Personal growth and discrete harmonic functions
"You are the average of the five people you spend the most time with."
A Google search says this quote is by Jim Rohn. I think other people have said similar things. I've heard it quoted many times. The implication is usually that you can improve your life by hanging around better people.
Here are three things it makes me think of.
- It sounds approximately true.
- It can't be literally true.
- It reminds me of harmonic functions.
There are numbers to back up the assertion that things like your income and even your weight approximately match the average of the people around you. Cause and effect go both ways: people like to hang out with people like themselves, and we become like the people we hang around.
It's an aphorism, not meant to be taken literally. But a moment's thought shows that it can't be literally true for everybody. In any social network, someone has the most money, for example. That person's net worth cannot be the average of the net worth of others in the group, unless everyone has the exact same net worth. The same applies to the poorest person in the network.
The reason I say that this reminds me of harmonic functions is the mean value theorem. If a function satisfies Laplace's equation in some region, at any point in the interior of the region, the value of the function equals the average of the function over a spherical region centered at the point. But this is only true in the interior. On the boundary, you might have a maximum or minimum. If the boundary is compact, you will have a maximum and a minimum, provided the function extends continuously to its boundary.
I think of the continuous case first because I spent years thinking about such things. But there's a discrete analog of harmonic functions that applies directly to the example above. If you have some network, such as a social network, and assign number to each node, you have a discrete harmonic function if the value at every node is the average of the values at its neighboring nodes. For a finite network, a function cannot be harmonic at every point unless it is constant, for reasons given above. But a function could be harmonic at all but two nodes of a graph, or approximately harmonic at all nodes.
Related posts:
- Discrete harmonic functions
- Boundary conditions are the hard part
- Life lessons from differential equations