Here we show, how paths from order 3 to order n  can be constructed in complete networks.

We use the Software program Mathematica for it.

We consider the numbers of the points in the network to be a mathematical set.

• if the network has n points, then the set is  M= { 1,2,3, …  n }
• each set has 2^n  sub sets, including the set itself and the void set.
•  The set M  with 4 points M = {1,2,3,4} has then 16 subsets in total, but we are interested only in subsets, which have 3 or more elements, up to n elements, because a path needs to have minimal 3 points or more.

Step 1: 

In Mathematica we can use the function:

KSubsets[ {1,2,3,4},3]  and we get the 4 subsets:

• {1,2,3}, {1,2,4},{1,3,4},{2,3,4}
• and we have to add the set itself, because each set is a subset of itself

Because a set has only different elements, feedback loops are excluded automatically by this approach.

 Step 2:

In the second step we must calculate  for each subset all permutations, for which we use the function Permutations in Mathematica:

Permutations [1,2,3]

Permutations[1,2,4]

Permutations[1,3,4]

Permutations[2,3,4]

Permutations[1,2,3,4]

We check, if these results agree with our  earlier  calculations,  in the table below:

and see, that a complete network of order n=4   has 24 paths of order 3 and 24 paths of order 4

With the help of Mathematica it is no problem, to calculate the paths of almost any order. The limit is only the RAM of the used Computer.

Networks of order higher than about 7 and especially all the paths in them, can only be analyzed with the help of Computer-Softwareprograms.

But we must check of course, if the calculations are correct, using networks of lower order, which we still can analyze with manual calculations