In this article, we go through some of the basic visualization
functionality in the nett package.
Visualizing a DCSBM
Let us sample a network from a DCSBM:
n = 1500
Ktru = 4
lambda = 15 # expected average degree
oir = 0.1
pri = 1:Ktru
set.seed(1234)
theta <- EnvStats::rpareto(n, 2/3, 3)
B = pp_conn(n, oir, lambda, pri=pri, theta)$B
z = sample(Ktru, n, replace=T, prob=pri)
# sample the adjacency matrix
A = sample_dcsbm(z, B, theta)
We can plot the network using community labels \(z\) to color the nodes:
original = par("mar")
gr = igraph::graph_from_adjacency_matrix(A, "undirected") # convert to igraph object
par(mar = c(0,0,0,0))
out = nett::plot_net(gr, community = z)
We can also plot the degree distribution:
summary(igraph::degree(out$gr))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 1.00 10.00 13.00 14.81 17.00 131.00
A latent variable model
Now consider a latent variable model with \(K\) communities as follows: The adjacency
matrix \(A = (A_{ij})\) is generated as
a symmetric matrix, with independent Bernoulli entries above the
diagonal with \[\begin{align}\label{eq:dclvm:def}
\mathbb E [\,A_{ij} \mid x, \theta\,] \; \propto \; \theta_i
\theta_j e^{- \|x_i - x_j\|^2} \quad \text{and} \quad
x_i = 2 e_{z_i} + \frac34 w_i
\end{align}\] where \(e_k\) is
the \(k\)th basis vector of \(\mathbb R^d\), \(w_i \sim N(0, I_d)\), \(\{z_i\} \subset [K]^n\) are multinomial
labels (similar to the DCSBM labels) and \(d =
K\). The proportionality constant in~\(\eqref{eq:dclvm:def}\) is chosen such that
the overall network has expected average degree \(\lambda\)
We can generate from this model using the
nett::sample_dclvm() function as follows:
d = Ktru
labels = sample(Ktru, n, replace = T, prob = pri)
labels = sort(labels)
mu = diag(Ktru)
x = 2*mu[labels, ] + 0.75*matrix(rnorm(n*d), n)
A = sample_dclvm(x, lambda, theta)
Visualizing the network and its degree distribution goes as
before:
original = par("mar")
gr = igraph::graph_from_adjacency_matrix(A, "undirected") # convert to igraph object
par(mar = c(0,0,0,0))
out = nett::plot_net(gr, community = labels)
nett::plot_deg_dist(gr)
#> Warning in nett::plot_deg_dist(gr): There are 0-degree nodes. Omitting them on
#> log scale.
summary(igraph::degree(out$gr))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 1.00 6.00 11.00 13.61 18.00 198.00
Visualizing Political Blogs network
Let us compare with Political Blogs network accessible via
polblogs.
original = par("mar")
par(mar = c(0,0,0,0))
out = nett::plot_net(polblogs, community = igraph::V(polblogs)$community)
nett::plot_deg_dist(polblogs)
#> Warning in nett::plot_deg_dist(polblogs): There are 0-degree nodes. Omitting
#> them on log scale.
summary(igraph::degree(polblogs))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.00 1.00 8.00 25.62 33.00 468.00