Heider平衡-持续的方法
The Heider balance - a continuous approach
摘要：
研究了由$N$结点组成的全连通图中的Heider平衡问题.链路由实对称阵列r（i，j）描述，i，j=1，.，N.在一个社会群体中，节点代表群体成员，链接代表他们之间的关系，积极的（友好的）或消极的（敌对的）。在平衡状态下，对于所有三元组（i，j，k），r（i，j）r（j，k）r（k，i）>0。根据卡特赖特和哈拉里的结构定理，在这种状态下，群被分成两个子群，子群之间有友好的内部关系和敌对的关系。在这里，系统的动力学建议由一组微分方程来确定。方程的形式保证了一旦达到HB，它就持续存在。此外，对于N=3的动力学正确地再现了系统的平衡状态的趋势。用数值方法求解了这些方程。最初，r（i，j）是以单位宽度的对称均匀分布围绕零分布的随机数。直到N=500的计算表明总是达到HB。获得平衡状态的时间随系统大小N而变化，为N^{-1/2}。关系的范围，最初很窄，在HB附近变得很宽。这意味着，这种关系是严重两极化的。在我们的计算中，这些关系被限制在零附近的给定范围内。在这种限制下，我们的结果可以帮助解释一些统计数据。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
