Learning to see the edges

(One of a series of posts on my 2020 learning goal to get deeper into graph theory and its applications. The top image is from a piece by British weaver Peter Collingwood. It's significance will become clear below.)

Graph theory models the world as edges and nodes. The nodes are the dots, the edges are the lines that connect them.

The above sketch is by Vaidehi Joshi. If you are looking for a gentle introduction to graph theory this ten minute read from Vaidehi Joshi on Medium is a charming introduction (she covers many other interesting threads as well, so this is worth an exploration).

Most of us, myself included, find it easier to focus on the nodes than the edges. The node is often the noun and the edge the verb. We focus on the nouns because we find it easier to see objects than actions or relations. It is easier for most of us to see and to think about edges when they are represented as nouns.

One of the most famous problems in graph theory and the source of many of its theorems is the 'Seven Bridges of K?nigsberg.' Can you walk through the city crossing each bridge once and only once? Before going on, sketch this and try to find the solution yourself.

Leonhard Euler answered this question in 1736. The answer is that it is impossible. (This is the same Euler that came up with Euler's identity: e raised to pi x i minus one equals zero.) It is easier to see this if we abstract the map of K?nigsberg into a graph.

The dots (nodes) on the left and right are the two islands in the river, the dots on the top and bottom are the two shores of the river, the edges are the bridges. I find it easier to think about the edges when they are nice, large solid things like bridges.

But most edges are more abstract. In the Ibbaka Talent Skill Graph, for example, skills are the nodes and the edges are the relationships between skills. The most common edges in the Skill Graph are probably 'used with' and 'used by.' These connections generally tell us more than the skill itself. The 'used with' relation helps one find skill cluster that are frequently used together. In most cases, more than one skill is needed to achieve a goal (there is another edge, from skills to goals) and understanding the cluster is critical to skill management. But rather than search out the relationships, most systems focus on managing unrelated lists of skills. This approach prevents real advances in skill and competency management, but it takes and effort to get beyond the nodes to think about the edges. Graph theory is how you think about the edges.

It can still be useful to have some tools to think about the edges more clearly. One of these is Linkography, as set out on the book of the same name by Gabriela Goldschmidt.

Goldschmidt is interested in understanding the design process. She understands the process as a series of design moves (small design decisions) executed over time (we are after all time beings) in which each move can be influenced by moves before it and can influence moves that come later. Design is often a collaborative process, so different moves can be made by different people.

As the process is time bound, the design moves are laid out on a timeline. There are many ways one could show the links (edges) that represent the forward (in time) 'influences' relationship and backwards in time 'influenced by' relationship. Perhaps the most obvious is by adding edges to the timeline like this (this is called an archiograph).

The problem is that this does not make it particularly easy to understand the relationships. Goldschmidt proposes something different. A linkograph turns the edges into nodes to make it easier to see and think about them. In their article 'Using linkography to compare creative methods for group ideation' Hatcher et al. give a good guide to the process. The key attributes of a linkograph are shown below.

The nodes below the process line of design moves represent the connections between design moves. They are called link nodes. A node representing a link. Opening the connections between design steps in this way makes all sorts of patterns visible. Identified above are chunks, webs, say tooths, forelinks and backlinks. It is worth thinking about what each of these means in a design process.

Chunk - usually a major piece of functionality, a module so to speak. Studying linkographs can help us understand how modules emerge in a design process.

Webs - tightly interconnected decisions, usually smaller than chunks but more densely connected.

Sawtooths - a series of iterated design decisions, often evolving or refining one aspect of the design.

Forelink - a design move that influences many later decisions.

Backlink - a design move that integrates many earlier decisions.

Linkographs have taken design research in some interesting directions. One of the most important of these is the application of information theory to the linograph. I doubt this would have happened without the innovation of making the link a node. Learning to see the edges, however you do it, generally brings new insights.

In this case, I am not sure that anyone would have applied information theory to a linear design process without the visual cues provided by the linograph. In 'Order, structure and disorder in space syntax and linkography: intelligibility, entropy and complexity measures' Tamer El-Khouly and Adam Penn took ideas from the space syntax community and applied them to linkographs to get a deeper understanding of the properties of different design processes. The below figures from this paper may inspire you to go deeper into this.

We have looked at two ways to help us to see the edges.

The first was finding a physical analog for the edge. A bridge in the city of K?nigsberg. The wires in an old fashioned telephone or power distribution system. Lines painted on the floor of an airport terminal or hospital to help you navigate.

The second was linkographs, that turn edges into nodes.

What else could there be?

I find that traversing the graph is another way to get a deeper appreciation of its links. Of course that is what the original K?nigsberg challenge was based on, you had to walk around the city crossing each bridge (edge) only once. Road networks are another form of graph (see ). Studying maps is one way to understand such a network, but moving around it physically gives a different, embodied, understanding.

This is fine for physical spaces, but we are not generally given user interfaces that let us traverse conceptual graphs, like the skill graph or value graph (another of the foundational data structures used at Ibbaka). This is a direction I would the Ibbaka interfaces to go. I want to be able to 'walk across' the different graphs we are creating. This would let us use approaches from computer science like depth first versus breadth first search in exploring a person's skills and how they connect to the skills of another person.

The learning pattern of wandering through Wikipedia articles (or any other hypertext) is a kind of semantic analog of this. One navigates by clicking on links, and quickly loses context, but exploration can be a powerful way to build an understanding of the connections between ideas.

Back in the 1990s, several of the people now at Ibbaka were part of ThoughtShare. This is a company we created to capture the paths people take when navigating the Internet and then giving them the ability to annotate and share the paths. I still think this is a powerful idea, and I have not yet seen anyone do it as well as what we implemented more than twenty years ago. Unfortunately, ThoughtShare got caught up in the tech wreck and the finance team that got control of the company pivoted it into blogging platforms.

So now I am trying to train myself to see the links. The textile used at the top of this piece is by British textile artist Peter Collingwood, I am fortunate to has this piece in the house where I live and get to study it from time to time. One of the many fascinating things about his work is the way he has the warp of a piece become the weft so that the traditional distinction between warp and weft blurs. One book about his work is called Woven: Unwoven and for me this is a good way of thinking about the play between edge and node and that the distinction between the two is fundamentally arbitrary. What is a node can be represented as an edge and vice versa. Warp can be weft. Learning to see the edges opens new ways of seeing the relations between things. And afterall, there is no independent origination.