I had a free choice of topics for today! Nobody had a suggestion for the letter ‘N’, so, I’ll take one of my own. If you did put in a suggestion, I apologize; I somehow missed the comment in which you did. I’ll try to do better in future.
Nearest Neighbor Model.
Why are restaurants noisy?
It’s one of those things I wondered while at a noisy restaurant. I have heard it is because restauranteurs believe patrons buy more, and more expensive stuff, in a noisy place. I don’t know that I have heard this correctly, nor that what I heard was correct. I’ll leave it to people who work that end of restaurants to say. But I wondered idly whether mathematics could answer why.
It’s easy to form a rough model. Suppose I want my brilliant words to be heard by the delightful people at my table. Then I have to be louder, to them, than the background noise is. Fine. I don’t like talking loudly. My normal voice is soft enough even I have a hard time making it out. And I’ll drop the ends of sentences when I feel like I’ve said all the interesting parts of them. But I can overcome my instinct if I must.
The trouble comes from other people thinking of themselves the way I think of myself. They want to be heard over how loud I have been. And there’s no convincing them they’re wrong. If there’s bunches of tables near one another, we’re going to have trouble. We’ll each by talking loud enough to drown one another out, until the whole place is a racket. If we’re close enough together, that is. If the tables around mine are empty, chances are my normal voice is enough for the cause. If they’re not, we might have trouble.
So this inspires a model. The restaurant is a space. The tables are set positions, points inside it. Each table is making some volume of noise. Each table is trying to be louder than the background noise. At least until the people at the table reach the limits of their screaming. Or decide they can’t talk, they’ll just eat and go somewhere pleasant.
Making calculations on this demands some more work. Some is obvious: how do you represent “quiet” and “loud”? Some is harder: how far do voices carry? Grant that a loud table is still loud if you’re near it. How far away before it doesn’t sound loud? How far away before you can’t hear it anyway? Imagine a dining room that’s 100 miles long. There’s no possible party at one end that could ever be heard at the other. Never mind that a 100-mile-long restaurant would be absurd. It shows that the limits of people’s voices are a thing we have to consider.
There are many ways to model this distance effect. A realistic one would fall off with distance, sure. But it would also allow for echoes and absorption by the walls, and by other patrons, and maybe by restaurant decor. This would take forever to get answers from, but if done right it would get very good answers. A simpler model would give answers less fitted to your actual restaurant. But the answers may be close enough, and let you understand the system. And may be simple enough that you can get answers quickly. Maybe even by hand.
And so I come to the “nearest neighbor model”. The common English meaning of the words suggest what it’s about. We get it from models, like my restaurant noise problem. It’s made of a bunch of points that have some value. For my problem, tables and their noise level. And that value affects stuff in some region around these points.
In the “nearest neighbor model”, each point directly affects only its nearest neighbors. Saying which is the nearest neighbor is easy if the points are arranged in some regular grid. If they’re evenly spaced points on a line, say. Or a square grid. Or a triangular grid. If the points are in some other pattern, you need to think about what the nearest neighbors are. This is why people working in neighbor-nearness problems get paid the big money.
Suppose I use a nearest neighbor model for my restaurant problem. In this, I pretend the only background noise at my table is that of the people the next table over, in each direction. Two tables over? Nope. I don’t hear them at my table. I do get an indirect effect. Two tables over affects the table that’s between mine and theirs. But vice-versa, too. The table that’s 100 miles away can’t affect me directly, but it can affect a table in-between it and me. And that in-between table can affect the next one closer to me, and so on. The effect is attenuated, yes. Shouldn’t it be, if we’re looking at something farther away?
This sort of model is easy to work with numerically. I’m inclined toward problems that work numerically. Analytically … well, it can be easy. It can be hard. There’s a one-dimensional version of this problem, a bunch of evenly-spaced sites on an infinitely long line. If each site is limited to one of exactly two values, the problem becomes easy enough that freshman physics majors can solve it exactly. They don’t, not the first time out. This is because it requires recognizing a trigonometry trick that they don’t realize would be relevant. But once they know the trick, they agree it’s easy, when they go back two years later and look at it again. It just takes familiarity.
This comes up in thermodynamics, because it makes a nice model for how ferromagnetism can work. More realistic problems, like, two-dimensional grids? … That’s harder to solve exactly. Can be done, though not by undergraduates. Three-dimensional can’t, last time I looked. Weirdly, four-dimensional can. You expect problems to only get harder with more dimensions of space, and then you get a surprise like that.
The nearest-neighbor-model is a first choice. It’s hardly the only one. If I told you there were a next-nearest-neighbor model, what would you suppose it was? Yeah, you’d be right. As long as you supposed it was “things are affected by the nearest and the next-nearest neighbors”. Mathematicians have heard of loopholes too, you know.
As for my restaurant model? … I never actually modelled it. I did think about the model. I concluded my model wasn’t different enough from ferromagnetism models to need me to study it more. I might be mistaken. There may be interesting weird effects caused by the facts of restaurants. That restaurants are pretty small things. That they can have echo-y walls and ceilings. That they can have sound-absorbing things like partial walls or plants. Perhaps I gave up too easily when I thought I knew the answer. Some of my idle thoughts end up too idle.
I should have my next Fall 2018 Mathematics A-To-Z post on Tuesday. It’ll be available at this link, as are the rest of these glossary posts.