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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High-throughput techniques are leading to an explosive growth in the size of biological databases and creating the opportunity to revolutionize our understanding of life and disease. Interpretation of these data remains, however, a major scientific challenge. Here, we propose a methodology that enables us to extract and display information contained in complex networks 1,2,3. Specifically, we demonstrate that one can (i) find functional modules 4,5 in complex networks, and (ii) classify nodes into universal roles according to their pattern of intra- and inter-module connections. The method thus yields a “cartographic representation ” of complex networks. Metabolic networks 6,7,8 are among the most challenging biological networks and, arguably, the ones with more potential for immediate applicability 9. We use our method to analyze the metabolic networks of twelve organisms from three different super-kingdoms. We find that, typically, 80 % of the nodes are only connected to other nodes within their respective modules, and that nodes with different roles are affected by different evolutionary constraints and pressures. Remarkably, we

We analyze the structuture of a large metabolic network, that of the energy and biosynthesis metabolism of Escherichia coli. This network is a paradigmatic case for the large genetic and metabolic networks that functional genomics efforts are beginning to elucidate. To analyze the structure of networks involving hundreds or thousands of components by simple visual inspection is impossible, and a quantitative framework is needed to analyze them. We propose a graph theoretical description of the E. coli metabolic network, a description that we hope will prove useful for other genetic networks. We find that this network is a small world graph, a type of graph observed in a variety of seemingly unrelated areas, such as friendship networks in sociology, the structure of electrical power grids, and the nervous system of C. elegans. Moreover, its connectivity follows a power law, another unusual but by no means rare statistical distribution. This architecture may serve to minimize trans...

We present statistical analyses of the large-scale structure of three types of semantic networks: word associations, WordNet, and Roget's thesaurus. We show that they have a small-world structure, characterized by sparse connectivity, short average path-lengths between words, and strong local clustering. In addition, the distributions of the number of connections follow power laws that indicate a scale-free pattern of connectivity, with most nodes having relatively few connections joined together through a small number of hubs with many connections. These regularities have also been found in certain other complex natural networks, such as the world wide web, but they are not consistent with many conventional models of semantic organization, based on inheritance hierarchies, arbitrarily structured networks, or high-dimensional vector spaces. We propose that these structures reflect the mechanisms by which semantic networks grow. We describe a simple model for semantic growth, in which each new word or concept is connected to an existing network by differentiating the connectivity pattern of an existing node. This model generates appropriate small-world statistics and power-law connectivity distributions, and also suggests one possible mechanistic basis for the effects of learning history variables (age-ofacquisition, usage frequency) on behavioral performance in semantic processing tasks.

A cell’s behavior is a consequence of the complex interactions between its numerous constituents, such as DNA, RNA, proteins and small molecules. Cells use signaling pathways and regulatory mechanisms to coordinate multiple processes, allowing them to respond to and adapt to an ever-changing environment. The large number of components, the degree of interconnectivity and the complex control of cellular networks are becoming evident in the integrated genomic and proteomic analyses that are emerging. It is increasingly recognized that the understanding of properties that arise from whole-cell function require integrated, theoretical descriptions of the relationships between different cellular components. Recent

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In the study of the spectra of power law graphs, there are basically two competing approaches. One is to prove analogues of Wigner’s semi-circle law while the other predicts that the eigenvalues follow a power law distributions. Although the semi-circle law and the power law have nothing in common, we will show that both approaches are essentially correct if one considers the appropriate matrices. We will prove that (under certain mild conditions) the eigenvalues of the (normalized) Laplacian of a random power law graph follow the semi-circle law while the spectrum of the adjacency matrix of a power law graph obeys the power law. Our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants. Of interest are a number of (new) values for the exponent β where phase transitions for eigenvalue distributions occur. The spectrum distributions have direct implications to numerous graph algorithms such as randomized algorithms that involve rapidly mixing Markov chains, for example.

Cellular phenotypes are determined by the differential activity of networks linking coregulated genes. Available methods for the reverse engineering of such networks from genome-wide expression profiles have been successful only in the analysis of lower eukaryotes with simple genomes. Using a new method called ARACNe (algorithm for the reconstruction of accurate cellular networks), we report the reconstruction of regulatory networks from expression profiles of human B cells. The results are suggestive a hierarchical, scale-free network, where a few highly interconnected genes (hubs) account for most of the interactions. Validation of the network against available data led to the identification of MYC as a major hub, which controls a network comprising known target genes as well as new ones, which were biochemically validated. The newly identified MYC targets include some major hubs. This approach can be generally useful for the analysis of normal and pathologic networks in mammalian cells. Cell phenotypes are determined by the concerted activity of thousands of genes and their products. This activity is coordinated by a complex network that regulates the expression of genes controlling common functions, such as the formation of a transcriptional complex or the availability of a signaling pathway. Understanding this organization is crucial to elucidate normal cell physiology as well as to dissect complex pathologic phenotypes. Studies in lower organisms indicate that the structure of both protein-protein interaction and metabolic networks is of a hierarchical scale-free nature 1,2 , characterized by an inverse relationship between the number of nodes and their connectivity (scale-free) and by a preferential interaction among highly connected genes, called hubs (hierarchical). Although scale-free networks may represent a common blueprint for all cellular constituents, evidence of scale-free topology in higher-order eukaryotic cells is currently limited to coexpression networks 3,4 , which tend to identify entire subpathways rather than individual interactions. Identifying the organizational network of eukaryotic cells is still a key goal in understanding cell physiology and disease. Genome-wide clustering of gene-expression profiles has provided an initial step towards the elucidation of cellular networks. But the organization of gene-expression profile data into functionally meaningful genetic information has proven difficult and so far has fallen short of uncovering the intricate structure of cellular interactions. This challenge, called network reverse engineering or deconvolution, has led to an entirely new class of methods aimed at producing high-fidelity representations of cellular networks as graphs, where nodes represent genes and edges between them represent interactions, either between the encoded proteins or between the encoded proteins and the genes (we use 'genetic interaction' to refer to both types of mechanisms). Available methods fall into four broad categories: optimization methods 5-7 , which maximize a scoring function over alternative network models; regression techniques Here we present the successful reverse engineering of geneexpression profile data from human B cells. Our study is based on ARACNe (algorithm for the reconstruction of accurate cellular networks), a new approach for the reverse engineering of cellular networks from microarray expression profiles. ARACNe first identifies statistically significant gene-gene coregulation by mutual information, an information-theoretic measure of relatedness. It then eliminates indirect relationships, in which two genes are coregulated through one or more intermediaries, by applying a well-known staple of data

How does the Web look? How could we tell an abnormal social network from a normal one? These and similar questions are important in many fields where the data can intuitively be cast as a graph; examples range from computer networks to sociology to biology and many more. Indeed, any M: N relation in database terminology can be represented as a graph. A lot of these questions boil down to the following: “How can we generate synthetic but realistic graphs? ” To answer this, we must first understand what patterns are common in real-world graphs and can thus be considered a mark of normality/realism. This survey give an overview of the incredible variety of work that has been done on these problems. One of our main contributions is the integration of points of view from physics, mathematics, sociology, and computer science. Further, we briefly describe recent advances on some related and interesting graph problems.