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The Heider Balance: A Continuous Approach

Author

Listed:
  • KRZYSZTOF KUŁAKOWSKI

    (Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, PL-30059 Kraków, Poland)

  • PRZEMYSŁAW GAWROŃSKI

    (Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, PL-30059 Kraków, Poland)

  • PIOTR GRONEK

    (Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, PL-30059 Kraków, Poland)

Abstract

The Heider balance (HB) is investigated in a fully connected graph ofNnodes. The links are described by a real symmetric arrayr (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) > 0for 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,${\bf \dot r =r\cdot r}$. The form of equations guarantees that once HB is reached, it persists. Also, forN =3the 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 toN =500show that HB is always reached. Timeτ(N)to get the balanced state varies with the system sizeNasN-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.

Suggested Citation

  • Krzysztof Kułakowski & Przemysław Gawroński & Piotr Gronek, 2005. "The Heider Balance: A Continuous Approach," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 16(05), pages 707-716.
  • Handle: RePEc:wsi:ijmpcx:v:16:y:2005:i:05:n:s012918310500742x
    DOI: 10.1142/S012918310500742X
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    Citations

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    Cited by:

    1. Krawczyk, Malgorzata J. & Kułakowski, Krzysztof, 2022. "Structural balance in one time step," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 606(C).
    2. Arabzadeh, S. & Sherafati, M. & Atyabi, F. & Jafari, G.R. & Kułakowski, K., 2021. "Lifetime of links influences the evolution towards structural balance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 567(C).
    3. Du, Haifeng & He, Xiaochen & Wang, Jingjing & Feldman, Marcus W., 2018. "Reversing structural balance in signed networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 780-792.
    4. Malarz, Krzysztof & Kułakowski, Krzysztof, 2021. "Heider balance of a chain of actors as dependent on the interaction range and a thermal noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 567(C).
    5. Pedro Cisneros-Velarde & Francesco Bullo, 2020. "Signed Network Formation Games and Clustering Balance," Dynamic Games and Applications, Springer, vol. 10(4), pages 783-797, December.
    6. Deng, Hongzhong & Qi, Mingze & Li, Mengjun & Ge, Bingfeng, 2021. "Limited cognitive adjustments in signed networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 584(C).
    7. Sajjad Salehi & Fattaneh Taghiyareh, 2020. "Stabilizing social structure via modifying local patterns," Journal of Combinatorial Optimization, Springer, vol. 39(4), pages 1079-1095, May.

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