P. Gawroński

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}

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

. 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.

Dynamics of interacting wires

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.

To cooperate or to defect? Altruism and reputation

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°.

Surface and Bulk Magnetic Hysteresis Loops of Co-Rich Glass Covered Microwires

Abstract The technique of coupled map lattices is applied to investigate the time dependence of magnetization of a set of bistable wires. Subsequent moments of time tk, between which the mapping is performed, are chosen when a domain wall within a wire starts to move or stops. Iterative maps are formed by means of the integration of the equations of motion of the domain walls. The system is proved to be piecewisely integrable, and it does not exhibit chaos in long time limit. However, the Lyapunov exponent determined numerically is positive during a transient time. For small amplitude of the applied magnetic field, more than one limit cycle is found. Observed random long-time behaviour can be assigned to thermal fluctuations of the switching field, which shift a trajectory from one limit cycle to another one. We show also some experimental results on the hysteresis loops of Fe77.5B15Si7.5 and on the fluctuation distribution of the switching field.

A Numerical Trip to Social Psychology: Long-Living States of Cognitive Dissonance

The basic difficulty in cooperation theory is to justify the cooperation. Here we propose a new approach, where players are driven by their altruism to cooperate or not. The probability of cooperation depends also on the co-player’s reputation. We find that players with positive altruism cooperate and meet cooperation. In this approach, payoffs are not relevant.

Gain and loss of esteem, direct reciprocity and Heider balance

The investigation of the magnetization reversal process in amorphous wires and microwires is one of the most important tasks related to the use of these wires in different devices. In particular the intensive study of magnetic properties of the nearly zero magnetostriction Co-rich wires have been performed in relation with the giant magnetoimpedance (GMI) effect. This GMI effect is of great interest in sensor application. The origin of the GMI effect is related with the penetration depth of the skin effect. Consequently, investigation on the magnetization reversal in the surface area of the wire becomes a particular importance. Therefore we used the magneto-optical Kerr effect (MOKE) for magnetization reversal study as a method, which provides a quite interesting information about the magnetic properties of the surface domain structure of the wire.

Perpendicular magnetisation from in-plane fields in nano-scaled antidot lattices

The Heider theory of cognitive dissonance in social groups, formulated recently in terms of differential equations, is generalized here for the case of asymmetric interpersonal ties. The space of initial states is penetrated by starting the time evolution several times with random initial conditions. Numerical results show the fat-tailed distribution of the time when the dissonance is removed. For small groups (N=3) we found some characteristic patterns of the long-living states. There, mutual relations of one of the pairs differ in sign. PACS numbers: 89.65.-s, 02.50.-r.

Micromagnetism of permalloy antidot arrays prepared from alumina templates

The effect of gain and loss of esteem is introduced into the equations of time evolution of social relations, hostile or friendly, in a group of actors. The equations allow for asymmetric relations. We prove that in the presence of this asymmetry, the majority of stable solutions are jammed states, i.e. the Heider balance is not attained there. A phase diagram is constructed with three phases: the jammed phase, the balanced phase with two mutually hostile groups, and the phase of so-called paradise, where all relations are friendly.