Title Page
Abstract
Contents
Chapter 1. Introduction 29
1.1. Complex Network 29
1.2. Complex contagion 30
1.3. Overview of dissertation 31
Chapter 2. Higher-order epidemics 33
2.1. Phase transition and critical phenomena of the simplicial susceptible-infected-susceptible (s-SIS) model 33
2.2. Static model of uniform hypergraph 35
2.3. Simplicial SIS model 37
2.4. Heterogeneous mean-field theory (annealed approximation) 39
2.4.1. Self-consistency equation 39
2.5. Phase transition and critical behavior 42
2.5.1. Order parameter 42
2.5.2. Susceptibility 46
2.5.3. Correlation size 47
2.6. Numerical simulations 50
2.6.1. Numerical methods 50
2.6.2. Numerical results 52
2.7. Degree distribution of static model 58
2.8. Asymptotic behavior of G′(Θ) 58
2.9. Susceptibility 59
2.10. Containment strategy for simplicial SIS model 60
2.10.1. Hypergraph popularity-similarity optimization (h-PSO) model 63
2.10.2. Individual- and pair-based mean-field theories 65
2.10.3. Immunization strategies 69
2.10.4. Numerical Results 73
2.11. Summary and conclusion 78
Chapter 3. Phase transition in vaccination strategy 81
3.1. Introduction 81
3.2. Susceptible-infected-recovered-dead (SIRD) model 83
3.3. Results 84
3.3.1. Fatality- and contact-based strategies 84
3.3.2. Transition and path-dependency of the optimal vaccination strategy 87
3.3.3. Real-world epidemic diseases 91
3.3.4. Complex epidemic stages, vaccine breakthrough infection, and reinfection 94
3.4. Conclusion 96
Chapter 4. Application of graph neural network (GNN) on spreading processes 98
4.1. Introduction 98
4.2. Prediction and mitigation of avalanche dynamics in power grids using graph neural network 98
4.2.1. Avalanche dynamics 100
4.2.2. Avalanche mitigation strategy 103
4.2.3. Graph neural network (GNN) 106
4.2.4. Conclusion 112
4.3. Epidemic control using graph neural network ansatz 114
4.3.1. Model 116
4.3.2. Vaccination strategy 119
4.3.3. Results 125
4.3.4. Conclusion 131
Chapter 5. Quantum spreading processes in complex networks 133
5.1. Introduction 133
5.2. Permutational symmetry 137
5.3. Quantum contact process 139
5.4. Dissipative Transverse Ising model 142
5.4.1. Transverse Ising model 142
5.4.2. Dissipative transverse Ising model 145
5.5. Comparison with quantum jump Monte Carlo simulation 155
5.6. Quantum contact process in scale-free networks 157
5.6.1. Annealed approximation and self-consistency equation 157
5.6.2. Phase transition and critical behavior 160
5.6.3. Numerical results 166
5.7. Summary and Discussion 169
Chapter 6. Conclusion 173
Bibliography 174
초록 216
Table 2.1. Analytic solutions of the critical exponents for the s-SIS model. 50
Table 2.2. Numerical list of critical exponents of the s-SIS model obtained by the FSS method. Theoretical values calculated in Sec. 2.5 are presented in parentheses. 57
Table 2.3. Dynamic critical exponents of s-SIS model obtained using the dynamical FSS method. 58
Table 4.1. Performance measure Rm of vaccination strategies: random vaccination, vaccination in descending order of degree, eigenvector centrality (EC), betweenness centrality (BC), avalanche size, failure fraction, avalanche centrality (AC), and GNN...[이미지참조] 108
Table 4.2. Hyper parameters used to train GNN. 110
Table 5.1. Summary of previous analytical results. We considered three problems with the quantum models. For each model, the system Hamiltonian (ˆHS) and Lindblad operators (ˆLℓ) were defined. Semiclassical, Weiss, and Keldysh field-theoretic approaches...[이미지참조] 134
Table 5.2. Critical exponent α values for different ω values. 142
Table 5.3. Summary of universality classes for classical, closed quantum, and open quantum Ising models. The system Hamiltonian (ˆHS) and Lindblad operators (ˆLℓ) were defined for each model. Our numerical results indicate that dissipation changes the upper...[이미지참조] 156
Figure 2.1. Degree distribution of the static model of (a) 2-uniform (graph) and (b) 3-uniform hypergraph generated with the fitness exponent 1/μ = 1.3. The system... 36
Figure 2.2. Schematic illustration of the simplicial contagion process through hyperedgesof size 3 in (a) and (b), and 4 in (c) and (d). The susceptible and infected nodes... 38
Figure 2.3. Self-consistency function G(Θ) of SF 3-uniform hypergraphs with degree exponent (a) λ = 2.2, (b) 2.5, and (c) 2.8, corresponding to cases i) λ 〈 λc, ii) λ = λc,...[이미지참조] 43
Figure 2.4. Density of infected nodes versus control parameter λ for various degree exponent values λ for (a) d = 3 and (c) d = 4. Susceptibility versus control parameter... 44
Figure 2.5. (a) Self-consistency function GN(Θ) in finite systems versus Θ for 3-uniform hypergraphs with λ = 2.8. (b) Deviation λc(N) − λc(∞) versus system size...[이미지참조] 50
Figure 2.6. Finite-size scaling analysis of the s-SIS model on SF 3-uniform hypergraphswith three degree exponents: λ = 2.1 〈 λc (a) and (b), λ = 2.9 〉 λc (c) and...[이미지참조] 53
Figure 2.7. Scaling plots of χ₁N−γ¹/¯ν versus (κ − κc)N1/¯ν with degree exponents (a) λ = 2.9 and (b) λ = 3.5, with (a) γ1 = 0.48 and ¯ν = 2.11, (b) γ1 = 0.50 and...[이미지참조] 54
Figure 2.8. Plots of κc(N) − κc(∞) versus N on double-logarithmic scale for (a) λ = 2.1, (b) λ = 2.9, and (c) λ = 3.5. Slope of each plot represents −1/¯ν.[이미지참조] 54
Figure 2.9. Scaling plots of the density of infection ρ(t) starting from the fully infected state versus tN−¯z (a) and (c) and t(κ − κc)ν∥ (b) and (d) for λ = 2.9 (a) and (b) and...[이미지참조] 56
Figure 2.10. (a) The degree distribution of the 3-uniform hypergraph popularity similarity optimization (h-PSO) model. The mean degree 〈k〉 = 6, temperature... 63
Figure 2.11. Random edge immunization (Random), H-eigenscore (H-ES), EI, and SIP-based strategies tested in various synthetic and empirical networks: (a, e, i) the... 72
Figure 2.12. Random hyperedge immunization (Random), H-eigenscore (H-ES), EI, and SIP-based strategies tested in 3-uniform hypergraphs: (a, c) the hypergraph static... 74
Figure 2.13. Random hyperedge immunization (Random) and SIP-based strategy tested in empirical hypergraphs: (a, c, e) the congressional bill cosponsorship (in 2000)... 75
Figure 3.1. (a-b) The mortality rate as a result of various vaccination strategies. The contagion rate is (a) η = 0.05 and (b) η = 0.4, while the recovery rate is normalized... 85
Figure 3.2. (a, d) Fraction of recovered population, (b, e) average fatality of the vaccinated population, and (c, f) average contact rate of the vaccinated population of the... 88
Figure 3.3. The mortality rate of a mixed strategy that combines the high-fatality and high-contact strategies. The contagion rate is η = 0.4, and vaccine supply is, from top... 89
Figure 3.4. The age contact matrix of the (a) United Kingdom and (b) United States, and (c) contact rate of each age group. The population is divided into 17 groups: aged... 92
Figure 3.5. Average age of vaccinated individuals for (a) TB, (b) COVID-19, and (c) COVID-19 with reinfection and finite vaccine efficacy. The contagion rates are... 93
Figure 3.6. The mortality rate of the mixed strategy of high-fatality and high-contact strategies for (a) TB and (b) COVID-19. The contagion rate is η = 0.25, and the vac-... 93
Figure 4.1. Computation time of ML model as a function of network size. The Schultz-Heitzig-Kurths (SHK) random power grid model was used. The computation time is... 100
Figure 4.2. Avalanche size distribution of SHK network with various numbers of nodes for the ML model. The control parameter α is taken as 0.25, at which the avalanche... 101
Figure 4.3. Paradoxical effect of reinforcement in electric power grid of France. Red square (indicated by A) represents the initial failure, red dots represent secondary fail-... 103
Figure 4.4. Performance of avalanche mitigation strategies in the SHK network of size N = 1000. Degree, eigenvector centrality (EC), betweenness centrality (BC),... 107
Figure 4.5. Structure of GNN. Only the adjacency matrix of the network is input, and the features of all nodes are initially constant. Batch normalization and ReLU activa-... 109
Figure 4.6. The cumulative fraction Cσ(n) of AC as a function of n on an SHK network of size N = 1000. The mean cumulative fraction 〈Cσ〉 is the area under the cumulative...[이미지참조] 111
Figure 4.7. Performance of the GNN prediction in large networks in the range N = 1000−8000, which is trained in various network sizes from 100 to 999. All the results... 113
Figure 4.8. Density of infection of SIS model calculated by MMCA as the function of weight parameter wr(ℓ) of GNNA. The plot illustrates the loss surface projected on the...[이미지참조] 124
Figure 4.9. Performance of GNNA compared to centrality-based vaccination strategies.(a) Largest connected component size of network dismantling and the density of... 126
Figure 4.10. Transition of the optimal vaccination strategy in the SIRD model. (a) Phi coefficient between the nodes vaccinated by GNNA and high-BC/IFR vaccination... 130
Figure 5.1. (a) Phase diagram of the QCP model on a fully connected graph in the parameter space (κ, ω), determined by direct numerical enumeration of the Liouville... 138
Figure 5.2. Plots of the order parameter 〈n(t)〉 of the QCP model as a function of time t for fixed κ = 1 but different (a) ω = 0.0, (b) 0.6, (c) 0.8, and (d) 1.0. As N is increased,... 141
Figure 5.3. FSS analysis for the transverse Ising model on fully connected graphs. (a) Plot of the order parameter m as a function of △c − △ for different system sizes. We...[이미지참조] 145
Figure 5.4. (a) Phase diagram of the DTI model on fully connected graphs in the parameterspace (△, Γ). A continuous transition occurs across the solid white curve. The... 150
Figure 5.5. FSS analysis for the DTI model at △ = 0.5 and J = 1 on fully connected graphs. (a) Plot of the order parameter m as a function of Γc − Γ for different system...[이미지참조] 151
Figure 5.6. (a) Phase diagram of the DTI model on fully connected graphs in the parameterspace (△, J). A continuous transition occurs across the solid white curve. The... 152
Figure 5.7. FSS analysis for the DTI model at △ = 0.2 and J = 1 on fully connected graphs. (a) Plot of the order parameter m as a function of J − Jc for different system...[이미지참조] 153
Figure 5.8. Comparison of the data sets obtained by the direct enumerations of the Lindblad equation based on the PI state (solid curve) and the quantum jump Monte... 155
Figure 5.9. (a)-(c) Plots of the ad hoc potential U(θ) as a function of θ for different ω values. (d)-(f) Plots of the self-consistency function G(θ) versus θ for different ω... 162
Figure 5.10. Numerical solutions of the self-consistency equation. (a) Solution θ of the self-consistency equation, (b) density of active sites, and (c) susceptibility of the steady... 162
Figure 5.11. The phase diagram of the QCP model on SF networks. The type of PT depends on the degree exponent λ. If λ 〈 λc = 3, the transition point becomes zero...[이미지참조] 165
Figure 5.12. Numerical simulation results for the PTs of the QCP model on the static model. Plots of (a) the density of active sites 〈n〉 and (d) the susceptibility versus the... 167