Lecture 18: Network Data

Brian J. Smith

2026-04-16

Networks

Networks are mathematical structures containing nodes and edges.

Code

# Create a random graph
set.seed(20260416)
g <- sample_gnm(60, 100) %>% 
  set_edge_attr(name = "weight",
                value = rexp(n = 100, rate = 0.9))

# Graph
ggraph(g, layout = "igraph", algorithm = "fr") +
  geom_edge_link(linewidth = 1) +
  geom_node_point(size = 5, color = "purple") +
  coord_equal() +
  theme_graph()

Networks

Nodes are the items in a network.
- Other terms for a node:
  - Vertex
  - Point
Edges specify relations between nodes.
- Other terms for an edge:
  - Link
  - Connection
  - Arc
  - Line

Networks

The branch of mathematics concerned with networks is known as graph theory.
- Uses the term graph for a network.
- Term dates back to Sylvester (1878, Nature).

Networks

Origins of graph theory date back to Leonhard Euler and the Seven Bridges of Königsberg problem.

Networks

Nodes can have attributes.

Code

# Create a random graph
set.seed(20260416)
g <- sample_gnm(10, 12) %>% 
  set_vertex_attr(name = "size", value = rpois(n = 10, lambda = 3) + 1)

ggraph(g, layout = "auto") +
  geom_edge_link() +
  geom_node_point(aes(size = size, color = size)) +
  scale_color_viridis_c() +
  theme_graph()

Networks

Edges can have attributes.

Code

# Create a random graph
set.seed(20260416)
g <- sample_gnm(8, 12) %>% 
  set_edge_attr(name = "size", value = rbinom(n = 12, size = 2, prob = 0.5))

ggraph(g, layout = "auto") +
  geom_edge_link(aes(linewidth = size, color = size)) +
  geom_node_point(size = 6) +
  theme_graph()

Networks

Edges can be undirected or directed.

Code

# Create a graph
g1 <- make_graph(edges = c(1, 2, 2, 3, 3, 4),
                 directed = FALSE)

g2 <- make_graph(edges = c(1, 2, 2, 3, 3, 4),
                 directed = TRUE)

p1 <- ggraph(g1, layout = "linear") +
  geom_edge_link(linewidth = 1,
                 color = "purple") +
  geom_node_point(size = 6) +
  coord_flip() +
  ggtitle("Undirected") +
  theme_graph() +
  theme(plot.title = element_text(hjust = 0.5))

p2 <- ggraph(g2, layout = "linear") +
  geom_edge_link(linewidth = 1, arrow = arrow(angle = 30, 
                                               length = unit(0.2, "inches"),
                                               type = "closed"),
                  end_cap = circle(0.11, "inches"),
                  color = "purple") +
  geom_node_point(size = 6) +
  coord_flip() +
  ggtitle("Directed") +
  theme_graph() +
  theme(plot.title = element_text(hjust = 0.5))

p1 + p2

Networks

When an edge connects a node to itself, it is called a loop.

Code

g <- make_graph(edges = c(1, 2, 2, 3, 3, 4, 4, 4),
                directed = TRUE)

ggraph(g, layout = "linear") +
  geom_edge_loop(linewidth = 1,
                arrow = arrow(angle = 30, 
                              length = unit(0.2, "inches"),
                              type = "closed"),
                end_cap = circle(0.11, "inches"), 
                color = "purple") +
  geom_edge_link(linewidth = 1,
                arrow = arrow(angle = 30, 
                              length = unit(0.2, "inches"),
                              type = "closed"),
                end_cap = circle(0.11, "inches"), 
                color = "purple") +
  geom_node_point(size = 6) +
  coord_cartesian(ylim = c(-0.5, 0.75)) +
  theme_graph()

Networks

Networks can have cycles where connections between nodes create a loop.
- A network with at least one cycle is called cyclic,
- and a network with no cycles is called acyclic.

Code

# Create a graph
g1 <- make_graph(edges = c(1, 2, 2, 3, 3, 4),
                 directed = FALSE)

g2 <- make_graph(edges = c(1, 2, 2, 3, 3, 4, 4, 2),
                 directed = FALSE)

p1 <- ggraph(g1, layout = "linear") +
  geom_edge_link(linewidth = 1,
                 color = "purple") +
  geom_node_point(size = 6) +
  coord_flip() +
  ggtitle("Acyclic") +
  theme_graph() +
  theme(plot.title = element_text(hjust = 0.5))

p2 <- ggraph(g2, layout = "auto") +
  geom_edge_link(linewidth = 1, 
                  color = "purple") +
  geom_node_point(size = 6) +
  coord_flip() +
  ggtitle("Cyclic") +
  theme_graph() +
  theme(plot.title = element_text(hjust = 0.5))

p1 + p2

Networks

A network with hierarchical structure is called a tree.
Trees cannot have cycles.

Code

# Random graph
set.seed(20260416)
tr <- sample_tree(n = 30, directed = TRUE, method = "lerw")

ggraph(tr, layout = 'dendrogram') + 
  geom_edge_elbow(linewidth = 1) +
  theme_graph()

Networks

A connected network has some way to get between any two nodes.
A disconnected graph has some gaps.

Code

# Create a graph
g1 <- make_graph(edges = c(1, 2, 2, 3, 3, 4, 4, 2),
                 directed = FALSE)

g2 <- make_graph(edges = c(1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 7, 5),
                 directed = FALSE)

p1 <- ggraph(g1, layout = "igraph", algorith = "fr") +
  geom_edge_link(linewidth = 1,
                 color = "purple") +
  geom_node_point(size = 6) +
  ggtitle("Connected") +
  theme_graph() +
  theme(plot.title = element_text(hjust = 0.5))

p2 <- ggraph(g2, layout = "igraph", algorith = "fr") +
  geom_edge_link(linewidth = 1, 
                  color = "purple") +
  geom_node_point(size = 6) +
  ggtitle("Disconnected") +
  theme_graph() +
  theme(plot.title = element_text(hjust = 0.5))

p1 + p2

Networks

A connected network might have a node or edge that acts as a bottleneck. If that node/edge is removed, the network becomes disconnected.
- Cut node or cut edge.

Code

# Create a graph
g1 <- make_graph(edges = c(1, 2, 2, 3, 3, 1, 
                           # Node 4 is cut node
                           3, 4, 4, 5,
                           5, 6, 6, 7, 7, 5),
                 directed = FALSE) %>% 
  set_vertex_attr(name = "cut", value = c((1:7) == 4))

g2 <- make_graph(edges = c(1, 2, 2, 3, 3, 1, 
                           # Cut vertex between 3 and 4
                           3, 4,
                           4, 5, 5, 6, 6, 7, 7, 4),
                 directed = FALSE) %>% 
  set_edge_attr(name = "cut", value = c((1:8) == 4))

p1 <- ggraph(g1, layout = "igraph", algorith = "fr") +
  geom_edge_link(linewidth = 1,
                 color = "black") +
  geom_node_point(size = 7, aes(color = cut)) +
  ggtitle("Cut Node") +
  theme_graph() +
  theme(plot.title = element_text(hjust = 0.5))

p2 <- ggraph(g2, layout = "igraph", algorith = "fr") +
  geom_edge_link(linewidth = 1, aes(color = cut)) +
  geom_node_point(size = 6) +
  ggtitle("Cut Edge") +
  theme_graph() +
  theme(plot.title = element_text(hjust = 0.5))

p1 + p2

Visualizing Networks

This part of the lecture is largely based on Chapter 9 of Visualization Analysis & Design.

“Arrange Networks and Trees”

We’ll take some ideas from the text and some other examples from the internet to talk about how to visualize networks.

Visualizing Networks

Node-Link Diagrams

Node-link diagrams are the most common visual encoding idiom for networks.
- Point marks for nodes.
- Line marks for links.
- Vertical/horizontal position may or may not be used to encode any information.

Node-Link Diagrams

Node-Link Diagrams

Example from WLF 2200: Principles of Ecology
This network is known as a life-cycle diagram and the projection matrix associated with it is known as a Leslie matrix.
Horizontal position encodes age.

Node-Link Diagrams

A spindle diagram shows phylogenetic relationships in a tree.
Horizontal width (of edges) encodes taxonomic diversity.
Vertical position encodes geological time.

Node-Link Diagrams

Trees can also be displayed with a radial layout.

Node-Link Diagrams

When spatial position is not used to encode an attribute, the layout of nodes can be arbitrary.
Many potential algorithms for deciding the placement of nodes for visual aesthetics or readability.

Node-Link Diagrams

A popular layout idiom is force-directed placement.
- Many variants on the basic idea.
- Position nodes with a simulation of physical forces.
  - Nodes repel each other.
  - Edges act like springs to draw each other together.
- Many algorithms are nondeterministic (stochastic).

Matrix Views

A network can be transformed into an adjacency matrix.
- An \(n \times n\) square matrix, where \(n\) is the number of nodes.
Visualization can be based on the matrix instead of using node-link diagrams.

Code

# Create a graph
g1 <- make_graph(edges = c(1, 2, 2, 3, 3, 1, 
                           # Node 4 is cut node
                           3, 4, 4, 5,
                           5, 6, 6, 7, 7, 5),
                 directed = FALSE) %>% 
  set_vertex_attr(name = "cut", value = c((1:7) == 4))

mat <- g1 %>% 
  as.matrix() %>% 
  as.matrix
colnames(mat) <- 1:7
rownames(mat) <- 1:7

par(mfrow = c(1, 2))
plot(g1, main = "Node-Link Diagram")
image(mat, asp = 1, axes = FALSE, main = "Adjacency Matrix", lwd = 0.5)
axis(1, at = seq(0, 1, length.out = 7), labels = 1:7)
axis(2, at = seq(0, 1, length.out = 7), labels = 1:7)
box()

Containment: Hierarchy Marks

Another approach is to show hierarchical structure with containment rather than connection.
- Treemaps
- Circular packing

Networks in R

Some useful R packages:
- igraph, go-to package for networks in R?
- ggraph, extension of ggplot2 for networks.
- tidygraph, a tidy API for network manipulation.
- Others?
R Graph Gallery
- Network graph

Questions?

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