This series of posts is a tour through of the design space of graph visualization. As I promised, I will do my best to objectively justify as many visualization decisions as I can. This means we will have to go slow; I won’t even draw anything today! In this post, I will only take the very first step: all we will do is think about graphs, and what might be interesting about them.

## What is in a graph?

A graph $G$ has two things: a set of vertices $V$, and a set of edges $E$, where each edge is represented by an ordered pair of distinct vertices (so in this definition we will not have multiple edges and “self-edges”). To denote that $(a, b)$ is in $E$, I will use $a \to b$.

Usually, we also have a mapping $v_\textrm{attr}$ from $V$ to some other space $V_A$. This gives us attributes of these vertices (names of the people in your social network, names of the computers in your intranet, etc.). A similar mapping $e_\textrm{attr}$ from $E$ to $E_A$ does the same for edges (is $b$ married to $c$ or does $b$ work for $c$? How far is $h$ from $j$?, etc.).

These define a graph, but they don’t say much of what is interesting about them. So let’s list some properties of (these very general) graphs. By explicitly thinking about them, we can see the impact they will have on our choices of pictures.

### Graphs are directed or undirected

One important characteristic of graphs is whether they are directed or undirected. When we say that a graph is undirected, we mean that whenever $(a,b) \in E$, it is implied that $(b,a) \in E$: in other words, the has-edge relation is symmetric, and $e_\textrm{attr}((a,b)) = e_\textrm{attr}((b,a))$. (For undirected graphs, I will write $a \unicode{x2013} b$ to mean that both $(a, b) \in E$ and $(b, a) \in E$ are true). Otherwise, we say that $G$ is directed.

This distinction is important because, remember, the first rule of visualization is “draw all there is, but no more”. If our graph is such that $a \to b$ does not imply $b \to a$, our visualization of it better not imply that the relationship between $a$ and $b$ look symmetric. Of course, “making the relationship look symmetric” is not a formal statement, and we might argue about what it really says. But this is what I meant about the difference between a formal systematization and an “informal” one: we should not disconsider the notion simply because we don’t know how to formalize it! And, as we will see, I believe this distinction does guide the visualization choice.

### Graphs have paths

An edge $a \to b$ in a graph implies some sort of connection between $a$ and $b$, and we typically think of these connections being transitive. So if $a \to b$ and $b \to c$ encode some relationship, we tend to think of there existing some relationship between $a$ and $c$ as well (we will say $a \leadsto b$ to say that there exists some path $a \to \ldots \to b$).

This reveals another interesting property of graphs. Let’s say you send the elements of $V$ into new sets, such that whenever $a \leadsto b$ and $b \leadsto a$, $a$ and $b$ must go into the same set. Then, every element of $V$ ends up in exactly one new set. These sets form a partition (into “strongly connected components”, SCCs). Natural partitions like this are your data’s way of telling you to consider divide-and-conquer. If you think paths are important (implying that SCCs are important as well), then your resulting visualization should be “partition-preserving” too: 1) your visualization should have the ability to visually represent a partition of vertices (call it a “visual partition”) and 2) iff $a$ and $b$ are in the same partition, then the visualization of $G$ should put $a$ and $b$ in the same “visual partition”.

### Paths have cycles

We will call a path $a \leadsto a$ which does not repeat internal vertices a cycle (and we will require that cycles in undirected graphs have at least three two internal vertices). A directed graph with no cycles is a dag (“directed acyclic graph”) and an undirected graph with no cycles is a tree.

Vertices of a dag can be assigned natural numbers such that for every pair of vertices $a$ and $b$ such that $a \leadsto b$, $f(a) < f(b)$. If your paths encode dependencies, this assignment of numbers ranks the dependencies, and is good information to have around.

### Many undirected graphs have a metric structure

The final structure I want to mention is the metric structure. For some undirected graphs, there is a very natural way to come up with a distance function between two vertices such it resembles the familiar distances in plain old two- and three-dimensional space. Our eyes are reasonably good at distance judgements (yes, that’s somewhat controversial because of optical illusions and such. But if we are sensitive to these issues, I believe we can use Cleveland to back the statement.)

Anyway, a function $d: V \times V \to R$ is a metric if:

• $d(a, b) \ge 0$, with equality iff $a = b$.
• $d(a, b) = d(b, a)$
• $d(a, c) \le d(a, b) + d(b, c)$, for all $b$

(Assume that the graph is connected for now; any pair of vertices (a,b) is such that $a \leadsto b$ or $b \leadsto a$.) If one of the attributes of undirected graph edges is a positive weight associated with each edge, then the standard metric to assign to a graph is the shortest-path metric, where we say that the distance $d(a, b)$ is given by the smallest cost of a path, this cost being the sum of the edge weights along the path.

“But what if my graph has negative edge attributes?”, you ask. Good question! Then you simply can’t use a metric to describe that particular attribute of your graph. And slightly less trivially, if your visualization technique implies that your graph obeys some metric, then it is telling a lie. As a preview of the next few posts, this “metric-friendliness” will be a crucial distinction between network diagrams and matrix diagrams.

Next up, I will talk about 2D space; a sheet of blank paper where we get to write. Then we will put those things together, and bam, visualization.