Calculates exposure to adoption over time via multiple different types of weight matrices. The basic model is exposure to adoption by immediate neighbors (outdegree) at the time period prior to ego’s adoption. This exposure can also be based on (1) incoming ties, (2) structural equivalence, (3) indirect ties, (4) attribute weighted (5) network-metric weighted (e.g., central nodes have more influence), and attribute-weighted (e.g., based on homophily or tie strength).
A dynamic graph (see netdiffuseR-graphs).
\(n\times T\) matrix. Cumulative adoption matrix obtained from
toa_mat
Either a character scalar (if graph is diffnet),
or a numeric matrix of size \(n\times T\). Weighting for each time, period (see details).
Either a graph that should be used instead of graph,
or "se" (see details).
Logical scalar. When TRUE, computed using outgoing ties.
Logical scalar. When TRUE weights will be considered. Otherwise non-zero values will be replaced by ones.
Logical scalar. When TRUE, the exposure will be between zero
and one (see details).
Passed to struct_equiv.
Logical scalar. When TRUE autolinks (loops, self edges) are allowed (see details).
Integer scalar. When different from 0, the resulting exposure matrix will be the lagged exposure as specified (see examples).
Further arguments passed to struct_equiv (only used when
alt.graph="se").
A matrix of size \(n\times T\) with exposure for each node.
Exposure is calculated as follows:
$$ % E_t = \left(S_t \times \left[x_t \circ A_t\right]\right) / (S_t \times x_t) % $$
Where \(S_t\) is the graph in time \(t\), \(x_t\) is an attribute vector of size \(n\) at time \(t\), \(A_t\) is the t-th column of the cumulative adopters matrix (a vector of length \(n\) with \(a_{ti}=1\) if \(i\) has adopted at or prior to \(t\)), \(\circ\) is the kronecker product (element-wise), and \(\times\) is the matrix product.
By default the graph used for this calculation, \(S\), is the social network. Alternatively,
in the case of diffnet objects, the user can provide an alternative
graph using alt.graph. An example of this would be using \(1/SE\),
the element-wise inverse of the structural equivalence matrix (see example below).
Furthermore, if alt.graph="se", the inverse of the structural equivalence
is computed via struct_equiv and used instead of the provided
graph. Notice that when using a valued graph the option valued should
be equal to TRUE, this check is run automatically when running the
model using structural equivalence.
If the alt.graph is static, then the function will warn about it
and will recycle the graph to compute exposure at each time point.
An important remark is that when calculating structural equivalence the
function assumes that this is to be done to the entire graph regardless of
disconnected communities (as in the case of the medical innovations
data set). Hence, structural equivalence for individuals for two different
communites may not be zero. If the user wants to calculate structural
equivalence separately by community, he should create different diffnet
objects and do so (see example below). Alternatively, for the case of
diffnet objects, by using the option groupvar (see struct_equiv), the user can provide
the function with the name of a grouping variable–which should one in the
set of static vertex attributes–so that the algorithm is done by group
(or community) instead of in an aggregated way.
If the user does not specifies a particular weighting attribute in attrs,
the function sets this as a matrix of ones. Otherwise the function will return
an attribute weighted exposure. When graph is of class diffnet,
attrs can be a character scalar specifying the name of any of the graph's
attributes, both dynamic and static. See the examples section for a demonstration using
degree.
When outgoing=FALSE, \(S\) is replaced by its transposed, so in the
case of a social network exposure will be computed based on the incoming ties.
If normalize=FALSE then denominator, \(S_t \times x_t\),
is not included. This can be useful when, for example, exposure needs to be
computed as a count instead of a proportion. A good example of this can be
found at the examples section of the function rdiffnet.
Burt, R. S. (1987). "Social Contagion and Innovation: Cohesion versus Structural Equivalence". American Journal of Sociology, 92(6), 1287. doi:10.1086/228667
Valente, T. W. (1995). "Network models of the diffusion of innovations" (2nd ed.). Cresskill N.J.: Hampton Press.
Other statistics:
bass,
classify_adopters(),
cumulative_adopt_count(),
dgr(),
ego_variance(),
hazard_rate(),
infection(),
moran(),
struct_equiv(),
threshold(),
vertex_covariate_dist()
# Calculating lagged exposure -----------------------------------------------
set.seed(8)
graph <- rdiffnet(20, 4)
#> Warning: 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.
expo0 <- exposure(graph)
expo1 <- exposure(graph, lags = 1)
# These should be equivalent
stopifnot(all(expo0[, -4] == expo1[, -1])) # No stop!
# Calculating the exposure based on Structural Equivalence ------------------
set.seed(113132)
graph <- rdiffnet(100, 4)
#> Warning: 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.
SE <- lapply(struct_equiv(graph), "[[", "SE")
SE <- lapply(SE, function(x) {
x <- 1/x
x[!is.finite(x)] <- 0
x
})
# These three lines are equivalent to:
expo_se2 <- exposure(graph, alt.graph="se", valued=TRUE)
# Notice that we are setting valued=TRUE, but this is not necesary since when
# alt.graph = "se" the function checks this to be setted equal to TRUE
# Weighted Exposure using degree --------------------------------------------
eDE <- exposure(graph, attrs=dgr(graph))
# Which is equivalent to
graph[["deg"]] <- dgr(graph)
eDE2 <- exposure(graph, attrs="deg")
# Comparing using incoming edges -------------------------------------------
eIN <- exposure(graph, outgoing=FALSE)
# Structral equivalence for different communities ---------------------------
data(medInnovationsDiffNet)
# Only using 4 time slides, this is for convenience
medInnovationsDiffNet <- medInnovationsDiffNet[, , 1:4]
# METHOD 1: Using the c.diffnet method:
# Creating subsets by city
cities <- unique(medInnovationsDiffNet[["city"]])
diffnet <- medInnovationsDiffNet[medInnovationsDiffNet[["city"]] == cities[1]]
diffnet[["expo_se"]] <- exposure(diffnet, alt.graph="se", valued=TRUE)
for (v in cities[-1]) {
diffnet_v <- medInnovationsDiffNet[medInnovationsDiffNet[["city"]] == v]
diffnet_v[["expo_se"]] <- exposure(diffnet_v, alt.graph="se", valued=TRUE)
diffnet <- c(diffnet, diffnet_v)
}
# We can set the original order (just in case) of the data
diffnet <- diffnet[medInnovationsDiffNet$meta$ids]
diffnet
#> Dynamic network of class -diffnet-
#> Name : Medical Innovation
#> Behavior : Adoption of Tetracycline
#> # of nodes : 125 (1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, ...)
#> # of time periods : 4 (1 - 4)
#> Type : directed
#> Final prevalence : 1.00
#> Static attributes : city, detail, meet, coll, attend, proage, length, ... (58)
#> Dynamic attributes : expo_se (1)
# Checking everything is equal
test <- summary(medInnovationsDiffNet, no.print=TRUE) ==
summary(diffnet, no.print=TRUE)
stopifnot(all(test[!is.na(test)]))
# METHOD 2: Using the 'groupvar' argument
# Further, we can compare this with using the groupvar
diffnet[["expo_se2"]] <- exposure(diffnet, alt.graph="se",
groupvar="city", valued=TRUE)
# These should be equivalent
test <- diffnet[["expo_se", as.df=TRUE]] == diffnet[["expo_se2", as.df=TRUE]]
stopifnot(all(test[!is.na(test)]))
# METHOD 3: Computing exposure, rbind and then adding it to the diffnet object
expo_se3 <- NULL
for (v in unique(cities))
expo_se3 <- rbind(
expo_se3,
exposure(
diffnet[diffnet[["city"]] == v],
alt.graph = "se", valued=TRUE
))
# Just to make sure, we sort the rows
expo_se3 <- expo_se3[diffnet$meta$ids,]
diffnet[["expo_se3"]] <- expo_se3
test <- diffnet[["expo_se", as.df=TRUE]] == diffnet[["expo_se3", as.df=TRUE]]
stopifnot(all(test[!is.na(test)]))
# METHOD 4: Using the groupvar in struct_equiv
se <- struct_equiv(diffnet, groupvar="city")
se <- lapply(se, "[[", "SE")
se <- lapply(se, function(x) {
x <- 1/x
x[!is.finite(x)] <- 0
x
})
diffnet[["expo_se4"]] <- exposure(diffnet, alt.graph=se, valued=TRUE)
test <- diffnet[["expo_se", as.df=TRUE]] == diffnet[["expo_se4", as.df=TRUE]]
stopifnot(all(test[!is.na(test)]))