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Statistical Physics of Coevolving Networks

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Statistical Physics of Coevolving Networks UNIVERSITY OF WARSAW DOCTORAL THESIS Statistical physics of coevolving networks Author: Supervisor: Tomasz RADUCHA prof. dr hab. Ryszard KUTNER Assistant supervisor: dr Tomasz GUBIEC A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Faculty of Physics Warsaw – Palma de Mallorca, May 2020 iii List of publications A significant part of results presented in this thesis has been published in the following articles: • T. Raducha and T. Gubiec, Coevolving complex networks in the model of so- cial interactions, Physica A: Statistical Mechanics and its Applications vol. 471: 427-435, 2017 • T. Raducha and T. Gubiec, Predicting language diversity with complex net- works, PloS one, vol. 13.4: e0196593, 2018 • T. Raducha, M. Wili´nski,T. Gubiec, and H. E. Stanley, Statistical mechan- ics of a coevolving spin system, Physical Review E, vol. 98.3: 030301, 2018 • T. Raducha, B. Min, and M. San Miguel, Coevolving nonlinear voter model with triadic closure, Europhysics Letters, vol. 124.3: 30001, 2018 • T. Raducha and M. San Miguel, Emergence of complex structures from non- linear interactions and noise in coevolving networks, arXiv:2004.06515, 2020 v “You don’t need something more to get something more.” Murray Gell-Mann vii Abstract UNIVERSITY OF WARSAW Faculty of Physics Doctor of Philosophy Statistical physics of coevolving networks by Tomasz RADUCHA Statistical physics has introduced key concepts for analyzing systems con- sisting of a large number of elements. It showed that problems seemingly out of reach for quantitative description can be actually treated in a strict manner. The essential achievement was a shift in the notion of prediction from deterministic to stochastic ground. The approach of statistical physics has been successfully applied in numerous branches of science, of which the most important one for this thesis is the theory of complex systems. Networks create the core of complexity science. When a studied object contains many interacting parts and the pattern of interaction is not trivial, networks are the most natural tool to describe it. Some recent publications even identify complex systems as coevolving multilayer networks. One can argue with such a rigorous definition, but the coevolution of structure and state on its own has proved to be a crucial feature in complex systems. Studies on the coevolution’s impact on the behavior of particular models are relatively new. Moreover, a general theory of coevolving networks is so far out of reach. In order to get closer to universal laws we need to under- stand single problems first. Thus, I seek to explore the effects of coevolution in different models, both analytic (or equilibrium) and algorithmic (or non- equilibrium) ones. The thesis explores outcomes of introducing coevolving mechanisms in three models, namely the voter model, the Axelrod model and the Ising model. The first one is a reference point in the quantitative description of social systems, however it was successfully applied also in physics, biol- ogy, and finance. The voter model is extended in this work to integrate coevolution, triadic closure, nonlinear interactions, and noise. The Axelrod model has a purely social interpretation - it’s a model of social interactions or dissemination of culture. It is studied in the thesis how different types of rewiring, during the coevolution of the network’s structure and state of the nodes, influence the final topology of the system. Results obtained using the Axelrod model are compared with empirical data. The model was improved for better agreement with the empirically observed scaling behavior. No- tably, previous extensions of the model didn’t resolve the contradiction with viii the empirical data, which is solved here. The Ising model was originally constructed to explain ferromagnetism. However, it gained much bigger at- tention than the first application might have suggested. It became a reference point for network models and has been studied in many variations, not only empirically implied but also theoretically thrilling. In this abstract context, the spin dynamics from the Ising model is combined in the thesis with topo- logical traits of the nodes to analyze the outcome of a coevolving equilibrium model with structural traits included in the Hamiltonian. In all models studied throughout this thesis new results are obtained. The most important ones are the following. In the nonlinear coevolving voter model with triadic closure a new shattered phase is observed together with high values of the clustering coefficient. When the noise is included two new phases are obtained. One of these phases persists in the thermody- namic limit. The other one contains topological communities driven by state of the nodes, what was not observed before for the coevolving voter model with noise. Additionally, a new analytical description of the model is pro- posed. As this model is the most general one, it contains previously studied limit cases like the coevolving voter model or the nonlinear coevolving voter model. Similarly, in the Axelrod model high clustering is generated, as well as a power-law degree distribution. But most importantly, due to the imple- mented changes the model displays a new scaling of the number of domains with the system size. This result, in contrast to results obtained with the orig- inal model, is consistent with empirical data. Finally, an equilibrium model of coevolving networks is proposed for the first time. More precisely, the Hamiltonian of the model includes not only states of the nodes and their mutual interactions, but also degree of the nodes, as a local topological trait. A rich phase diagram obtained in this way is described analytically. The ob- served configurations coincide with those obtained in non-equilibrium mod- els, suggesting a possibility of their equilibrium description. The results of the thesis provide a new insight into the behavior of coe- volving networks from a statistical physics perspective. The theory of com- plex systems is not yet complete and new building blocks are being discov- ered. Hopefully, after analyzing and understanding enough separate parts a more universal theory will emerge. The content of this thesis contributes to the development of such theory by providing several new building blocks. Above all, these blocks take into account coevolution of network’s structure and state, as one of the crucial properties of complex systems is their adap- tive behavior. The obtained results surely represent advancement in partic- ular models that were studied within the scope of the thesis, but can be also seen as another step towards the theory of complex systems. ix Acknowledgements I would like to thank my supervisor prof. Ryszard Kutner for the support and advice throughout my PhD studies. I am also grateful for his patience in reading and correcting this thesis. I am grateful to my assistant supervisor dr Tomasz Gubiec with whom I conducted my first scientific research. I appreciated his company and sense of humor in any circumstances. I am in debt to my friends in this amazing journey – Mateusz Wili´nski and Jarosław Klamut. I would like to thank Mateusz for the encouragement without which I would probably not finish my PhD and Jarek for invaluable help at the faculty whenever I was abroad. I would like to thank prof. Maxi San Miguel for giving me an opportunity to work with him. Results of this work create a significant part of this thesis. Finally, I would like to thank all people who I had the pleasure to inter- act with during my scientific work. Intellectual stimulus from such interac- tions built up my scientific awareness and influenced the shape of this thesis. Surely, a big part of the satisfaction from scientific work comes from the pos- sibility to collaborate with wise people. xi Contents List of publications iii Abstract vii Acknowledgements ix List of Abbreviations xiii List of Symbols xv 1 Introduction1 1.1 Physics of networks.........................1 1.1.1 The concept of network..................1 1.1.2 Empirical examples.....................3 1.1.3 Foundations of theory...................5 1.2 Outline of this work.........................9 2 Basics of network science 11 2.1 Graph theory............................ 11 2.1.1 Coefficients and measures................. 12 2.1.2 Types of graphs....................... 16 2.2 Fundamental models........................ 21 2.2.1 Random networks..................... 22 2.2.2 Exponential random graphs................ 25 2.2.3 Small-world networks................... 28 2.2.4 Scale-free networks..................... 31 2.2.5 Constant rewiring..................... 35 2.3 Coevolving networks........................ 37 3 The voter model 41 3.1 Static network............................ 42 3.1.1 Homogeneous case..................... 42 3.1.2 Complex networks..................... 47 3.2 Coevolving voter model...................... 50 3.3 Nonlinear coevolving voter model................ 55 3.3.1 Global rewiring....................... 55 3.3.2 Local rewiring........................ 61 3.4 Nonlinear coevolving voter model with noise.......... 67 xii 4 The Axelrod model 83 4.1 Static network............................ 84 4.1.1 Original definition..................... 84 4.1.2 Selected extensions..................... 91 4.2 Coevolving Axelrod model.................... 94 4.2.1 Random rewiring...................... 94 4.2.2 Preferential attachment and triadic closure....... 98 4.3 A solution of the problem with empirical data........
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