Network entropy

Network entropy

If networks are indeed created spontaneously, it seems to warrant that they will tend to reach a state of equilibrium. But what is “equilibrium” in a network? If we try to relate it to Planck’s formula, we can conclude that in equilibrium, the distribution of the links in the junctions will give the largest number of different ways to construct the network.

Paradoxically, the asymmetry of the links simplifies the mathematical calculations required to calculate the entropy of networks. Let us say that we have a network of N persons, with P asymmetrical links. The essence of a one-way link is that one person’s connection to others is not influenced by the number of links that they have to him. In other words, one might say that a unilateral link is associated to a specific person the same way that a particle belongs to a box. To calculate the distribution of links in equilibrium, we have to find the distribution of the links among the junctions that maximizes Shannon’s entropy. The obtained result is a generalization of Benford’s Law, which was discussed in the previous chapter.

The maximum entropy distribution of P links among N junctions (persons), (for the derivation, see Appendix B-3) is given by,

r(n)=ln(1+1/n)/ln(N+1) were n=1,2…P

Here, n can be any integer, provided that it does not exceed the total number of links, P. It is seen that the relative density of the links,  is a decreasing function of n. n expresses the ratio between the number of links (or particles) and the number of junctions (or boxes).

PP.144 -179.