Krasnytska M. Collective behavior on complex networks: fundamental aspects and applications

The present thesis focuses on studying the collective behavior of complex systems, specifically exploring the characteristics of phase transitions in magnets with non-trivial architecture, the phenomenon of ordering in classical models of statistical physics, and their modifications on complex networks. The critical behavior of the Potts model with invisible states on graphs of various topologies was analyzed, demonstrating that a large number of invisible states can change the nature of the phase transition. In particular, the critical number of invisible states capable of altering the transition type to first-order, even in percolation (in the limit q → 1), was determined. A novel generalization of the Ising model with variable spin length was proposed. For this model on different types of graphs, universal critical characteristics of second-order phase transitions were calculated, including critical exponents, logarithmic correction exponents, scaling functions, and critical amplitude ratios. Two new universality classes for logarithmic corrections were identified for the model on annealed scale-free networks. For the same model on a three-dimensional lattice, a representation of the effective Hamiltonian (belonging to the universality class of the diluted Ising model) was obtained. Additionally, using renormalization group theory, asymptotic and effective critical exponents were determined. It was proven that impurities do not necessarily have to be non-magnetic to produce similar effects on critical behavior. For the Blume-Capel model, the critical behavior on a complete graph was analyzed for the first time using the formalism of partition function zeros. Methods of complex network theory were applied for the analysis and visualization of the co-authorship social network of journal authors and the semantic network of concepts. The study demonstrated how network growth mechanisms and their dynamics are realized in practice, what they lead to, and how real networks are described by the proposed models when investigating collective effects and the structure of complex networks themselves.