Visualization

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:

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:

nett::plot_deg_dist(gr)

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:

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.

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