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

Complex networks are studied across many fields of science. To uncover their structural design principles, we defined “network motifs,” patterns of interconnections occurring in complex networks at numbers that are significantly higher than those in randomized networks. We found such motifs in networks from biochemistry, neurobiology, ecology, and engineering. The motifs shared by ecological food webs were distinct from the motifs shared by the genetic networks of Escherichia coli and Saccharomyces cerevisiae or from those found in the World Wide Web. Similar motifs were found in networks that perform information processing, even though they describe elements as different as biomolecules within a cell and synaptic connections between neurons in Caenorhabditis elegans. Motifs may thus define universal classes of networks. This

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Network Motifs: Simple Building Blocks of Complex Networks

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

In the past few years, the discovery of small-world and scale-free properties of many natural and artificial complex networks has stimulated a great deal of interest in studying the underlying organizing principles of various complex networks, which has led to dramatic advances in this emerging and active field of research. The present article reviews some basic concepts, important progress, and significant results in the current studies of various complex networks, with emphasis on the relationship between the topology and the dynamics of such complex networks. Some fundamental properties and typical complex network models are described; and, as an example, epidemic dynamics are analyzed and discussed in some detail. Finally, the important issue of robustness versus fragility of dynamical synchronization in complex networks is introduced and discussed.

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Complex networks: small-world, scale-free and beyond

Background: Many technological, biological, social, and information networks fall into the broad class of 'small-world' networks: they have tightly interconnected clusters of nodes, and a shortest mean path length that is similar to a matched random graph (same number of nodes and edges). This semi-quantitative definition leads to a categorical distinction ('small/not-small') rather than a quantitative, continuous grading of networks, and can lead to uncertainty about a network's small-world status. Moreover, systems described by small-world networks are often studied using an equivalent canonical network model-the Watts-Strogatz (WS) model. However, the process of establishing an equivalent WS model is imprecise and there is a pressing need to discover ways in which this equivalence may be quantified.

Methodology/Principal Findings: We defined a precise measure of 'small-world-ness' S based on the trade off between high local clustering and short path length. A network is now deemed a 'small-world' if S. 1-an assertion which may be tested statistically. We then examined the behavior of S on a large data-set of real-world systems. We found that all these systems were linked by a linear relationship between their S values and the network size n. Moreover, we show a method for assigning a unique Watts-Strogatz (WS) model to any real-world network, and show analytically that the WS models associated with our sample of networks also show linearity between S and n. Linearity between S and n is not, however, inevitable, and neither is S maximal for an arbitrary network of given size. Linearity may, however, be explained by a common limiting growth process.

Conclusions/Significance: We have shown how the notion of a small-world network may be quantified. Several key properties of the metric are described and the use of WS canonical models is placed on a more secure footing.

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Network 'small-world-ness': a quantitative method for determining canonical network equivalence.

Recent theoretical studies and extensive data analyses have revealed a common feature displayed by biological, social, and technological networks: the presence of small world patterns. Here we analyze this problem by using several graphs obtained from one of the most common technological systems: electronic circuits. It is shown that both analogic and digital circuits exhibit small world behavior. We conjecture that the small world pattern arises from the compact design in which many elements share a small, close physical neighborhood plus the fact that the system must define a single connected component (which requires shortcuts connecting different integrated clusters). The degree distributions displayed are consistent with a conjecture concerning the sharp cutoffs associated to the presence of costly connections [Amaral et al., Proc. Natl. Acad. Sci. USA 97, 11 149 (2000)], thus providing a limit case for the classes of universality of small world patterns from real, artificial networks. The consequences for circuit design are outlined.

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Topology of technology graphs: small world patterns in electronic circuits.

Recent theoretical studies and extensive data analyses have revealed a common feature displayed by biological, social and technological networks: the presence of small world patterns. Here we analyse this problem by using several graphs obtained from one of the most common technological systems: electronic circuits. It is shown that both analogic and digital circuits exhibit SW behavior. We conjecture that the SW pattern arises from the compact design in which many elements share a small, close physical neighborhood plus the fact that the system must define a single connected component (which requires shortcuts connecting different integrated clusters). The degree distributions displayed are consistent with a conjecture concerning the sharp cutoffs associated to the presence of costly connections [Amaral et al., Proc. Natl. Acad. Sci. USA 97 , 11149 (2000)] thus providing a limit case for the classes of universality of small world patterns from real, artificial networks. The consequences for circuit design are outlined.

Christiaan Janssen

Papers

Topology of technology graphs: small world patterns in electronic circuits.

The Topology of Technology Graphs: Small World Patterns in Electronic Circuits