5 research outputs found
κ³ μ°¨ μνΈμμ©νλ 볡μ‘κ³μ λ μ€λ¦ νμ
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Όλ¬Έ(λ°μ¬) -- μμΈλνκ΅λνμ : μμ°κ³Όνλν 물리·μ²λ¬ΈνλΆ(물리νμ 곡), 2022.2. λ°±μ©μ£Ό.κ΅¬μ± μμλ€μ λ―Έμμ λμνλ§μΌλ‘λ μ€λͺ
ν μ μλ 볡μ‘κ³μ λ§μ νμλ€μ κ΅¬μ± μμλ€μ 볡μ‘ν μνΈμμ©μ κ·Έ κ·Όμμ λκ³ μλ€. μ΄κ΄μ κ΄μ μμ 볡μ‘κ³μ μ±μ§μ ν΄μνκ³ μ΄ν΄νλ λ°©λ²μΌλ‘ μ£Όλͺ©μ λ°μλ λ€νΈμν¬ κ³Όνμ ν΅ν΄ μ€λͺ
ν΄ μλ 물리μ νμλ€ μ€μλ μ§ μνΈμμ©μΌλ‘ νμλμ§ μλ μ±μ§μ μνΈμμ©λ€μ΄ μ‘΄μ¬νλ€. λ μ΄μμ λ§μ μμλ€μ λμλ€λ°μ μνΈμμ©μ μ μ νκ² νννκΈ° μν΄ μ¬μ©λλ λͺ κ°μ§ μνμ λκ΅¬κ° λμ
λμλ€. μ΄ νμλ
Όλ¬Έμμλ κ³ μ°¨ μνΈμμ©μ ν¬ν¨ν κ΅¬μ‘°λ‘ νμ΄νΌκ·Έλνμ λ¨μ²΄ 볡ν©μ²΄λ₯Ό μ¬μ©νμ¬ μ΄ μμμ κΈ°μ‘΄ μ§ μνΈμμ© κ΅¬μ‘°μμ λνλλ 보νΈμ±κ³Ό λμνλ€μ΄ μ΄λ»κ² λ¬λΌμ§λμ§λ₯Ό ν΄μμ , μμΉμ λ°©λ²μ ν΅ν΄ νꡬνλ€.
λ¨Όμ κ³ μ°¨ μνΈμμ©μΌλ‘μ ννμ΄ νμν μ¬ν νμμΈ κ³΅μ μ λ€νΈμν¬ λ°μ΄ν°λ₯Ό λΆμνλ€. μ°λ¦¬λ 곡μ μ λ€νΈμν¬ λ°μ΄ν°λ₯Ό λ¨μ²΄ 볡ν©μ²΄λ‘ νννμ¬ κ·Έ μμμ νΉμ±μ μκ° μ§νλ₯Ό νμΈνμλ€. μ΄κ²μ μ±μ₯ κ³Όμ μμ μμμ μΈ λΆλ³λμΈ λ² ν° μκ° μ°¨μμ λ°λΌ μμ°¨μ μΌλ‘ μ°½λ°μ΄ λνλλ€λ κ²μ νμΈνμλ€. μ΄λ¬ν μμμ μμκ³Ό ꡬ쑰μ νΉμ±μ μ±μ₯ κ³Όμ μμ λͺ¨λ°©ν μ μλ λ¨μ²΄ 볡ν©μ²΄ λͺ¨νμ ꡬμ±νμ¬ κ·Έ νΉμ±λ€μ ν΅κ³μ μΌλ‘ νμΈν κ²°κ³Ό, λ² ν° μμ μ°½λ°μ΄ 무νμ°¨μ μμ μ΄μ νΉμ§μ 보μΈλ€λ κ²μ λ°κ²¬νμλ€. μ΄λ₯Ό μ€λͺ
ν μ μλ μκ° μμ‘΄ λΉμ¨λ°©μ μμ μ μμνλ₯Ό μμ±ν¨μ λ°©λ²μΌλ‘ μ κ·Όνμ¬ λ¬΄νμ°¨μ μμ μ΄λ₯Ό μ΄λ‘ μ μΌλ‘ κ·λͺ
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νμ΄νΌκ·Έλνλ μ¬ν μ°κ²°λ§μμ μ¬λ¬ μ¬λμ΄ ν¨κ» νλ ν μ¬μ΄μ μνΈμμ©μ νννλλ°μλ μ¬μ©ν μ μλ€. λ§€κ° μ€μ¬μ±μ νμ΄νΌκ·Έλνμμ μΈ‘μ ν μ μ° μκ³ λ¦¬μ¦μ μ μνκ³ , μ¬ν μ°κ²°λ§μ λ¬μ¬ν μ²λμλ νμ΄νΌκ·Έλν λͺ¨νμ ꡬμ±νμλ€. μ΄λ‘λΆν° νμ 맀κ°μ€μ¬μ± λΆν¬λ κ°κ°μΈμ 맀κ°μ€μ¬μ± λΆν¬μλ μμ΄νκ² κ±°λμ κ³± λΆν¬κ° μ곑λμ΄ μ§μν¨μμ κ°μλ₯Ό 보μμ νμΈνκ³ , μ΄ κ°μμ μ λκ° νμ ν¬κΈ°μ κ΄λ ¨λ¨μ 보μλ€. νμ§λ§ μ€μ λ°μ΄ν°μ λ°μλλ μΆκ° μ 보λ₯Ό νμ κ°μ€μΉλ‘μ¨ λμ
νλ©΄ 맀κ°μ€μ¬μ± λΆν¬μ κ±°λμ κ³± λ²μΉμ΄ μ¬κ΅¬μ±λ¨μ νμΈνμλ€. νμ§λ§ νμ΄ κ°μ§λ κ°μ€μΉκ° μλλΌ νμ΄νΌκ·Έλν ꡬ쑰μμ μ‘΄μ¬νλ μμΉκ° νμ μ±κ³Όμ λμ± μ€μν μμλΌλ λ°μ§κ΄μ μΈ κ²°κ³Όλ₯Ό μ»μλ€.
λ§μ§λ§μΌλ‘ 볡μ‘κ³μμ 보μ΄λ λμν κ³Όμ μ λ―ΈμΉλ κ³ μ°¨ μνΈμμ©μ μν₯μ λκΈ°ν νμμ μλ₯Ό ν΅ν΄ μ΄ν΄λ³΄μλ€. λκΈ°ν κ³Όμ μ μμ° λ° μΈκ³΅ μμ€ν
μ κ΄λ²μν κΈ°λ₯μ μ€μν μν μ νκΈ° λλ¬Έμ κ·Έ μ§λ¨ νλμ νμ±μ λ―ΈμΉλ λ―Έμμ μνΈμμ© κ΅¬μ‘°μ λν μ΄ν΄λ νμλΆκ°κ²°νλ€κ³ ν μ μλ€. μ°λ¦¬λ μ²λμλ νμ΄νΌκ·Έλν μμμμ ꡬλΌλͺ¨ν λͺ¨νμ μ°μ λ°©μ μμ μ€νΈ-μν€μΌ κ°μ€ νμ΄λ²κ³Ό νκ· μ₯ μ΄λ‘ μ μ΄μ©νμ¬ μ‘°μ¬νμλ€. κ·Έ κ²°κ³Όλ‘ μ°κ²° ꡬ쑰μ λΆκ· μΌν¨μ λ°λΌ μ°μ μμ μ΄μμ λΆμ°μ μμ μ΄λ‘μ λ³νκ° λνλ¨μ νμΈνμλ€. νΉν λΆμ°μ μμ μ΄λ κ·Έ μ μ΄μ μμ μκ³ νμμ 보μ΄λ νμ΄λΈλ¦¬λ μμ μ΄μμ νμΈνμκ³ , μ΄ μκ³ μ§μλ₯Ό ν΄μμ , μμΉμ μΌλ‘ κ²°μ νμλ€.Rooted in the complex interactions of components, diverse aspects in complex systems cannot be explained by the microscopic dynamics of components. Among the physical phenomena that have been described through network science, which have received attention as a way of interpreting and understanding the properties of complex systems from a holistic point of view, are intrinsic interactions that cannot be reduced to pairwise interactions. In this dissertation, we analytically and numerically explore the emergent phenomena that appear in these higher-order interactions. As structures include these simultaneous interactions of two or more elements, we use hypergraphs and simplicial complexes.
First, the coauthorship data, which is a social relationship that requires the expression of higher-order interactions, is analyzed. We confirmed the time evolution of the topological features by expressing the coauthorship data as a simplicial complex. In the process of its growth, we find that the Betti numbers, topological invariants, sequentially appeared according to the dimension. As a result of statistical confirming the properties by constructing a random simplicial complex model that can imitate these topological aspects and structural characteristics in the growing process, we reveal that the development of the Betti number showed the infinite-order phase transition. By generating function, the steady-state analysis of the time-dependent rate equation that mimics the model algorithms suggests the theoretical explanations of the infinite-order transitions.
Hypergraphs also are widely used to express interactions between teams with multiple people in a social network. Here we propose a computational algorithm to measure betweenness centralities in a hypergraph. Furthermore, a scale-free hypergraph model is constructed to describe the features of social networks. From this, we confirm that the distribution of the team's betweenness centrality, showing exponential decaying, is different from the power-law distribution of individual betweenness centrality. Interestingly, the decaying rate is related to the size of the team. However, if additional information reflected in the actual data is introduced as the team's weight, the power law of the betweenness centrality distribution is reconstructed. Counterintuitively, we observe that the location of a team in the hypergraph structure, such as whether the team is near the hub, not the weight of the team, is a more crucial factor in the team's performance.
Finally, the influence of higher-order interactions on the dynamics detected in complex systems is examined through examples of synchronization phenomena. Synchronization appears in a wide range of natural and artificial systems. Thus, an understanding of the microscopic group interaction structure is indispensable. We investigated the Kuramoto model on a scaleless hypergraph using the Ott-Antonsen ansatz of continuity equations and the mean-field theory. As a result, we find that the non-uniformity of the connection structure resulted in a change from continuous phase transition to discontinuous phase transition. In particular, we observe that discontinuous phase transition is indeed a hybrid phase transition, showing a critical phenomenon at its transition point. The critical exponents are determined both analytically and numerically.Abstract i
Contents iii
List of Figures vii
List of Tables ix
1 Introduction 1
1.1 Complex systems in nature 1
1.2 Representations of complex systems 3
1.3 Structure and goal of the dissertation 6
2 Representations of interactions 10
2.1 Interaction 10
2.2 Pairwise represenation: graph theory 11
2.2.1 Definitions and concepts 11
2.2.2 Random graph models 17
2.3 Higher-order representation: concepts of hypergraphs 19
2.3.1 Mathematical definitions 19
2.4 Algebraic topology: special case of higher-order interactions 21
2.4.1 Simplicial complexes 21
2.4.2 Homology groups 22
3 Basics of percolation transitions 25
3.1 What is percolation 25
3.2 Percolation on complex networks 27
4 Homological percolation transitions: numerical approach 29
4.1 Introduction 29
4.2 Results 33
4.2.1 Homological percolation transitions 33
4.2.2 Facet degree distribution 37
4.2.3 Minimal model 39
4.2.4 Kahle localization 42
4.3 Conclusion 44
4.4 Discussion 45
5 Homological percolation transitions: analytical approach 46
5.1 Introduction 46
5.2 Model 49
5.3 Percolation transition 49
5.3.1 Cluster-size distribution, giant cluster, and mean cluster size 49
5.3.2 Graph and facet degree distributions 55
5.4 Homological percolation transition 58
5.4.1 Model generalization and Betti number 58
5.4.2 Rigorous description of the first Betti number 60
5.5 Discussion 63
5.6 Conclusion 64
6 Criticality from shortest path dynamics on hypergraphs: Betweenness centraility distribution 65
6.1 Betweenness centrality in hypergraphs 65
6.2 Random hypergraph model with preferential attachment 68
6.3 Real data analysis 72
6.4 Summary and discussion 76
7 Synchronization of coupled oscillators 81
7.1 Synchronization 81
7.2 Kuramoto model 82
7.2.1 Order parameter 83
7.2.2 Self-consistency equation approaches 84
8 Synchronization transitions in hypergraphs 86
8.1 Introduction 86
8.2 Model 87
8.3 Self-consistency equations 89
8.4 Results 92
8.5 Summary and discussion 92
9 Conclusion 95
Appendices 97
Appendix A Topological data analysis 98
A.1 Persistent homology 98
Appendix B Appendices for chapter 4 100
B.1 Derivation of master equation of joint generating function 100
Appendix C Appendices for chapter 5 103
C.1 BC distributions for the BA-II hypergraphs 103
C.2 Features of our coauthorship data 104
C.3 Parameter analysis 105
C.3.1 Relations among the BCes and parameters 105
C.3.2 Influential parameters for top BCes 107
C.4 Analysis of the small size dataset 108
Appendix D Appendices for chapter 6 118
D.1 Heterogeneous mean-field theory 118
D.2 Critical behavior 120
D.3 Correlation size 121
Bibliography 123
Abstract in Korean 135λ°