The degrees of freedom of a higher order network are not computed correctly when the data is not complete

[user1246638]

I made a few corrections to the computation of the degrees of freedom of a higher order network, so that it works now, if the size of the data is large enough. If it isn't, the degrees of freedom will be overestimated.

The calculation goes as follows: we need to compute the number of links of a higher order network, and the number of nodes with a non-zero out-degree. The first, we infer from the first order network (sum of elements of the power of the adjacency matrix). The second we calculate by iterating through higher order nodes, and counting the ones with the non-zero out-degree.

The degrees of freedom are the difference of the two numbers:
< dof > = < number of higher-order links > - < number of higher order non-zero out-degree nodes >

Since the first order network is recovered whole with not much data, the first quantity is determined correctly. The second quantity, however, is underestimated when the data is insufficient, because the higher-order network is not complete.

This means that we overestimate the degrees of freedom of an order when we do not have enough data for the order. It finds the correct value when the data is large enough to visit all the nodes, however.