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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    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${2}^{-{N}^{2}/2}$$\end{document}2    −    N   2   /  2. Our finding can explain a lack of the Heider balance in many social systems, where multilayer structures mediate social interactions.

Destructive influence of interlayer coupling on Heider balance in bilayer networks