在物理學、生物學、電腦科學、社會學和工程學領域，出現的許多隨機圖的數學處理中，我們常常得到多維不動點方程，其問題的解取決於不動點方程的根。通常情況下，不動點在其領域內總是有一個解。問題在於是否存在第二個解。在這篇論文中，針對這個問題，我們提出了只有一個解的條件和存在第二個解的條件，同時對不動點方程研究得出的結果，提出了可以應用的條件。

對不動點方程研究的結果，我們探討了兩面的應用。在第一個應用中，我們對經典構型模型進行了推廣。與經典配置模型一樣，廣義配置模型允許使用者指定任意的度分佈。在我們的廣義配置模型中，將配置模型的存根劃分為大小相等的b區塊，並選擇一個置換函數h，將一個區塊與另一個區塊聯繫起來。在每個區塊中，我們隨機指定一個與q成比例的數量，作為1類存根，其中q是一個在[0;1]範圍內的參數。其他的存根被指定為類型2存根。為了構建一個網路，隨機選擇一個未連接的存根。假設這個存根在i塊中，如果它是一個1類型存根，將這個存根與h(i)塊中隨機選擇的未連接的1型存根相連。如果它是一個2類型存根，就把它和一個隨機選擇的未連接的2 類型存根連接起來。我們重複這個過程，直到所有的存根都被連接起來。在一個假設下，我們得到了所構建的圖中兩個隨機鄰接頂點的聯合度分佈的封閉形式。基於這個聯合度分佈，我們表明，對於任何固定的b，Pearson度相關函數在q中是線性的。通過適當地選擇置換函數h，使得我們所構建的演算法，可以創建同種異構的網路，以及異種異構的網路。我們通過對這個模型進行滲濾分析，匯出了一個多維的不動點方程，其滿足主要結果可以應用的條件，通過大量的電腦類比驗證了我們的研究結果。

在第二個應用中，我們提出了一個易感-感染-恢復（SIR）模型，其中有戴口罩的個體和不戴口罩的個體。在該模型中，疾病傳播率、康復率和戴口罩的個體比例都是隨時間變化的。我們構建了一個疾病傳播率的漸進式估計方法，基於約翰-霍普金斯大學發佈的COVID-19資料得到了感染率與康復率，通過最大似然估計來確定戴口罩的個體比例。利用概率估計來確定戴口罩的個體比例，使得隨機的SIR型的過渡概率最大化。如果感染者的數量很大，過渡概率在數值上很難計算，在此我們利用中心極限定理和均值場理論，得到過渡概率的近似值。通過數值研究表明，我們的近似方法效果良好。同時我們開發了鍵滲流分析法來預測最終被感染的人口比例，在假設SIR模型的參數不再發生變化，被感染的人口比例不發生任何變化，鍵滲流分析匯出了二維的不動點方程。我們表明，這個不動點方程滿足了我們所研究的結果條件，滲濾閾值正是流行病的基本繁殖係數。

In many mathematical treatments of random graphs that arise in physics, biology, computer science, sociology and engineering, we often end up with multi-dimensional fixed point equations. The solution of the problems hinges on the roots of the fixed point equations. Typically, the fixed point equations always have a solution in their domain. The question is whether there is a second solution. In this thesis, we address this question. We give conditions, in which there is only one solution and conditions, in which there exists a second solution. We also give conditions on the fixed point equations that our result can be applied.

We present two applications of this result. In the first application, we present a generalization of the classical configuration model. Like the classical configuration model, the generalized configuration model allows users to specify an arbitrary degree distribution. In our generalized configuration model, we partition the stubs in the configuration model into b blocks of equal sizes and choose a permutation function h to associate one block with another. In each block, we randomly designate a number proportional to q of stubs as type 1 stubs, where q is a parameter in the range [0;1]. Other stubs are designated as type 2 stubs. To construct a network, randomly select an unconnected stub. Suppose that this stub is in block i. If it is a type 1 stub, connect this stub to a randomly selected unconnected type 1 stub in block h(i). If it is a type 2 stub, connect it to a randomly selected unconnected type 2 stub. We repeat this process until all stubs are connected. Under an assumption, we derive a closed form for the joint degree distribution of two random neighboring vertices in the constructed graph. Based on this joint degree distribution, we show that the Pearson degree correlation function is linear in q for any fixed b. By properly choosing h, we show that our construction algorithm can create assortative networks as well as disassortative networks. We present a percolation analysis of this model. This analysis leads to a multi-dimensional fixed point equation. We show that this fixed point equation satisfies the conditions, in which our main result can be applied. We verify our results by extensive computer simulations.

In the second application, we present a susceptible-infected-recovered(SIR) model with individuals wearing facial masks and individuals who do not. The disease transmission rates, the recovering rates and the fraction of individuals who wear masks are all time dependent in the model. We develop a progressive estimation of the disease transmission rates and the recovering rates based on the COVID-19 data published by Johns Hopkins University. We determine the fraction of individual who wear masks by a maximum likelihood estimation, which maximizes the transition probability of a stochastic susceptible-infected-recovered model. The transition probability is numerically difficult to compute if the number of infected individuals is large. We develop an approximation for the transition probability based on central limit theorem and mean field approximation. We show through numerical study that our approximation works well. We develop a bond percolation analysis to predict the eventual fraction of population who are infected, assuming that parameters of the SIR model do not change any more. We show that the bond percolation analysis leads to a two dimensional fixed point equation. We show that this fixed point equation satisfies the conditions in which our main result can be applied. The percolation threshold is exactly the basic reproduction number of the epidemic.

中文摘要 i
Abstract iii
Acknowledgements vi
List of Figures x
List of Tables xii
1 Introduction 1
1.1 Application I - Generalized Configuration Model 6
1.2 Application II - Epidemic Networks with Face-MaskWearers 8
1.3 Organization of this Thesis 9
2 A Multi-Dimensional Fixed Point Equation 12
2.1 Attracting and Repelling Fixed Points 12
2.2 Proof of Theorem 5 17
3 The Generalized Configuration Model 24
3.1 Construction of a Random Network 29
3.2 Joint Distribution of Degrees 35
3.3 Assortativity and Disassortativity 42
3.4 An Application: Percolation 46
3.5 Numerical and Simulation Results 49
3.6 Conclusions 54
4 Epidemic Spreading in a Social Network with Facial Masks
Wearing Individuals 56
4.1 Time Dependent SIR Model 60
4.2 Maximum Likelihood Estimation of p(t) 65
4.3 Clustering Coefficient 79
4.4 Percolation Analysis and Percolation Threshold 84
4.5 Numerical and Simulation Results 89
4.6 Conclusions 97
5 Future Work 99
Appendix 101
Bibliography 108

[1] M. Newman, Networks: An Introduction. New York: Oxford University Press, 2010.
[2] D. Easley and J. Kleinberg, Networks, crowds and markets reasoning about a highly connected world. Cambridge University Press, 2010.
[3] A. Granas and J. Dugundji, Fixed Point Theory. New York: Springer, 2003.
[4] R. P. Agarwal, M. Meehan, and D. O’regan, Fixed point theory and applications. Cambridge university press, 2009, vol. 141.
[5] K. Goebel and W. Kirk, Topics in metric fixed point theory. Cambridge University Press, 2009.
[6] R. Cohen, K. Erez, D. ben Avraham, and S. Havlin, “Resilence of the internet to random breakdowns,” Phys. Rev. Lett., vol. 85, pp. 4626–4628, 2000.
[7] S. Karlin and H. M. Taylor, A first course on stochastic processes. New York: Academic Press, 1966.
[8] E. A. Bender and E. R. Canfield, “The asymptotic number of labelled graphs with given degree sequences,” J. Comb. Theory Series A, vol. 24, pp. 296–307, 1978.
[9] Béla Bollobás, “A probabilistic proof of an asymptotic formula for the number of labelled regular graphs,” European Journal of Combinatorics, vol. 1, pp. 311–316, 1980.
[10] M. Molloy and B. Reed, “A critical point for random graphs with a given degree sequence,” Random Struct. Alg., vol. 6, pp. 161–179, 1995.
[11] M. Molloy and B. Reed, “The size of the giant component of a random graph with a given degree sequence,” Combinatorics, Probability and Computing, vol. 7, pp. 295–306, 1998.
[12] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, “Random graphs with arbitrary degree distributions and their applications,” Physical Review E, vol. 64, p. 026118, 2001.
[13] A. C. K. Lai, C. K. M. Poon, and A. C. T. Cheung, “Effectiveness of facemasks to reduce exposure hazards for airborne infections among general populations,” Journal of the Royal Society Interface, vol. 9, no. 70, p. 938, 2012.
[14] N. C. J. Brienen, A. Timen, J. Wallinga, J. E. V. Steenbergen, and P. F. M. Teunis, “The effect of mask use on the spread of influenza during a pandemic,” Risk Analysis, vol. 30, no. 8, pp. 1210–1218, 2010.
[15] A. Davies, K.-A. Thompson, K. Giri, G. Kafatos, J. Walker, and A. Bennett, “Testing the efficacy of homemade masks: Would they protect in an influenza pandemic?” Disaster medicine and public health preparedness, vol. 7, pp. 413–418, August 2013.
[16] R. B. Patel, S. D. Skaria, M. M. Mansour, and G. C. Smaldone,“Respiratory source control using a surgical mask: An in vitro study,”Journal of Occupational & Environmental Hygiene, 2016.
[17] CSSEGISandData and J. H. University, Covid-19, 2020. [Online]. Available: https://github.com/CSSEGISandData/COVID-19
[18] D. Kempe, J. Kleinberg, and É. Tardos, “Maximizing the spread of influence through a social network,” in Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, 2003, pp. 137–146.
[19] R. L. Devaney, A First Course in Chaotic Dynamic Systems: Theory. United States of America: Addison-Wesley, 1992.
[20] C. Meyer, Matrix analysis and applied linear algebra. Philadelphia, USA: SIAM, 2000.
[21] W. Kulpa, “The Poincaré-Miranda theorem,” The American Mathematical Monthly, vol. 104(6), no. 6, pp. 545–550, 1997.
[22] W. Rheinboldt and J. Vandergraft, “A simple approach to the Perron–Frobenius theory for positive operators on general partiallyordered finite-dimensional linear spaces,” Math. Comput., vol. 27,p. 139–145, 1973.
[23] C. Guiver, “On the strict monotonicity of spectral radii for classes of bounded positive linear operators,” Positivity, vol. 22, pp. 1173–1190, 2018.
[24] A. Barabási and R. Albert, “Emergence of scaling in random networks,”Science, vol. 286, pp. 509–512, 1999.
[25] D. J. Watts and S. H. Strogatz, “Collective dynamics of ’smallworld’networks,” Nature, vol. 393, pp. 440–442, 1998.
[26] M. Newman, “Mixing patterns on networks,” Phys. Rev. E, vol. 67,p. 026126, 2003.
[27] D. N. Litvak and R. van der Hofstad, “Degree-degree correlations in random graphs with heavy-tailed degrees,” Enschede, the Netherlands, October 2012. [Online]. Available: http://doc.utwente.nl/84367/
[28] R. Xulvi-Brunet and I. M. Sokolov, “Changing correlations in networks: assortativity and dissortativity,” Acta Physica Polonica B, vol. 36, no. 5, pp. 1431–1455, 2005.
[29] Z. Nikoloski, N. Deo, and L. Kucera, “Degree-correlation of a scale-free random graph process,” in Proc. Discrete Mathematics and Theoretical Computer Science, vol. AE, 2005, pp. 239–244.
[30] M. Boguñá, R. Pastor-Satorras, and A. Vespignani, “Absence of epidemic threshold in scale-free networks with degree correlations,” Physical Review Letters, vol. 90, p. 028701, 2003.
[31] Y. Moreno, J. B. Gómez, and A. F. Pacheco, “Epidemic incidence in correlated complex networks,” Physical Review E, vol. 68, p. 035103, 2003.
[32] M. Boguñá and R. Pastor-Satorras, “Epidemic spreading in correlated complex networks,” Physical Review E, vol. 66, p. 047104, 2002.
[33] V. M. Eguíluz and K. Klemm, “Epidemic threshold in structured scale-free networks,” Physical Review Letters, vol. 89, no. 10, 108701, 2002.
[34] M. Schläpfer and L. Buzna, “Decelerated spreading in degreecorrelated networks,” Physical Review E, vol. 85, p. 015101, 2012.
[35] S. Johnson, J. J. Torres, J. Marro, and M. A. Munoz, “The entropic origin of disassortativity in complex networks,” Physical Rev. Letter, vol. 104, p. 108702, 2010.
[36] D. Braha and Y. Bar-Yam, “The statistical mechanics of complex product development: Empirical and analytical results,” Management Science, vol. 53, no. 7, pp. 1127–1145, 2007.
[37] D. Braha, “The complexity of design networks: Structure and dynamics,”in Experimental Design Research, P. Cash, T. S. Mario, and Štorga, Eds., 2016, pp. 129–151.
[38] A. Pomerance, E. Ott, M. Girvan, and W. Losert, “The effect of network topology on the stability of discrete state models of genetic control,” Proceedings of the National Academy of Sciences, vol. 106, pp. 8209–8214, 2009.
[39] A. Vázquez and Y. Moreno, “Resilence to damage of graphs with degree correlations,” Physical Review E, vol. 67, p. 015101, 2003.
[40] D. S. Callaway, J. E. Hopcroft, J. M. Kleinberg, M. E. J. Newman, and S. H. Strogatz, “Are randomly grown graphs really random?” Physical Review E, vol. 64, p. 041902, 2001.
[41] M. Catanzaro, G. Caldarelli, and L. Pietronenero, “Assortative model for social networks,” Physical Review E, vol. 70, p. 037101,2004.
[42] J. Zhou, X. Xu, J. Zhang, J. Sun, M. Small, and J.-A. Lu, “Generating an assortative network with a given degree distribution,” Intern. J. of Bifuration and Chaos, vol. 18, no. 11, pp. 3495– 3502, 2008.
[43] A. Ramezanpour, V. Karimipour, and A. Mashaghi, “Generating correlated networks from uncorrelated ones,” Physical Review E, vol. 67, p. 046107, 2005.
[44] K. E. Bassler, C. I. D. Genio, Péter L Erd˝os, István Miklós, and Zoltán Toroczkai, “Exact sampling of graphs with prescribed degree correlations,” New Journal of Physics, vol. 17, p. 083052, 2015.
[45] J. Cardy and P. Grassberger, “Epidemic models and percolation,” J. Phys. A: Math. Gen., vol. 18, no. 6, 1985.
[46] C. Moore and M. Newman, “Epidemics and percolation in smallworld networks,” Phys. Rev. E, vol. 61, p. 5678, 2000.
[47] L. Sander, C. Warren, I. Sokolov, C. Simon, and J. Koopman,“Percolation on heterogeneous networks as a model for epidemics, Mathematical Biosciences, vol. 80, pp. 293–305, November-December 2002.
[48] N. Schwartz, R. Cohen, D. ben Avraham, A.-L. Barabasi, and S. Havlin, “Percolation in directed scale-free networks,” Phys. Rev.E., vol. 66, p. 015104, 2002.
[49] M. Newman, “Assortative mixing in networks,” Physical Review Letters, vol. 89, p. 208701, 2002.
[50] Erd˝os and Rényi , “On random graphs,” Publicationes Mathematicicae, vol. 6, pp. 290–297, 1959.
[51] H. Klein-Hennig and A. K. Hartmann, “Bias in generation of random graphs,” Physical Review E, vol. 85, p. 026101, 2012.
[52] P. Bratley, B. L. Fox, and L. E. Schrage, A Guide to Simulation, 2nd ed. New York: Springer-Verlag, 1987.
[53] A. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications. New York: Springer, 2011.
[54] W. Li, J. Zhou, and J.-a. Lu, “The effect of behavior of wearing masks on epidemic dynamics,” Nonlinear Dynamics, vol. 101, no. 3, pp. 1995–2001, 2020.
[55] O. Damette, “Zorro versus covid-19: fighting the pandemic with face masks,” medRxiv, 2021.
[56] C. Betsch, L. Korn, P. Sprengholz, L. Felgendreff, S. Eitze, P. Schmid, and R. Böhm, “Social and behavioral consequences of mask policies during the covid-19 pandemic,” Proceedings of the National Academy of Sciences, vol. 117, no. 36, pp. 21 851–21 853, 2020.
[57] M. Opuszko and J. Ruhland, “Impact of the network structure on the sir model spreading phenomena in online networks,” in Conference: The Eighth International Multi-Conference on Computing in the Global Information Technology ICCGI, Nice, France, 2013.
[58] D. F. Bernardes, M. Latapy, and F. Tarissan, “Relevance of sir model for real-world spreading phenomena: Experiments on a largescale p2p system,” in 2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining. IEEE, 2012, pp. 327–334.
[59] Z.Wang, Q. Guo, S. Sun, and C. Xia, “The impact of awareness diffusion on sir-like epidemics in multiplex networks,” Applied Mathematics and Computation, vol. 349, pp. 134–147, 2019.
[60] C. Zheng, C. Xia, Q. Guo, and M. Dehmer, “Interplay between sirbased disease spreading and awareness diffusion on multiplex networks,” Journal of Parallel and Distributed Computing, vol. 115, pp. 20–28, 2018.
[61] E. Coupechoux and M. Lelarge, “How clustering affects epidemics in random networks,” Advances in Applied Probability, vol. 46, no. 4, pp. 985–1008, 2014.
[62] P. Trapman, “On analytical approaches to epidemics on networks,”Theoretical population biology, vol. 71, no. 2, pp. 160–173, 2007.
[63] T. House and M. J. Keeling, “Deterministic epidemic models with explicit household structure,” Mathematical biosciences, vol. 213, no. 1, pp. 29–39, 2008.
[64] T. House and M. Keeling, “Household structure and infectious disease transmission,” Epidemiology & Infection, vol. 137, no. 5, pp.654–661, 2009.
[65] B. Rockx, T. Kuiken, S. Herfst, T. Bestebroer, M. M. Lamers,B. B. O. Munnink, D. de Meulder, G. van Amerongen, J. van den Brand, N. M. Okba et al., “Comparative pathogenesis of covid-19, mers, and sars in a nonhuman primate model,” Science, vol. 368, no. 6494, pp. 1012–1015, 2020.
[66] Z. Du, X. Xu, Y. Wu, L. Wang, B. J. Cowling, and L. A. Meyers, “Serial interval of covid-19 among publicly reported confirmed cases,” Emerging infectious diseases, vol. 26, no. 6, p. 1341, 2020.
[67] S. Ghahramani, Fundamentals of Probability with Stochastic Processes, 3rd ed. Pearson Prentice Hall, 2005.
[68] A. Wipf, “Statistical approach to quantum field theory : An introduction,” Lect.Notes Phys., vol. 864, pp. 1–390, Nov 2012.
[69] M. Newman and D. Watts, “Scaling and percolation in the smallworld network model,” Phys. Rev. E, vol. 60, pp. 7332–7342, 2000.
[70] Y. Hu, B. Ksherim, R. Cohen, and S. Havlin, “Percolation in interdependent and interconnected networks: Abrupt change from second- to first-order transitions,” Phys. Rev. E., vol. 84, p. 066116, 2011.
[71] M. Á. Serrano and M. Boguná, “Percolation and epidemic thresholds in clustered networks,” Physical review letters, vol. 97, no. 8, p. 088701, 2006.
[72] The death toll in Wuhan was revised from 2579 to 3869 (4/17), 2020. [Online]. Available: http://www.china.org.cn/china/Off_the_Wire/2020-04/17/content_75943843.htm
[73] Wikipedia, COVID-19 pandemic in Beijing, 2020. [Online].Available: https://en.wikipedia.org/wiki/COVID-19_pandemic_in_Beijing#June
[74] Wikipedia, COVID-19 pandemic in Xinjiang, 2020. [Online]. Available: https://en.wikipedia.org/wiki/COVID-19_pandemic_in_Xinjiang#July_2020
[75] en.people.cn, Officials rule out domestic transmission as origin of Dalian COVID-19 cluster, 2020. [Online]. Available: http://en.people.cn/n3/2020/0803/c90000-9717687.html
[76] T. of India, Coronavirus recovery: What do India’s high COVID recovery numbers really mean? We explain, 2020. [Online]. Available: https://timesofindia.indiatimes.com/life-style/health-fitness/health-news/coronavirus-recovery-what-do-indias-high-covid-recovery-numbers-really-mean-we-explain/photostory/78102510.cms
[77] H. Times, India’s Covid-19 recovery rate surges past 78%, one of the highest globally: Govt, 2020. [Online]. Available: https://www.hindustantimes.com/india-news/india-s-covid-19-recovery-rate-surges-past-78-one-of-the-highest-globally-govt/story-wU8YaDW6qG6WEBieTzoD7J.html
[78] Wikipedia, COVID-19 pandemic in South Korea, 2020, 2020.[Online]. Available: https://en.wikipedia.org/wiki/COVID-19pandemicinSouthKorea
[79] Wikipedia, COVID-19 pandemic in Japan, 2020, 2020. [Online].Available: https://en.wikipedia.org/wiki/COVID-19pandemicinJ
apan
[80] J. Leskovec and A. Krevl, “SNAP Datasets: Stanford large network dataset collection,” http://snap.stanford.edu/data, Jun. 2014.