We analyze the formation of the Heider balance [2,3] in a bilayer network forming a link multiplex, i.e. when interlayer connections exist only between copies of the same link in different layers. Strengths of interlayer coupling are described by a pair of positive or negative parameters (β₁, β₂). The system dynamics is a link dynamics. Each link is described by a continuous variable between (-1) and (+1) and its evolution depends on neighbor links from the same layer and its replica from the other layer.

There are two processes driving system dynamics: a tendency to achieve Heider balance between links in a given layer and a tendency to follow the same or the opposite state of a corresponding link in the other layer. As a result, three types of solutions are possible: a non-stationary solution, a stationary solution with Heider balance and a stationary solution without Heider balance. For a given pair (β₁, β₂) there are phases where only one solution type is achieved and there are intermediate regions where, depending on the initial conditions, two solution types are possible. For instance, when both coupling coefficients are strong enough and of different signs the system always experiences oscillations. The obtained results comprise rich diagrams of model parameters that allow us to identify the areas of coupling coefficients leading to high probability of attaining Heider balance.

[1] F. Heider, The Psychology of Interpersonal Relations (Psychology Press, 1958).

[2] K. Kulakowski, P. Gawroński, and P. Gronek, Int. J. Mod. Phys. C 16, 707 (2005)

[3] P.J. Górski, K. Kulakowski, P. Gawroński, and J.A. Hołyst, (to be published)

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