Displaying pagerank is quite complex, since most end users don't know what it is and how it's defined.
Furthermore, Pagerank follows a power-law, which is inherently hard for humans to understand.

An easier way to understand the pagerank p of an entity is to display the % of the network it is part of. Something like "entity is part of the top 1%". However sometimes we wish to display the pagerank as a 0-100 score. The linear mapping $f(p) = 100p$ is really really poor to understand for humans.

Considering the power law model with exponent b and a graph of N nodes, the highest j-th node by pagerank has

$p(j) = \frac{(1-b)}{N} \cdot N^b \cdot j^{-b}$

by inverting the formula one get the position of a node with pagerank p

$j(p) = N \cdot (\frac{(1-b)}{Np})^{1/b}$

and the % of the network with pagerank lower than p is simply

$l(p) = \frac{N - j(p)}{N} = 1-(\frac{1-b}{Np})^{1/b}$

To remove the singularity in 0 we add to the denominator $(1-b)$

$F(p) = 1-(\frac{1-b}{Np + (1-b)})^{1/b}$

This way $F(0) = 0$

This is close, but the values are too skewed towards the high scores. We can lower the exponent to something like $a = 0.3$ or $a = 0.4$ to make them feel more "natural".

// Score returns a "natural" score from 0-100 given the following inputs:
// - pagerank: the pagerank of the entity
// - nodes: the number of nodes in the graph
export const Score = (pagerank, nodes) => {
  const b = 0.76
  const a = 0.38     // curvature control
  const C = 1 - b    // base constant

  const denom = nodes * pagerank + C
  const value = 1 - (C / denom) ** a
  return 100 * value
}