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## This tutorial

This tutorial introduces several contagion or diffusion processes and how they operate across various networks. Network structures are key to how diffusion processes play out.

By the end of this tutorial, you will be able to:

- ‘play’ a diffusion process over any kind of network
- print, plot, graph and interpret the results
- understand cascade models
- understand the role of network structure on diffusion
- understand threshold models
- understand the role of seeds for network intervention
- understand compartmental models
- understand the role of vaccination strategies
- understand learning models
- understand the role of influential nodes

## Influence cascade models

The simplest kind of diffusion process is an independent cascading model, where any contact with the attribute (disease or innovation) is enough to cause infection/adoption. If the network is connected, the attribute will ‘cascade’ across the network until all nodes are ‘infected’/have adopted.

### Diffusing across a lattice

Let us begin with a lattice network that shows us exactly how such a cascade works. A lattice network resembles, say, people standing on stones outlined on coordinates; each person can touch the person next to them. That’s why this structure is often used to stand in for spatial structure. Remember, to create a lattice network, all you need to do is run `create_lattice()`, together with the number of nodes (of each mode) and the maximum number of neighbors (width) as the arguments. Let’s create a one-mode lattice with 32 nodes, maximum width 4, and graph it using `graphr()`:

    lat <- create_lattice(32, width = 4)
    graphr(lat)

Ok, great! That’s made a nice little lattice network. Next we want to play a diffusion process on this network. To do this, we need to run `play_diffusion()`. This function plays a simple contagion process on a network, where any contact with an infected/adopter node is enough to cause susceptible nodes to adopt/become infected by the following timepoint. The function returns a network object that contains information about how the diffusion played out. Then we can investigate the diffusion process by extracting this information using `as_changelist()` and `as_diffusion()`. The first simply extracts the list of adoption/infection events, while the second summarises the diffusion process by time point.

    lat_w_diff <- play_diffusion(lat)
    (diff_events <- as_changelist(lat_w_diff))
    (diff_summ <- as_diffusion(lat_w_diff))

The changelist presents a list of the events at each time point. In the event variable, the ‘I’ indicates that these are all infection/adoption events. Where ‘t’ is 0, that means that these are the seeds producing the starting condition for the diffusion. The diffusion summary shows the number of nodes that don’t (yet) have the attribute (S for susceptible, but can also be non-adopter) and those that do have the attribute (I for infected, or adopter) at each time point (t). We can see a steady growth here, except for a slower initialisation and winding down.

### Visualising cascades

We have several different options for visualising diffusions. The first visualisation option that we have is to plot the diffusion result itself.

    plot(diff_summ)
    # plot(diff_summ, all_steps = FALSE) # To plot only 'where the action is', use the argument `all_steps = FALSE`.

This plot effectively visualises what we observed from the print out of the `diff_summ` object above. The red line traces the proportion of infected; the blue line the (inverse) proportion of susceptible nodes in the network. The grey histogram in the plot shows how many nodes are newly ‘infected’ at each time point, or the so-called ‘force of infection’ (\(F = \beta I\)).

But maybe we want to also/instead view the diffusion on the actual network. Here we can use all the three main graphing techniques offered in `{autograph}`. First, `graphr()` will graph a static network where the nodes are coloured according to how far through the diffusion process the node adopted. Note also that any seeds are indicated with a triangle.

    graphr(lat_w_diff, node_size = 0.3)

Second, `graphs()` visualises the stages of the diffusion on the network. By default it will graph the first and last wave, but we can change this by specifying which waves to graph.

    graphs(lat_w_diff)
    graphs(lat_w_diff, waves = c(1,4,7,10))

We can see here exactly how the attribute in question (ideas, information, disease?) is diffusing across the network. It’s like a cascade of red sweeping across the space!

Lastly, `grapht()` animates this diffusion process to a gif. It can take a little time to encode, but it is worth it to see exactly how the attribute is diffusing across the network! Note that if you run this code in the console, you get a calming progress bar; in the tutorial you will just need to be patient.

> NB: It is not currently working (for diffusions) at the moment, but a refactoring soon will fix this and other related bugs.

    # grapht(lat_w_diff, node_size = 10)

### Varying network structure

While a lattice structure is one way of representing spatially governed diffusion, social and political networks can have quite different structures. We will get into some examples in later sections, but let’s clarify the impact of network structure here by visualising how the diffusion process differs when everything but the structure stays the same. We’re going to create (deterministically) and generate (stochastically) a range of different network structures with the same number of nodes and see the difference the structure makes.

- `create_ring()`: Creates a ring or chord graph of the given dimensions that loops around and is of a specified width or thickness.
- `generate_random()`: Generates a random network with a particular probability.
- `generate_scalefree()`: Generates a small-world structure following the lattice rewiring model.
- `generate_smallworld()`: Generates a scale-free structure following the preferential attachment model.

    graphr(play_diffusion(create_ring(32, width = 2)))
    graphr(play_diffusion(generate_random(32, 0.15)))
    graphr(play_diffusion(generate_scalefree(32, 0.025)))
    graphr(play_diffusion(generate_smallworld(32, 0.025)))

Which diffusion process completed first? `graphr()` only colors nodes’ relative adoption, and `graphs()` (at least by default) only graphs the first and last step. But there is an easier way. Play these same diffusions again, this time nesting the call within `net_infection_complete()`.

    net_infection_complete(play_diffusion(create_ring(32, width = 2)))
    net_infection_complete(play_diffusion(generate_random(32, 0.15)))
    net_infection_complete(play_diffusion(generate_scalefree(32, 0.025)))
    net_infection_complete(play_diffusion(generate_smallworld(32, 0.025)))

### Free play: US States

Run an influence cascade on US states’ geographic contiguity in `irps_usgeo`. You can start the infection in California by specifying `seeds = 5`.

    us_diff <- play_diffusion(irps_usgeo, seeds = 5)
    plot(us_diff)
    graphr(us_diff)
    net_infection_complete(us_diff)

What’s happening here? Can you interpret this?

## Linear threshold models

So far, we’ve been using a simple cascading diffusion model where each node needs only to be in contact with one infectious individual to be infected. But what if nodes have higher thresholds , or only some nodes have higher thresholds? This is known as a linear threshold model, where if infection/influence on a node through some (potentially weighted) network exceeds some threshold, then they will adopt/become infected.

### Threshold rising

Let’s use the ring network again this time to illustrate the impact of rising thresholds on network diffusion. Remember that a ring network is similar to a lattice network, but typically thinner (use `width = 2`) and is connected at each end. That it is connected at each end should ensure that the diffusion completes, right? Let’s see what the results are if you play four different diffusions:

- `seeds = 1, thresholds = 1`
- `seeds = 1, thresholds = 2`
- `seeds = 1:2, thresholds = 2`
- `seeds = c(1,16), thresholds = 2`

    rg <- create_ring(32, width = 2)
    plot(play_diffusion(rg, seeds = 1, thresholds = 1))
    plot(play_diffusion(rg, seeds = 1, thresholds = 2))
    plot(play_diffusion(rg, seeds = 1:2, thresholds = 2))
    plot(play_diffusion(rg, seeds = c(1,16), thresholds = 2))

Whereas a threshold of one will result in complete infection, a threshold of two will not lead to any diffusion process unless there are two seeds and they are both in another nodes neighbourhood. Note also that the nodes need to be adjacent to overcome the higher threshold.

### Varying degrees

In a ring network, all nodes have the same degree. But many typical social networks include some variation in degree. A threshold of 2 would be easy to surpass for particularly well connected nodes, but impossible for pendants. Let’s see what happens when we use this threshold on a scale-free network instead.

    plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1, thresholds = 2))

That’s because a scale-free network includes many nodes with only one or two ties, and a few very high degree nodes that act as critical brokers to the rest of the network.

### Complex thresholds

Let’s try again, but this time we’re going to specify the threshold as a proportion of contacts that should be infected before the node will become infected. This is sometimes called a fractional threshold model or complex diffusion. Try thresholds of 0.1, 0.25, and 0.5 on two seeds and 10 steps on the scale-free networks we have been using here.

    plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))
    plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))
    plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))

    plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.1, steps = 10))
    plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.25, steps = 10))
    plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.5, steps = 10))

What’s happening here is that the high degree nodes in this scale-free network are obstructing the diffusion process because it is unlikely that many of their branches are already infected.

### Varying thresholds

Lastly, note that it may be that thresholds vary across the network. Let’s use an example network to explore this: `fict_lotr`.

    graphr(fict_lotr, node_color = "Race")

Something is going around Middle-Earth, but different races have different resistances (i.e. thresholds). Let us say that there is a clear ordering to this.

    lotr_resist <- fict_lotr %>% mutate(resistance = dplyr::case_when(Race == "Dwarf" ~ 2,
                                                       Race == "Elf" ~ 4,
                                                       Race == "Ent" ~ 5,
                                                       Race == "Hobbit" ~ 3,
                                                       Race == "Human" ~ 1,
                                                       Race == "Maiar" ~ 6))
    graphs(play_diffusion(lotr_resist, thresholds = "resistance"))

Fun! Now how would you interpret what is going on here? Can you rewrite the code above so that fractional thresholds are used?

### Free play: Lord of the Rings

## Make it start

Let’s say that you have developed an exciting new policy and you are keen to maximise how quickly and thoroughly it is adopted. We are interested here in network intervention .

### Choosing where to seed

Since the ring network we constructed is cyclical, then no matter where the ‘infection’ starts, it should diffuse throughout the whole network. To see whether this is true, try seeding the innovation at the first and sixteenth (middle) node and see whether the result is any different.

    plot(play_diffusion(create_ring(32, width = 2), seeds = 1))
    plot(play_diffusion(create_ring(32, width = 2), seeds = 16))

Now what if we seed the network with more than one infected node? Choosing the first four nodes we can see that the process is jump-started, but doesn’t really conclude that much faster.

    # Remember we want to see the first four nodes.

    plot(play_diffusion(create_ring(32, width = 2), seeds = ____))

    rg_d3 <- play_diffusion(create_ring(32, width = 2), seeds = 1:4)
    plot(rg_d3)

    # graph the diffusion within the network
    graphs(play_diffusion(create_ring(32, width = 2), seeds = 1:4),
           layout = "stress")

But what if we seed the network at three different places? Here we can use `node_is_random()` to randomly select some nodes to seed. Try it with four randomly-selected nodes and see what you get.

    # We will be using the node_is_random() within the seed argument to random select
    # 4 nodes

    plot(play_diffusion(create_ring(32, width = 2),
                        seeds = ____(create_ring(32, width = 2), ____)))

    plot(play_diffusion(create_ring(32, width = 2),
                        seeds = node_is_random(create_ring(32, width = 2), 4)))

Where the innovation/disease is optimally seeded to accelerate or decelerate diffusions is a crucial question in network intervention studies.

Now let’s see whether where the infection is seeded matters when the network has a different structure. Here let’s play and plot two diffusion on the lattice network, one with the first node as seed and again one on the middle.

    plot(play_diffusion(lat, seeds = 1))
    plot(play_diffusion(lat, seeds = 16))
    lat %>%
      add_node_attribute("color", c(1, rep(0, 14), 2, rep(0, 16))) %>%
      graphr(node_color = "color")

    # visualise diffusion in lattice graph
    graphs(play_diffusion(lat, seeds = 16), layout = "grid")

Let’s try one more network type, this time the scale-free network. Play and plot the results over ten steps for node 10, random, maximum, and minimum nodes as seeds.

Similar to the previous examples, we will be using the following functions within the seed argument:

- `node_is_random()`: Returns a logical vector indicating a random selection of nodes as TRUE.
- `node_is_max()`: Returns logical of which nodes hold the maximum of some measure.
- `node_is_min()`: Returns logical of which nodes hold the minimum of some measure.

We could use these on degree centrality, or perhaps some other kind of centrality?

    sf <- generate_scalefree(32, 0.025)
    sf %>%
      as_tidygraph() %>%
      mutate(degree = ifelse(node_is_max(node_degree(sf)) == TRUE, "max",
                          ifelse(node_is_min(node_degree(sf)) == TRUE, "min", "others"))) %>%
      graphr(node_color = "degree") + guides(color = "legend") + labs(color = "degree")

    plot(play_diffusion(sf, seeds = 10, steps = 10))
    plot(play_diffusion(sf, seeds = node_is_random(sf), steps = 10))
    plot(play_diffusion(sf, seeds = node_is_max(node_degree(sf)), steps = 10))
    plot(play_diffusion(sf, seeds = node_is_min(node_degree(sf)), steps = 10))

    # visualise diffusion in scalefree network
    graphs(play_diffusion(sf, seeds = node_is_min(node_degree(sf)), steps = 10))
    graphs(play_diffusion(sf, seeds = 16, steps = 10), waves = 1:3)

### Other strategies

There are many other strategies considered in the diffusion literature. Many of them are implemented here and might be considered as strategies:

- `node_is_independent()` identifies nodes belonging to the largest independent set
- `node_is_cutpoint()` identifies nodes at the articulation points in a network
- `node_is_core()` identifies nodes that are members of the network’s core
- `node_is_fold()` identifies nodes that are in a structural fold between two or more triangles
- `node_is_mentor()` identifies high indegree nodes as mentors

    sf %>% mutate_nodes(ni = node_is_independent()) %>% graphr(node_color = "ni")
    plot(play_diffusion(sf, seeds = node_is_independent(sf), steps = 10))

### Advertising adoption

Another strategy often employed is that of advertising how much adoption there already is in the network globally. The mechanism is that as individuals see the global prevalence of adoption increasing, they might be inclined to adopt even if they are not (yet) in direct contact. For example, these lawyers might accept word-of-mouth directly from two colleagues, but are also attuned to common practice irrespective of whether direct colleagues have adopted the innovation.

    plot(play_diffusion(ison_lawfirm, thresholds = 2, prevalence = 0.005),
         all_steps = FALSE)

## Compartmental models

So far we’ve been looking at variations on a pretty straight-forward diffusion process where nodes can only belong to one of two states or ‘compartments’, Susceptible and Infected (the basic SI model). This has been useful, but sometimes what we are interested in, whether disease, innovation, or some other behaviour, has more complicated and probabilistic dynamics. But before we get into that, let’s see how we can play and plot several simulations to see what the range of outcomes might be like.

### Running multiple simulations

To do this, we need to use `play_diffusions()` (note the plural). It has all the same arguments as its singular counterpart, along with a couple of additional parameters to indicate how many simulations it should run, e.g. `times = 50`, whether it should use `strategy = "multisession"` to run the simulations across multiple cores instead of the default `strategy = "sequential"`, and `verbose = TRUE` if it should inform you of computational progress. Try this out with our well-mixed random network, 10 steps, 5 times, and with a `transmissibility` parameter set to 0.5 to indicate that in only 1/2 cases is contagion successful.

    rando <- generate_random(32, 0.1)
    graphr(rando)
    plot(play_diffusions(rando, transmissibility = 0.5, times = 5, steps = 10))

Note that in this plot the number of new infections is not plotted because this might vary a bit each time the simulation is run. Instead, the loess line smooths over the varying trajectories and a (hardly distinguishable for this call) grey border to the line represents the standard error. The blue line is the proportion of nodes in the Susceptible compartment, and the red line is the proportion of nodes in the Infected compartment.

### SIR models

Let’s start off with an SIR model in which, after some period in which an infected node is themselves infectious, they recover and can no longer infect or become reinfected. To add a recovered component to the model, specify the `recovery` argument. Let’s try a rate of recovery of 0.20, which means that it’ll take an infected node on average 5 steps (days?) to recover.

    # Remember, we are still looking at the random network, "rd", with a
    # recovery rate of 20 percent.

    plot(play_diffusions(____, recovery = ____))

    plot(play_diffusions(rando, recovery = 0.2))

What we see in these kinds of models is typically a spike in infections towards the start, but as these early infections recover and become immune, then they can provide some herd immunity to those who remain susceptible. If you get moderately different results each time, try increasing the number of `times` the simulation is run, which should average out these differences and make the results more reliable.

    plot(play_diffusions(rando, recovery = 0.2, times = 100))

### SIRS models

That’s great, but maybe the immunity conferred from having recovered from the contagion doesn’t last forever. In this kind of model, add an additional `waning` parameter of 0.05. This means that after twenty steps (on average), a recovered node may lose its recovered status and become susceptible again. Play a single diffusion so that you can see what’s going on in a particular run.

    plot(play_diffusion(rando, recovery = 0.2, waning = 0.05))

Depending on your particular simulation, there might be some variation, so let’s run this same diffusion but multiple (100?) times.

    plot(play_diffusions(rando, recovery = 0.2, waning = 0.05, times = 100))

### SEIR models

Lastly, we’ll consider a compartment for nodes that have been Exposed but are not yet infectious. This kind of an incubation period is due to some `latency` (\(\sigma\)). This should also be specified as a proportion, but note that this is inverted internally. This means that a latency of 0 means that exposure immediately renders the node infectious. A latency of 0.75 means that it will take the node approximately 4 days (1/1-0.75 = 1/0.25 = 4) to become infectious. Play a single diffusion so that you can see what’s going on in a particular run.

    set.seed(123)
    plot(play_diffusion(rando, seeds = 10, latency = 0.25, recovery = 0.2))

    # visualise diffusion with latency and recovery
    graphs(play_diffusion(rando, seeds = 10, latency = 0.25, recovery = 0.2), waves = c(1,5,10))

## Make it stop

In this section, we are interested in how to most effectively halt a diffusion process.

An attribute’s reproduction number, or \(R_0\), is a measure of the rate of infection or how quickly that attribute will reproduce period over period. It is calculated as \(R_0 = \min\left(\frac{T}{1/L}, \bar{k}\right)\), i.e. the transmissibility (proportion of susceptible nodes that are infected at each time period) over the average recovery or infection length (average length of time nodes remain infected).

It can be interpreted as follows: - Where \(R_0\) > 1, the ‘disease’ will ‘infect’ more and more nodes in the network.
- Where \(R_0\) < 1, the ‘disease’ will not sustain itself and eventually die out.
- Where \(R_0\) = 1, the ‘disease’ will continue as endemic, if conditions allow.

So how can we establish the \(R_0\) here? We can use `net_reproduction()`.

    rd_diff <- play_diffusion(rando, transmissibility = 0.25, recovery = 0.05)
    plot(rd_diff)
    # R-nought
    net_reproduction(rd_diff)
    net_infection_total(rd_diff)

Ok, so there’s (probably: you may have slightly different results) a danger here. The relationship between the reproduction number and the observed infection should be clear. The \(R_0\) is over 1 and the ‘disease’ seems to burn through the network infecting almost everyone. How can we try to resist or halt such a disease?

Ok, let’s start with option 1. After all, those who built up natural immunity (recovered) may have already protected some parts of the network from complete infection. But then…

### How many people do we need to vaccinate?

We can identify how many people need to be vaccinated through the Herd Immunity Threshold or HIT. HIT indicates the threshold at which the reduction of susceptible members of the network means that infections will no longer keep increasing, allowing herd immunity to be achieved. `net_immunity()` gives us the proportion of the population that would need to be recovered or vaccinated for the network to have herd immunity.

    # Herd Immunity Threshold
    net_immunity(rd_diff)
    net_immunity(rd_diff, normalized = FALSE)

In this model, the HIT score indicates a good proportion of nodes in the network would need to be vaccinated or otherwise protected to achieve herd immunity. The unnormalised version gives the number of nodes that would need to be vaccinated. Ok, so let’s try this strategy.

    rd_diff_vacc <- play_diffusion(rando, transmissibility = 0.25, recovery = 0.05,
                                   immune = 2:9)
    plot(rd_diff_vacc)
    net_infection_total(rd_diff_vacc)

We can see that we more rapidly reach a situation in which vaccinated and naturally recovered nodes protect many more susceptible nodes. (Your results may vary a bit since these are stochastic models).

### Free play: Grey’s Anatomy

Let’s try this out on the `fict_greys` dataset, a dataset of character ‘hook-ups’ in the Grey’s Anatomy TV show. You can just concentrate on the giant component (which is plenty incestuous!). We could say that there’s a new, highly infectious disease transmittable through hooking up and, I have it on authority, it all starts with Mark Sloan.

    greys <- to_giant(fict_greys)
    graphr(greys)
    graphs(play_diffusion(greys, seeds = "Mark Sloan", transmissibility = 0.25,
                          latency = 0.25, recovery = 0.2, waning = 0.2),
           waves = c(1,5,10,15), labels = FALSE)

## Learning models

The three strategies for hindering contagion mentioned earlier usually rely on nodes’ voluntary participation: they must accept that vaccination, medication, or behavioral change is necessary to combat the contagion.

Lastly, we’re going to consider a rather simple type of learning model: a DeGroot learning model. A question often asked of these kinds of models is whether, despite heterogeneous initial beliefs, those beliefs will converge through (network) interaction. As you will recall, a network that is strongly connected and aperiodic will converge to a consensus of (any) beliefs entered.

### Expectations of convergence and consensus

Let’s try this out on the `ison_networkers` dataset of communication among network academics. Perhaps we wish to get them to change their local networks/behavior, but they have different beliefs about whether this will make a difference. To see whether this network will converge to consensus, check whether the network is connected and aperiodic :

- `is_connected()` marks whether network is weakly connected if the network is undirected or strongly connected if directed.
- `is_aperiodic()` marks whether network is aperiodic, meaning there is no integer k > 1 that divides the length of every cycle of the graph.

    # By default is_connected() will check whether a directed network
    # is strongly connected.

    is_connected(ison_networkers)
    is_aperiodic(ison_networkers) # this can take a few seconds

### Playing the DeGroot learning model

Now let’s see whether you are right. We want to see whether some random distribution of beliefs converges to a consensus in this network (`ison_networkers`). Let’s play the DeGroot learning game on this network with a vector of random belief probabilities (the same length as the nodes in the network) drawn from the binomial distribution with probability 0.25. Create the distribution of beliefs and graph the network to show where they have been distributed. Then play the learning model with these beliefs, and plot the result.

    beliefs <- rbinom(net_nodes(____), 1, prob = 0.25)
    ____ %>% mutate(____ = beliefs) %>% graphr(node_color = "____")
    netlearn <- play_learning(____, ____)
    plot(____)

    beliefs <- rbinom(net_nodes(ison_networkers), 1, prob = 0.25)
    ison_networkers %>% mutate(beliefs = beliefs) %>% graphr(node_color = "beliefs")
    (netlearn <- play_learning(ison_networkers, beliefs))
    plot(netlearn)

Each line in this plot represents the belief trajectory of a single node at each step. About a quarter of the nodes begin believing, and the other three quarters do not. Then we can see how responsive these nodes are to the random distribution of beliefs across the network. Some revise their beliefs more significantly than others.

### Free play: Networkers

The most influential nodes in this network are those that have the highest eigenvector centrality. Which are the highest eigenvector centrality nodes in this network?

    node_eigenvector(ison_networkers)
    ison_networkers %>%
      mutate(who_to_convince = node_is_max(node_eigenvector(ison_networkers))) %>%
      graphr(node_color = who_to_convince)
    beliefs2 <- rep(0, net_nodes(ison_networkers))
    beliefs2[node_is_max(node_eigenvector(ison_networkers))] <- 1
    ison_networkers %>% mutate(beliefs = beliefs2) %>% graphr(node_color = "beliefs")
    (netlearn2 <- play_learning(ison_networkers, beliefs2))
    plot(netlearn2)

## Glossary

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

Aperiodic
  An aperiodic network is a network where the greatest common divisor of the lengths of its cycles is one.
Connected
  A connected network is one with a single (strong) component.
Hit
  A herd immunity threshold is the proportion of the population that would need to be immune for a disease to cease being endemic.
Intervention
  A network intervention is a process to accelerate behavioural change or improve performance in a network.
Ltm
  A linear threshold model is a diffusion model where nodes are resistant to activation up to some threshold.
Threshold
  A threshold is a limit over which behaviour is expected to vary.
Undirected
  An undirected or line network is one in which tie direction is undefined.

# Diffusion and Learning

#### by James Hollway