Aims

By the end of this tutorial, you should be able to:

•   Graph any compatible network with graphr() and read what the defaults show you
•   Map node and tie attributes to colour, shape, size, and groups
•   Set a consistent theme across all your plots, and tailor palettes for accessibility or print
•   Add titles, labels, and legends that help others read your graph
•   Choose an appropriate layout, and know when the distances between nodes can be interpreted
•   Arrange multiple graphs together with graphs() and animate change over time with grapht()
•   Plot centrality measures and other results with consistent plot() methods
•   Export publication-ready figures with ggsave()

Choose your own data: The worked examples below mostly use fict_lotr, a fictional network of affinities among Lord of the Rings characters bundled with {manynet}, plus a couple of classical datasets. But wherever there is an exercise box, you are encouraged to swap in a network that interests you. Remember the three flavours of bundled data as a rough difficulty ladder — Classic (ison_*, small & tidy), Fiction (fict_*, mid-sized & fun), Real-world (irps_*, larger & realistic) — and that you can browse the full list with table_data().

Plotting approaches

To understand graph and network visualisation with {autograph}, it is useful to review the different approaches already taken in R. Plotting in R is typically based around two main approaches:

• the ‘base’ approach in R by default, and
• the ‘grid’ approach made popular by the famous and very flexible {ggplot2} package.¹

In the case of base R graphics, plots are essentially written straight to the plotting device. This means that they are not easily modified after the fact: you would need to replot the whole thing to change something. Moreover, while there is an admirably clean aesthetic to base R graphics, it can be difficult to modify or extend them to your needs.

In the case of grid graphics, plots are built up in layers, and thus can be modified after the fact. That is, you can initialise a plot using ggplot2::ggplot(), specifying the data and mapping variables to various aesthetic features, and then add layers to it using + to add further points and lines, but also titles, legends, etc.

The following figure illustrates the difference between these two approaches.² Run the code to compare the two plots. (There are buttons to run the code you have entered, to start over, and — where available — to receive hints and solutions. You will use these throughout the tutorial.)

plot(mtcars$hp, mtcars$mpg,
     main = "Base R: MPG vs Horsepower",
     xlab = "Horsepower",
     ylab = "Miles per Gallon",
     pch = 19,
     col = "blue")
ggplot(mtcars, aes(x = hp, y = mpg)) +
  geom_point(color = "blue") +
  labs(title = "ggplot2: MPG vs Horsepower",
       x = "Horsepower",
       y = "Miles per Gallon")

Graphing approaches

Approaches to plotting graphs or networks in R can be similarly divided:

• two classic packages, {igraph} and {sna}, both build upon the ‘base’ R graphics engine,
• newer packages {ggnetwork} and {ggraph} build upon a ‘grid’ approach.³

Let’s see how the fict_lotr network would be plotted using {igraph} and {ggraph}, adding a title to each to facilitate comparison, but otherwise relying on default behaviour.

plot(as_igraph(fict_lotr),
     main = "igraph: fict_lotr")
ggraph::ggraph(as_tidygraph(fict_lotr)) +
  ggraph::geom_edge_link() +
  ggraph::geom_node_point() +
  ggtitle("ggraph: fict_lotr")

We can see here that {igraph} plots the network in a fairly basic way, straight to the plotting device (window). By default, it uses a force-directed layout (see the Layouts section below),⁴ colors the nodes orange, and prints node labels if they have them. However, the layout is not optimised for the size of the plotting window, the node labels are regularly overlapping, and the orange color with black borders is not particularly appealing or helpful for label legibility. It only works with ‘igraph’ objects.

In contrast, {ggraph} offers the trademark flexibility of the grammar of graphics approach. However, it requires the user to build up a plot from the ground up, which can be daunting for new users and fiddly even for experienced ones. Four lines are required to get even a basic plot, with an additional line required if a grey background is not desired. No labels or other information are added by default, and would also require additional lines. It works with ‘tidygraph’ objects, which are an additional layer on top of ‘igraph’ objects.

Your first graph

{autograph} builds upon these packages, but takes a somewhat different approach. It builds upon the ‘grid’ approach of {ggplot2} and {ggraph}, lending itself to the additional layering and flexibility of those packages. Because it depends on the coercion routines available in {manynet}, it can be used with network-related objects from most common network analysis packages — igraph, network, tidygraph, matrices, edgelists, and more — without you needing to convert anything first. Unlike those packages, though, it offers concise and easy-to-use functions with sensible defaults for most common use cases, using the information that is available in the network object.

The first thing you will want to do when you import or create a new network dataset is draw it. Compared to the {igraph} and {ggraph} examples above, autograph::graphr() offers a much more concise way to draw the network. Try it now.

graphr(fict_lotr)

Note everything that happened without being asked: graphr() recognised that the network is labelled and printed the node labels, chose a deterministic layout (so you get the same picture every time), sized and spaced the labels to minimise overlap, and dropped the axes and grey background that mean nothing for networks. Because the network is undirected , there are no arrowheads; for a directed network, graphr() would draw them automatically.

The package also offers methods for plotting statistics related to networks (e.g. degree distributions ) and models of them (e.g. goodness-of-fit plots). We will get to some of these later in this tutorial, and others are demonstrated in the tutorials of other {stocnet} packages. {autograph} also offers consistent theming across graphs and plots, so that you do not need to keep specifying the same options over and over again.

In the following pages, we’re going to go through a number of different ways of taking control of the graphing process. Click ‘Next Topic’ to continue.

Shaping nodes

One of the first things we might be interested in doing is understanding better the distribution of some categorical variable. Our fict_lotr dataset contains a variable called Race, so let’s try and change the shape of the nodes by this variable. Following the syntax shown in the table above, we just need to reference the variable name in the node_shape argument. Print the network first to check the attribute’s (case-sensitive) name, then graph it.

fict_lotr
graphr(fict_lotr, node_shape = "Race")

We can see here that there are six different races present.⁵ Unfortunately, this is a few too many different categories to be effectively distinguished by shape: at a glance, can you quickly find the triangles among the squares? Shape works best for two or three categories at most.

One place where shape excels, though, is distinguishing the node sets of a multimodal network — and there graphr() does it for you. For a two-mode network, nodes in the first mode are drawn as circles and nodes in the second mode as squares, with a “Mode” legend added automatically; were a third node set mapped to shape, it would be drawn as triangles. Graph the ison_southern_women network, where the women (first mode) appear as circles and the events they attended (second mode) as squares.

graphr(ison_southern_women)

Colouring nodes

Let’s try instead colouring the nodes by this “Race” variable. It is very similar to the shape example above. Can you complete the code yourself?

# Use the same syntax as with node_shape, but with the node_color argument.
# Remember to name the attribute in quotation marks.

graphr(fict_lotr, node_color = "Race")

That’s much easier to read. Note how a legend has been added automatically, using the colours of whatever theme is currently set (more on themes soon).

How should we interpret this graph? Since the same colours seem to be clustered together, with the humans and hobbits each clustered together in the centre of the graph, and the elves clustered towards the left, we might infer that there is some homophily going on here — that characters tend to be connected to others of the same race — a hypothesis to test properly in another tutorial. Interpreting an attribute-coloured graph like this is often the first, informal step toward a more formal analysis.

An alternative to colouring the nodes is to use the node_group argument to highlight groups in a network. This puts a shaded area around nodes of the same group. For rather spatially clustered distributions, this can be a very effective way to show groupings, but it is sensitive to the layout used: if nodes of the same group are not close together, the shaded areas can overlap and make the graph harder to read.

graphr(fict_lotr, node_group = "Race")

Note that node_color and node_group can be used together, either to highlight different groupings, or to emphasise group assignment where the groups interpenetrate, as described above.

Sizing nodes

What about if we’re interested in a continuous variable instead of a categorical variable? While the fict_lotr dataset does not contain any continuous nodal variables, we can create one rather easily from the network itself. Let’s use each node’s degree , which is the number of ties incident/connecting to the node.

fict_lotr |>
  mutate(Degree = node_by_deg(fict_lotr)) |>
  graphr(node_size = "Degree")

Larger nodes are now the better-connected characters, and a size legend has been added. Who turns out to be the most connected character in the fellowship?

Tying up loose ends

All this works similarly with ties/edges. Just replace node_ with edge_ in the arguments above, and you can control edges’ size and color. In the following example, we add two tie attributes: a continuous variable measuring how ‘close’ each tie is to others, and a binary variable indicating whether the tie is part of a triangle or not, and then colour the ties by the latter. Run the code, then try colouring or sizing the ties by "weight" instead.

fict_lotr |>
  mutate_ties(weight = tie_by_closeness(fict_lotr),
              is_tri = tie_is_triangular(fict_lotr)) |>
  graphr(edge_color = "is_tri")

Note also that some tie attributes are recognised automatically: if a network contains a tie attribute called weight, ties will be sized by weight without you asking, and a tie attribute called type will be used to distinguish tie types. Naming your attributes accordingly can save you some typing.

Pointing arrows

So far our example network has been undirected. For directed networks, graphr() adds arrowheads automatically, pointing from the sender to the receiver of each tie, and trims them back so they are not swallowed by the receiving node. Arrowheads are also scaled automatically with the width of the ties: thin ties get small arrowheads, thick ties get larger (but capped) ones, and ties of width zero lose their arrowheads entirely. This means arrowheads stay proportionate even when tie width is mapped from a weight attribute, as in the ison_networkers network of messages exchanged among early network researchers. You can also scale them manually: because the arrowheads follow the tie width, setting edge_size yourself resizes both together. Compare the automatic sizing with a manually thickened version.

(graphr(ison_networkers) + ggtitle("Automatic") |
   graphr(ison_networkers, edge_size = 1) + ggtitle("Manual (edge_size = 1)"))

Taming dense or disconnected networks

Two arguments help when a network is too dense, or too sparse, to read at a glance.

Larger, denser networks can turn into a ‘hairball’, where the sheer number of ties obscures everything. edge_bundle pulls ties that travel in similar directions into shared paths — like cabling them together — so that the main ‘highways’ of the network stand out. It is off by default; set edge_bundle = TRUE (or name a specific algorithm: "force", "path", or "minimal") to switch it on. Compare a dense random network with and without bundling.

rand <- manynet::generate_random(40, 0.1)
(graphr(rand) + ggtitle("Unbundled") |
   graphr(rand, edge_bundle = TRUE) + ggtitle("Bundled"))

At the other extreme, many networks contain isolates — unconnected nodes — which, under a force-directed layout, drift to the margins and squeeze the connected core into a clump. The isolates argument decides what happens to them: "legend" (the default) drops them from the drawing but records how many there were in the legend, "caption" notes them in a caption instead, and "keep" leaves them in place. Add two unconnected characters to fict_lotr and compare keeping them with noting them in the legend.

lotr_iso <- fict_lotr |>
  add_nodes(2, list(name = c("Tom Bombadil", "Goldberry")))
(graphr(lotr_iso, isolates = "keep") + ggtitle("keep") |
   graphr(lotr_iso, isolates = "legend") + ggtitle("legend"))

For very large real-world networks such as irps_blogs, the two work well together: edge_bundle = TRUE untangles the connected core while isolates = "legend" keeps its several hundred unconnected blogs from crowding that core out.

Free play

Your turn: choose another network and illustrate something about it. Pick a dataset with interesting node attributes — here is one suggestion per flavour:

Print the network first to see which attributes are available, then map one or two of them to colour, shape, size, or groups.

Setting a theme

Perhaps you are preparing a presentation, representing your institution, department, or research centre at home or abroad. In this case, you may wish to theme the whole network with institutional colors and fonts. Indeed, you may even want to set a theme that is then reused across all your graphs and plots. {autograph} offers a number of themes that can be set using the stocnet_theme() function. Once set, the theme applies to every subsequent graph and plot in your session — no need to repeat yourself.

stocnet_theme("default")
graphr(fict_lotr, node_color = "Race")
stocnet_theme("iheid")
graphr(fict_lotr, node_color = "Race")
stocnet_theme("default")

Currently available themes include a number of institutional themes ("iheid", "ethz", "uzh", "rug", "unibe", "oxf", "unige", "cmu", "iast", "hwu") as well as stylistic ones ("default", "bw", "crisp", "neon", "rainbow"). Run stocnet_theme() without arguments to see which theme is currently set. More institutional scales and themes can be implemented upon pull request.

Who’s hue?

By default, graphr() will use a color palette that offers fairly good contrast and better accessibility. However, a different hue might offer a better aesthetic or identifiability for some nodes. Because the graphr() function is based on the grammar of graphics, it’s easy to extend or alter aesthetic aspects. Here let’s try and change the colors assigned to the different races in the fict_lotr dataset. Note that despite the argument being node_color, when overwriting the colors please use functions of the type ggplot2::scale_fill_*(), as it is the “fill” aesthetic that is being mapped to the variable in this case.

graphr(fict_lotr,
           node_color = "Race")

graphr(fict_lotr,
           node_color = "Race") +
  ggplot2::scale_fill_hue()

At this stage, it is worth noting that not everyone experiences colours in the same way. Some people are colour-blind, whether by deuteranomaly, deuteranopia, protanomaly, or protanopia, and so it is worth checking that your visualisations are accessible to them.⁶ Others are less sensitive to colour distinctions. The old trope is that males are less sensitive to colour distinctions:⁷

Grayscale

Other times color may not be desired. Some publications require grayscale images. To use a grayscale color palette, replace _hue from above with _grey (note the ‘e’ spelling):

graphr(fict_lotr,
           node_color = "Race") +
  ggplot2::scale_fill_grey()

As you can see, grayscale is more effective for continuous variables or for very few discrete categories than for the six categories used here. If you need to distinguish several categories in print, consider combining grayscale with node_shape, or use the "bw" theme, which is designed for this purpose.

Manual override

Or we may want to choose particular colors for each category. This is pretty straightforward to do with ggplot2::scale_fill_manual(). Some common color names are available, but otherwise hex color codes can be used for more specific colors. Unspecified categories are coloured (dark) grey.

graphr(fict_lotr,
           node_color = "Race") +
  ggplot2::scale_fill_manual(
    values = c("Dwarf" = "red",
               "Hobbit" = "orange",
               "Maiar" = "#DEC20B",
               "Human" = "lightblue",
               "Elf" = "lightgreen",
               "Ent" = "darkgreen")) +
  labs(fill = "Color")

Labels

With our fict_lotr example above, because the network is itself labelled, graphr() automatically adds the node labels. If you do not want these labels, you can remove them from the network before passing it on to graphr(), or more simply use the argument labels = FALSE.

graphr(fict_lotr, labels = FALSE)

Without the labels, the structure of the network is clearer and easier to interpret, though we lose the information about which node is which character. Which you prefer depends on what the graph is for: exploring who-is-who, or communicating overall structure.

Titles

{autograph} works well with both {ggplot2} and {ggraph} functions that can be appended to create more tailored visualisations. Let’s try this by adding a title to a plot. Append (with a +) labs(title = ) to add a title to a plot, say “My graph”, and then add also a subtitle (an argument to that function), say “I did this”.

# Fill in the blanks (this is a template, not runnable code):
# graphr(fict_lotr) +
#   labs(title = _____, subtitle = _____)

graphr(fict_lotr) +
  labs(title = "My visualisation",
       subtitle = "I did this")

Note that you can also use ggtitle() to do the same thing, but if you just remember labs() you can also use it to add labels for x and y axes, and legends (see below).

Legends

While {autograph} attempts to provide legends where necessary, in some cases the legends offer insufficient detail, or are absent, such as in the following figure, where we highlight the node with the highest betweenness centrality.

fict_lotr |>
  mutate(maxbet = node_is_max(node_by_betweenness(fict_lotr))) |>
  graphr(node_color = "maxbet")

Which node is highlighted here, and why might that be? Without a legend title, a reader cannot know what the colour signifies. {autograph} supports the {ggplot2} way of adding legends after the main plot has been constructed, using guides() to add in the legends, and labs() for giving those legends particular titles. Note that we can use "\n" within the legend title to make the title span multiple lines.

fict_lotr |>
  mutate(maxbet = node_is_max(node_by_betweenness(fict_lotr))) |>
  graphr(node_color = "maxbet") +
  guides(color = "legend") +
  labs(color = "Maximum\nBetweenness")

To change the position of the legend, add the theme() function from {ggplot2}. The legend can be positioned at the top, bottom, left, or right, or removed using “none”.

Force-directed layouts

Force-directed layouts update some initial placement of vertices through the operation of some system of metaphorically-physical forces. These might include attractive and repulsive forces.

(graphr(ison_southern_women, layout = "kk") + ggtitle("Kamada-Kawai") |
   graphr(ison_southern_women, layout = "fr") + ggtitle("Fruchterman-Reingold") |
   graphr(ison_southern_women, layout = "stress") + ggtitle("Stress Minimisation"))

The Kamada-Kawai (KK) method inserts a spring between all pairs of vertices that is the length of the graph distance between them. This means that edges with a large weight will be longer. KK offers a good layout for lattice -like networks, because it will try to space the network out evenly.

The Fruchterman-Reingold (FR) method uses an attractive force between directly connected vertices, and a repulsive force between all vertex pairs. The attractive force is proportional to the edge’s weight, thus edges with a large weight will be shorter. FR offers a good baseline for most types of networks.

The Stress Minimisation (stress) method is related to the KK algorithm, but offers better runtime, quality, and stability and so is generally preferred. Indeed, {autograph} uses it as the default for most networks. It has the advantage of returning the same layout each time it is run on the same network.

Other force-directed layouts available include:

• Simulated annealing (Davidson and Harel 1993): "dh"
• Graph embedder (Frick et al. 1995): "gem"
• Graphopt (Schmuhl): "graphopt"
• Distributed recursive graph layout (Martin et al. 2008): "drl"

Layered layouts

Layered layouts arrange nodes into horizontal (or vertical) layers, positioning them so that they reduce crossings. These layouts are best suited for directed acyclic graphs, two-mode networks, or other data with a natural hierarchy or ordering.

graphr(ison_southern_women, layout = "bipartite") + ggtitle("Bipartite")
graphr(ison_southern_women, layout = "hierarchy") + ggtitle("Hierarchy")
graphr(ison_southern_women, layout = "railway") + ggtitle("Railway")

Note that "hierarchy" and "railway" use a different algorithm to {igraph}’s "bipartite", and generally perform better, especially where there are multiple layers. Whereas "hierarchy" tries to position nodes to minimise overlaps, "railway" sequences the nodes in each layer to a grid so that nodes are matched as far as possible. For the "hierarchy" layout you can also steer which set sits where by passing a center argument — "events" or "actors" for a two-mode network, or the name of a particular node — which helps when the default places the less interesting set on top.

graphr(ison_southern_women, layout = "hierarchy", center = "events")

If you want to flip the horizontal and vertical, you could flip the coordinates, or use something like the following layout.

graphr(ison_southern_women, layout = "alluvial") + ggtitle("Alluvial")

Other layered layouts include:

• Tree: "tree"
• Dominance layouts

Circular layouts

Circular layouts arrange nodes around (potentially concentric) circles, such that crossings are minimised and adjacent nodes are located close together. In some cases, location or layer can be specified by attribute or mode.

graphr(ison_southern_women, layout = "concentric") + ggtitle("Concentric")

The "concentric" layout can also place nodes on rings by a grouping you choose: pass a membership argument (a node attribute name, or a vector the same length as the number of nodes). Ring the Lord of the Rings characters by their race.

graphr(fict_lotr, layout = "concentric", membership = "Race")

Other such layouts include:

• circular: "circle"
• sphere: "sphere"
• star: "star"
• arc or linear layouts: "linear"

Spectral layouts

Spectral layouts arrange nodes according to the eigenvalues of the Laplacian matrix of a graph. These layouts tend to exaggerate the clustering of like-nodes and the separation of less similar nodes in two-dimensional space.

graphr(ison_southern_women, layout = "eigen") + ggtitle("Eigenvector")

Somewhat similar are multidimensional scaling (MDS) techniques, which visualise the similarity between nodes in terms of their proximity in a two-dimensional (or more) space.

graphr(ison_southern_women, layout = "mds") + ggtitle("Multidimensional Scaling")

Other such layouts include:

• Pivot multidimensional scaling: "pmds"

Grid layouts

Grid layouts arrange nodes based on some Cartesian coordinates. These can be useful for making sure all nodes’ labels are visible, but horizontal and vertical lines can overlap, making it difficult to distinguish whether some nodes are tied or not.

graphr(ison_southern_women, layout = "grid") + ggtitle("Grid")

Other grid layouts include:

• orthogonal layouts for e.g. printed circuit boards
• grid snapping for other layouts

That last point deserves a demonstration. Rather than committing to a full grid, graphr()’s snap = TRUE argument keeps whatever layout you asked for but snaps its coordinates onto a grid — trading a little positional accuracy for the label legibility of a grid. Compare the stress layout with its snapped version.

(graphr(fict_lotr) + ggtitle("stress") |
   graphr(fict_lotr, snap = TRUE) + ggtitle("stress + snap"))

Manual layouts

Whatever their differences, all these layout algorithms do the same job: they return a table of node coordinates. Nothing stops you computing that table yourself, inspecting it, adjusting a coordinate or two, and handing the result back to graphr() via its x and y arguments. This is handy when a layout is almost right — say one label sits awkwardly, or you want a particular node set apart — or when you need the same hand-tuned positions across several figures. Compute a stress layout for fict_lotr, inspect the coordinate table, banish Gollum to the top-right corner, and re-graph.

lo <- ggraph::create_layout(as_tidygraph(fict_lotr), layout = "stress")
head(lo[, c("name", "x", "y")])
lo$x[lo$name == "Gollum"] <- max(lo$x) + 1
lo$y[lo$name == "Gollum"] <- max(lo$y) + 1
graphr(fict_lotr, x = lo$x, y = lo$y)

The same trick lets you reuse a layout across plots (compute once, pass the same x/y to each call), which keeps node positions identical between figures — useful when readers need to compare them.

Arrangements

{autograph} uses the {patchwork} package for arranging graphs together, e.g. side-by-side or above one another. The syntax is quite straightforward and is used throughout these vignettes/tutorials. Basically, you just use + or | to put graphs side-by-side, and / to put them above one another. Parentheses can be used to group graphs together. Try graphing fict_lotr and ison_algebra side-by-side, and then one above the other.

# Fill in the blanks (this is a template, not runnable code):
# graphr(_____) + graphr(_____)
# graphr(_____) / graphr(_____)

graphr(fict_lotr) + graphr(ison_algebra)
graphr(fict_lotr) / graphr(ison_algebra)

Sets

graphr() is not the only graphing function included in {autograph}. To graph sets of networks together, graphs() makes sure that two or more networks are plotted together, using a consistent layout and theme across the panels so that they can be compared. This might be a set of ego networks, subgraphs , or waves of a longitudinal network.

graphs(to_subgraphs(fict_lotr, "Race"),
       waves = c(1,2,3,4))

What is happening here is that to_subgraphs() is creating a list of subgraphs — one per race — and then graphs() is plotting them together at once with the same set of aesthetic parameters. The waves argument selects which networks in the list to plot — here the first four of the six race subgraphs. Left to its own devices, graphs() plots just the first and last networks of longer lists, which suits before-and-after comparisons of longitudinal networks.

When the panels share the same nodes, graphs() computes a single layout and reuses it across panels so that positions line up and can be compared; by default it uses the "first" network’s layout, but based_on = "last" or "both" are available. Sharing a layout means every panel has to draw every node, so in that case isolates are kept in place.

Dynamics

grapht() is another alternative to graphr(), this time rendering network changes over time as an animated gif. Longitudinal networks (with discrete waves) and dynamic networks (with dated changes) are both supported. Nodes appear, move, and fade as they enter and exit the network, and node positions transition smoothly between waves. Run the following to animate a randomly-evolving version of our Lord of the Rings network. (Be patient — rendering an animation takes considerably longer than drawing a static graph, and requires the suggested {gganimate} and {gifski} packages.)

fict_lotr |>
  mutate_ties(wave = sample(2001:2012, manynet::net_ties(fict_lotr), replace = TRUE)) |>
  to_waves(cumulative = TRUE) |>
  grapht()

Note that here, as with weight and type in the previous section, attribute naming matters a little: a time attribute called wave marks the network as longitudinal for {manynet}, so to_waves() (and grapht() itself, passed such a network directly) will split it without being told which attribute to use. From {manynet} 2.2.2, any other name (say, year) works just as well — it only needs declaring via to_waves()’s attribute argument.

Glossary

Here are some of the terms that we have covered in this tutorial:

Betweenness

The betweenness centrality of a node is the proportion of shortest paths between all pairs of nodes that pass through that node.

Closeness

The closeness centrality of a node is the reciprocal of the sum of its distances to all other nodes.

Community

A community is a set of nodes more densely connected to one another than to other nodes in the network.

Complex

A complex network is one that includes or can include loops or self-ties.

Degree

The degree of a node is the number of connections it has.

Directed

A directed network is a network where the ties have a direction, from a sender to a receiver.

Distribution

A degree distribution is the frequency distribution of the degrees of the nodes in a network.

Homophily

A tendency for nodes to connect to similar nodes.

Isolate

An isolate is a node with degree equal to zero.

Label

A labelled network includes unique labels for each node (or ties) in the network.

Lattice

A network that can be drawn as a regular tiling.

Longitudinal

A longitudinal network is one observed in two or more discrete waves or panels over time.

Network

A network comprises one or more sets of nodes, one or more sets of ties among them, and potentially some node, tie, or network-level attributes.

Node

A node or vertex is an entity or actor within a network.

Reciprocity

A measure of how often nodes in a directed network are mutually linked.

Signed

A signed network is one where ties are marked as positive or negative, such as friendship and enmity or alliance and conflict.

Subgraph

A subgraph comprises a subset of the nodes and ties in a network.

Tie

A tie, edge, or link is a connection or relationship between two nodes.

Triangle

A cycle of length three in a network.

Twomode

A two-mode (or bipartite) network is a network with two different sets of nodes, where ties connect only nodes from different sets, such as people and the events they attend.

Undirected

An undirected or line network is one in which tie direction is undefined.

Weighted

A weighted network is where the ties have been assigned weights.