What's a good weighting function?

Question

I'm trying to perform some calculations on a non-directed, cyclic, weighted graph, and I'm looking for a good function to calculate an aggregate weight.

Each edge has a distance value in the range [1,∞). The algorithm should give greater importance to lower distances (it should be monotonically decreasing), and it should assign the value 0 for the distance ∞.

My first instinct was simply 1/d, which meets both of those requirements. (Well, technically 1/∞ is undefined, but programmers tend to let that one slide more easily than do mathematicians.) The problem with 1/d is that the function cares a lot more about the difference between 1/1 and 1/2 than the difference between 1/34 and 1/35. I'd like to even that out a bit more. I could use √(1/d) or ∛(1/d) or even ∜(1/d), but I feel like I'm missing out on a whole class of possibilities. Any suggestions?

(I thought of ln(1/d), but that goes to -∞ as d goes to ∞, and I can't think of a good way to push that up to 0.)

Later:

I forgot a requirement: w(1) must be 1. (This doesn't invalidate the existing answers; a multiplicative constant is fine.)

Answer 1

perhaps:

exp(-d)

edit: something along the lines of

exp(k(1-d)), k real

will fit your extra requirement (I'm sure you knew that but what the hey).

Answer 2

How about 1/ln (d + k)?

Answer 3

How about this?

w(d) = (1 + k)/(d + k) for some large k

d = 2 + k would be the place where w(d) = 1/2

Answer 4

Some of the above answers are versions of a Gaussian distribution which I agree is a good choice. The Gaussian or normal distribution can be found often in nature. It is a B-Spline basis function of order-infinity.

One drawback to using it as a blending function is its infinite support requires more calculations than a finite blending function. A blend is found as a summation of product series. In practice the summation may stop when the next term is less than a tolerance.

If possible form a static table to hold discrete Gaussian function values since calculating the values is computationally expensive. Interpolate table values if needed.

Answer 5

It seems you are in effect looking for a linear decrease, something along the lines of infinity - d. Obviously this solution is garbage, but since you are probably not using a arbitrary precision data type for the distance, you could use yourDatatype.MaxValue - d to get a linear decreasing function for this.

In fact you might consider using (yourDatatype.MaxValue - d) + 1 you are using doubles, because you could then assign the weight of 0 if your distance is "infinity" (since doubles actually have a value for that.)

Of course you still have to consider implementation details like w(d) = double.infinity or w(d) = integer.MaxValue, but these should be easy to spot if you know the actual data types you are using ;)