Sarkanych P. Universality of complex systems: partition function zeros and complex networks

The goal of the dissertation is to study the universality of complex systems. The notion of universality - the independence of the characteristic behaviour of a macroscopic system, consisting of many interacting parts, from the details of the structure of this system - is one of the basic concepts of statistical physics. By complex systems we mean those characterized by collective behaviour that is not a simple consequence of the properties of their constituent parts. The concept of complex system is applied in many traditional disciplines of science and is the subject of a new interdisciplinary field of knowledge - the complexity science. Methods and concepts of statistical physics are among the important components of this branch as is concept of universality that has become one of the basic concepts of the complex systems science. Since complex systems are characterized by the collective behaviour of many interacting components, the theory of phase transitions provides a natural tool for their study. The dissertation consists of four sections. The first section of the dissertation provides a literature review. The main focus of this section is on the models and methods of their description used in the dissertation. In particular, two models are used in the thesis: the Ising model with dipole interactions and the Potts model with invisible states. The Ising model with dipole interactions is used to describe patterns formation in complex systems, and the Potts model with invisible states is used to study how entropy affects the universality. Among the methods used in the thesis, two are described in detail. The first method is the analysis of the zeros of a partition function in a complex plane. Based on the properties of the location of these zeros, it is possible to obtain the critical properties of the system. The second is the method of complex networks. Within this method, a system of many particles (agents) is represented in the form of a graph, where the nodes are interacting particles, and the edges denote the interaction between them. The second section of the dissertation uses the method of analysis of partition function zeros to study the critical behaviour of a 2d Ising model with dipole interaction and a 1d Potts model with invisible states. For 2d Ising model with dipole interactions, the region of the phase diagram is analysed and it is shown that the critical exponents depend continuously on the ratio of the short-range and dipole interaction constants $ \delta $. The values of the critical exponents obtained in the partition function zeros density approach, agree well with the results of short-time Monte-Carlo simulations. For the 1d Potts model with $ q $ visible and $ r $ invisible states, the exact solution was found using the transfer matrix method. Two conditions were found to shift a phase transition to the positive temperature. The first condition is the consideration of external complex magnetic fields. The second mechanism is to consider the negative number of invisible states. These two mechanisms were shown to be connected by the duality relation. The third section of the dissertation uses an approximation of a nonuniform mean field to investigate a Potts model with invisible states on an arbitrary graph. Two partial cases are considered: a complete graph and a scale-free network. For the complete graph it is shown that in the region $ 1 \leq q <2 $ the phase diagram is characterized by two marginal values, $ r_{c1} $ and $ r_{c2} $. Below $ r_{c1} $, only the second-order phase transition occurs in the system. There is only a first-order phase transition above $ r_{c2} $. And in the area $ r_{c1} <r <r_{c2} $ there are two phase transitions: a first-order transition at lower temperature and a second-order at higher temperature. In the Ising model case $ q = 2 $, on the complete graph, the two marginal values coincide at $ r_c \approx 3.62 $. On a scale-free network with the node degree distribution $ P(k)\sim k^{-\lambda} $ even in the case $q=2$ two $ \lambda-$dependent marginal values $r_ {c1}(\lambda)$ and $ r_{c2}(\lambda)$ were obtained. These quantities play the same role as their counterparts in the previous case. It is also shown that wherever there is a second-order phase transition, the number of invisible states $ r $ does not affect the value of the critical exponents. The fourth section of the dissertation uses the complex networks approach to analyse the social network of characters of Bylyny. This network possesses a number of properties, common with the properties of social networks of other epics. These properties remain unchanged for networks that characterize epic narratives of different cultures and have been created at different times. Thus, epics have universal properties, which allows to obtain additional classification based on their quantitative analysis.