Many complex systems display a surprising degree of tolerance against errors. For example, relatively simple organisms grow, persist and reproduce despite drastic pharmaceutical or environmental interventions, an error tolerance attributed to the robustness of the underlying metabolic network. Complex communication networks display a surprising degree of robustness: although key components regularly malfunction, local failures rarely lead to the loss of the global information-carrying ability of the network. The stability of these and other complex systems is often attributed to the redundant wiring of the functional web defined by the systems' components. Here we demonstrate that error tolerance is not shared by all redundant systems: it is displayed only by a class of inhomogeneously wired networks, called scale-free networks, which include the World-Wide Web, the Internet, social networks and cells. We find that such networks display an unexpected degree of robustness, the ability of their nodes to communicate being unaffected even by unrealistically high failure rates. However, error tolerance comes at a high price in that these networks are extremely vulnerable to attacks (that is, to the selection and removal of a few nodes that play a vital role in maintaining the network's connectivity). Such error tolerance and attack vulnerability are generic properties of communication networks.

Lethality and centrality in protein networksCell biology traditionally identifies proteins based on their individual actions as catalysts, signaling molecules, or building blocks of cells and microorganisms. Currently, we witness the emergence of a post-genomic view that expands the protein's role, regarding it as an element in a network of proteinprotein interactions as well, with a 'contextual' or 'cellular' function within functional modules 1, 2 . Here we provide quantitative support for this paradigm shift by demonstrating that the phenotypic consequence of a single gene deletion in the yeast, S. cerevisiae, is affected, to a high degree, by the topologic position of its protein product in the complex, hierarchical web of molecular interactions.The S. cerevisiae protein-protein interaction network we investigate has 1870 proteins as nodes, connected by 2240 identified direct physical interactions, and is derived from combined, nonoverlapping data 3, 4 obtained mostly by systematic two-hybrid analyses 3 . Due to its size, a complete map of the network (Fig. 1a), while informative, in itself offers little insight into its large-scale characteristics. Thus, our first goal was to identify the architecture of this network, determining if it is best described by an inherently uniform exponential topology with proteins on average possessing the same number of links, or by a highly heterogeneous scale-free topology with proteins having widely different connectivities 5 . As we show in Fig. 1b, the probability that a given yeast protein interacts with k other yeast proteins follows a power-law 5 with an exponential cutoff 6 at k c ≅ 20, a topology that is also shared by the protein-protein interaction network of the bacterium, H. pylori 7 . This indicates that the network of protein interactions in two separate organisms forms a highly inhomogeneous scale-free network in which a few highly connected proteins play a central role in mediating interactions among numerous, less connected proteins.An important known consequence of the inhomogeneous structure is the network's simultaneous tolerance against random errors coupled with fragility against the removal of the most connected nodes 8 . Indeed, we find that random mutations in the genome of S. cerevisiae, -modeled by the removal of randomly selected yeast proteins-, do not affect the overall topology of the network. In contrast, when

An important goal in biology is to uncover the fundamental design principles that provide the common underlying structure and function in all cells and microorganisms [6][7][8][9][10][11][12][13] . For example, it is increasingly appreciated that the robustness of various cellular processes is rooted in the dynamic interactions among its many constituents [14][15][16] , such as proteins, DNA, RNA, and small molecules.Recent scientific developments improve our ability to identify the design principles that integrate these interactions into a complex system. Large-scale sequencing projects have not only provided complete sequence information for a number of genomes, but also allowed the development of integrated pathway-genome databases [17][18][19] that provide organism-specific connectivity maps of metabolic-and,

The co-authorship network of scientists represents a prototype of complex evolving networks. By mapping the electronic database containing all relevant journals in mathematics and neuro-science for an eight-year period (1991-98), we infer the dynamic and the structural mechanisms that govern the evolution and topology of this complex system. First, empirical measurements allow us to uncover the topological measures that characterize the network at a given moment, as well as the time evolution of these quantities. The results indicate that the network is scale-free, and that the network evolution is governed by preferential attachment, affecting both internal and external links. However, in contrast with most model predictions the average degree increases in time, and the node separation decreases. Second, we propose a simple model that captures the network's time evolution. Third, numerical simulations are used to uncover the behavior of quantities that could not be predicted analytically.Comment: 14 pages, 15 figure

No abstract