The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. 
Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. 
The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. 
The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.

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Statistical mechanics of complex networks

Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

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The Structure and Function of Complex Networks

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

We review the recent rapid progress in the statistical physics of evolving networks. Interest has focused mainly on the structural properties of complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of this kind have recently been created, which opens a wide field for the study of their topology, evolution, and the complex processes which occur in them. Such networks possess a rich set of scaling properties. A number of them are scale-free and show striking resilience against random breakdowns. In spite of the large sizes of these networks, the distances between most of their vertices are short - a feature known as the 'small-world' effect. We discuss how growing networks self-organize into scale-free structures, and investigate the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation and disease spread on these networks. We present a number of models demonstrat...

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Evolution of networks

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.

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Critical phenomena in complex networks

The small-world transition is a first-order transition at zero density p of shortcuts, whereby the normalised shortest-path distance = /L undergoes a discontinuity in the thermodynamic limit. On finite systems the apparent transition is shifted by Δp ~ L−d. Equivalently a "persistence size" L* ~ p−1/d can be defined in connection with finite-size effects. Assuming L* ~ p−τ, simple rescaling arguments imply that τ = 1/d. We confirm this result by extensive numerical simulation in one to four dimensions, and argue that τ = 1/d implies that this transition is first order.

First-order transition in small-world networks

We discuss shortest-path lengths $\mathcal{l}(r)$ on periodic rings of size L supplemented with an average of $\mathrm{pL}$ randomly located long-range links whose lengths are distributed according to ${P}_{l}\ensuremath{\sim}{l}^{\ensuremath{-}\ensuremath{\mu}}.$ Using rescaling arguments and numerical simulation on systems of up to ${10}^{7}$ sites, we show that a characteristic length $\ensuremath{\xi}$ exists such that $\mathcal{l}(r)\ensuremath{\sim}r$ for $rl\ensuremath{\xi}$ but $\mathcal{l}(r)\ensuremath{\sim}{r}^{{\ensuremath{\theta}}_{s}(\ensuremath{\mu})}$ for $r\ensuremath{\gg}\ensuremath{\xi}.$ For small p we find that the shortest-path length satisfies the scaling relation $\mathcal{l}(r,\ensuremath{\mu},p)/\ensuremath{\xi}=f(\ensuremath{\mu},r/\ensuremath{\xi}).$ Three regions with different asymptotic behaviors are found, respectively: (a) $\ensuremath{\mu}g2$ where ${\ensuremath{\theta}}_{s}=1,$ (b) $1l\ensuremath{\mu}l2$ where $0l{\ensuremath{\theta}}_{s}(\ensuremath{\mu})l1/2,$ and (c) $\ensuremath{\mu}l1$ where $\mathcal{l}(r)$ behaves logarithmically, i.e., ${\ensuremath{\theta}}_{s}=0.$ The characteristic length $\ensuremath{\xi}$ is of the form $\ensuremath{\xi}\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}=1/(2\ensuremath{-}\ensuremath{\mu})$ in region (b), but depends on L as well in region (c). A directed model of shortest paths is solved and compared with numerical results.

https://arxiv.org/pdf/cond-mat/0201083.pdf

Shortest paths on systems with power-law distributed long-range connections.

We use detrended fluctuation analysis (DFA) to study the dynamics of blood pressure oscillations and its feedback control in rats by analyzing systolic pressure time series before and after a surgical procedure that interrupts its control loop. We found, for each situation, a crossover between two scaling regions characterized by exponents that reflect the nature of the feedback control and its range of operation. In addition, we found evidences of adaptation in the dynamics of blood pressure regulation a few days after surgical disruption of its main feedback circuit. Based on the paradigm of antagonistic, bipartite (vagal and sympathetic) action of the central nerve system, we propose a simple model for pressure homeostasis as the balance between two nonlinear opposing forces, successfully reproducing the crossover observed in the DFA of actual pressure signals.

Detrended Fluctuation Analysis of Systolic Blood Pressure Control Loop

Generalized simulated annealing method in the analysis of atom–atom interaction

We use detrended fluctuation analysis (DFA) to study the dynamics of blood pressure oscillations and its feedback control in rats by analyzing systolic pressure time series before and after a surgical procedure that interrupts its control loop. We found, for each situation, a crossover between two scaling regions characterized by exponents that reflect the nature of the feedback control and its range of operation. In addition, we found evidence of adaptation in the dynamics of blood pressure regulation a few days after surgical disruption of its main feedback circuit. Based on the paradigm of antagonistic, bipartite (vagal and sympathetic) action of the central nerve system, we propose a simple model for pressure homeostasis as the balance between two nonlinear opposing forces, successfully reproducing the crossover observed in the DFA of actual pressure signals.

https://iopscience.iop.org/article/10.1088/1367-2630/11/10/103005/pdf

M. Argollo de Menezes

Papers

First-order transition in small-world networks

Shortest paths on systems with power-law distributed long-range connections.

Detrended Fluctuation Analysis of Systolic Blood Pressure Control Loop

Generalized simulated annealing method in the analysis of atom–atom interaction

Detrended fluctuation analysis of a systolic blood pressure control loop