<p>Individual proteins, and small collections of proteins, have been extensively studied for at least two hundred years. Today, more than 350 genomes have been completely sequenced and the proteomes of these genomes have been at least partially mapped. The inventory of protein coding genes is the first step toward understanding the cellular machinery. Recent studies have generated a comprehensive data set for the physical interactions between the proteins of <i>Saccharomyces cerevisiae</i>, in addition to some less extensive proteome interaction maps of higher eukaryotes. Hence, it is now becoming feasible to investigate important questions regarding the evolution of protein-protein networks. For instance, what is the evolutionary relationship between proteins that interact, directly or indirectly? Do interacting proteins co-evolve? Are they often derived from each other? In order to perform such proteome-wide investigations, a top-down view is necessary. This is provided by network (or graph) theory.</p><p>The proteins of the cell may be viewed as a community of individual molecules which together form a society of proteins (nodes), a network, where the proteins have various kinds of relationships (edges) to each other. There are several different types of protein networks, for instance the two networks studied here, namely metabolic networks and protein-protein interaction networks. The metabolic network is a representation of metabolism, which is defined as the sum of the reactions that take place inside the cell. These reactions often occur through the catalytic activity of enzymes, representing the nodes, connected to each other through substrate/product edges. The indirect interactions of metabolic enzymes are clearly different in nature from the direct physical interactions, which are fundamental to most biological processes, which constitute the edges in protein-protein interaction networks.</p><p>This thesis describes three investigations into the evolution of metabolic and protein-protein interaction networks. We present a comparative study of the importance of retrograde evolution, the scenario that pathways assemble backward compared to the direction of the pathway, and patchwork evolution, where enzymes evolve from a broad to narrow substrate specificity. Shifting focus toward network topology, a suggested mechanism for the evolution of biological networks, preferential attachment, is investigated in the context of metabolism. Early in the investigation of biological networks it seemed clear that the networks often display a particular, 'scale-free', topology. This topology is characterized by many nodes with few interaction partners and a few nodes (hubs) with a large number of interaction partners. While the second paper describes the evidence for preferential attachment in metabolic networks, the final paper describes the characteristics of the hubs in the physical interaction network of <i>S. cerevisiae</i>.</p>

The study of complex networks presented a huge development in last decades. In this dissertation we want to analyze the epidemic spread in scale-free networks through the Susceptible - Infected - Susceptible (SIS) model. We review the fundamental concepts to describe complex networks and the classical epidemiological models. We implement an algorithm that produces a scale-free network and explore the Quenched Mean-Field (QMF) dynamics in a scale-free network. Moreover, we simulate a change on the topology of the network according to the states of the nodes, and it generates a positive epidemic threshold. We show analytically that the fraction of infected vertices follows a power-law distribution in the vicinity of this critical point / O estudo de redes complexas tem se desenvolvido muito nos últimos anos. Nesta dissertação queremos analisar o processo de propagação de epidemia em redes livres de escala através do modelo Suscetível - Infectado - Suscetível (SIS). Apresentamos uma revisão de redes e as principais características dos modelos epidemiológicos clássicos. Implementamos um algoritmo que produz uma rede livre de escala dado um expoente e exploramos a dinâmica do modelo Quenched Mean-Field (QMF) inserido em uma rede livre de escala. Além disso, foi simulada uma possível alteração na topologia da rede, devido aos estados dos vértices infectados, que gerou um limiar epidêmico positivo no modelo e a probabilidade de vértices infectados seguiu uma lei de potência na vizinhança desse ponto crítico

The study of complex networks presented a huge development in last decades. In this dissertation we want to analyze the epidemic spread in scale-free networks through the Susceptible - Infected - Susceptible (SIS) model. We review the fundamental concepts to describe complex networks and the classical epidemiological models. We implement an algorithm that produces a scale-free network and explore the Quenched Mean-Field (QMF) dynamics in a scale-free network. Moreover, we simulate a change on the topology of the network according to the states of the nodes, and it generates a positive epidemic threshold. We show analytically that the fraction of infected vertices follows a power-law distribution in the vicinity of this critical point / O estudo de redes complexas tem se desenvolvido muito nos últimos anos. Nesta dissertação queremos analisar o processo de propagação de epidemia em redes livres de escala através do modelo Suscetível - Infectado - Suscetível (SIS). Apresentamos uma revisão de redes e as principais características dos modelos epidemiológicos clássicos. Implementamos um algoritmo que produz uma rede livre de escala dado um expoente e exploramos a dinâmica do modelo Quenched Mean-Field (QMF) inserido em uma rede livre de escala. Além disso, foi simulada uma possível alteração na topologia da rede, devido aos estados dos vértices infectados, que gerou um limiar epidêmico positivo no modelo e a probabilidade de vértices infectados seguiu uma lei de potência na vizinhança desse ponto crítico

Individual proteins, and small collections of proteins, have been extensively studied for at least two hundred years. Today, more than 350 genomes have been completely sequenced and the proteomes of these genomes have been at least partially mapped. The inventory of protein coding genes is the first step toward understanding the cellular machinery. Recent studies have generated a comprehensive data set for the physical interactions between the proteins of Saccharomyces cerevisiae, in addition to some less extensive proteome interaction maps of higher eukaryotes. Hence, it is now becoming feasible to investigate important questions regarding the evolution of protein-protein networks. For instance, what is the evolutionary relationship between proteins that interact, directly or indirectly? Do interacting proteins co-evolve? Are they often derived from each other? In order to perform such proteome-wide investigations, a top-down view is necessary. This is provided by network (or graph) theory. The proteins of the cell may be viewed as a community of individual molecules which together form a society of proteins (nodes), a network, where the proteins have various kinds of relationships (edges) to each other. There are several different types of protein networks, for instance the two networks studied here, namely metabolic networks and protein-protein interaction networks. The metabolic network is a representation of metabolism, which is defined as the sum of the reactions that take place inside the cell. These reactions often occur through the catalytic activity of enzymes, representing the nodes, connected to each other through substrate/product edges. The indirect interactions of metabolic enzymes are clearly different in nature from the direct physical interactions, which are fundamental to most biological processes, which constitute the edges in protein-protein interaction networks. This thesis describes three investigations into the evolution of metabolic and protein-protein interaction networks. We present a comparative study of the importance of retrograde evolution, the scenario that pathways assemble backward compared to the direction of the pathway, and patchwork evolution, where enzymes evolve from a broad to narrow substrate specificity. Shifting focus toward network topology, a suggested mechanism for the evolution of biological networks, preferential attachment, is investigated in the context of metabolism. Early in the investigation of biological networks it seemed clear that the networks often display a particular, 'scale-free', topology. This topology is characterized by many nodes with few interaction partners and a few nodes (hubs) with a large number of interaction partners. While the second paper describes the evidence for preferential attachment in metabolic networks, the final paper describes the characteristics of the hubs in the physical interaction network of S. cerevisiae.

Wealth distribution is an important issue in Economics, especially wealth inequality. There are no absolutely perfect solutions to this issue long times ago. Despite we are in 21 century, the situation are getting worse and hard to resolve. We focus on developing an agent simulation model based on Evolutionary game and Scale-free network. From this model, we observed some phenomena in wealth distribution by changing the network structure. And we experiments 5 strategies to increase the connectivity of agents in network, which increasing the edges to fully-connected network or increasing the edges between two different groups. After these experiments, we find that the increasing of connectivity in the network is positive to the agents¡¦ wealth accumulation which means we build more relationship with the other people is benefit to our wealth accumulation. Furthermore, we divide the agents in the simulation to three groups: the poor, the middle class and the rich. In the group simulation, we find that increasing the connectivity inside the poor group is the best way to decrease the Gini coefficient and wealth inequality.

Phan (2003)等人的研究首先採用網路結構的觀點，來進行在獨占市場下的消費行為分析。本文延伸Phan等人的研究，採用多樣性的不同網路結構，特別是以「無標度網路」(scale-free)作為市場背後的網路結構，藉此來探討網路對於市場需求行為的影響。同時，為了有效地進行量化分析，我們建立了一些數學公式，以便能夠精準地比較不同網路對於需求面的影響程度，其中包含了消費者剩餘、雪崩效果和磁滯效果等。在實驗結果中，我們發現網路結構的確會影響到消費行為的表現，並且當市場規模變大時，仍存在著一些因網路不同而有的差異性。 / The economic implications of network topologies are studied via a monopolist's model of market networks originally proposed by Phan, et al. (2003). By embedding the market into a larger collection of network topologies, in particular, a class of scale-free networks, we extend the early analysis built upon a class of ring networks. To facilitate the study of the impacts of network topologies upon market demand, various measures concerning social welfare (the consumer's surplus), the avalanche effect, and the hysteresis effect, are formally established. Comparisons based on these measures show that network topologies matter, and their implied differences will remain even when the network size becomes large.