Analyze mixture of data attributes

Question

When I have a list of persons (nodes) with the attributes:

REVENUE | LINKS

I can search for the people with most links and the people with most revenue. But when I want to know who has the highest "mixture" of both attribues: How can I do this?

That is: We have a ramdon variable $X_1$ for revenue and a random variable $X_2$ for links. I can order the data $\hat{X_1}$ and I can order the data $\hat{X_2}$. But I do not know how to order the list $(\hat{X_1}|\hat{X_2})$.

I have read that the Principal component analysis can be helpful, that is to say, that one can work with eigenvalues etc. to obtain a result. Nonetheless I do not know which steps I would need to follow.

I am new in that area and would like to know which is the best model for that purpose.

Thank you in advance for your help.

Answer 1

Another way of looking at you question is:

given two points $\vec p,\vec q$ in the plane $\mathbb{R}^2$, is there a way of deciding which one is greater or equal then the other? The answer is tricky.

Formally, there is no way of defining a relation such that $\vec p$ is either $>$, $=$ or $<$ then $\vec q$.

But this is not a new problem. Despite this fact, what is commonly done to solve this kind of issues is to exploit functions like norms and say that $\vec p$ is 'bigger' than $\vec q$ if $||\vec p||_2\geq||\vec q||_2$.

Other example of function like the Euclidean norm above might be: $$f(\vec p)=p_1+\alpha\cdot p_2$$
where you can weight the relevance of the two variables according to your purpose.

In the end, there is no common way, it is very dependent on the task, and it's up to the designer decide.