When we think to a network the first thing that rise in our mind probably are computers connected to servers by ethernet cables / wifi.

Our society can also be seen as a network where the computers are the people, the ethernet cables / wifi are the trust relations and the servers are the representers.

The Penrose method (or square-root method) is a method devised in 1946 by Professor Lionel Penrose for allocating the voting weights of delegations (possibly a single representative) in decision-making bodies proportional to the square root of the population represented by this delegation.

http://en.wikipedia.org/wiki/Penrose_method

Metcalfe's law states that the value of a telecommunications network is proportional to the square of the number of connected users of the system (n²).

Even if this law is a little rough, because it considers only the “nodes” not the “arcs” (connections) and works properly in open systems because it does not consider the borders of closed systems, it represent the “first” intuition about the gravity effect of networks.

http://en.wikipedia.org/wiki/Metcalfe's_law

As well as the gravity law that decreases by the square of the distance in social relations we can consider distance as the hops between two nodes.

The studies of networking and social phenomena have been going more over the Metcalfe's law.

First of all the scale-free network model, is able to describe social interactions of humans like social networks, working collaborations, trasmitted diseases, etc.

Here the power law degree distribution of nodes has an exponent between 2 and 3:

A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as

where C is a normalization constant and γ is a parameter whose value is typically in the range 2 < γ < 3, although occasionally it may lie outside these bounds.

http://en.wikipedia.org/wiki/Scale-free_network

According to the theoretical behaviours of networks, larger groups will dominate the entire network, because of their huge “gravity” and marginal utility.

This explain the big power of such companies like MS, Apple, Google that crunches any other small one.

Let's immagine a non weighted liquidfeedback system where the 30.000 pirates of germany and the 10.000 pirates of the rest of the world can vote togheter. Of course germany would dominate the world!

The solution to balance the groups, avoiding greedy behaviours, is to smooth the power by using the inverse function of the square: the square root.