# Visualization

library(nett)
library(igraph)

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)  par(mar = original) 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:

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)


par(mar = original)
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)


par(mar = original)
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