Systems as diverse as genetic networks or the world wide web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature is found to be a consequence of the two generic mechanisms that networks expand continuously by the addition of new vertices, and new vertices attach preferentially to already well connected sites. A model based on these two ingredients reproduces the observed stationary scalefree distributions, indicating that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems. * alb@nd.edu 1

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.

A major achievement in the study of complex networks is the observation that diverse systems, from sub-cellular biology [1][2][3] to social networks [4][5][6], exhibit universal topological characteristics [7][8][9][10][11][12][13][14]. Yet this universality does not naturally translate to the dynamics of these systems [15][16][17][18], hindering our progress towards a general theoretical framework of network dynamics. The source of this theoretical gap is the fact that the behavior of a complex system cannot be uniquely predicted from its topology, but rather depends also on the dynamic mechanisms of interaction between the nodes [19], hence systems with similar structure may exhibit profoundly different dynamic behavior. To bridge this gap, we derive here the patterns of network information transmission, indeed, the essence of a network's behavior [20][21][22], by offering a systematic translation of topology into the actual spatio-temporal propagation of perturbative signals. We predict, for an extremely broad range of nonlinear dynamic models, that the propagation rules condense around three highly distinctive dynamic universality classes, characterized by the interplay between network paths, degree distribution and the interaction dynamics. Our formalism helps us leverage the major advances in the mapping of real world networks, into predictions on the actual dynamic propagation, from the spread of viruses in social networks [23][24][25][26][27] to the diffusion of genetic information in cellular systems [28,29].The spread of information in a complex system is mediated by its underlying topology, with the metric of network paths commonly assumed to be the main determinant of the propagation [24,[29][30][31]. This rationale has motivated a widespread effort to retrieve the structure of many real world networks [32][33][34], which in turn emerged as a powerful tool to visualize and predict information propagation, such as epidemic spreading via air-traffic [24,35] or neuronal activity patterns along the pathways of the connectome [36]. In all these cases, the network topology exposes the natural geometry of the propagation, with network distance being the main predictor of the spreading behavior. Yet, network topology does not always capture information propagation in such a transparent fashion, due to the diverse forms of nonlinear interactions that may take arXiv:1801.08854v1 [physics.soc-ph] 26 Jan 2018 of the dynamics (α = 1/2) leads to different behavior, as now the signal seems to skip the most adjacent nodes and appear first at more distant neighbors (red). To deepen our observation of the different response patterns, we focus on a specific pair of target nodes, highlighted in grey and black. In case α = 1 (blue) we find that these two nodes exhibit similar behavior, featuring an almost synchronous response (Fig. 1g). The picture dramatically changes, however, when α = 1/2 (red), in which case the signal impacts the black node at a much later time (Fig. 1h). Strikingly, the sequence of responses is reve...

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

A network of disorders and disease genes linked by known disordergene associations offers a platform to explore in a single graphtheoretic framework all known phenotype and disease gene associations, indicating the common genetic origin of many diseases. Genes associated with similar disorders show both higher likelihood of physical interactions between their products and higher expression profiling similarity for their transcripts, supporting the existence of distinct disease-specific functional modules. We find that essential human genes are likely to encode hub proteins and are expressed widely in most tissues. This suggests that disease genes also would play a central role in the human interactome. In contrast, we find that the vast majority of disease genes are nonessential and show no tendency to encode hub proteins, and their expression pattern indicates that they are localized in the functional periphery of the network. A selection-based model explains the observed difference between essential and disease genes and also suggests that diseases caused by somatic mutations should not be peripheral, a prediction we confirm for cancer genes.biological networks ͉ complex networks ͉ human genetics ͉ systems biology ͉ diseasome