Simulation of diffusion networks: rdiffnet

Author

Thomas W. Valente and George G. Vega Yon

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:

# 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.
  )
# Warning in (function (graph, p, algorithm = "endpoints", both.ends = FALSE, :
# The option -copy.first- is set to TRUE. In this case, the first graph will be
# treated as a baseline, and thus, networks after T=1 will be replaced with T-1.

  • 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:

      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 \(i\)

Rumor spreading

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)
  )
# Warning in (function (graph, p, algorithm = "endpoints", both.ends = FALSE, :
# The option -copy.first- is set to TRUE. In this case, the first graph will be
# treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
summary(diffnet_rumor)
# 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         78       103 (0.21)          0.16      0.01  0.01 (0.00) *** 
#        3        187       290 (0.58)          0.47      0.01  0.01 (0.00) *** 
#        4        183       473 (0.95)          0.87      0.01  0.01 (0.00) *** 
#        5         27       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_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

set.seed(09)
diffnet_complex <- 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"
)
plot_adopters(diffnet_rumor, 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_rumor, 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 80.00 183.0000 338.0000000 470.0000000
# prop  0.05  0.16   0.3660   0.6760000   0.9400000
# rate  0.00  2.20   1.2875   0.8469945   0.3905325
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=))

This can actually be simplified by using the function rdiffnet_multiple. The following lines of code accomplish the same as the previous code avoiding the for-loop (from the user’s perspective). 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):

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
  )
boxplot(ans_1and2, col="ivory", xlab = "Time", ylab = "Threshold")
boxplot(ans_2and3, col="tomato", add=TRUE)
boxplot(t(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? (solution script and solution plot)