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I am trying to develop a peer rated IQ network which can be implemented in the real world. To summarise the problem statement, consider an ecosystem of 100 people. The aim is to rank the 100 people in decreasing order of IQ.

Data is available in the following format. Consider a 100x100 table. Each entry A[i][j] in the table denotes what the i'th person thinks of j'th person's IQ. There is no entry for A[i][i]. Also, there may be a null entry in some A[i][j], if the two people don't know each other.

Using this data, I need to sort the 100 people in decreasing order of IQ.

[PS: There is no correct answer to the problem. I am looking for different approaches.]

Vishal Patil
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1 Answers1

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Some reflections...

You may know that the ranking a certain individual has done of the 100 persons is called a permutation of the set $1,2,...100$.

An important idea is to express how two "permutations" differ.

There are different ways to express a distance between two different permutations $\pi_1$ and $\pi_2$.

One of the most interesting is Cayley distance which is the minimal number of transpositions (elementary exchanges between the values at two different positions) one has to apply to $\pi_1$ in order to obtain $\pi_2$. See this.

Once you have obtained all the mutual distances between the different "rankings", you should observe ideally a cluster of coherent answers, and outliers that should hopefully be eliminated (people who are not reliable...). Afterwhile, a mean must be found to "average" the good ones...

Another track: Spearman rank correlation.

The Cayley distance can be computed using the decomposition of a permutation into a product of cycles ; see here.

An interesting text.

Jean Marie
  • 81,803
  • I will get back in a few days. I am reading more on it. I was wondering if this is a scalable solution - say if instead of 100 people we have 1 million people.

    I was also trying to modify the page-rank algorithm to fit it in the context of the above problem.

     – Vishal Patil Feb 11 '21 at 15:39