*(In which I am inspired by an offhand statement my psychology professor makes, prompting me to daydream for a while about graph theory instead of paying attention to the lecture like I should have been doing.)*

You remember that thing from the end of the last post that looked like a screenshot from a video game of some sort? It looked like this:

Yeah, remember that? That's not what this post is about.

Instead, I'm going to tell a story. It starts with me in psychology class.

We were learning about the factors that cause friendships to form. One of them, unsurprisingly, is shared opinions and interests. It seems that if person X and person Y feel the same way about issue Z, they are more likely to be friends, and if their opinions differ, they are less likely to be friends. (Sounds simple enough so far.)

My psychology professor gave an example of this which came from a Seinfeld episode. I don't remember the names of the characters involved (and I never watched Seinfeld, so I don't remember the episode), but there's a female main character who gets a boyfriend, and finds out after that he likes hunting. She does not like hunting, so the relationship between them looks like this:

The blue lines represent positive relationships, and the red line represents a negative relationship.

According to psychologists who spend a lot of time thinking about these things, people's desire to be consistent makes a situation like the one above into an unstable situation. It is likely that one of the following three things will happen: the main character will learn to like hunting, the boyfriend will learn to dislike it, or the two of them will stop liking each other. Here are depictions of these three possible endings:

Of course, stronger forces (such as the power of true love) can and do hold unstable situations like this together, but, in the absence of outside forces, things like the first diagram tend not to persist--they tend to resolve themselves in one of the above three ways.

It works not just for two people and an abstract idea, but also for three people. In this case, the rules that people follow can be summarized like this: "the enemy of my enemy is my friend" and "the friend of my friend is my friend." If there's a place where this rule does not apply, the situation is unstable.

It might not be clear at first why the one on the left (three red lines of animosity) is unstable. It's because the enemy of my enemy is supposed to be my friend. Here, though, the enemy of my enemy is also my enemy. There's untapped potential for an alliance to form around a mutual foe.

So here's where the math bit comes in. My psychology professor said: "There's a neat trick to check whether a situation like this is stable: write a positive sign next to the positive relationships and a negative sign next to the negative relationships. Multiply the three signs together. If the result is positive, it's stable, and if it's negative, it's not."

This is the sort of thing that makes a math major sit up and pay attention.

It's true, of course, at least when you have just three people/things. The two unstable situations are above. Multiply the signs on the first one and you get -1*-1*-1 = -1. In the second situation, 1*1*-1 = -1.

In the other possible situations (the stable ones) multiplying the signs yields positive 1.

What I was wondering was: what about larger graphs?

Well, consider this graph:

This one is unstable. A and B like each other, B and C like each other... but A can't stand C, and C gets irritable just thinking about A. Liking should be transitive- in a completely stable world, at least. So this isn't stable. But multiply all the signs, and you get 1.

The multiply-the-signs thing was a neat trick for degree-3 graphs, but it doesn't work for larger-scale situations, when the graph has more vertices (vertices are the people or ideas in the graph).

So how *do* we tell if a situation with multiple vertices is stable? Well, in order to follow the "enemy of my enemy" rule and the "friend of my friend" rule, no three vertices may have a love-love-hate relationship or a hate-hate-hate relationship, so neither of the two forbidden unstable triangles from above may even form *part* of the graph of a stable situation.

So what sort of graph doesn't include either of those two types of triangles?

Well, one thing that must be true is that friends come in clumps, where everyone in the clump is friends with everyone else in the clump. Otherwise the "friend of my friend is my friend" rule is violated.

Also, if person A is in a clump, and person B is not in the clump, person A cannot be friends with person B, which is sadly reminiscent of the way middle school works. This is not a sadistic extra rule I'm imposing- it follows from the other rules. Here's how: because person B is not in the clump, there must be some person C in the clump who is enemies with person B (otherwise everyone in the clump is friends with person B and person B is in the clump. But we know person B isn't in the clump.) A likes C, since they are in the same clump. So we have A likes C, C hates B, B likes A. This is not stable.

So, if everything is stable, everybody is divided into clumps which all hate each other, but *within* which everyone is friends.

(Remember, ideas or actions like "hunting" can be part of a clump too, not just people. I'm not sure what the relationship between two *ideas* means, but I guess two ideas can have a positive relationship if society associates them with each other- like being smart and having poor eyesight.)

*How many* clumps can there be, though?

There actually can't be more than 3, since if there are, we can pick one person from each clump, and they all must hate each other, which is unstable (the enemy of my enemy is supposed to be my friend).

So there are actually only two possible stable states of this system:

- Everyone is friends with everyone else, and shares exactly the same opinions on everything.
- Everyone is divided into two opposing camps, which internally agree on everything but mutually detest each other and cannot come to consensus on anything.

If you want a visual, the second option looks like this (the red portion is what graph theorists call a bipartite graph):

Now what does this mean?

The first option sounds either like Utopia or like a horrible totalitarian state, depending on how you read it. They both kind of fit the description. Unfortunately, this superficial similarity between Utopia and totalitarian states has occasionally lead people to mistake one for the other, with disastrous results.

The second one sounds kind of like an unflattering description of our own two-party system. Of course, the US is not *really* divided into two parties which internally agree on everything but cannot agree with each other on anything at all. There *is* room for dissent within parties, and there *is* cooperation and compromise between parties.

There's just less of it than we'd like.

But as long as:

- there are only two acceptable ways to view an issue - for or against - and
- people try to be consistent by continually agreeing with their friends and taking the opposite stance from their enemies, and
- people try not to believe or do things which the public opinion deems incompatible, like owning a rifle and driving a Prius...

...then reality will, depressingly, probably look a lot like the graph above.

If you don't like that, then you must fight the system- by being ambivalent and inconsistent!

Okay, that's not actually what I believe. I believe that people shouldn't be afraid to have opinions which *other people* will call ambivalent, or to do or believe things which *society* will call inconsistent. Because, really, the ideas of the world don't come sorted into exactly two groups, one of which you must accept and the other of which you must reject.

But the idea of an ambivalent rebel *is* kind of funny.

Anyway, that occupied my thoughts for a while in psychology class. Eventually I decided to be a good student and turn my attention back to the lecture.

But I still wonder: what happens if you *do* add an ambivalence relationship? Maybe it would be a purple line, or a grey line, or something. Then we could make rules governing it, like "If more than half of my friends are ambivalent about something, I am ambivalent about it too" or "If my enemy is ambivalent about something, I make a point of having a definite opinion about it, just to annoy my enemy." What would the stable graphs look like then?

## Comments

I really liked this post, it made more sense seeing the visuals than when you tried to explain it to me last week :)

So for ambivalence, what would the corresponding math look like? For instance, red yields a positive number, and blue a negative one. Would purple (ambivalence) be 0 and just mess everything up because it would just zero the entire 'equation'? Then to un-zero it, you would have to have another relationship that would multiply by 1/0 (aka infinity) to yield one again. What would the inverse of ambivalence be? Apathy?

I actually think apathy would be the 0 and ambivalence the infinity in my system. Then for it to be adequately balanced you would just have to have an equal number of ambivalent and apathetic people as well as an even number of negative relationships. Maybe I'm thinking about this too much, but it was certainly thought-provoking!

Fascinating blog post, Collin! I was a double major in bio and music (class of '00), and your post makes me wish I'd taken more math. If you do figure out stable solutions for graphs containing ambivalence, it could have some ramifications in fields beyond social network theory-- ecology, for one. Keep us posted!

Ambivalent rebels of the world unite!

(With probability p, where 0<p<1)

Becca ~ The multiplication rule didn't really scale up past degree-3 graphs, even when there were just two possible stances people were allowed to take toward each other and toward ideas. A negative value for the product of all the signs just means there is an odd number of negative relationships, and a positive value means there is an even number. But it is interesting to think about weird variations on the multiplication rule, like your zero and infinity suggestion. (What if we had two other edge colors which have values of i and -i?)

Susan ~ If this kind of math shows up in ecology, then I am SUPER interested in ecology. Not that I wasn't before, mind you. Hmmm... they're offering a class in population ecology next term. Maybe I should look into this.

Silva ~ YEAH!!! ...I mean, MAYBE!!!

I approve! Your tangentially related discourse is quite fascinating. :D

A bit interesting, except the connection to politics doesn't really hold water. Obviously, there are many other variables at work than the simple concept of friendship, since it also deals not just with common likes but also with common shared goals towards obtaining power, in the context of politics. Also, this entire theory completely makes the assumption, everywhere, that people with contrasting opinions cannot be friends. Which, regardless of general prevalence, is obviously not true in every situation, so it's worth mentioning. This simple fact actually completely complicates and essentially deconstructs this entire theory of relationships because it shows that it fails to take into account the true complexity of human interaction, which (at this point) cannot even be reliably approximated in every context with simple observations or math - math even farther abstracted into theory because it is likely incapable of a "proof". Yay for friendship though.

Good point, Stephen. I'm definitely not claiming that friendship or politics actually works like this. It doesn't. This is a ridiculously simplified

modelof friendship and politics. If we tried to account for every nuance of of human relationships, we would wind up with an unwieldy and complex model that still wouldn't even begin to approach the complexity of real human relationships.That's the beauty of mathematical modeling, though - we simplify our assumptions, and end up with a manageable system that we can actually study. The model isn't perfect, certainly, but its behavior may nevertheless tell us something about the way the real-world system works.

Also, this model doesn't assume that people with contrasting opinions can't be friends. It just assumes that that situation is less "stable" than the situation where they feel the same way about an issue, or another person. When we introduce time into the model, the unstable triangles will be the ones most likely to change.

As for the connection of this model of friendship to politics, you may be right that it doesn't hold water, but there are some notable mathematicians who think that it DOES. Steven Strogatz from Princeton actually studies this model too, and he recently published a paper where he used a version of this model (with time involved) to "predict" the collapse of the balance of powers going in to World War II. The model actually correctly predicted which countries would join the Allies and Axis Powers, with only one mistake - it predicted that Portugal would join the Axis. I'll leave it to you to decide whether or not that is impressive, or useful. =)