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

A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known—a collaboration network and a food web—and find that it detects significant and informative community divisions in both cases.

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Community structure in social and biological networks

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

Complex networks: Structure and dynamics

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.

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Power-Law Distributions in Empirical Data

A solution for the time- and age-dependent connectivity distribution of a growing random network is presented. The network is built by adding sites that link to earlier sites with a probability A(k) which depends on the number of preexisting links k to that site. For homogeneous connection kernels, A(k) approximately k(gamma), different behaviors arise for gamma 1, and gamma = 1. For gamma 1, a single site connects to nearly all other sites. In the borderline case A(k) approximately k, the power law N(k) approximately k(-nu) is found, where the exponent nu can be tuned to any value in the range 2<nu<infinity.

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Connectivity of growing random networks.

Scaling theory and exactly solved models in the kinetics of irreversible aggregation

Reaction-limited cluster aggregation is modeled with the kinetic rate (Smoluchowski) equations, using a kernel determined intrinsically by the clusters fractal geometry. The kernel scales with cluster mass as ${\mathrm{M}}_{1}$${\mathrm{M}}_{2}^{\ensuremath{\lambda}\mathrm{\ensuremath{-}}1}$ (${\mathrm{M}}_{1}$\ensuremath{\gg}${\mathrm{M}}_{2}$), and ${\mathrm{M}}_{1}^{\ensuremath{\lambda}}$ (${\mathrm{M}}_{1}$\ensuremath{\approxeq}${\mathrm{M}}_{2}$), with \ensuremath{\lambda}=1 in three dimensions, resulting in exponential kinetics and a cluster mass distribution ${\mathrm{C}}_{\mathrm{M}}$\ensuremath{\sim}${\mathrm{M}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\tau}}}$, with \ensuremath{\tau}=(3/2), in excellent accord with experiments. The singular nature of this solution forces the adjustment of the cluster fractal dimension, ${\mathrm{d}}_{\mathrm{f}}$, thereby determining its value.

Universal kinetics in reaction-limited aggregation.

The understanding of complex systems has become a central issue because such systems exist in a wide range of scientific disciplines. We here focus on financial markets as an example of a complex system. In particular we analyze financial data from the S&P 500 stocks in the 19-year period 1992-2010. We propose a definition of state for a financial market and use it to identify points of drastic change in the correlation structure. These points are mapped to occurrences of financial crises. We find that a wide variety of characteristic correlation structure patterns exist in the observation time window, and that these characteristic correlation structure patterns can be classified into several typical "market states". Using this classification we recognize transitions between different market states. A similarity measure we develop thus affords means of understanding changes in states and of recognizing developments not previously seen.

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Identifying States of a Financial Market

The authors consider a system of substances Ai reacting according to the following scheme: Ak+Al-RKl to k+l, (Rkl=Rlk>or=0). (The reaction is taken as irreversible.) They discuss the existence of global solutions of the kinetic equations derived for the concentrations. It is shown that one cannot expect the total number of monomers to remain constant. Rather, it can decrease as the result of the formation of infinite clusters (gelation). With this restriction, they obtain that a physically reasonable global solution exists if Rkl<or=rkrl and rk=o(k). It is further conjectured that no gelation will take place if rk=o( square root k). The case Rjk=(Aj+B)(Ak+B) is also solved explicitly and shown to exhibit gelation at t=1/A(A+B).

http://iopscience.iop.org/article/10.1088/0305-4470/14/12/030/pdf

Francois Leyvraz

Papers

Connectivity of growing random networks.

Scaling theory and exactly solved models in the kinetics of irreversible aggregation

Universal kinetics in reaction-limited aggregation.

Identifying States of a Financial Market

Singularities in the kinetics of coagulation processes