The Heider balance - a continuous approach

Abstract

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. 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 are solved numerically. Initially, r(i,j) are 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 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 somestatistical data.