at specific values of key predictors.
In our simple Bernoulli model, we're modelling only the number of edges,
and so we only need to condition our predicted probabilities on this one variable.
We can calculate the probability of a tie forming between any two actors as follows,
indicating the addition of a tie by increasing the edges count by 1
(later there is an example of how to do this for more complex models with multiple variables).
However, we can also obtain this information by using the handy `predict()` function.
We can simply pass data on our independent variables, 
either in-sample, out-of-sample, or totally-made-up,
to `predict()` and see what probability of a 1 would be predicted by the model.

```{r predprob, exercise=TRUE, exercise.setup = "flombern"}
exp(coef(flom.bern)) / (1 + exp(coef(flom.bern)))
adj <- predict(flom.bern, output = "matrix")
adj[is.na(adj)] <- 1
adj * t(adj)
```
  
We can see here an estimated probability of a tie forming between any 
two actors of 0.1666667.
Wait, where have we seen this number before?
This is the same number that we obtained for the density of the network (0.1666667).

### Summary
  
So, interpreting ERGM results can be a little tricky, but for now remember three points:

1. Nearly all of these approaches require one to be cognizant of “where we are on the curve”.
2. When it comes to interpretation, a picture really is often more valuable than text or tables,
and can help guard against misinterpretation.
3. With very rare exceptions, never a good idea to present quantities of interest without their associated measures of uncertainty.

## Assessing goodness-of-Fit

### Simulating

So we have a Bernoulli model for the Florentine marriage network.
Once we have estimated the coefficients of an ERGM, 
the model is completely specified. 
It defines a probability distribution across all networks	of this size. 
But is this a _good_ model?
If the model is a good fit to the observed data, 
then networks drawn from this distribution will be 
likely to "resemble" the observed data. 
To see examples of networks drawn from this distribution we use `simulate()`:	

```{r flom-sim, exercise=TRUE, exercise.setup = "flombern", fig.width=9}
flom.bern.sim <- simulate(flom.bern)
graphr(flomarriage) + ggtitle("Observed") +
  graphr(flom.bern.sim) + ggtitle("Simulated")
```

So using `simulate()`, we get a network drawn from the distribution of networks
generated by our Bernoulli model.
As such, we can expect it to look similar to the observed network in some ways.
Since the Bernoulli model is generating networks of similar density on
the same number of nodes, these features will be fairly well captured.
Yet you may see the number of isolates, degree distribution, geodesic distances,
and appearances of triangles all change. 

### GOFing

What happens if you run the chunk above again? And again? Do they look the same?
Because each simulation looks a bit different, we should try and simulate many,
and then examine whether features of the observed network are being well-captured _in general_.
We could just simulate many networks,
`simulate(flom.bern, nsim=100)`,
and then analyse them according to various network statistics of interest,
but `{ergm}` also has a function -- `gof()` -- that does this for us.
Let's test for all the normal structural features:

```{r flom-gof, exercise=TRUE, exercise.setup = "flombern", fig.width=9}
(flom.bern.gof <- gof(flom.bern,  GOF = ~ distance + degree + espartners))
plot(flom.bern.gof, statistic = "dist") # geodesic distances
plot(flom.bern.gof, statistic = "deg") # degree distribution
plot(flom.bern.gof, statistic = "espart") # edgewise shared partners
# plot(flom.bern.gof, statistic = "triadcensus") # geodesic distances
```

These plots compare the observed and simulated networks on various auxiliary 
network statistics such as the degree distribution, geodesic distances, 
and edgewise shared partners.
The thick red (or otherwise highlighted) line annotated with numbers represent 
the observed statistics.
Box/whisker and violin plots show the distribution of the statistic across the simulated networks.

Looking at the plots, we see that the model did not do a very good job at
predicting the network structure. 
While the geodesic distances were fairly well captured 
(this is easy in a small network like this),
we can see some over- and under-estimation of the degree distribution
and edgewise shared partners.

What does this mean for our Bernoulli model?
This model was not great. We only had one covariate (edges) and so 
just modelled the unconditional probability of a tie appearing (anywhere...).
We should add covariates that may better explain the degree level and 
triangle formation/clustering we see in our network. 
We need additional modelling!

## Markov model

```{r visflo2, echo=FALSE, purl = FALSE, fig.width=9, setup = "flom-gof"}
p1 <- as_tidygraph(flomarriage) |> mutate_nodes(Degree = node_by_deg()) |>  
  mutate_ties(Triangles = tie_is_triangular()) |> 
  graphr(node_size = "Degree", edge_color = "Triangles") + 
  ggplot2::theme(legend.position = 'none')
p2 <- (plot(flom.bern.gof, statistic = "deg") / plot(flom.bern.gof, statistic = "espart"))
p1 + p2
```

Let's review what we know about the marriage network to get a better fit.
There are some nodes with more activity (higher degree) in this network
than others, and some triangle configurations appear.
Yet our Bernoulli model (unsurprisingly) did not capture these features well.
It overestimated nodes with a degree of two (there are actually only two, but the model is expecting around four), 
and underestimated nodes with degrees of one (there are four pendants, whereas the model expects around three) or three (the model expects around 4, but six are observed).
While the Bernoulli model got the density perfectly right, duh,
it seems that it is not capturing other features of the network of interest.
Indeed, most social network analysts would find this unsurprising, 
as they consider social ties more than just random chance.

### Convergence

Since the Bernoulli model was misjudging "clustering",
let's add a term often thought to be a measure of "clustering": 
the number of completed `triangle`s.
Because this model involves broader dependencies, 
the estimation draws on MCMC and is thus stochastic.
This means that your output may differ slightly.

```{r flom-markov, exercise=TRUE, exercise.setup = "ergm-data"}
flom.mark1 <- ergm(flomarriage ~ edges + triangle) 
# Add verbose=T to watch for the convergence test p-value and 
# log-likelihood improvement to go down!
plot(flom.mark1)
tidy(flom.mark1)
glance(flom.mark1)
```

First, the console output reports that the estimation procedure has converged
on stable estimates with 99% confidence (p-value < 0.0001).
You can also visually check convergence by plotting the model object.
In the plot, we see that the log-likelihood stabilises over time,
indicating convergence.

### Interpretation

How do we interpret the coefficients now that the model is more complex?
First, we can (seemingly) ignore the edges/intercept now, even if significant,
as it will just be counterbalancing the other effects in the model.
In the results, we see that the MCML estimate for triangles is 
not statistically significant.
But if we try and interpret it anyway, for the practice,
we see that we cannot completely ignore the edges coefficient,
_because_ it is part of the story of when and where ties form.
For endogenous effects relating to the structure of the network,
the coefficients need to be interpreted _with_ the coefficient of the edges (the intercept).
For exogenous effects (eg. the monadic covariate 'wealth' later),
the coefficients can be interpreted separately.
An extra tie is perhaps not _just_ an extra tie, but could also create
one or more triangles, and an extra triangle will always create an extra tie, etc.
So it depends...:

- if the tie considered will not add any triangles to the network, log-odds are: Edges  
- if it will add one triangle to the network, log-odds are: Edges + Triangle  
- if it will add two triangles to the network, log-odds are: Edges + Triangle x 2
and so on.

So we need to think about working out the probabilities based on the tie's context:

```{r markov-predict, exercise=TRUE, exercise.setup = "flom-markov"}
# for edges + triangle
exp(coef(flom.mark1)[[1]] + coef(flom.mark1)[[2]])/(1 + exp(coef(flom.mark1)[[1]] + coef(flom.mark1)[[2]]))
# for edges + 2 triangles
exp(coef(flom.mark1)[[1]] + 2*coef(flom.mark1)[[2]])/(1 + exp(coef(flom.mark1)[[1]] + 2*coef(flom.mark1)[[2]]))
probs <- predict(flom.mark1, output = "matrix")
probs["Barbadori","Ridolfi"]
probs["Albizzi","Tornabuoni"]
```

### Is it any good?

So now we have a properly converged model. 
But does it fit? (These two things are independent)
Let's test for all the normal structural features.
We want to check fit against auxiliary statistics because we want to know
if we might be missing anything.

```{r flom-mark1-gof, exercise=TRUE, exercise.setup = "flom-markov", fig.width=9}
flom.mark1.gof <- gof(flom.mark1,  GOF = ~ degree + espartners)
(plot(flom.bern.gof, statistic = "deg")/
plot(flom.mark1.gof, statistic = "deg")) | 
(plot(flom.bern.gof, statistic = "esp")/
plot(flom.mark1.gof, statistic = "esp"))
```

So far we have (mostly) fitted a model to the observed network using only structural effects.
But we are often interested in these effects on top of traditional explanations,
or as controls for those traditional explanations.
Below consider the effect of two potential explanations: 
money (the nodal covariate `nodecov(wealth)`) and the related business network (`dyadcov(flobusiness)`).

```{r markov-freeplay, exercise = TRUE, fig.width=9}

```

For more effects and more flexible uses of such effects, consult the `{ergm}` manual: `help('ergm-terms')`  
For a more complete discussion of these terms see the paper
`Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects'
in J Stat Software v. 24. (link available online at http://www.statnet.org)  

## Social circuit models

### New dataset

The Florentine dataset was relatively easy to fit.
Sometimes social networks can be considerably more difficult to model.
Let's try and model a larger, more complex network, `faux.magnolia.high`
from the `{ergm}` package.
Load it, call up the documentation, and visualise the giant component.

```{r magnolia-data, exercise = TRUE, fig.width=9}
# ?faux.magnolia.high
data('faux.magnolia.high')
faux.magnolia.high
magnolia <- to_giant(faux.magnolia.high) 
(graphr(magnolia, labels = FALSE, node_color = "Grade", node_size = 0.2) + ggtitle("Grade")) +
(graphr(magnolia, labels = FALSE, node_color = "Sex", node_size = 0.2) + ggtitle("Sex"))
```

So we now have 1461 vertices and 974 edges
We also have four attributes: grade, race, sex and vertex.names
Let's pick up where we left off, with fitting a covariate-based model:

```{r magncov1, exercise = TRUE}
magn.covr1 <- ergm(magnolia ~ edges + nodematch('Grade', diff = T) + 
                     nodematch('Race', diff = T))
summary(magn.covr1)	
```

### Troubleshooting

Note that the nodematch coefficients for Other and Hispanic are estimated as -Inf. 
Why is this?	

```{r magnracefreq, exercise = TRUE}
table(magnolia %v% 'Race') # Frequencies of race	
mixingmatrix(magnolia, "Race")	
```

Looks like there are some of every race, what's the problem?
Ah, so the problem is that there are very few students in the "Other" category, 
and these students form no homophilous (within-group)	ties.
Same with "Hispanic" students.
The empty cells are what produce the `-Inf` estimates.
To avoid this, you can remove this category from the `nodematch()` call.

```{r magncov2, exercise = TRUE}
magn.covr2 <- ergm(magnolia ~ edges + nodematch('Grade', diff = T) + 
                     nodematch('Race', diff = T, keep = c(1:2,4,6)))  
summary(magn.covr2)
```

### Degeneracy

There are also more serious issues: When a model is a particularly poor 
representation of the observed network, estimation may be affected. 
In the worst case scenario,	simulated networks will be so different 
from the observed network	that the algorithm fails altogether. 
This can occur for two general reasons. 

1. the simulation algorithm may fail to converge, and	
the sampled networks are thus not from the specified distribution.	
2. the model parameters used to simulate the networks are too	
different from the MLE, so even though the simulation algorithm is	
producing a representative sample of networks, this is not the sample	
that would be produced under the MLE.	

For more detailed discussions of model degeneracy in the ERGM context,	
see  papers in e.g. *Social Networks* v.29 and *J Stat Software* v. 24.

Now let's try to fit a triangle model like with the Florentine dataset above:

```{r magnmark1, exercise = TRUE, purl=FALSE}
magn.mark1 <- ergm(magnolia ~ edges + triangle)
```

Did you get an error? 
That's because in the process of trying to fit this model, the algorithm
heads off into simulating networks much denser and clustered than the observed network.
This is a big problem, and such a clear indicator of degeneracy that the algorithm 
cuts off to avoid heading off into areas that would cause memory issues.
Degeneracy is a feature of how well the suggested model compares to the observed network

If you have a degenerate model, there is some 'witchcraft' (art + science)
that can be done to tweak the way the simulations are generated,
and I can offer some assistance with that if you show me your `?mcmc.diagnostics`,
but more often than not it's a problem of how the model is specified.

A good rule of thumb, if facing such an issue, is to:

a) make sure you include degree- or centrality-based effects 
in addition to closure-based effects like triangles;
b) use geometrically weighted or alternating forms of both
centrality (`gwdegree`) and closure/triangle (`gwesp`) effects.

So let's try that now:

```{r magnstr, exercise = TRUE}
magn.str <- ergm(magnolia ~ edges + gwdegree(0.25, fixed = T) + 
                     gwesp(0.25, fixed = T))
plot(magn.str)
```

Excellent, now we're getting somewhere!
It converged, not bad. 
Everything is highly significant,
and both gwdeg and gwesp are positive.
A positive gwdegree effect generally means that preferential attachment is operating.
A positive gwesp effect generally means that there is closure.
By fixing the alpha for each at 0.25, we're saying that successive contributions
to the statistic matter less though (which helps us avoid suuper dense networks, etc.)

But we don't want to just see whether there is an aggregation of ties
around particular nodes (positive gwdegree) and collections of ties (positive gwesp),
we want to know whether e.g. this is really preferential attachment 
-- an endogenous effect -- or could it be just that the ties are collating around
particular nodes because of their attributes.
That's why we also need to model these as a multivariate model together
with these possibly confounding variables/alternative explanations.

```{r magnattr, exercise = TRUE}
magn.attr <- ergm(magnolia ~ edges + gwesp(0.25,fixed=T) + gwdegree(0.25,fixed=T) + 
                     nodematch('Grade') +	nodematch('Race') + nodematch('Sex'))
plot(magn.attr)
```

Success!
(though one could maybe increase the MCMC sample size and/or interval)
In practice it might take more trial and error to find this solution...


So now we have a final model for (the main component of the) Magnolia high school.
What does the model say?
Everything is significant and positive,
suggesting that all this matters.
But that there are still significant effects for gwdegree and gwesp,
even as we control for grade, race, and sex homophily,
suggests that these statistically significant effects may represent
endogenous effects over and beyond what can be explained by the other variables
and indicating that there may be such network mechanisms as ties collecting
around already popular nodes or joining two-paths operating here.

Now that you have a fitted model, let's see how well it fits:  

```{r magnfit, exercise = TRUE, fig.width=9}
magn.attr.gof <- gof(magn.attr,  GOF = ~ degree + distance + triadcensus + espartners - model)
plot(magn.attr.gof, statistic = "deg")
```

Looks ok, though there's still some over and underestimation,
so there's room for improvement...
Let's also compare a simulated network with the observed network

```{r magsim, exercise = TRUE, fig.width=9}
magn.sim1 <- simulate(magn.attr)
graphr(magnolia, node_color = "Grade", node_size = 0.2, labels = FALSE) + 
  ggtitle("Observed") +
  graphr(magn.sim1, node_color = "Grade", node_size = 0.2, labels = FALSE) + 
  ggtitle("Simulated")
```

Note: if all else fails, simulated networks can help diagnose problems

Task 3: Run an ERGM on the Florentine business network with terms capturing 
centrality, cohesion, and the influence of monadic (nodal) and dyadic 
covariates all at once. Check convergence and fit and interpret the results.

```{r flob, exercise = TRUE}
flob <- ergm(flobusiness ~ edges + gwdegree(0.25, fixed = T) + gwesp(0.25, fixed = T) + 
       nodecov('wealth') + dyadcov(flomarriage))
tidy(flob)
glance(flob)
```

What can we tell from this?
Model convergence: 
Convergence test p-value: 0.0080. Converged with 99% confidence.
Edges (-), gwesp (+), and marriage (+) were statistically significant.
The probability of seeing a tie is much higher when nodes are already
connected through a marriage tie!

### Let's analyse GOF

```{r flobgof, exercise = TRUE, fig.width=9}
(flob.gof <- gof(flob,  GOF = ~ degree + triadcensus + espartners - model))
plot(flob.gof)
```

## Free play


```{r dependq, echo=FALSE, purl = FALSE}
question("What kind of (statistical) dependencies do network analysts often consider?",
  answer("Independence", message = "A model with full independence across observations would be considered too simplistic for network data."),
  answer("Dyad independence", correct = TRUE),
  answer("Markov", correct = TRUE),
  answer("Social circuit", correct = TRUE),
  answer("Partial conditional", correct = TRUE),
  answer("Trade", message = "This is not a type of statistical dependency."),
  answer("Environment", message = "This is not a type of statistical dependency."),
  answer("Health", message = "This is not a type of statistical dependency."),
  random_answer_order = TRUE,
  allow_retry = TRUE
)
```

While these are the conclusions from this 'play' data, 
you may have more and more interesting data at hand.
Take a look, for example, at the `fict_actually` data in the `{manynet}` package.
How would you go about specifying such a model?
Why is such an approach more appropriate for network data than linear
or logistic regression?

```{r freeplay, exercise = TRUE, fig.width=9}

```

