How Pairs Of Partners Emerge In An Initially Fully Connected Society

A social group is represented by a graph, where each pair of nodes is connected by two oppositely directed links. At the beginning, a given amountp(i)of resources is assigned randomly to each nodei. Also, each linkr(i,j)is initially represented by a random positive value, which means the percentage of resources of nodeiwhich is offered to nodej. Initially, the graph is fully connected, i.e., all nondiagonal matrix elementsr(i,j)are different from zero. During the simulation, the amounts of resourcesp(i)change according to the balance equation. The nodes reorganize their activity with time to give more resources to those which give them more. This is the rule of varying the coefficientsr(i,j). The result is that after some transient time, only some pairs(m,n)of nodes survive with nonzerop(m)andp(n), each pair with symmetric and positiver(m,n)=r(n,m). Other coefficientsr(m,i≠n)vanish. Unpaired nodes remain with no resources, i.e., theirp(i)=0, and they cease to be active as they have nothing to offer. The percentage of survivors (i.e., those withp(i)positive) increases with the velocity of varying the numbersr(i,j), and it slightly decreases with the size of the group. The picture and the results can be interpreted as a description of a social algorithm leading to marriages.