dcsbm {fastRG} | R Documentation |
Create an undirected degree corrected stochastic blockmodel object
Description
To specify a degree-corrected stochastic blockmodel, you must specify
the degree-heterogeneity parameters (via n
or theta
),
the mixing matrix (via k
or B
), and the relative block
probabilities (optional, via pi
). 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_degree
or expected_density
argument
to avoid large memory allocations associated with
sampling large, dense graphs.
Usage
dcsbm(
n = NULL,
theta = NULL,
k = NULL,
B = NULL,
...,
pi = rep(1/k, k),
sort_nodes = TRUE,
force_identifiability = FALSE,
poisson_edges = TRUE,
allow_self_loops = TRUE
)
Arguments
n |
(degree heterogeneity) The number of nodes in the blockmodel.
Use when you don't want to specify the degree-heterogeneity
parameters |
theta |
(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 |
k |
(mixing matrix) The number of blocks in the blockmodel.
Use when you don't want to specify the mixing-matrix by hand.
When |
B |
(mixing matrix) A |
... |
Arguments passed on to
|
pi |
(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 dimensions of |
sort_nodes |
Logical indicating whether or not to sort the nodes
so that they are grouped by block and by |
force_identifiability |
Logical indicating whether or not to
normalize |
poisson_edges |
Logical indicating whether or not
multiple edges are allowed to form between a pair of
nodes. Defaults to |
allow_self_loops |
Logical indicating whether or not
nodes should be allowed to form edges with themselves.
Defaults to |
Value
An undirected_dcsbm
S3 object, a subclass of the
undirected_factor_model()
with the following additional
fields:
-
theta
: A numeric vector of degree-heterogeneity parameters. -
z
: The community memberships of each node, as afactor()
. The factor will havek
levels, wherek
is the number of communities in the stochastic blockmodel. There will not always necessarily be observed nodes in each community. -
pi
: Sampling probabilities for each block. -
sorted
: Logical indicating where nodes are arranged by block (and additionally by degree heterogeneity parameter) within each block.
Generative Model
There are two levels of randomness in a degree-corrected
stochastic blockmodel. First, we randomly chose a block
membership for each node in the blockmodel. This is
handled by 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 depending on your desired
graph representation.
Block memberships
Let z_i
represent the block membership of node i
.
To generate z_i
we sample from a categorical
distribution (note that this is a special case of a
multinomial) with parameter \pi
, such that
\pi_i
represents the probability of ending up in
the ith block. Block memberships for each node are independent.
Degree heterogeneity
In addition to block membership, the DCSBM also allows
nodes to have different propensities for edge formation.
We represent this propensity for node i
by a positive
number \theta_i
. Typically the \theta_i
are
constrained to sum to one for identifiability purposes,
but this doesn't really matter during sampling (i.e.
without the sum constraint scaling B
and \theta
has the same effect on edge probabilities, but whether
B
or \theta
is responsible for this change
is uncertain).
Edge formulation
Once we know the block memberships z
and the degree
heterogeneity parameters theta
, we need one more
ingredient, which is the baseline intensity of connections
between nodes in block i
and block j
. Then each edge
A_{i,j}
is Poisson distributed with parameter
\lambda[i, j] = \theta_i \cdot B_{z_i, z_j} \cdot \theta_j.
See Also
Other stochastic block models:
directed_dcsbm()
,
mmsbm()
,
overlapping_sbm()
,
planted_partition()
,
sbm()
Other undirected graphs:
chung_lu()
,
erdos_renyi()
,
mmsbm()
,
overlapping_sbm()
,
planted_partition()
,
sbm()
Examples
set.seed(27)
lazy_dcsbm <- dcsbm(n = 1000, k = 5, expected_density = 0.01)
lazy_dcsbm
# sometimes you gotta let the world burn and
# sample a wildly dense graph
dense_lazy_dcsbm <- dcsbm(n = 500, k = 3, expected_density = 0.8)
dense_lazy_dcsbm
# explicitly setting the degree heterogeneity parameter,
# mixing matrix, and relative community sizes rather
# than using randomly generated defaults
k <- 5
n <- 1000
B <- matrix(stats::runif(k * k), nrow = k, ncol = k)
theta <- round(stats::rlnorm(n, 2))
pi <- c(1, 2, 4, 1, 1)
custom_dcsbm <- dcsbm(
theta = theta,
B = B,
pi = pi,
expected_degree = 50
)
custom_dcsbm
edgelist <- sample_edgelist(custom_dcsbm)
edgelist
# efficient eigendecompostion that leverages low-rank structure in
# E(A) so that you don't have to form E(A) to find eigenvectors,
# as E(A) is typically dense. computation is
# handled via RSpectra
population_eigs <- eigs_sym(custom_dcsbm)