In weighed random graph where the edge weight is restricted to $[0,1]$, what are the usual assumptions of edge weight distribution?

Question

In classic ER random graph, the edge distribution is Bernoulli. Given a weighted random graph where the edge weight is restricted in $[0,1]$, is there a canonical assumption of the weight distribution in researches? If so, it would be very helpful if you could also help provide some reference.

Answer 1

The only choices of weight distributions that I would consider natural (choices that do not need further motivation) are the uniform distribution on $[0,1]$, and the exponential distribution (but the exponential is not bounded, which is the case you're interested in).

The idea is that a research problem is more interesting if it is simple to describe. If you can only prove your result for weights following the $\text{Beta}(3,17)$ distribution, that is less interesting, unless you motivate it:

• Maybe there's an application for which this distribution is natural.
• Maybe this distribution gives strange and unexpected results, and other distributions give boring results.

The simplest distribution on $[0,1]$ to describe is the uniform, so it should be the default.

The motivation for considering the exponential distribution is from results such as One, Two and Three Times $\frac{\log n}{n}$ for Paths in a Complete Graph with Random Weights by Svante Janson. This paper considers a large class of weight distributions (which includes the uniform) and proves that all of them behave the same, because when you take the minimum of many random variables (under some assumptions such as the PDF is approximately linear near $0$) the result is exponentially distributed in the limit.

So in other similar problems, we can assume that the weight distribution is exponential without much loss of generality. (I expect that most problems which involve finding minimum-weight whatevers in the graph will behave like this.)

Of course, a result that works for many distributions is stronger than a result which only holds for one specific distribution.