Is the Hop Plot a Logistic Distribution?

The hop plot is a visualization of the distribution of pairwise distances in a network. It shows the number of nodes one can reach (on average) by going N steps in a network.  Here’s an example for the US electrical power grid network:

The X axis shows the numbers of steps taken and the Y axis shows the number of nodes reached, as a percentage.

What kind of distribution do these hop plots follow?  Maybe a normal distribution? A logistic distribution?  We could do all kinds of statistical tests to find out, but I want to do something different. I want to answer that question visually. Therefore, we’re going to use different ways of drawing the Y axis of the plot that correspond to each distribution. Let me explain: If we apply the inverse cumulative distribution function to the values on the Y axis, the plot should show a straight line if the values follows that distribution.  Since the values on the Y axis are between 0 and 1 and have the shape of a sigmoid function (see Wikipedia), we’re going to try out the various sigmoid functions that we know.

(1) Normal distribution – Inverse error function

The cumulative distribution function of the normal distribution is the error function. Let’s try it:

Hmm… does not quite look like a line.

(2) Logistic distribution

The cumulative distribution function of the logistic distribution is the logistic function:

Not that good either.

(3) The tangent sigmoid

We’ll use the arctangent function as our (scaled) cumulative distribution function:

(4) Hyperbolic tangent

We’ll use the hyperbolic tangent as our (scaled) cumulative distribution function:

This too does not fit at all.

Conclusion

In conclusion, none of the functions fit. Therefore, the distances in the US power grid do not follow any of the simple sigmoid models.

Two notes are in order though:

  • A plot is not a statistical test. You should do an actual statistic test when you write a paper!
  • Does anyone known what the usual network models (Erdős–Rényi, Barabási–Albert, Watts–Strogatz, etc.) result in?

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References

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