This series of posts is a tour of the design space of graph visualization. I’ve written about graphs and their properties, and how the encoding of data into a visual representation is crucial. In this post, I will use those ideas to justify the choices behind a classic algorithm for laying out directed, mostly-acyclic graphs.

## Putting what we know to work

In a way, this sequence of posts is an attempt to understand how we turn the process of designing visualizations from mostly art with some craft, to mostly craft with some art. We do know a few rules which work, but we don’t really know how we know them.

In part 1, I asked: “what is in a graph?”. In part 2, I asked “what is in a sheet of paper?”. It is my view that every visualization design should start answering those two questions: what are the structures which we care about in our data? what can we work with?

These questions have little to do with the actual design of a visualization, but they lead right away to what in my mind is the fundamental axiom of visualization design: A visualization design must match structures in data to structures in visual encodings. From here on out, I will call this The Axiom (tongue possibly in cheek).

Following The Axiom means we identify important structures about the data domain, study the medium in which we are going to encode them, and create an encoding in which these match: the graph properties should be represented by the closest matching properties in a sheet of paper.

This notion is of course central in Cleveland and McGill’s classic paper, and you can also see something like it driving much of what Bertin wrote about in his seminal book. For example, Cleveland and McGill famously showed that positions along a common scale are a very effective way to portray a set of real numbers. So if you have a single set of numbers you want your users to understand, following The Axiom means using a dot plot.

Big whoop, right?

But there are other types of structures we would like to portray in our visualizations. In this post, we will see some of them.

## Looking at a DAG

To make this exercise simpler, assume that our visualizations will be node-link diagrams. So your mission is simply to specify positions of the nodes on a cartesian plane, and edges will be drawn as straight lines (directed edges will use traditional arrowheads).

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You’re given a graph $G$ which happens to have a “mostly acyclic structure”. That is, there are either no cycles, or relatively few of them, such that you expect that the majority of the interesting properties of $G$ come from its acyclicity. If your mission were to portray this graph faithfully on paper, what would you do? As the simplest possible “null plot”, this is a graph whose coordinates are just positions on the unit square chosen uniformly at random. Aside from (maybe) being able to tell which vertices appear to have more edges than others, we cannot see much. But we didn’t really expect much to begin with.

Following The Axiom, we already identified one property we want to preserve, namely the “directionality” of the graph. This is good, but it is not very actionable: how do we design a visualization around that? Going back to part 1, remember that we learned that the vertices of acyclic graphs can be ranked: we can give every vertex $v$ an integer $r(v)$ such that if there is a path from vertex $v_1$ to $v_2$, then $r(v_1) < r(v_2)$.

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Let’s set aside the fact that there are many such possible rankings, and accept for now that a rank ordering is a good representation of the notion of acyclicity. The question is then: can we design a visualization that uses it? One possibility is to use the rank as one of the coordinates of the node positions. As a direct consequence, all graph edges will point in the same direction. We’re still using random numbers for the horizontal position of the vertices, but it’s clear that the ranks are encoding some decent amount of structure in the graph (of course, if we wanted to know for sure, we should be using formal inference methods, but that’s another story).

To decide the horizontal positions of the nodes, there are many possible solutions. To begin with, we are going to use the Axiom to state that positions should be unique (“every node is different” becomes “different nodes should be drawn in different places”).

In addition, we will need a new postulate that sounds a little like the contrapositive of The Axiom. Let’s call it The Other Axiom: Everything shown by a visualization should exist in the data. That is, if something “looks like a feature”, then it had better exist in the data. Of course, what is a feature and what is not a feature can only be determined by psychological experiments, but let’s ignore that important point for now.

This notion is obviously close to Tufte’s principle of maximizing data ink. From my reading, Tufte advocates maximizing data ink as economy in service of aesthetics. I, however, want to use the Other Axiom to try and keep a bijective mapping between data and visual representation: if the Axiom is violated, then two different datasets will look the same, and that’s a problem. But if the Other Axiom is violated, then even if the visual mapping is unique, inspecting the resulting visualization might make you think the data is different from what it actually is. The Axiom tries to prevent blurred vision; the Other Axiom, hallucinations.

Since we are using one of the coordinates for the node ranks, the plot naturally grows an axis perpendicular to the rank coordinate. We need to assign this remaining coordinate, but we want to be careful to avoid giving the impression that our visualization is encoding information in how the edges point one way or another (when ranks are drawn vertically, edges will generally point left or right; we want to avoid implying that this direction is meaningful). In addition, edge crossings are an obvious visual feature, and since adjacent edges meet at a node, we don’t want to give the impression of “ghost nodes” by introducing unnecessary edge crossings. Preferring vertical edges seems to be a sensible way to convey “no additional information”, so we will try to position nodes so that edges are “mostly vertical”. It also prevents edge crossings.

There are two main problems in this solution. First, the goal of vertical edges clashes directly with the goal of unique node positions. So we need to compromise somehow (and the devil is in the algorithmic details). But just as importantly, what constitutes a “natural arrangement” with “no additional information” is a mix of cultural and innate characteristics about which we know very little (but there has been recent work in the area).

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Leaving aside all those important details, this is what an algorithm which (roughly) encodes the above principles generates. Aside from one important step, this is a result from graphviz’s classic “dot” algorithm. Visually, the main problem with this approach is in the edges. For simplicity, I have completely ignored the problem of edge occlusion, but that has led to a violation of The Axiom: in a subgraph of the kind $a \to b, b \to c, a \to c$, the principles I set above mean that the edge $a \to c$ will necessarily be obscured. One can explicitly route edges around nodes, and this is precisely what “dot” does; I left that out in my crude drawing.

The paper describing “dot” is worth, at the very least, skimming over; I like that it sets forth a few visual principles and then the algorithm itself (which is, alas, fairly complicated) is designed around those simple principles. The same kinds of statements can be found in the classic Reingold-Tilford tree drawing algorithm. I don’t think I see this structure in more recent visualization papers. Should we be asking ourselves why?

So there you have it, a very basic graph drawing algorithm distilled to its very basics: Find some structure you care about (in this case, acyclicity); find a way to encode it visually, and make sure your encoding is effective (the Axiom) and faithful (the Other Axiom).

## Next, undirected graphs

Of course, defining the entirety of graph visualization as choosing node positions in the plane is a gross simplification, and one which lets us apply the Axioms straightforwardly. It will not always be this simple, and in this series we will get to more delicate cases. But before we go there, the next post will discuss (albeit in less detail) a few popular graph drawing algorithms for undirected graphs.