Krzysztof Kulakowski

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,

Crowd dynamics - being stuck

{\bf \dot r =r\cdot r}

Opinion polarization in the Receipt–Accept–Sample model

. 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 τ(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.

Evaluation of the saturation magnetostriction in nearly zero magnetostrictive glass-coated amorphous microwires

We consider a crowd of N persons trying to leave some area trough a small exit. The probability is calculated that an individual is able to withdraw from the crowd under oneʼs own steam. The problem is simulated within the generalized force model (D. Helbing, et al., Nature 407 (2000) 487), and all model parameters are taken from this paper. The results indicate, that in a crowd of 150 persons, this probability is not greater than ten percent. We also evaluate the number of helpers necessary to get the above probability of fifty percent.

Some Recent Attempts to Simulate the Heider Balance Problem

The Zaller theory of opinion formation is reformulated with one free parameter μ, which measures the largest possible ideological distance which can be made by a citizen in one mental step. Our numerical results show the transient effects: (i) the political awareness, measured by the number of received messages, increases with time first exponentially, later linearly; (ii) for small μ correlations are present between previously and newly received messages; (iii) these correlation lead to a hyperdiffusion effect in the space of attitudes of messages. Citizens with small μ are more prone to extremal opinions.

Magnetization Reversal in Spin Patterns with Complex Geometry

Measurements of the saturation magnetostriction λs of glass-coated covered (Co1−xMnx)75Si10B15 (x=0.09, 0.10 and 0.105) and Co56.5Fe6.5Ni10B16Si11 amorphous microwires carried out by means of the stress dependence of initial magnetic susceptibility method are reported. The conditions of applicability of this method to the microwires have been analyzed, taking into account the domain structure expected for very low negative magnetostriction of the metallic nucleus. The experimental data of λs obtained in those amorphous microwires are in agreement with those reported in amorphous ribbons with similar compositions.

Clustering in random line graphs

This article traces how research into the Heider balance has moved forward in the past 10 years, from a sociological problem about group divisions to a computational one.

Relation between surface magnetization reversal and magnetoimpedance in Co-rich amorphous microwires

We study field-driven dynamics of spins with antiferromagnetic interactions along the links of a complex substrate geometry, which is modeled by graphs of a controlled connectivity distribution. The magnetization reversal occurs in avalanches of spin flips, which are pinned by the topological constraints of the underlying graph. The hysteresis loop and avalanche sizes are analyzed and classified in terms of the graphs connectivity and clustering. The results are relevant for magnets with a hierarchical spatial inhomogeneity and for design of nanoscale magnetic devices.

Dynamics of interacting wires

Abstract We investigate the degree distribution P ( k ) and the clustering coefficient C of the line graphs constructed on the Erdos–Renyi networks, the exponential and the scale-free growing networks. We show that the character of the degree distribution in these graphs remains Poissonian, exponential and power law, respectively, i.e. the same as in the original networks. When the mean degree 〈 k 〉 increases, the obtained clustering coefficient C tends to 0.50 for the transformed Erdos–Renyi networks, to 0.53 for the transformed exponential networks and to 0.61 for the transformed scale-free networks. These results are close to theoretical values, obtained with the model assumption that the degree–degree correlations in the initial networks are negligible.

Stress dependence of bistability in a zero-magnetostrictive amorphous wire

We report the investigations on the giant magnetoimpedance (GMI) effect and the surface magnetic structure performed in the series of glass covered Co-rich amorphous microwires with different thicknesses of glass covering. The calculation of the surface hysteresis loops has been performed in the frames of the model which takes into account the helical magnetic anisotropy. The correlation between the value of the GMI ratio, the jump of the circular magnetization, and the value of the angle of helical anisotropy has been established. It was found that the angle of the surface helical anisotropy, for which the highest value of GMI, is close to 60°.