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, [Formula: see text]. 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.

The crossover from antidot to dot magnetic behavior on arrays patterned in a ferromagnetic thin film has been achieved by modifying only the geometry. A series of antidot arrays has been fabricated on cobalt with fixed diameter d and by reducing the period of the array p from p d to p < d. A dramatic change in the coercivity dependence with p, correlated with a significant modification in the magnetic domain structure observed by x-ray photoemission electron microscopy, evidences the crossover. An intermediate regime has been found between the superdomain structure present in antidot arrays and the array of astroid-state noncorrelated dots. The study has been reproduced for a different ferromagnetic material, permalloy, and supported by micromagnetic simulations.

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.

We consider the problem of Heider balance in a link multiplex, i.e. a special multiplex where coupling exists only between corresponding links. Numerical simulations and analytical calculations demonstrate that the presence of such interlayer connections hinders the emergence of the Heider balance. The effect is especially pronounced when the interactions between layers are negative, similarly as in antiferromagnetically coupled spin layers. The larger is the network, the narrower is the region of coupling parameters where the Heider balance can exist. If the interlayer couplings are of opposite signs and are strong enough, then the link dynamics can be reduced to the system of weakly coupled harmonic oscillators. For large strongly-coupled networks and randomly chosen initial conditions the probability of attaining the Heider balance decreases with the network size N as . Our finding can explain a lack of the Heider balance in many social systems, where multilayer structures mediate social interactions.

Hexagonally ordered Py antidot arrays were prepared by sputtering onto anodic alumina membrane templates, with varying antidot diameter and lattice constant parameter. Experimental magnetic characterization together with micromagnetic simulations was performed to unveil the coercivity mechanism. Experimental measurements show that coercivity monotonically increases with the antidot diameter in reasonable agreement with simulations. This is understood considering the presence of geometrical micrometric domains with perfect hexagonal order. Contrarily, simulations for a single crystal sample predict that the coercivity should decrease with the antidots diameter.