To specify a degree-corrected stochastic blockmodel, you must specify
the degree-heterogeneity parameters (via n or theta_out and
theta_in), the mixing matrix
(via k_out and k_in, or B), and the relative block
probabilities (optional, via p_out and pi_in).
We provide defaults for most of these
options to enable rapid exploration, or you can invest the effort
for more control over the model parameters. We strongly recommend
setting the expected_out_degree, expected_in_degree,
or expected_density argument
to avoid large memory allocations associated with
sampling large, dense graphs.
directed_dcsbm(
n = NULL,
theta_out = NULL,
theta_in = NULL,
k_out = NULL,
k_in = NULL,
B = NULL,
...,
pi_out = rep(1/k_out, k_out),
pi_in = rep(1/k_in, k_in),
sort_nodes = TRUE,
force_identifiability = TRUE,
poisson_edges = TRUE,
allow_self_loops = TRUE
)A directed_dcsbm S3 object, a subclass of the
directed_factor_model() with the following additional
fields:
theta_out: A numeric vector of incoming community
degree-heterogeneity parameters.
theta_in: A numeric vector of outgoing community
degree-heterogeneity parameters.
z_out: The incoming community memberships of each node,
as a factor(). The factor will have k_out levels,
where k_out is the number of incoming
communities in the stochastic blockmodel. There will not
always necessarily be observed nodes in each community.
z_in: The outgoing community memberships of each node,
as a factor(). The factor will have k_in levels,
where k_in is the number of outgoing
communities in the stochastic blockmodel. There will not
always necessarily be observed nodes in each community.
pi_out: Sampling probabilities for each incoming
community.
pi_in: Sampling probabilities for each outgoing
community.
sorted: Logical indicating where nodes are arranged by
block (and additionally by degree heterogeneity parameter)
within each block.
(degree heterogeneity) The number of nodes in the blockmodel.
Use when you don't want to specify the degree-heterogeneity
parameters theta_out and theta_in by hand. When n is specified,
theta_out and theta_in are randomly generated from
a LogNormal(2, 1) distribution.
This is subject to change, and may not be reproducible.
n defaults to NULL. You must specify either n
or theta_out and theta_in together, but not both.
(degree heterogeneity) A numeric vector
explicitly specifying the degree heterogeneity
parameters. This implicitly determines the number of nodes
in the resulting graph, i.e. it will have length(theta_out) nodes.
Must be positive. Setting to a vector of ones recovers
a stochastic blockmodel without degree correction.
Defaults to NULL. You must specify either n
or theta_out and theta_in together, but not both. theta_out
controls outgoing degree propensity, or, equivalently,
row sums of the adjacency matrix.
(degree heterogeneity) A numeric vector
explicitly specifying the degree heterogeneity
parameters. This implicitly determines the number of nodes
in the resulting graph, i.e. it will have length(theta) nodes.
Must be positive. Setting to a vector of ones recovers
a stochastic blockmodel without degree correction.
Defaults to NULL. You must specify either n
or theta_out and theta_in together, but not both. theta_in
controls incoming degree propensity, or, equivalently, column
sums of the adjacency matrix.
(mixing matrix) The number of outgoing blocks in the blockmodel.
Use when you don't want to specify the mixing-matrix by hand.
When k_out is specified, the elements of B are drawn
randomly from a Uniform(0, 1) distribution.
This is subject to change, and may not be reproducible.
k_out defaults to NULL. You must specify either k_out and
k_in together, or B. You may specify all three at once, in which
case k_out is only used to set pi_out (when pi_out is
left at its default argument value).
(mixing matrix) The number of incoming blocks in the blockmodel.
Use when you don't want to specify the mixing-matrix by hand.
When k_in is specified, the elements of B are drawn
randomly from a Uniform(0, 1) distribution.
This is subject to change, and may not be reproducible.
k_in defaults to NULL. You may specify all three at once, in which
case k_in is only used to set pi_in (when pi_in is
left at its default argument value).
(mixing matrix) A k_out by k_in matrix of block connection
probabilities. The probability that a node in block i connects
to a node in community j is Poisson(B[i, j]).
matrix and Matrix objects are both acceptable.
Defaults to NULL. You must specify either k_out and
k_in together, or B, but not both.
Arguments passed on to directed_factor_model
expected_in_degreeIf specified, the desired expected in degree
of the graph. Specifying expected_in_degree simply rescales S
to achieve this. Defaults to NULL. Specify only one of
expected_in_degree, expected_out_degree, and expected_density.
expected_out_degreeIf specified, the desired expected out degree
of the graph. Specifying expected_out_degree simply rescales S
to achieve this. Defaults to NULL. Specify only one of
expected_in_degree, expected_out_degree, and expected_density.
expected_densityIf specified, the desired expected density
of the graph. Specifying expected_density simply rescales S
to achieve this. Defaults to NULL. Specify only one of
expected_in_degree, expected_out_degree, and expected_density.
(relative block probabilities) Relative block
probabilities. Must be positive, but do not need to sum
to one, as they will be normalized internally.
Must match the rows of B, or k_out. Defaults to
rep(1 / k_out, k_out), or a balanced outgoing blocks.
(relative block probabilities) Relative block
probabilities. Must be positive, but do not need to sum
to one, as they will be normalized internally.
Must match the columns of B, or k_in. Defaults to
rep(1 / k_in, k_in), or a balanced incoming blocks.
Logical indicating whether or not to sort the nodes
so that they are grouped by block. Useful for plotting.
Defaults to TRUE. When TRUE, rows of the expected adjacency matrix
are first sorted by outgoing block membership, and then by
incoming degree-correction parameters within each incoming block.
A similar sorting procedure occurs independently from the columns,
according to the incoming blocks.
Additionally, pi_out and pi_in are sorted in increasing order,
and the columns of the B matrix are permuted to match the
new orderings.
Logical indicating whether or not to
normalize theta_out such that it sums to one within each incoming
block and theta_in such that it sums to one within each outgoing
block. Defaults to TRUE.
Logical indicating whether or not
multiple edges are allowed to form between a pair of
nodes. Defaults to TRUE. When FALSE, sampling proceeds
as usual, and duplicate edges are removed afterwards. Further,
when FALSE, we assume that S specifies a desired between-factor
connection probability, and back-transform this S to the
appropriate Poisson intensity parameter to approximate Bernoulli
factor connection probabilities. See Section 2.3 of Rohe et al. (2017)
for some additional details.
Logical indicating whether or not
nodes should be allowed to form edges with themselves.
Defaults to TRUE. When FALSE, sampling proceeds allowing
self-loops, and these are then removed after the fact.
There are two levels of randomness in a directed degree-corrected
stochastic blockmodel. First, we randomly chose a incoming
block membership and an outgoing block membership
for each node in the blockmodel. This is
handled by directed_dcsbm(). Then, given these block memberships,
we randomly sample edges between nodes. This second
operation is handled by sample_edgelist(),
sample_sparse(), sample_igraph() and
sample_tidygraph(), depending on your desired
graph representation.
Let \(x\) represent the incoming block membership of a node and \(y\) represent the outgoing block membership of a node. To generate \(x\) we sample from a categorical distribution with parameter \(\pi_out\). To generate \(y\) we sample from a categorical distribution with parameter \(\pi_in\). Block memberships are independent across nodes. Incoming and outgoing block memberships of the same node are also independent.
In addition to block membership, the DCSBM also nodes to have different propensities for incoming and outgoing edge formation. We represent the propensity to form incoming edges for a given node by a positive number \(\theta_out\). We represent the propensity to form outgoing edges for a given node by a positive number \(\theta_in\). Typically the \(\theta_out\) (and \(theta_in\)) across all nodes are constrained to sum to one for identifiability purposes, but this doesn't really matter during sampling.
Once we know the block memberships \(x\) and \(y\)
and the degree heterogeneity parameters \(\theta_{in}\) and
\(\theta_{out}\), we need one more
ingredient, which is the baseline intensity of connections
between nodes in block i and block j. Then each edge forms
independently according to a Poisson distribution with
parameters
$$ \lambda = \theta_{in} * B_{x, y} * \theta_{out}. $$
Other stochastic block models:
dcsbm(),
mmsbm(),
overlapping_sbm(),
planted_partition(),
sbm()
Other directed graphs:
directed_erdos_renyi()
set.seed(27)
B <- matrix(0.2, nrow = 5, ncol = 8)
diag(B) <- 0.9
ddcsbm <- directed_dcsbm(
n = 1000,
B = B,
k_out = 5,
k_in = 8,
expected_density = 0.01
)
ddcsbm
population_svd <- svds(ddcsbm)
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