Introduction

Before start, a recapt of concepts that we will be using here

1. Exposure: What proportion/number of your neighbors has adopted an innovation.
2. Threshold: What was the proportion/number of your neighbors had adopted by the time you 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 on 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 set of early adopters will be random,
4. Early adopters will be a 10% of the network,
5. The graph 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 such diffusion network we can use the rdiffnet function included in the package:

# 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 is established,

  3. Se set of t networks is created (if required), and

  4. Simulation starts at t=2, assigning adopters based on exposures and thresholds:

     1. For each \(i \in N\), if its exposure at \(t-1\) is greater than its threshold, then adopts, otherwise continue without change.

     2. next \(i\)

Disease spreading

library(netdiffuseR)

set.seed(09)
diffnet_disease <- 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)
  )

summary(diffnet_disease)

# Diffusion network summary statistics
# Name     : A diffusion network
# Behavior : Random contagion
# -----------------------------------------------------------------------------
#  Period   Adopters   Cum Adopt. (%)   Hazard Rate   Density   Moran's I (sd)  
# -------- ---------- ---------------- ------------- --------- ---------------- 
#        1         25        25 (0.05)             -      0.01 -0.00 (0.00)     
#        2        100       125 (0.25)          0.21      0.01  0.01 (0.00) *** 
#        3        228       353 (0.71)          0.61      0.01  0.01 (0.00) *** 
#        4        141       494 (0.99)          0.96      0.01  0.00 (0.00) *** 
#        5          6       500 (1.00)          1.00      0.01               -  
# -----------------------------------------------------------------------------
#  Left censoring  : 0.05 (25)
#  Right centoring : 0.00 (0)
#  # of nodes      : 500
# 
#  Moran's I was computed on contemporaneous autocorrelation using 1/geodesic
#  values. Significane levels  *** <= .01, ** <= .05, * <= .1.

plot_diffnet(diffnet_disease, slices = c(1, 3, 5))

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

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

Complex contagion

set.seed(09)
diffnet_complex <- rdiffnet(
  seed.graph = diffnet_disease$graph,
  seed.nodes = which(diffnet_disease$toa == 1),
  rewire = FALSE,
  threshold.dist = function(i) rbeta(1, 3, 10),
  name = "Complex Sim",
  behavior = "More complex than contact"
)

plot_adopters(diffnet_disease, what = "cumadopt", include.legend = FALSE)
plot_adopters(diffnet_complex, bg="tomato", add=TRUE, what = "cumadopt")
legend("topleft", legend = c("Disease", "Complex"), col = c("lightblue", "tomato"),
       bty = "n", pch=19)

Mentor Matching

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

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

summary(diffnet_mentored)

# Diffusion network summary statistics
# Name     : Diffusion using Mentors
# Behavior : Random contagion
# -----------------------------------------------------------------------------
#  Period   Adopters   Cum Adopt. (%)   Hazard Rate   Density   Moran's I (sd)  
# -------- ---------- ---------------- ------------- --------- ---------------- 
#        1         25        25 (0.05)             -      0.01 -0.00 (0.00)     
#        2         92       117 (0.23)          0.19      0.01  0.01 (0.00) *** 
#        3        152       269 (0.54)          0.40      0.01  0.01 (0.00) *** 
#        4        150       419 (0.84)          0.65      0.01  0.01 (0.00) *** 
#        5         73       492 (0.98)          0.90      0.01 -0.00 (0.00) **  
# -----------------------------------------------------------------------------
#  Left censoring  : 0.05 (25)
#  Right centoring : 0.02 (8)
#  # of nodes      : 500
# 
#  Moran's I was computed on contemporaneous autocorrelation using 1/geodesic
#  values. Significane levels  *** <= .01, ** <= .05, * <= .1.

cumulative_adopt_count(diffnet_complex)

#          1      2          3           4           5
# num  25.00 88.000 229.000000 395.0000000 485.0000000
# prop  0.05  0.176   0.458000   0.7900000   0.9700000
# rate  0.00  2.520   1.602273   0.7248908   0.2278481

cumulative_adopt_count(diffnet_mentored)

#          1       2          3           4           5
# num  25.00 117.000 269.000000 419.0000000 492.0000000
# prop  0.05   0.234   0.538000   0.8380000   0.9840000
# rate  0.00   3.680   1.299145   0.5576208   0.1742243

Example by changing threshold

# 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)))

# 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.")
}
# Simulation 50 of 500 done.
# Simulation 100 of 500 done.
# Simulation 150 of 500 done.
# Simulation 200 of 500 done.
# Simulation 250 of 500 done.
# Simulation 300 of 500 done.
# Simulation 350 of 500 done.
# Simulation 400 of 500 done.
# Simulation 450 of 500 done.
# Simulation 500 of 500 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.")
}
# Simulation 50 of 500 done.
# Simulation 100 of 500 done.
# Simulation 150 of 500 done.
# Simulation 200 of 500 done.
# Simulation 250 of 500 done.
# Simulation 300 of 500 done.
# Simulation 350 of 500 done.
# Simulation 400 of 500 done.
# Simulation 450 of 500 done.
# Simulation 500 of 500 done.

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

boxplot(ans_1and2, col="ivory")
boxplot(ans_2and3, col="tomato", add=TRUE)

• 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?