The Heider balance: A continuous approach
The Heider balance (HB) is investigated in a fully connected graph of N nodes. The links are described by a real symmetric array r(i,j), i, j = 1, ...,N. In a social group, nodes represent group members and links represent relations between them, positive (friendly) or negative (hostile). At the balanced state, r(i,j)r(j,k)r(k,i) > 0 for all the triads (i, j, k). As follows from the structure theorem of Cartwright and Harary, at this state the group is-divided into two subgroups, with friendly internal relations and hostile relations between the subgroups. Here the system dynamics is proposed to be determined by a set of differential equations, (r) over dot = r center dot r. The form of equations guarantees that once HB is reached, it persists. Also, for N = 3 the dynamics reproduces properly the tendency of the system to the balanced state. The equations axe solved numerically. Initially, r(i, j) axe random numbers distributed around zero with a symmetric uniform distribution of unit width. Calculations up to N = 500 show that HB is always reached. Time tau(N) to get the balanced state varies with the system size N as N-1/2. The spectrum of relations, initially narrow, gets very wide near HB. This means that the relations are strongly polarized. In our calculations, the relations are limited to a given range around zero. With this limitation, our results can be helpful in an interpretation of some statistical data.
sociophysics interpersonal relations fully connected network