Nowadays, with the rapid expansion of increasing utilization of renewable energy sources, power grid is evolving into a much complex man-made system in the technological age. Under the new circumstances, traditional methodologies for power system security analysis are facing a serious challenge. For the past decade, many countries have experienced large blackouts, which expose potential problems of current models and methodologies in power industry. On the other hand, since the 9.11 event and frequent suicide bombing attacks in some countries, terrorism has become a major threat for national security. With the extensive growth of terrorism activities, power system, the significant critical infrastructure, probably becomes the target of terrorists. If this happens, the impact is dramatically severe and may yield more frequent blackouts. This Ph.D. thesis aims at developing some advanced models and methodologies for exploring the vulnerability of power system and protecting it against potential terrorism threat. The dissertation mainly consists of the following four parts. (1)Complex network theory based power system security and vulnerability analysis methodologies are introduced. Mathematically, an interconnected complex power grid can be described as a complex network of nodes connected by edges. Generally speaking, topology parameters of network structure include important information of the structure. That is to say, some critical nodes and lines can have significant impact on large-scale blackouts. The thesis will present a new methodology to recognize those critical nodes and lines in power grids. (2)Complex system theory based power grid security and vulnerability analysis methodologies are presented. Power grid is a complex dynamic evolutionary system over years with continuous expansion so as to underpin the ongoing increase of power demand. Some properties of complex systems may have important relationship with large-scale blackouts. In other words, there may be some stages of evolutionary power systems that would be more likely to cause large blackouts. The thesis will investigate the relationship to identify those critical stages of power grids. (3)Game theory is applied to methodologies for power system security and vulnerability analysis. Terrorists are often considered as fully intelligent and strategic actors who can even hire scientists and power engineers to seek the vulnerability of power systems and then launch a vital attack. Game theory does treat actors as fully strategic players and has been successfully applied to many disciplines including economics, political science and military. The thesis will present new models and analysis methods for protecting power systems under terrorism attacks. (4)Cyber security technology is considered in power system security and vulnerability analysis. It is known that information technology plays an import role in today and next generation grid. In this situation, cyber security should be an important issue. If it is vulnerable to malicious threats such as hackers and cyber-terrorists, power grid will not reach its full capabilities. The thesis will present an initial framework to reduce the vulnerability of power grid against potential cyber attack.

Nowadays, with the rapid expansion of increasing utilization of renewable energy sources, power grid is evolving into a much complex man-made system in the technological age. Under the new circumstances, traditional methodologies for power system security analysis are facing a serious challenge. For the past decade, many countries have experienced large blackouts, which expose potential problems of current models and methodologies in power industry. On the other hand, since the 9.11 event and frequent suicide bombing attacks in some countries, terrorism has become a major threat for national security. With the extensive growth of terrorism activities, power system, the significant critical infrastructure, probably becomes the target of terrorists. If this happens, the impact is dramatically severe and may yield more frequent blackouts. This Ph.D. thesis aims at developing some advanced models and methodologies for exploring the vulnerability of power system and protecting it against potential terrorism threat. The dissertation mainly consists of the following four parts. (1)Complex network theory based power system security and vulnerability analysis methodologies are introduced. Mathematically, an interconnected complex power grid can be described as a complex network of nodes connected by edges. Generally speaking, topology parameters of network structure include important information of the structure. That is to say, some critical nodes and lines can have significant impact on large-scale blackouts. The thesis will present a new methodology to recognize those critical nodes and lines in power grids. (2)Complex system theory based power grid security and vulnerability analysis methodologies are presented. Power grid is a complex dynamic evolutionary system over years with continuous expansion so as to underpin the ongoing increase of power demand. Some properties of complex systems may have important relationship with large-scale blackouts. In other words, there may be some stages of evolutionary power systems that would be more likely to cause large blackouts. The thesis will investigate the relationship to identify those critical stages of power grids. (3)Game theory is applied to methodologies for power system security and vulnerability analysis. Terrorists are often considered as fully intelligent and strategic actors who can even hire scientists and power engineers to seek the vulnerability of power systems and then launch a vital attack. Game theory does treat actors as fully strategic players and has been successfully applied to many disciplines including economics, political science and military. The thesis will present new models and analysis methods for protecting power systems under terrorism attacks. (4)Cyber security technology is considered in power system security and vulnerability analysis. It is known that information technology plays an import role in today and next generation grid. In this situation, cyber security should be an important issue. If it is vulnerable to malicious threats such as hackers and cyber-terrorists, power grid will not reach its full capabilities. The thesis will present an initial framework to reduce the vulnerability of power grid against potential cyber attack.

This thesis develops a methodology for designing resilient system-of-systems (SoS) networks. This methodology includes a capability-based resilience assessment framework, used to quantify SoS resilience. A complex networks approach is used to generate potential SoS network designs, focusing on scale-free and random network topologies, degree-based and random rewiring adaptation, and targeted and random node removal threats. Statistical design methods, specifically response surface methodology, are used to evaluate SoS networks and provide an understanding of the advantages and disadvantages of potential designs. Linear regression is used to model a continuous representation of the network design space, and determine optimally resilient networks for particular threat types. The methodology is applied to an information exchange (IE) network model (i.e., a message passing network model) and military command and control (C2) model. Results show that optimally resilient IE network topologies are random for networks with adaptation, regardless of the threat type. However, the optimally resilient adaptation method sharply transitions from being fully random to fully degree-based as threat randomness increases. These findings suggest that intermediately defined networks should not be considered when designing for resilience. Cost-benefit analysis of C2 networks suggests that resilient C2 networks are more cost-effective than robust ones, as long as the cost of rewiring network links is less than three-fourths the cost of creating new links. This result identifies a threshold for which a resilient network design approach is more cost-effective than a robust one.This thesis develops a methodology for designing resilient system-of-systems (SoS) networks. This methodology includes a capability-based resilience assessment framework, used to quantify SoS resilience. A complex networks approach is used to generate potential SoS network designs, focusing on scale-free and random network topologies, degree-based and random rewiring adaptation, and targeted and random node removal threats. Statistical design methods, specifically response surface methodology, are used to evaluate SoS networks and provide an understanding of the advantages and disadvantages of potential designs. Linear regression is used to model a continuous representation of the network design space, and determine optimally resilient networks for particular threat types. The methodology is applied to an information exchange (IE) network model (i.e., a message passing network model) and military command and control (C2) model. Results show that optimally resilient IE network topologies are random for networks with adaptation, regardless of the threat type. However, the optimally resilient adaptation method sharply transitions from being fully random to fully degree-based as threat randomness increases. These findings suggest that intermediately defined networks should not be considered when designing for resilience. Cost-benefit analysis of C2 networks suggests that resilient C2 networks are more cost-effective than robust ones, as long as the cost of rewiring network links is less than three-fourths the cost of creating new links. This result identifies a threshold for which a resilient network design approach is more cost-effective than a robust one.

Orientador: Andriana Susana Lopes de Oliveira Campanharo / Resumo: Os Modelos Baseados em Indivíduos (MBI’s) têm sido crescentemente empregados na modelagem de processos infecciosos. Um MBI consiste de uma estrutura na qual ocorrem interações entre um certo número de indivíduos, cujo comportamento é determinado por um conjunto de características que evoluem estocasticamente no tempo. Estudos recentes têm mostrado que as redes complexas constituem um suporte natural para o estudo da propagação de uma doença. Redes complexas são descritas por um conjunto de vértices (nós), arestas (conexões, ligações ou links) e algum tipo de interação entre os mesmos. Na formulação original do MBI e em modelos SIR (Suscetível, Infectado e Recuperado) e SEI (Suscetível, Exposto e Infectado), as relações entre os indivíduos são representadas por grafos completos, ou seja, todos os indivíduos estão conectados entre si. Como a topologia de uma rede real não pode ser descrita por uma rede puramente aleatória, nesse trabalho o MBI foi implementado de forma a incorporar modelos mais realísticos de redes de contato na propagação de uma doença infecciosa. De maneira geral, observou-se que redes complexas com diferentes topologias resultam em curvas de indivíduos suscetíveis, infectados e recuperados (ou suscetíveis, expostos e infectados) com diferentes comportamentos, e desta forma, que a evolução de uma dada doença, em particular a tuberculose, é altamente sensível à topologia de rede utilizada. Mais especificamente, observou-se que quanto maior o valor do comprimen... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: Individual-Based Models have been increasingly employed in the modeling of an infectious process. An IBM consists of a structure in which interactions occur between a certain number of individuals, whose behavior is determined by a set of characteristics that evolve stochastically in time. Recent studies have shown that complex networks are a natural framework for the study of a disease spread. Complex networks are described by a set of vertices (or nodes), edges (connections or links) and some type of interactions between them. In the original IBM approach and in SIR (Susceptible, Infected, Recovered) and SEI (Susceptible, Exposed and Infected) models, the relations between individuals are represented by complete graphs, that is, all individuals are connected to each other. Since the topology of a real network can not be described by a purely random network, in this work an IBM has been implemented in order to incorporate some realistic contact networks xvii models. In general, it was observed that complex networks with different topologies correspond to curves of susceptible, infected and recovered individuals (or susceptible, exposed and infected) with different behaviors, and thus, that the evolution of a given disease, in particular tuberculosis, is highly sensitive to a network topology. More specifically, it was observed that the higher the value of the mean jump length is, the faster the disease spreads and consequently, the higher is the number of infected individual... (Complete abstract click electronic access below) / Mestre

Orientador: Luiz Fernando Bovolato / Resumo: As Redes Complexas podem descrever vários tipos de sistemas importantes através da representação dos grafos. Com o aumento da capacidade de processamento e armazenamento dos computadores, tornou-se possível o acesso e a análise de várias bases de dados de diversas áreas, o que permitiu a comparação de redes do mundo real com os modelos de redes já existentes. Essas redes complexas apresentam propriedades que são úteis nas análises dos mais diversos aspectos das redes e com os mais variados propósitos. O presente trabalho tem por objetivo investigar as propriedades estruturais e de funcionamento das redes de distribuição de energia, considerando suas diferentes topologias, a fim de definir um modelo através do uso de Redes Complexas com o propósito de analisar o comportamento da rede, considerando, características de desempenho, resiliência e identificação de falhas. Nesta tese foi utilizada uma rede padrão chamada de Rede Modelo de distribuição de energia elétrica a qual foi modelada e simulada para servir de referência para comparar as métricas da Rede de Distribuição da cidade de Ilha Solteira. Nestes modelos analisados, os transformadores referem-se aos vértices da rede enquanto que as ligações entre eles representam as arestas do grafo. Um fato importante na utilização dos modelos é a possibilidade de estudar e detectar qualquer característica dos relacionamentos e assim direcionar recursos para uma proposta. Os alimentadores analisados em questão comparados a rede padrã... (Resumo completo, clicar acesso eletrônico abaixo) / Doutor

Esse trabalho consiste em um estudo sobre a aplicabilidade da medida de centralidade de intermediação (betweenness centrality) combinada com demandas de tráfego em redes viárias com o objetivo de identificar os principais pontos dessas redes. Como principais pontos refere-se aqui aos que aparecem com maior frequência entre os caminhos utilizados pelos motoristas que se deslocam pela rede viária. Trata-se de um estudo exploratório, que se inicia com a aplicação da centralidade de intermediação sobre redes simples, estendendo-se até simulações sobre redes baseadas em mapas reais. Nesse trabalho é analisado o comportamento da medida de centralidade sobre a topologia da rede - i.e. tanto sem considerar uma demanda, como considerando demandas de diversas magnitudes. Para ilustrar a proposta, os resultados são confrontados com valores de centralidade de intermediação calculados sobre as taxas de ocupação das vias extraídas de simulação microscópica. Ao final, foram apresentadas evidências de que o método proposto consegue explicar os fluxos de tráfego com melhor desempenho do que a centralidade de intermediação original. No entanto, o método mostrou-se muito sensível à função de custo utilizada na atribuição da demanda de tráfego ao grafo da rede. Os melhores resultados demonstrados pela abordagem proposta foram obtidos em experimentos sobre redes não regulares e com demandas de tráfego não uniformes. No caso de redes regulares com demanda uniforme, o melhor desempenho foi obtido pelo cálculo da centralidade sem considerar a demanda, mas atribuindo-se o custo unitário às arestas do grafo representativo da rede. / This work consists of a study of applicability of betweenness centrality combined with traffic demands in road networks with the objective of identifying their central points. By central points we refer to those which appear with high frequency among the paths used by drivers that move along the road network. It is an exploratory study, which begins with the application of the betweenness centrality on simple networks, extending to simulations on networks based on real maps. In this study we have analyzed the behavior of the metric over the network topology - i.e. without considering demand, as well as experiments considering demands with several magnitudes. To illustrate the proposed method, the results are compared with betweenness centrality values calculated over roadways occupation rates extracted from microscopic simulation. At the end, evidence that the proposed method can explain traffic flows with better performance than the original betweenness centrality were presented. However, the proposed method was shown to be very sensitive to the cost function used in the allocation of the graph network traffic demand. The best results demonstrated by the proposed approach were obtained in experiments on nonregular networks and non-uniform traffic demands. In the case of regular networks with uniform demand, the best performance was obtained by calculating the betweenness centrality without considering the demand, but assigning the unitary cost to the edges of the network graph.

In a complex adaptive system, diverse agents perform various actions without adherence to a predefined structure. The achievement of collaborative actions will be the result of continual interactions among them that shape a dynamic network. Agents may form an ad hoc organization based on the dynamic network of interactions for the purpose of achieving a long-term objective, which we termed a Network Organization (NO). Fervent and agile communication on social networking sites provides opportunities for potential issues to trigger individuals into individual actions as well as the attraction and mobilization of like-minded individuals into an NO that is both physically and virtually emergent. Examples are the rapid pace of Arab Spring proliferation and the diffusion rate of the Occupy Movement. We are motivated by a spontaneously formed NO as well as the quality of plasticity that enables the organization to change rapidly to describe an NO. Thus, we present a paradigm that serves as a reference model for organizations of socially networked individuals. This paradigm suggests modular components that can be combined to form an ad hoc network organization of agents. We touch on how this model accounts for external change in an environment through internal adjustment. For the predominant influences of the network substrate in an NO, multiple effects of it have an impact on the NO behaviors and directions. We envisioned several dimensions of such effects to include synergy, social capital, externality, influence, etc. A special focus in this work is measuring synergy and social capital as two predominant network effects. Synergy is perceived as different modalities of compatibility among agents when performing a set of coherent and correspondingly different actions. When agents are under no structural obligation to contribute, synergy is quantified through multiple forms of serendipitous agent chosen benevolence among them. The approach is to measure four types of benevolence and the pursuant synergies stemming from agent interactions. Social capital is another effect of networking that describes the accumulation of positive values of social flow and perceived trust plus abundance of communication over the common topic of NO. We provide measurement of social capital based on an agents’ expected benevolence. We examine those two effects in two different case studies — one case of a virtual organization and another of a real world terrorist organization — that best illustrate the main tenets of our conceptualization.

PAULA, Demétrius Ribeiro de. Dinâmica de redes neurais e formação de agregados em redes complexas. 2006. 90 f. Dissertação (Mestrado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2006. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2014-11-03T18:54:59Z No. of bitstreams: 1 2006_dis_drpaula.pdf: 978829 bytes, checksum: 6f46a6d334b9c066ec1350d74f8beb77 (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2014-11-03T19:00:07Z (GMT) No. of bitstreams: 1 2006_dis_drpaula.pdf: 978829 bytes, checksum: 6f46a6d334b9c066ec1350d74f8beb77 (MD5) / Made available in DSpace on 2014-11-03T19:00:08Z (GMT). No. of bitstreams: 1 2006_dis_drpaula.pdf: 978829 bytes, checksum: 6f46a6d334b9c066ec1350d74f8beb77 (MD5) Previous issue date: 2006 / The process by which news trends and ideas propagate in social communities can have a profound impact in the life of individuals. To understand thi process, we introduce a competitive cluster growth model in complex networks. In our model, each cluster represents the set of individuals with a certain opinion or preference. We investigate how the cluster size distribution depends on the topology of the network and how it is affected by the number of initial seeds dispersed in the structure. We study our model using different network models, namely, the Erdos-Renyi geometry, the preferential attachment model, and the so-called Apollonian network. This last complex geometry displays a cluster size distribution that follows a power-law with an exponent 1.0. Similar results have been obtained for the distributions of number of votes per candidate in the proportional elections for federal representation in Brazil. In the second part of this work, we investigate the temporal behavior of neural networks with small world topology and in networks built according to the preferential attachment model. In the first case we study the effect of the range of connections on the behavior of the time series. In both topologies, we detect the existence of cycles and investigate how their periods depend on the size of the system. / Este dissertação foi dividida em duas partes, na primeira parte nós propomos um modelo de crescimento competitivo de gregados em redes complexas para simular a propagação de idéias ou opiniões em comunidades. Investigamos como as distribuições de tamanhos de agregados variam com a topologia de construção da rede e com o número de sementes aleatoriamente dispersas na estrutura. Para tal, analisamos redes do tipo de Erdos-Rényi, redes de contato preferencial e a chamada rede Apoloniana. Esta última apresenta distribuições de tamanho de agregado em forma de uma lei de potência com um expoente aproximadamente 1. Resultados similares são observados com as distribuições obtidas para as frações de votos por candidato às eleições proporcionais para deputados no Brasil. Na segunda parte, analisamos o comportamento temporal da atividade neural em redes com características de mundo pequeno e em redes construídas segundo o modelo do contato preferencial. Nesta primeira topologia, estudamos como a série temporal se comporta com a variação do alcance das conexões. Em ambas as topologias, observamos a formação de períodos e investigamos como estes variam com o tamanho da rede.

REIS, Saulo Davi Soares e. Navegação em redes espacialmente correlacionadas. 2009. 72 f. Dissertação (Mestrado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2009. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2015-06-16T20:25:53Z No. of bitstreams: 1 2009_dis_sdsreis.pdf: 1915908 bytes, checksum: f38f094759742ce9d7cc35ad2d50209c (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2015-06-18T18:21:10Z (GMT) No. of bitstreams: 1 2009_dis_sdsreis.pdf: 1915908 bytes, checksum: f38f094759742ce9d7cc35ad2d50209c (MD5) / Made available in DSpace on 2015-06-18T18:21:10Z (GMT). No. of bitstreams: 1 2009_dis_sdsreis.pdf: 1915908 bytes, checksum: f38f094759742ce9d7cc35ad2d50209c (MD5) Previous issue date: 2009 / A significant number of real networks have well-defined spatial characteristics. We studied how network with spatially correlated topolgies can influence the processes of navigation through them. For this, we study the behavior of the average shortest-path length to networks defined within Kleinberg’s model [1, 2] to analyze the navigation dictated by rules of global knowledge. The Kleinberg’s model is characterized by allowing long-range connections between two vertices u and v distributed by a power-law probability distribution. For a better understanding of the topological characteristics presented by this family of networks, we applied the epidemic model susceptible-infected-susceptible (SIS) and we found that, we see that the Kleinberg’s model presents the small-world phenomenon only for a certain range of values of the clustering exponent α. We introduced a model of spatially embedded networks, conceptually based on the Kleinberg’s model. This model consist in introduction of a constrain to the distribution of long-range connections. We associate his constrain to a possible cost involved in the process of adding new long-range connections to the network. We studied how this cost constrain affects the navigation through the system, taking as a basis for comparison the work of Kleinberg for navigation with local knowledge, and our results conserning for navigation with global knowledge. / Um número significativo de redes reais apresentam características espaciais bem definidas. Nós estudamos como topologias de redes espacialmente correlacionadas podem influenciar processos de navegação através das mesmas. Para isso estudamos o comportamento do mínimo caminho médio para redes definidas dentro de modelo de Kleinberg para analisar a navegação ditada por regras de conhecimento global. O modelo que Kleinberg caracteriza-se por permitir conexões de longo alcance entre dois vértices u e v distribuídas por uma distribuição de probabilidade em lei de potência. Para um melhor entendimento das características topológicas apresentadas por essa família de redes, nós aplicamos o modelo epidêmico suscetível-infectado-suscetível (SIS), e com isso verificamos que o modelo de Kleinberg apresenta fenômeno de mundo pequeno apenas para uma determinada faixa de valores assumidos pelo expoente de agregação α. Em seguida, introduzimos um modelo de redes espacialmente embutidas, conceitualmente inspirado no modelo de Kleinberg. Este traduz-se na introdução de um vínculo para a distribuição das conexões de longo alcance. Associamos este vínculo a um possível custo envolvido no processo de adição de novas conexões de longo alcance à rede. Estudamos como esse vínculo no custo afeta a navegação na rede, tendo como base de comparação os trabalhos de Kleinberg para a navegação com conhecimento local da topologia, e nossos resultados considerando a navegação com conhecimento global.

JÁCOME, Samyr Silva Bezerra. Aspectos dinâmicos e estruturais em modelos de redes para sistemas complexos. 2009. 95 f. Tese (Doutorado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2009. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2015-06-18T18:53:05Z No. of bitstreams: 1 2009_tese_ssbjacome.pdf: 1272180 bytes, checksum: 4aaa7877f516c1dfd5dfe603734e397c (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2015-06-18T19:18:21Z (GMT) No. of bitstreams: 1 2009_tese_ssbjacome.pdf: 1272180 bytes, checksum: 4aaa7877f516c1dfd5dfe603734e397c (MD5) / Made available in DSpace on 2015-06-18T19:18:21Z (GMT). No. of bitstreams: 1 2009_tese_ssbjacome.pdf: 1272180 bytes, checksum: 4aaa7877f516c1dfd5dfe603734e397c (MD5) Previous issue date: 2009 / In this thesis we study systems where some form of disorder or non-homogeneity has a significant role at the complexity of the structural building or of the dynamics regulation of the system. First, we study the dynamics of Boolean networks, where the rules to update the state of the nodes are randomly chosen and control the global behavior of the system. At the critical threshold, and near to it, we propose that the transition to the critical regime can be characterized by the divergence of the relaxation time Tr. Based on simple scaling arguments, we show that the cumulative probability distribution of Tr decays as a power-law , with exponent iqual -1, for the annealed model at the critical region. Then, we study a novel method for network decomposition, which we apply to scale-free networks, that have the broad degree distribution as a fundamental feature. This method consists in a simultaneous and iterative remotion of groups of nodes with degree K until there are no more nodes with this degree in the network. Thus, we define new variables that characterize the process of decomposition and we obtain a set of well define exponents and parameters. From the behavior of these variables we can see, through some mathematical manipulations, that our method is self-consistent, serving as a useful tool for the study of the structural features of the network. At last, we study the backbones of the percolation cluster, where we use a network model with layers arranged in a disorderly way to represent some kind of anisotropy resistance to the percolation. Our numerical results indicate a break at the universality class on the fractal dimension and on the mass distribution of the backbones. / Nesta tese estudaremos sistemas onde alguma forma de desordem ou não-homogeneidade tem um importante papel na complexidade da formação estrutural ou da regulagem dinâmica do sistema. Primeiramente estudaremos a dinâmica das redes Booleanas, onde as regras de atualização escolhidas aleatoriamente controlam o comportamento global do sistema. Na condição crítica e próximo dela, propomos que a transição para o regime crítico pode ser caracterizado pela divergência do tempo de relaxação Tr. Baseados em simples argumentos de escalonamento, mostramos, além de outros resultados, que a probabilidade acumulativa da distribuição de Tr decai como uma lei de potência, com o expoente igual a -1, para o modelo annealed na região crítica. Em seguida estudamos um novo método de decomposição de redes aplicado às redes livres de escalas, onde a ampla distribuição de conectividade é um aspecto fundamental. O método consiste basicamente na retirada simultânea e iterativa de grupos de vértices com um determinado grau K de conectividade até que não haja mais sítios com este mesmo grau de conectividade na rede. Deste modo, definimos algumas variáveis que caractarizam o processo de decomposição e obtemos uma série de expoentes e parâmetros bem definidos. A partir do comportamento destas variáveis pudemos constatar por meio de algumas manipulações matemáticas que nosso método é auto-consistente, servindo como ótima ferramenta para estudo dos aspectos estruturais de uma rede. Por fim, estudamos os backbones, onde utilizamos um modelo de rede em que a desordem está no arranjo aleatório de camadas fáceis e difíceis à percolação. Os resultados numéricos indicam a quebra na classe de universalidade da geometria fractal e da distribuição de tamanhos de massa do backbones e também um comportamento assintótico da dimensão fractal no limite de grandes valores de massa e/ou anisotropia.