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The Metrics of Space: Molecule Design

January 15, 2013 Article Start Page 1 of 5 Next


Game spaces provide a context for the game's rules and systems, and a space for the game agents to perform mechanics. When we go about designing game spaces, sometimes thinking in pure spatial terms clouds what a designer needs to achieve with a certain game space.

For FPS games, sitting yourself down with your favorite prototyping tool kit and drawing corridors and rooms is a recipe for disaster. It is difficult to design interesting spatial puzzles when you are creating game spaces using the rules of reality. How many office blocks are fun to navigate?

Molecule design is a way of applying graphing theory for concepting and fine-tuning of various types of game spaces. This rational approach to design is a means to design spaces without thinking about the representational elements of space itself. This article still accepts the importance of planar maps; however, we need better tools to help us create these first.

This article will examine some useful tools gathered from the field of graphing theory that designers can use to conceptualize various game components. The latter half of this article will examine a real-world application of these tools. By doing so, we will examine how iterations of a level design benefited from this abstracted means of realizing space.

The Basics of Graphing

Graphing theory is a broad and diverse field of mathematics; however, this article discusses graphs that can explain spatial relationships. Core to graphs that explain spatial relationships are nodes and edges (Figure 1). Nodes can represent game spaces / rooms, pickups, spawn points and AI pathing nodes. Edges define relationships between nodes.

Figure 1

Figure 2 is a simple molecule consisting of several nodes, linked by edges. In this example, we have defined a set of tokens around the players spawn point. This is a literal depiction of space using a graphing approach. Nodes become linked by edges, and these define the shortest possible distance between the player and other node. The more powerful a token is, the longer the edge should become.

This approach works well for PvP games -- to create a game space with roughly similar distributions of pickups, to achieve game balance. Repeating and rotating a molecule leads to symmetrical distributions throughout the game space. Edges are abstract ways of defining relationships but not necessarily hallways or any other level geometry. To explain this further, we need to look at weighted and directed graphs.

Figure 2

We can manipulate the physical appearance of our edges to help communicate different types of relationships between the nodes. In Figure 3, the edge between nodes A and C is thicker than the rest. If we are using graphing theory to create spaces, and the nodes represent particular game spaces, then the larger edge does not imply a bigger space between the two nodes, but rather a more direct route.

Figure 3

Figure 4 takes our molecule from Figure 3 and uses weighted edges as a guideline to place out level geometry. In this example, heavy weighted edges create a path between nodes A and C that is direct and unimpeded. Alternatively, the thin edge connecting nodes A and B results in a meandering pathway that is complex in nature. This example shows that edges do not depict geometry, but rather the relationship between nodes.

Figure 4

We can further increase the information that an edge communications by adding direction. Figure 5 is an example of a graph that has directed and weighted edges. Figure 5 uses directed and weighted edges to communicate two different ways to get between node A and node B. The thicker edge is more direct than the other. Linking nodes B and C is an indirect one-way gate. The thick edge linking nodes A and C is another one-way gate. The thickness of this edge shows a direct and unimpeded relationship between the nodes.

Figure 5

Nodes and edges can represent nearly any feature of game level design. For example, we could use a system whereby the weight of the lines also tells us about the difficulty of getting between nodes. By using edges to depict vertical space, we could say that node C is the highest point of the map. Node C is then transitive in the sense that it can only be accessed from node B. The one-way direction between nodes B and C might be achieved by having a "jump pad" at node B, pointing towards node C, but not in the opposite direction. It is really at the discretion of the designer and their team to define a key for their particular molecule system.

To further explain the concept of using spatial molecules to create play spaces, let us consider one example molecule and how it should and should not be implemented. The molecule represented in Figure 6 is a simple spatial molecule that defines a linear level progression, suitable for single player type maps. Weighted edges have not been used in this example; however directed edges have been used to create interesting spatial puzzles.

Figure 6

Figure 7 is an example of what not to do with a spatial molecule. The reason to use a molecule-based approach is to free your creative process from thinking in purely spatial terms, and instead think about creating interesting spatial relationships. Although the planar map in Figure 7 does follow the spatial relationships of the molecule, it is a boring, linear space.

There are also a number of other flaws that demonstrate why designing maps from a planar perspective is problematic. First, the linear, room-by-room layout of the map is a direct product of drawing maps out in planar space. When your imaginative space is two-dimensional, your maps will be two-dimensional also. As such, there are no interesting vertical spaces and, more importantly, the objective is not clearly visible from the beginning of the map.

Figure 7

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James Castile
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Don't mind me. I'm still only half way through my first coffee...

A valuable perspective to have in the tool bag. Thank you. :cheers:

Simon Ludgate
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Random thought, but when I saw that map on page 4 of this article, I instantly thought of the arena layout on page 4 of Yureka chapter 204 (a Korean Manhua). I wonder how that fictional layout would survive under your analysis?

Luke McMillan
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That's not one I have ever seen before. I'll take a look and get back to you :)

Raph Koster
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Love love love. :)

As a note on further research, there is a good case to be made for the idea that a given dominion space can have a formal, quantifiable difficulty rating assigned to it based on metrics and completion rates -- in fact, multiple axes of analysis could give playstyle differentiators as well. With that, you could at a minimum get a good way to assess your levels' difficulty, and given sufficient data and some AI programming, even in theory build agents that can tell you that same info without needing to spend playtesters on it.

I am fairly sure some of those ideas, as well as some of those in the article, were presented by Andrew McLennan at GDC a few years ago.

Luke McMillan
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Thanks for the kind words Raph - it means a lot!

I have been taking an interest in some of the 3D engineering applications and how they are used to model and analyze thermal dynamics; specifically things like exhaust flow and heat distributions.

I would really like to try and see if game levels can be analyzed in the same way, basically simulating what would happen under "normal" conditions and leaving the play testers for the emergent stuff :)

I'll be sure to write a follow-up to this if when I figure out how to do this!

Raymond Ortgiesen
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Excellent article. It's refreshing to see talk about the technical aspect of level design rather than just discussing environment art and spatial relations. Both are important but I think you've helped develop an area that is woefully under studied by most level designers.

Daneel Filimonov
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This really is an awesome article! I love the fact that you can visualize the gameplay and flow of a level without wasting valuable time play-testing every single revision when all you need is just a pen and some paper (and some patience)! This has opened a new avenue for me when designing levels, thanks Luke! :)

Kenneth Blaney
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You've played it fast and loose with the math and notation of graph theory, but it is all in service of a larger point aimed at a non-mathematical audience so it is forgivable. If you are interested in fleshing this out more in a significantly more concrete and rigorous way I could recommend a few texts for you. (Teaching math and being strict about people's use of language in math proofs is my day job. Please pardon me if that came off rude in any way.)

I'm not sure if they are aware of it, but the development of Portal 2 heavily depended on the type of level design you are working with here. Specifically, they created individual levels with the idea that certain areas would be linked (either physically or just visually). The advantage they gained by designing
They then used portals to link the areas together. The final step in the level design was to stitch the various pieces together, add bridging geometry where appropriate and remove the portals.

I'm currently working on a project for which I've taken a similar level design strategy as what you mention here. That is, using directed graphs that represent the moment to moment player experience (and not just the level geometry). We too are working in UDK and I've found extensive use of the UTPortal class for this purpose. The directed graphs have been helpful to estimate player performance within a setting thus defining a good baseline for some of the goals and game balance issues well before testing has begun.

Finally, this method lends itself to a fractal-like expansion fairly readily. That is, since each of your nodes is representative of a player experience, it follows that the entire graph is representative of a larger player experience. Those larger player experiences can then, just as easily, be linked in a directed graph. If you were really clever, you could then force a certain consistency to the experience of your game by having the large experience graph be similar to the smaller experience graph which in turn is similar to an instantaneous experience graph. I recently did a study of Cyan's "Myst" along these lines which I would be happy to share with you if you are interested in doing some follow up to this.

This is way too long a response, but you hit upon a combination of subjects I'm intensely interested in. :)

Luke McMillan
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Hi Kenneth, thanks for reply.

In regards to the math - accessibility of these models was really high on my agenda. Plus, I rely on Jesse Schell's 10th rule of what game designers need to know about math and probability :)

What you mention in regards to your UDK project sounds really interesting. One of my students ( who used this model created a recursive space death match map. What I really like about it is for all intensive purposes it is a single, arena space. But as you mentioned, the portal class in UDK allows this space to be much, much more.

I'll add more of a reply when I get a chance to use a real keyboard!

Luke McMillan
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Actually, there was a section from this article which I needed to cut because I couldn't concisely explain it's application. I was discussing node distribution based on Voronoi Diagrams & Delaunay Triangulation. If you know of a good layman's explanation for Delaunay triangulation and how it can be used for better node distribution then I would love to hear your thoughts.

Kenneth Blaney
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If you want to talk about triangulation I'm not sure if there is anyway to get around talking about the circumcircle of a triangle. That said most people will follow you if you tell them that any three points not in a line forms a unique circle (the visual explanation of which would be to show that you can draw two circles with 0 points in common, 1 point in common or 2 points in common, but 3 just won't happen). You might then want to suggest that a designer will want to place nodes and edges that try to obey the Delaunay condition: when you scribe the circumcircles no nodes end up inside of a circle.

This sort of goes in with what you were saying about the edge lengths earlier in the paper which confused me earlier. That is, normally in graph theory we don't care too much about edge lengths, however in computational geometry we do. (More properly, computational geometry falls into computer science, but we are being interdisciplinary here so I'll step a little bit out of my comfort zone.) If the length of an edge represents something like the difficulty of an area in the level setting up a graph this way will help balance the experience. A graph that obeys the Delaunay condition will have a property that all of its triangles will tend towards equilateral (it maximizes the minimum angle of the triangles and the maximum value the smallest angle can have is 60 degrees, which only happens in the equilateral case) and so avoid stretched triangles. Avoiding stretched triangles minimizes long edges, which in turn minimizes inequity in paths through a level. If an area of the game is too hard and optional, players will quickly learn to avoid that area unless you make it worth the time. (For instance, speed runners of Super Metroid always avoid Spore Spawn because killing it takes a long time and it gives a useless reward. So that miniboss would be represented by a long edge and a fairly unattractive node if we were to create a graph of Super Metroid.) That said, there is nothing to prevent a good designer from breaking this rule and creating a truly grueling set of paths if the game design allows for it. (Example there could be the purely optional Red Knight in the first level of Demon's Souls.)

As I'm sure you know because you grouped them, calculating a Voronoi diagram is functionally similar to confirming that a graph obeys Delaunay. That is, connect the centers of the circumcircles you drew and then, where necessary, bisect the outer edge of the triangle (much easier to explain with a drawing). Now, the only thing that springs to mind about how a Voronoi diagram would be useful for level design would be it would start to inform you where level geometry might need to go to restrict player movement and more clearly define the zones of play from your Half-Life 2 example. You can see that Valve did this to an extent by using bridges, tunnels, curved roads and water giving you your figure 12 from figure 11. That said, this feels a little more like something you'd end up doing to fix problems with the pacing of the map after the fact.

I'd be interested to hear more about how you use Voronoi diagrams in level design. I really don't understand it at a deep enough level to do anything great with it.

Elif Bugdaycioglu
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I'm not designing a PvP game right now, but you gave me a whole new vision to use a modified molecular design in a single player adventure game! Thanks for this thorough article. :)

David OConnor
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Very interesting and worthwhile, thank you for sharing Luke! :)

Frank Washburn
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Excellent article. Do you have any caveats for designing level flow for a game that is literally in 2D space, like a metroidvania? Trying to keep in mind verticality is obviously very helpful for the in-depth examples you've listed, but mentally I'm having a difficult time trying to avoid any of these "linear, boring level traps" that you illustrated. Any notable examples of games and levels in 2D games that would you say use this kind of methodology in a 2D plane?

Raymond Ortgiesen
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It sounds like using nodes to represent major areas of the world, then distributing nodes that represent each of the types of pick ups you get and mapping where those allow you to go and where you can't go would be really helpful for establishing the flow of that type of game. In a way the items functions as gates or keys and you could represent that using the molecule.

Luke McMillan
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Thanks for the comments Frank. In one of the lectures I do about this topic, I apply the methodology to Sonic the Hedgehog II from an analytical perspective to examine the level layout and the notion of public vs. private spaces. I have the materials in lecture format so it is no problem for me to put together a quick blog post.

In regards to games which do this well in 2D space one comes immediately to mind however I haven't played it in about 10 years so I might be remembering it to be better than it was! The game is Another World. It was a really slow paced platformer with rotoscoped animations. The game makes good use of the space by not using scrolling. Instead, each section of the level is represented in a new window. The game space is classified as "Adjacent spaces displayed one room at a time" (Mark Wolf - The Medium of the Video Game).

What is cool about Another world is that each screen is not a node, but rather contains several spatial nodes. For instance, you might be in one screen with two levels, but there is no way to get to the other level in the screen you are currently in. This means that you will need to go to other screens and figure out the spatial relationships for your self. The game play is largely based around this concept, especially in the later levels.

Raymond is also on the money. According to Koster, humans are pattern recognition monsters! We get great pleasure from discovering patterns and using them to preempt risk & reward scenarios. This is why I really like the molecule approach because it immediately lends itself to this human desire.

Frank Washburn
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Thanks for the thoughtful response Luke. I haven't heard of Another World but I will surely have to look it up. If you were to put together a blog post about public vs. private spaces I would be eager to see what you came up with! Looking forward to any more of your analyses! :)

Axel Cholewa
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Excellent article on a very interesting topic!

But I have one little problem with the AOEs and their possible overlap in figs. 10+. The AOE you defined to represent "intensity of play". Meaning, the bigger the circle, the more intense the challenge. But this means that, when you combine it with a physical map like in the HF2 maps, the overlap of AOE doesn't mean anything. The radius of the circle is a measure of play intensity, while the distance between the circles is a measure of physical distance. You can just draw smaller circles without changing their meaning, because you can anyway only compare the AOEs with each other, but not with the edges.

Of course, if you use the AOE (as you probably did here) to show the actual, physical areas of activity (maybe that would be a good name for it: AOA), then your method works.

I hope I made myself clear :)

Axel Cholewa
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Oh, and since your using an abstract method, you can also use this in completely different areas. For example, a game like FTL doesn't have much level design, but it contains challenges which can be represented by graphs. Teachers could use a method like this to design exercises for their students, with the edges representing the time it takes for a student to go from one problem to next (if they have a choice).

Robert Casey
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Great article.

Lewis Wakeford
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Awesome article. I wonder, has anyone tried incorporating these theories into procedural level generation?

Nick Harris
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Exactly my thoughts on reading this article. However, I feel that this "London Underground Map" has no relevance to the automatic creation of interesting maps for a first-person shooter because it fails to represent sight-lines. It really is only a refinement of the old adventure games drawn out on grid, or hex paper for late night Dungeons & Dragons sessions. Leaning out from behind minimal cover for a rapid expert headshot just isn't simply represented here.

The article did make me think about the problem of procedurally generated combat maps though and my initial notion involves pitting AI driven teams against each other on a variety of maps generated for their atmosphere and credibility - i.e. the algorithm's sole concern is the creation of, say, multiple war-torn Beirut-style maps. Due to the inevitable asymmetry to these (as a bunch of ruined streets wouldn't happen to have convenient reflectional symmetry for the purposes of deathmatch balance), you would have to limit your battles to Attrition, Bomb, Conquest, Duel, or Espionage with one team being the attackers and the other attempting to hold onto their defensive position. The game profiler could adjust the equipment, armament, transportation and number of players independently for each team in order to rebalance the game if the defenders used their "hill" (or somesuch topographical, or topological, advantage to their strategic advantage), it could make the attackers more numerous, or allow for fewer 'tickets' on the defenders side, or none: letting them get permanently eliminated by only giving them one life, whilst the attackers drew on a substantial pool of respawn tickets, Alamo style, some of which could be taken by deceased defenders who were tired of being neutral observers.

Another, perhaps more pragmatic course is to leverage User Generated Content, something which has already proven itself with the Halo 3 Forge and over seven million maps made for Little Big Planet.

Lewis Wakeford
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Well no they don't take complex geometry into consideration, but I was thinking they would be the very starting point of generation. So you build a rough plan for the level first with molecule design which just outlines how intense certain areas will be and ensures they are enough calm areas in between. It also specifies the minimum number of exits/entrances to an area and in which general direction they are. Then a different algorithm and/or prefabs are used to generate those areas as mostly self contained challenges.

Max Quin
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Hi Luke this is a very late post but you mentioned this.

"When your imaginative space is two-dimensional, your maps will be two-dimensional also. As such, there are no interesting vertical spaces and, more importantly, the objective is not clearly visible from the beginning of the map."

I'm wondering if you could extrapolate on that. Especially concerning something like a 2D Game, and how that can be made spatially interesting given it is literally two-dimensional. Also regarding the visibility of objectives, what is the relavance and importance of this? I'd like to know more I'm intrigued.

Also I find it funny that designers arive at similair ideas, I'd always used graph theory and this idea of connections aswell but never formalized it in anyway. Also that's partially because Valve utilized the concept in many of their maps, they called it a 'Bounce' I believe in HL2 where one area connects back to another and creates interesting and flowing spatial arrangements (which makes games more fun).

I think by far the best use of connectivity is the mod Minerva: Metastasis, it's no wonder that guy was hired as a Level Designer by Valve.

Sorry for blathering on thank you very much for your insightful article and your time.