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

The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems-be they neurons, power stations or lasers-will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.

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Exploring complex networks

A key aim of postgenomic biomedical research is to systematically catalogue all molecules and their interactions within a living cell. There is a clear need to understand how these molecules and the interactions between them determine the function of this enormously complex machinery, both in isolation and when surrounded by other cells. Rapid advances in network biology indicate that cellular networks are governed by universal laws and offer a new conceptual framework that could potentially revolutionize our view of biology and disease pathologies in the twenty-first century.

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Network biology: understanding the cell's functional organization

We find that scale-free random networks are excellently modeled by simple deterministic graphs. Our graph has a discrete degree distribution (degree is the number of connections of a vertex), which is characterized by a power law with exponent $\ensuremath{\gamma}=1+\mathrm{ln}3/\mathrm{ln}2.$ Properties of this compact structure are surprisingly close to those of growing random scale-free networks with \ensuremath{\gamma} in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For a large network $(\mathrm{ln}N\ensuremath{\gg}1)$ the distribution tends to a Gaussian of width $\ensuremath{\sim}\sqrt{\mathrm{ln}N}$ centered at $\mathcal{l}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\sim}\mathrm{ln}N.$ We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent $2+\ensuremath{\gamma}.$

Pseudofractal scale-free web.

We study the growth of a network with aging of sites. Each new site of the network is connected to some old site with probability proportional (i) to the connectivity of the old site as in the Barab\'asi-Albert's model and (ii) to ${\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\alpha}},$ where $\ensuremath{\tau}$ is the age of the old site. We find both from simulation and analytically that the network shows scaling behavior only in the region $\ensuremath{\alpha}l1.$ When $\ensuremath{\alpha}$ increases from $\ensuremath{-}\ensuremath{\infty}$ to $0,$ the exponent $\ensuremath{\gamma}$ of the distribution of connectivities $[P(k)\ensuremath{\propto}{k}^{\ensuremath{-}\ensuremath{\gamma}}$ for large $k]$ grows from $2$ to the value for the network without aging. The ensuing increase of $\ensuremath{\alpha}$ to $1$ causes $\ensuremath{\gamma}$ to grow to $\ensuremath{\infty}.$ For $\ensuremath{\alpha}g1,$ the distribution $P(k)$ is exponentional.

Evolution of networks with aging of sites

In many real growing networks the mean number of connections per vertex increases with time. The Internet, the Word Wide Web, collaboration networks, and many others display this behavior. Such a growth can be called {\em accelerated}. We show that this acceleration influences distribution of connections and may determine the structure of a network. We discuss general consequences of the acceleration and demonstrate its features applying simple illustrating examples. In particular, we show that the accelerated growth fairly well explains the structure of the Word Web (the network of interacting words of human language). Also, we use the models of the accelerated growth of networks to describe a wealth condensation transition in evolving societies.

Accelerated growth of networks

We present an exact description of a crossover between two different regimes of simple analogies of small-world networks. Each of the sites chosen with a probability $p$ from $n$ sites of an ordered system defined on a circle is connected to all other sites selected in such a way. Every link is of a unit length. Thus, while $p$ changes from 0 to 1, an averaged shortest distance between a pair of sites changes from $\bar{\ell} \sim n$ to $\bar{\ell} = 1$. We find the distribution of the shortest distances $P(\ell)$ and obtain a scaling form of $\bar{\ell}(p,n)$. In spite of the simplicity of the models under consideration, the results appear to be surprisingly close to those obtained numerically for usual small-world networks.

Exactly solvable analogy of small-world networks

We note that the model discussed in the communication of S. Bornholdt and H. Ebel (World Wide Web scaling exponent from Simon's 1955 model, cond-mat/0008465) is the particular case of the model considered and solved exactly in our paper, cond-mat/0004434. These models may be used for estimation of the order of the deviation of the scaling exponent from 2 both for the distributions of incoming links and links coming out from nodes but not for the obtaining some specific values of the exponents from the WWW growth data. We emphasize that, unlike the statement of Bornholdt and Ebel, both the network under consideration and the model of Barab\'{a}si and Albert provide quite equal possibilities for individual growth. There is no great difference between them in this respect. The resulting distributions for individual nodes and arising scaling relations have been obtained in our paper, cond-mat/0004434. We discuss briefly the modern state of art in the physics of the evolving networks and the great role of the general principle -- {\em popularity is attractive} -- in the self-organization of complex communications networks, in physics of nonequilibrium phenomena, and in Nature.

S. N. Dorogovtsev

Papers

Pseudofractal scale-free web.

Evolution of networks with aging of sites

Accelerated growth of networks

Exactly solvable analogy of small-world networks

WWW and Internet models from 1955 till our days and the ``popularity is attractive'' principle