3-sim.qmd

---
title: "Simulation of diffusion networks: rdiffnet"
author: "Thomas W. Valente and George G. Vega Yon"
---

```{r setup, echo=FALSE, message=FALSE, warning=FALSE}
library(netdiffuseR)
knitr::opts_chunk$set(comment = "#")

```

# Introduction

Before we start, a review of the concepts we will be using here

1. Exposure: Proportion/number of neighbors that has adopted an innovation at each point in time.
2. Threshold: The proportion/number of your neighbors who had adopted at or one time period before ego (the focal individual) adopted.
3. Infectiousness: How much $i$'s adoption affects her alters.
4. Susceptibility: How much $i$'s alters' adoption affects her.
5. Structural equivalence: How similar are $i$ and $j$ in terms of position in the network.

# Simulating diffusion networks

We will simulate a diffusion network with the following parameters:

1.  Will have 1,000 vertices,
2.  Will span 20 time periods,
3.  The initial adopters (seeds) will be selected random,
4.  Seeds will be a 10\% of the network,
5.  The graph (network) will be small-world,
6.  Will use the WS algorithmwith $p=.2$ (probability of rewire).
7.  Threshold levels will be uniformly distributed between [0.3, 0.7\]

To generate this diffusion network, we can use the `rdiffnet` function included in the package:


```{r Generating the random graph}
# Setting the seed for the RNG
set.seed(1213)

# Generating a random diffusion network
net <- rdiffnet(
  n              = 1e3,                         # 1.
  t              = 20,                          # 2.
  seed.nodes     = "random",                    # 3.
  seed.p.adopt   = .1,                          # 4.
  seed.graph     = "small-world",               # 5.
  rgraph.args    = list(p=.2),                  # 6.
  threshold.dist = function(x) runif(1, .3, .7) # 7.
  )
```


*   The function `rdiffnet` generates random diffusion networks. Main features:
    
    1.  Simulating random graph or using your own,
    
    2.  Setting threshold levels per node,
    
    3.  Network rewiring throughout the simulation, and
    
    4.  Setting the seed nodes.
    
    
*   The simulation algorithm is as follows:
    
    1.  If required, a baseline graph is created,
    
    2.  Set of initial adopters and threshold distribution are established,
    
    3.  The set of t networks is created (if required), and
    
    4.  Simulation starts at t=2, assigning adopters based on exposures and thresholds:
    
        a.  For each $i \in N$, if its exposure at $t-1$ is greater than its threshold, then 
            adopts, otherwise continue without change.
            
        b.  next $i$
    
# Rumor spreading

```{r sim-rumor}
library(netdiffuseR)

set.seed(09)
diffnet_rumor <- rdiffnet(
  n = 5e2,
  t = 5, 
  seed.graph = "small-world",
  rgraph.args = list(k = 4, p = .3),
  seed.nodes = "random",
  seed.p.adopt = .05,
  rewire = TRUE,
  threshold.dist = function(i) 1L,
  exposure.args = list(normalized = FALSE)
  )
```

```{r summary-rumor}
summary(diffnet_rumor)
```


```{r plot-rumor, fig.align='center', cache=TRUE}
plot_diffnet(diffnet_rumor, slices = c(1, 3, 5))

# We want to use igraph to compute layout
igdf <- diffnet_to_igraph(diffnet_rumor, slices=c(1,2))[[1]]
pos <- igraph::layout_with_drl(igdf)

plot_diffnet2(diffnet_rumor, vertex.size = dgr(diffnet_rumor)[,1], layout=pos)
```


# Difussion

```{r sim-disease}
set.seed(09)
diffnet_disease <- rdiffnet(
  seed.graph = diffnet_rumor$graph,
  seed.nodes = which(diffnet_rumor$toa == 1),
  rewire = FALSE,
  threshold.dist = function(i) rbeta(1, 3, 10),
  name = "Diffusion",
  behavior = "Some social behavior"
)

```

```{r plot-disease-and-disease}
plot_adopters(diffnet_rumor, what = "cumadopt", include.legend = FALSE)
plot_adopters(diffnet_disease, bg="lightblue", add=TRUE, what = "cumadopt")
legend(
  "topleft",
  legend = c("Disease", "Rumor"),
  col = c("lightblue", "tomato"),
  bty = "n", pch=19
  )
```


# Multi-diffusion models (TBD)

# Mentor Matching

```{r mentor-match, cache = TRUE}

# Finding mentors
mentors <- mentor_matching(diffnet_rumor, 25, lead.ties.method = "random")

# Simulating diffusion with these mentors
set.seed(09)
diffnet_mentored <- rdiffnet(
  seed.graph = diffnet_disease,
  seed.nodes = which(mentors$`1`$isleader),
  rewire = FALSE,
  threshold.dist = diffnet_disease[["real_threshold"]],
  name = "Diffusion using Mentors"
)

summary(diffnet_mentored)
```

```{r toa_mat-mentors}
cumulative_adopt_count(diffnet_disease)
cumulative_adopt_count(diffnet_mentored)
```


# Example by changing threshold

The following block of code runs multiple diffnet simulations. Before we proceed, we will generate a scale-free homophilic network:

```{r}
#| label: scale-free-homophilic
# Simulating a scale-free homophilic network
set.seed(1231)
X <- rep(c(1,1,1,1,1,0,0,0,0,0), 50)
net <- rgraph_ba(t = 499, m=4, eta = X)

# Taking a look in igraph
ig  <- igraph::graph_from_adjacency_matrix(net)
plot(ig, vertex.color = c("azure", "tomato")[X+1], vertex.label = NA,
     vertex.size = sqrt(dgr(net)))
```

Besides of the usual parameters passed to `rdiffnet`, the `rdiffnet_multiple` function requires `R` (number of repetitions/simulations), and `statistic` (a function that returns the statistic of insterst). Optionally, the user may choose to specify the number of clusters to run it in parallel (multiple CPUs):

```{r rdiffnet-multiple}
nsim <- 500L

ans_1and2 <- rdiffnet_multiple(
  # Num of sim
  R              = nsim,
  # Statistic
  statistic      = function(d) cumulative_adopt_count(d)["prop",], 
  seed.graph     = net,
  t              = 10,
  threshold.dist = sample(1:2, 500L, TRUE),
  seed.nodes     = "random",
  seed.p.adopt   = .1,
  rewire         = FALSE,
  exposure.args  = list(outgoing=FALSE, normalized=FALSE),
  # Running on 4 cores
  ncpus          = 4L
  ) |> t()

ans_2and3 <- rdiffnet_multiple(
  # Num of sim
  R              = nsim,
  # Statistic
  statistic      = function(d) cumulative_adopt_count(d)["prop",], 
  seed.graph     = net,
  t              = 10,
  threshold.dist = sample(2:3, 500, TRUE),
  seed.nodes     = "random",
  seed.p.adopt   = .1,
  rewire         = FALSE,
  exposure.args  = list(outgoing=FALSE, normalized=FALSE),
  # Running on 4 cores
  ncpus          = 4L
  ) |> t()

ans_1and3 <- rdiffnet_multiple(
  # Num of sim
  R              = nsim,
  # Statistic
  statistic      = function(d) cumulative_adopt_count(d)["prop",], 
  seed.graph     = net,
  t              = 10,
  threshold.dist = sample(1:3, 500, TRUE),
  seed.nodes     = "random",
  seed.p.adopt   = .1,
  rewire         = FALSE,
  exposure.args  = list(outgoing=FALSE, normalized=FALSE),
  # Running on 4 cores
  ncpus          = 4L
  ) |> t()

```

```{r sim-sim-results}
boxplot(ans_1and2, col="ivory", xlab = "Time", ylab = "Proportion of Adopters")
boxplot(ans_2and3, col="tomato", add=TRUE)
boxplot(ans_1and3, col = "steelblue", add=TRUE)
legend(
  "topleft",
  fill = c("ivory", "tomato", "steelblue"),
  legend = c("1/2", "2/3", "1/3"),
  title = "Threshold range",
  bty ="n"
)
```


*   Example simulating a thousand networks by changing threshold levels.
    The final prevalence, or hazard as a function of threshold levels.

# Problems

1.  Given the following types of networks: Small-world, Scale-free, Bernoulli,
    what set of $n$ initiators maximizes diffusion?
    (<a href="files/sim-solutions.r" target="_blank">solution script</a> and <a href="files/sim-solutions.png" target="_blank">solution plot</a>)
    
# Appendix

The following is example code that can be used to run multiple simulations like it is done using the `rdiffnet_multiple` function. We do not recommend this approach but it may be useful for some users:

```{r sim-sim, cache = TRUE, collapse = TRUE}
# Now, simulating a bunch of diffusion processes
nsim <- 500L
ans_1and2 <- vector("list", nsim)
set.seed(223)
for (i in 1:nsim) {
  # We just want the cum adopt count
  ans_1and2[[i]] <- 
    cumulative_adopt_count(
      rdiffnet(
        seed.graph = net,
        t = 10,
        threshold.dist = sample(1:2, 500L, TRUE),
        seed.nodes = "random",
        seed.p.adopt = .10,
        exposure.args = list(outgoing = FALSE, normalized = FALSE),
        rewire = FALSE
        )
      )
  
  # Are we there yet?
  if (!(i %% 50))
    message("Simulation ", i," of ", nsim, " done.")
}

# Extracting prop
ans_1and2 <- do.call(rbind, lapply(ans_1and2, "[", i="prop", j=))

ans_2and3 <- vector("list", nsim)
set.seed(223)
for (i in 1:nsim) {
  # We just want the cum adopt count
  ans_2and3[[i]] <- 
    cumulative_adopt_count(
      rdiffnet(
        seed.graph = net,
        t = 10,
        threshold.dist = sample(2:3, 500L, TRUE),
        seed.nodes = "random",
        seed.p.adopt = .10,
        exposure.args = list(outgoing = FALSE, normalized = FALSE),
        rewire = FALSE
        )
      )
  
  # Are we there yet?
  if (!(i %% 50))
    message("Simulation ", i," of ", nsim, " done.")
}

ans_2and3 <- do.call(rbind, lapply(ans_2and3, "[", i="prop", j=))
```