Statistical mechanics of complex networks

Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links. While traditionally these systems were modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks is governed by robust organizing principles. Here we review the recent advances in the field of complex networks, focusing on the statistical mechanics of network topology and dynamics. After reviewing the empirical data that motivated the recent interest in networks, we discuss the main models and analytical tools, covering random graphs, small-world and scale-free networks, as well as the interplay between topology and the network's robustness against failures and attacks.Comment: 54 pages, submitted to Reviews of Modern Physic

Comparing Brain Networks of Different Size and Connectivity Density Using Graph Theory

Graph theory is a valuable framework to study the organization of functional and anatomical connections in the brain. Its use for comparing network topologies, however, is not without difficulties. Graph measures may be influenced by the number of nodes (N) and the average degree (k) of the network. The explicit form of that influence depends on the type of network topology, which is usually unknown for experimental data. Direct comparisons of graph measures between empirical networks with different N and/or k can therefore yield spurious results. We list benefits and pitfalls of various approaches that intend to overcome these difficulties. We discuss the initial graph definition of unweighted graphs via fixed thresholds, average degrees or edge densities, and the use of weighted graphs. For instance, choosing a threshold to fix N and k does eliminate size and density effects but may lead to modifications of the network by enforcing (ignoring) non-significant (significant) connections. Opposed to fixing N and k, graph measures are often normalized via random surrogates but, in fact, this may even increase the sensitivity to differences in N and k for the commonly used clustering coefficient and small-world index. To avoid such a bias we tried to estimate the N,k-dependence for empirical networks, which can serve to correct for size effects, if successful. We also add a number of methods used in social sciences that build on statistics of local network structures including exponential random graph models and motif counting. We show that none of the here-investigated methods allows for a reliable and fully unbiased comparison, but some perform better than others

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com

Dynamics and steady-state properties of adaptive networks

Tese de doutoramento, Física, Universidade de Lisboa, Faculdade de Ciências, 2013Collective phenomena often arise through structured interactions among a system's constituents. In the subclass of adaptive networks, the interaction structure coevolves with the dynamics it supports, yielding a feedback loop that is common in a variety of complex systems. To understand and steer such systems, modeling their asymptotic regimes is an essential prerequisite. In the particular case of a dynamic equilibrium, each node in the adaptive network experiences a perpetual change in connections and state, while a comprehensive set of measures characterizing the node ensemble are stationary. Furthermore, the dynamic equilibria of a wide class of adaptive networks appear to be unique, as their characteristic measures are insensitive to initial conditions in both state and topology. This work focuses on dynamic equilibria in adaptive networks, and while it does so in the context of two paradigmatic coevolutionary processes, obtained results easily generalize to other dynamics. In the rst part, a low-dimensional framework is elaborated on using the adaptive contact process. A tentative description of the phase diagram and the steady state is obtained, and a parameter region identi ed where asymmetric microscopic dynamics yield a symmetry between node subensembles. This symmetry is accounted for by novel recurrence relations, which predict it for a wide range of adaptive networks. Furthermore, stationary nodeensemble distributions are analytically generated by these relations from one free parameter. Secondly, another analytic framework is put forward that detects and describes dynamic equilibria, while assigning to them general properties that must hold for a variety of adaptive networks. Modeling a single node's evolution in state and connections as a random walk, the ergodic properties of the network process are used to extract node-ensemble statistics from the node's long-term behavior. These statistical measures are composed of a variety of stationary distributions that are related to one another through simple transformations. Applying this fully self-su cient framework, the dynamic equilibria of three di erent avors of the adaptive contact process are subsequently described and compared. Lastly, an asymmetric variant of the coevolutionary voter model is motivated and proposed, and as for the adaptive contact process, a low-dimensional description is given. In a parameter region where a dynamic equilibrium lets the in nite system display perpetual dynamics, this description can be further reduced to a one-dimensional random walk. For nite system sizes, this allows to analytically characterize longevity of the dynamic equilibrium, with results being compared to the symmetric variant of the process. A nontrivial parameter combination is identi ed for which, in the low-dimensional description of the process, the asymmetric coevolutionary model emulates symmetric voter dynamics without topological coevolution. This emerging symmetry is partially con rmed for the full system and subsequently elaborated on. Slightly varying the original asymmetric model, an additional asymptotic regime is shown to occur that coexists with all others and complicates system description.A estrutura das interacções entre os constituintes elementares de um sistema está frequentemente na origem de comportamentos colectivos não triviais. Em redes adaptativas, esta estrutura de interacção evolui a par com a dinâaica que nela assenta, traduzindo uma retroacção que de comum encontrar em vários sistemas complexos. Resultados analíticos sobre os estados assimptóticos destes sistemas são uma peça essencial para a sua compreensão e controlo. O equilíbrio dinâmico de um caso particular de estado assimptótico em que cada nodo da rede adaptativa vai sempre mudando o seu estado e as suas ligações a outros nodos, enquanto que um conjunto de medidas que caracterizam estatisticamente o ensemble dos nodos mantêm valores fixos. Alémm disso, uma classe muito geral de redes adaptativas apresenta equilíbrios dinâmicos que parecem ser únicos, no sentido em que aqueles valores estacionários não dependem das condições iniciais, quer em termos do estados dos nodos quer em termos da topologia da rede.Este trabalho incide no estudo do equilíbrio dinâmiico de redes adaptativas no contexto particular de dois modelos paradigmáticos de coevolação, mas os principais resultados podem ser facilmente generalizados a outros processos. Na primeira parte, revisita-se e desenvolve-se uma abordagem da variante adaptativa do processo de contacto baseada num modelo de baixa dimensão. Obtem-se uma descrição aproximada do diagrama de fases do sistema e do equilíbrio dinâmico, e identifica-se nessa fase uma combinação de parâmetros para a qual a dinâmica microscópica, que de assimétrica nos estados dos nodos, da origem a uma simetria entre os dois subconjuntos de nodos. Esta simetria é explicada através da derivação de relações de recorrência para as distribuições de grau, que a preveêm para uma ampla classe de redes adaptativas. Estas relações permitem também gerar analiticamente as distribuições de grau estacionárias de cada subconjunto de nodos a partir de um parâmetro livre.Na segunda parte, desenvolve-se uma outra abordagem analítica que permite detectar e descrever o equilíbrio dinâmico, a partir de propriedades gerais que se têm que verificar em muitas redes adaptativas. Na base desta abordagem está a descrição do processo estocástico associado à evolução do estado e das ligações de cada nó, e as propriedades ergódicas que permitem obter as estatísticas de ensemble na rede a partir do comportamento a longo termo de um nó. Estas medidas estatísticas podem ser calculadas a partir de várias distribuições estacionárias que se relacionam umas com as outras através de transformações simples. Como aplicação desta abordagem completa, os equilíbrios dinâmicos de três diferentes variantes do processo de contacto adaptativo são descritos e comparados. Finalmente, motiva-se e propõe-se uma variante assimétrica do voter model coevolutivo. A fase activa metastável é tentativamente descrita como uma random walk ao longo de uma variedade lenta, à semelhan ca do que foi feito na literatura para o modelo simétrico, e os resultados para os dois casos são comparados.É identicada uma combinação de parâmetros particular para a qual este modelo assim etrico emula o modelo simétrico em rede fixa, o que é mais um exemplo da simetria emergente prevista pelas relações de recorrência estabelecidas na primeira parte. Considera-se ainda uma outra variante assimétrica, mais complexa, do voter model co-evolutivo, que apresenta um diagrama de fases essencialmente diferente, e cuja descrição se mostra requerer novas abordagens.Fundação para a Ciência e a Tecnologia (FCT, SFRH/BD/45179/2008

Computation in Complex Networks

Complex networks are one of the most challenging research focuses of disciplines, including physics, mathematics, biology, medicine, engineering, and computer science, among others. The interest in complex networks is increasingly growing, due to their ability to model several daily life systems, such as technology networks, the Internet, and communication, chemical, neural, social, political and financial networks. The Special Issue “Computation in Complex Networks" of Entropy offers a multidisciplinary view on how some complex systems behave, providing a collection of original and high-quality papers within the research fields of: • Community detection • Complex network modelling • Complex network analysis • Node classification • Information spreading and control • Network robustness • Social networks • Network medicin

Proclivity or Popularity? Exploring Agent Heterogeneity in Network Formation

The Barabasi-Albert model (BA model) is the standard algorithm used to describe the emergent mechanism of a scale-free network. This dissertation argues that the BA model, and its variants, rarely take agent heterogeneity into account in the analysis of network formation. In social networks, however, people\u27s decisions to connect are strongly affected by the extent of similarity. In this dissertation, the author applies an agent-based modeling (ABM) approach to reassess the Barabasi-Albert model. This study proposes that, in forming social networks, agents are constantly balancing between instrumental and intrinsic preferences. After systematic simulation and subsequent analysis, this study finds that agents\u27 preference of popularity and proclivity strongly shapes various attributes of simulated social networks. Moreover, this analysis of simulated networks investigates potential ways to detect this balance within real-world networks. Particularly, the scale parameter of the power-distribution is found sensitive solely to agents\u27 preference popularity. Finally, this study employs the social media data (i.e., diffusion of different emotions) for Sina Weibo—a Chinese version Tweet—to valid the findings, and results suggest that diffusion of anger is more popularity-driven