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Popularity is attractive -- this is the formula underlying preferential attachment, a popular explanation for the emergence of scaling in growing networks. If new connections are made preferentially to more popular nodes, then the resulting distribution of the number of connections that nodes have follows power laws observed in many real networks. Preferential attachment has been directly validated for some real networks, including the Internet. Preferential attachment can also be a consequence of different underlying processes based on node fitness, ranking, optimization, random walks, or duplication. Here we show that popularity is just one dimension of attractiveness. Another dimension is similarity. We develop a framework where new connections, instead of preferring popular nodes, optimize certain trade-offs between popularity and similarity. The framework admits a geometric interpretation, in which popularity preference emerges from local optimization. As opposed to preferential attachment, the optimization framework accurately describes large-scale evolution of technological (Internet), social (web of trust), and biological (E.coli metabolic) networks, predicting the probability of new links in them with a remarkable precision. The developed framework can thus be used for predicting new links in evolving networks, and provides a different perspective on preferential attachment as an emergent phenomenon.

We provide a simple proof that graphs in a general class of self-similar networks have zero percolation threshold. The considered self-similar networks include random scale-free graphs with given expected node degrees and zero clustering, scale-free graphs with finite clustering and metric structure, growing scale-free networks, and many real networks. The proof and the derivation of the giant component size do not require the assumption that networks are treelike. Our results rely only on the observation that self-similar networks possess a hierarchy of nested subgraphs whose average degree grows with their depth in the hierarchy. We conjecture that this property is pivotal for percolation in networks., Comment: 4 pages, 3 figures

The Internet infrastructure is severely stressed. Rapidly growing overheads associated with the primary function of the Internet---routing information packets between any two computers in the world---cause concerns among Internet experts that the existing Internet routing architecture may not sustain even another decade. Here we present a method to map the Internet to a hyperbolic space. Guided with the constructed map, which we release with this paper, Internet routing exhibits scaling properties close to theoretically best possible, thus resolving serious scaling limitations that the Internet faces today. Besides this immediate practical viability, our network mapping method can provide a different perspective on the community structure in complex networks.

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as non-interacting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures. Finally, we show that targeted transport processes without global topology knowledge, made possible by our geometric framework, are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure.

Network motifs are small building blocks of complex networks. Statistically significant motifs often perform network-specific functions. However, the precise nature of the connection between motifs and the global structure and function of networks remains elusive. Here we show that the global structure of some real networks is statistically determined by the probability of connections within motifs of size at most 3, once this probability accounts for node degrees. The connectivity profiles of node triples in these networks capture all their local and global properties. This finding impacts methods relying on motif statistical significance, and enriches our understanding of the elementary forces that shape the structure of complex networks.

A large number of complex systems find a natural abstraction in the form of weighted networks whose nodes represent the elements of the system and the weighted edges identify the presence of an interaction and its relative strength. In recent years, the study of an increasing number of large scale networks has highlighted the statistical heterogeneity of their interaction pattern, with degree and weight distributions which vary over many orders of magnitude. These features, along with the large number of elements and links, make the extraction of the truly relevant connections forming the network's backbone a very challenging problem. More specifically, coarse-graining approaches and filtering techniques are at struggle with the multiscale nature of large scale systems. Here we define a filtering method that offers a practical procedure to extract the relevant connection backbone in complex multiscale networks, preserving the edges that represent statistical significant deviations with respect to a null model for the local assignment of weights to edges. An important aspect of the method is that it does not belittle small-scale interactions and operates at all scales defined by the weight distribution. We apply our method to real world network instances and compare the obtained results with alternative backbone extraction techniques.

We show that heterogeneous degree distributions in observed scale-free topologies of complex networks can emerge as a consequence of the exponential expansion of hidden hyperbolic space. Fermi-Dirac statistics provides a physical interpretation of hyperbolic distances as energies of links. The hidden space curvature affects the heterogeneity of the degree distribution, while clustering is a function of temperature. We embed the Internet into the hyperbolic plane, and find a remarkable congruency between the embedding and our hyperbolic model. Besides proving our model realistic, this embedding may be used for routing with only local information, which holds significant promise for improving the performance of Internet routing.

We study the dynamics of the contact-process, one of the simplest nonequilibrium stochastic processes, taking place on a scale-free network. We consider the network topology as annealed, i.e. all links are rewired at each microscopic time step, so that no dynamical correlation can build up. This is a practical implementation of the absence of correlations assumed by mean-field approaches. We present a detailed analysis of the contact process in terms of a Langevin equation, including explicitly the effects of stochastic fluctuations in the number of particles in finite networks. This allows us to determine analytically the survival time for spreading experiments and the density of active sites in surviving runs. The fluctuations in the topological structure induce anomalous scaling effects with respect to the system size when the degree distribution has an "hard" upper bound. When the upper bound is soft, the presence of outliers with huge connectivity perturbs the picture even more, inducing an apparent shift of the critical point. In light of these findings, recent theoretical and numerical results in the literature are critically reviewed.

Random scale-free networks are ultrasmall worlds. The average length of the shortest paths in networks of size N scales as lnlnN. Here we show that these ultrasmall worlds can be navigated in ultrashort time. Greedy routing on scale-free networks embedded in metric spaces finds paths with the average length scaling also as lnlnN. Greedy routing uses only local information to navigate a network. Nevertheless, it finds asymptotically the shortest paths, a direct computation of which requires global topology knowledge. Our findings imply that the peculiar structure of complex networks ensures that the lack of global topological awareness has asymptotically no impact on the length of communication paths. These results have important consequences for communication systems such as the Internet, where maintaining knowledge of current topology is a major scalability bottleneck., Comment: 4 pages, 2 figures

We show that complex (scale-free) network topologies naturally emerge from hyperbolic metric spaces. Hyperbolic geometry facilitates maximally efficient greedy forwarding in these networks. Greedy forwarding is topology-oblivious. Nevertheless, greedy packets find their destinations with 100% probability following almost optimal shortest paths. This remarkable efficiency sustains even in highly dynamic networks. Our findings suggest that forwarding information through complex networks, such as the Internet, is possible without the overhead of existing routing protocols, and may also find practical applications in overlay networks for tasks such as application-level routing, information sharing, and data distribution.

We demonstrate that the self-similarity of some scale-free networks with respect to a simple degree-thresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Clustering, i.e., cycles of length three, plays a crucial role in this framework as a topological reflection of the triangle inequality in the hidden geometry. We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the self-similarity properties that we measured in the real networks. Our findings indicate that hidden geometries underlying these real networks are a plausible explanation for their observed topologies and, in particular, for their self-similarity with respect to the degree-based renormalization.

Routing information through networks is a universal phenomenon in both natural and manmade complex systems. When each node has full knowledge of the global network connectivity, finding short communication paths is merely a matter of distributed computation. However, in many real networks nodes communicate efficiently even without such global intelligence. Here we show that the peculiar structural characteristics of many complex networks support efficient communication without global knowledge. We also describe a general mechanism that explains this connection between network structure and function. This mechanism relies on the presence of a metric space hidden behind an observable network. Our findings suggest that real networks in nature have underlying metric spaces that remain undiscovered. Their discovery would have practical applications ranging from routing in the Internet and searching social networks, to studying information flows in neural, gene regulatory networks, or signaling pathways.

The large-scale organization of the world economies is exhibiting increasingly levels of local heterogeneity and global interdependency. Understanding the relation between local and global features calls for analytical tools able to uncover the global emerging organization of the international trade network. Here we analyze the world network of bilateral trade imbalances and characterize its overall flux organization, unraveling local and global high-flux pathways that define the backbone of the trade system. We develop a general procedure capable to progressively filter out in a consistent and quantitative way the dominant trade channels. This procedure is completely general and can be applied to any weighted network to detect the underlying structure of transport flows. The trade fluxes properties of the world trade web determines a ranking of trade partnerships that highlights global interdependencies, providing information not accessible by simple local analysis. The present work provides new quantitative tools for a dynamical approach to the propagation of economic crises.

We develop a statistical theory to characterize correlations in weighted networks. We define the appropriate metrics quantifying correlations and show that strictly uncorrelated weighted networks do not exist due to the presence of structural constraints. We also introduce an algorithm for generating maximally random weighted networks with arbitrary $P(k,s)$ to be used as null models. The application of our measures to real networks reveals the importance of weights in a correct understanding and modeling of these heterogeneous systems., Comment: 4 pages, 2 figures

The percolation properties of clustered networks are analyzed in detail. In the case of weak clustering, we present an analytical approach that allows to find the critical threshold and the size of the giant component. Numerical simulations confirm the accuracy of our results. In more general terms, we show that weak clustering hinders the onset of the giant component whereas strong clustering favors its appearance. This is a direct consequence of the differences in the $k$-core structure of the networks, which are found to be totally different depending on the level of clustering. An empirical analysis of a real social network confirms our predictions., Comment: Updated reference list

We develop a full theoretical approach to clustering in complex networks. A key concept is introduced, the edge multiplicity, that measures the number of triangles passing through an edge. This quantity extends the clustering coefficient in that it involves the properties of two --and not just one-- vertices. The formalism is completed with the definition of a three-vertex correlation function, which is the fundamental quantity describing the properties of clustered networks. The formalism suggests new metrics that are able to thoroughly characterize transitive relations. A rigorous analysis of several real networks, which makes use of the new formalism and the new metrics, is also provided. It is also found that clustered networks can be classified into two main groups: the {\it weak} and the {\it strong transitivity} classes. In the first class, edge multiplicity is small, with triangles being disjoint. In the second class, edge multiplicity is high and so triangles share many edges. As we shall see in the following paper, the class a network belongs to has strong implications in its percolation properties.

We develop a theoretical approach to percolation in random clustered networks. We find that, although clustering in scale-free networks can strongly affect some percolation properties, such as the size and the resilience of the giant connected component, it cannot restore a finite percolation threshold. In turn, this implies the absence of an epidemic threshold in this class of networks extending, thus, this result to a wide variety of real scale-free networks which shows a high level of transitivity. Our findings are in good agreement with numerical simulations., Comment: 4 Pages and 3 Figures. Final version to appear in PRL

The understanding of the immense and intricate topological structure of the World Wide Web (WWW) is a major scientific and technological challenge. This has been tackled recently by characterizing the properties of its representative graphs in which vertices and directed edges are identified with web-pages and hyperlinks, respectively. Data gathered in large scale crawls have been analyzed by several groups resulting in a general picture of the WWW that encompasses many of the complex properties typical of rapidly evolving networks. In this paper, we report a detailed statistical analysis of the topological properties of four different WWW graphs obtained with different crawlers. We find that, despite the very large size of the samples, the statistical measures characterizing these graphs differ quantitatively, and in some cases qualitatively, depending on the domain analyzed and the crawl used for gathering the data. This spurs the issue of the presence of sampling biases and structural differences of Web crawls that might induce properties not representative of the actual global underlying graph. In order to provide a more accurate characterization of the Web graph and identify observables which are clearly discriminating with respect to the sampling process, we study the behavior of degree-degree correlation functions and the statistics of reciprocal connections. The latter appears to enclose the relevant correlations of the WWW graph and carry most of the topological information of theWeb. The analysis of this quantity is also of major interest in relation to the navigability and searchability of the Web., Comment: 10 pages 19 figures. Values in Table 2 and Figure 1 corrected. Figure 7 updated. Minor changes in the text

PageRank has become a key element in the success of search engines, allowing to rank the most important hits in the top screen of results. One key aspect that distinguishes PageRank from other prestige measures such as in-degree is its global nature. From the information provider perspective, this makes it difficult or impossible to predict how their pages will be ranked. Consequently a market has emerged for the optimization of search engine results. Here we study the accuracy with which PageRank can be approximated by in-degree, a local measure made freely available by search engines. Theoretical and empirical analyses lead to conclude that given the weak degree correlations in the Web link graph, the approximation can be relatively accurate, giving service and information providers an effective new marketing tool., Comment: 8 pages, 7 figures, 2 tables

We model the Internet as a network of interconnected Autonomous Systems which self-organize under an absolute lack of centralized control. Our aim is to capture how the Internet evolves by reproducing the assembly that has led to its actual structure and, to this end, we propose a growing weighted network model driven by competition for resources and adaptation to maintain functionality in a demand and supply ``equilibrium''. On the demand side, we consider the environment, a pool of users which need to transfer information and ask for service. On the supply side, ASs compete to gain users, but to be able to provide service efficiently, they must adapt their bandwidth as a function of their size. Hence, the Internet is not modeled as an isolated system but the environment, in the form of a pool of users, is also a fundamental part which must be taken into account. ASs compete for users and big and small come up, so that not all ASs are identical. New connections between ASs are made or old ones are reinforced according to the adaptation needs. Thus, the evolution of the Internet can not be fully understood if just described as a technological isolated system. A socio-economic perspective must also be considered., Comment: Submitted to the Proceedings of the 3rd International Conference NEXT-SigmaPhi

We present a generator of random networks where both the degree-dependent clustering coefficient and the degree distribution are tunable. Following the same philosophy as in the configuration model, the degree distribution and the clustering coefficient for each class of nodes of degree $k$ are fixed ad hoc and a priori. The algorithm generates corresponding topologies by applying first a closure of triangles and secondly the classical closure of remaining free stubs. The procedure unveils an universal relation among clustering and degree-degree correlations for all networks, where the level of assortativity establishes an upper limit to the level of clustering. Maximum assortativity ensures no restriction on the decay of the clustering coefficient whereas disassortativity sets a stronger constraint on its behavior. Correlation measures in real networks are seen to observe this structural bound.

Uncorrelated random scale-free networks are useful null models to check the accuracy an the analytical solutions of dynamical processes defined on complex networks. We propose and analyze a model capable to generate random uncorrelated scale-free networks with no multiple and self-connections. The model is based on the classical configuration model, with an additional restriction on the maximum possible degree of the vertices. We check numerically that the proposed model indeed generates scale-free networks with no two and three vertex correlations, as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree $k$, respectively.

We present a detailed analytical study of the $A+A\to\emptyset$ diffusion-annihilation process in complex networks. By means of microscopic arguments, we derive a set of rate equations for the density of $A$ particles in vertices of a given degree, valid for any generic degree distribution, and which we solve for uncorrelated networks. For homogeneous networks (with bounded fluctuations), we recover the standard mean-field solution, i.e. a particle density decreasing as the inverse of time. For heterogeneous (scale-free networks) in the infinite network size limit, we obtain instead a density decreasing as a power-law, with an exponent depending on the degree distribution. We also analyze the role of finite size effects, showing that any finite scale-free network leads to the mean-field behavior, with a prefactor depending on the network size. We check our analytical predictions with extensive numerical simulations on homogeneous networks with Poisson degree distribution and scale-free networks with different degree exponents., Comment: 9 pages, 5 EPS figures

We model the evolution of the Internet at the Autonomous System level as a process of competition for users and adaptation of bandwidth capability. We find the exponent of the degree distribution as a simple function of the growth rates of the number of autonomous systems and the total number of connections in the Internet, both empirically measurable quantities. This fact place our model apart from others in which this exponent depends on parameters that need to be adjusted in a model dependent way. Our approach also accounts for a high level of clustering as well as degree-degree correlations, both with the same hierarchical structure present in the real Internet. Further, it also highlights the interplay between bandwidth, connectivity and traffic of the network., Comment: Minor content changes and inset of fig.2

We analyze the degree distribution's cut-off in finite size scale-free networks. We show that the cut-off behavior with the number of vertices $N$ is ruled by the topological constraints induced by the connectivity structure of the network. Even in the simple case of uncorrelated networks, we obtain an expression of the structural cut-off that is smaller that the natural cut-off obtained by means of extremal theory arguments. The obtained results are explicitly applied in the case of the configuration model to recover the size scaling of tadpoles and multiple edges., Comment: 5 pages, 2 figures. To appear in EPJB

We propose a simple model of social network formation that parameterizes the tendency to establish acquaintances by the relative distance in a representative social space. By means of analytical calculations and numerical simulations, we show that the model reproduces the main characteristics of real social networks: non- vanishing clustering coefficient, assortative degree correlations, and the emergence of a hierarchy of communities. Our results highlight the importance of communities in the understanding of the structure of social networks.

We study a class models of correlated random networks in which vertices are characterized by \textit{hidden variables} controlling the establishment of edges between pairs of vertices. We find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices. The expressions obtained are checked by means of numerical simulations in a particular example. The general model is extended to describe a practical algorithm to generate random networks with an \textit{a priori} specified correlation structure. We also present an extension of the class, to map non-equilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system.

We review the behavior of epidemic spreading on complex networks in which there are explicit correlations among the degrees of connected vertices., Comment: Contribution to the Proceedings of the XVIII Sitges Conference "Statistical Mechanics of Complex Networks", eds. J.M. Rubi et, al (Springer Verlag, Berlin, 2003)

Economy, and consequently trade, is a fundamental part of human social organization which, until now, has not been studied within the network modelling framework. Networks are mathematical tools used in the modelling of a wide variety of systems in social and natural science. Examples of these networks range from metabolic and cell networks to technological webs. Here we present the first empirical characterization of the world trade web, that is, the network built upon the trade relationships between different countries in the world. This network displays the typical properties of complex networks, namely, scale-free degree distribution, the {\it small world} property, a high clustering coefficient and, in addition, degree-degree correlation between different vertices. All these properties make the world trade web a complex network, which is far from being well-described through a classical random network description.

We study a recently introduced class of scale-free networks showing a high clustering coefficient and non-trivial connectivity correlations. We find that the connectivity probability distribution strongly depends on the fine details of the model. We solve exactly the case of low average connectivity, providing also exact expressions for the clustering and degree correlation functions. The model also exhibits a lack of small world properties in the whole parameters range. We discuss the physical properties of these networks in the light of the present detailed analysis., Comment: 10 pages, 9 figures

Random scale-free networks have the peculiar property of being prone to the spreading of infections. Here we provide an exact result showing that a scale-free connectivity distribution with diverging second moment is a sufficient condition to have null epidemic threshold in unstructured networks with either assortative or dissortative mixing. Connectivity correlations result therefore ininfluential for the epidemic spreading picture in these scale-free networks. The present result is related to the divergence of the average nearest neighbors connectivity, enforced by the connectivity detailed balance condition.

We study a dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node's connectivities. For the case of Markovian complex networks, showing only correlations between pairs of nodes, we find an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links that from a node with connectivity $k$ go to nodes with connectivity $k'$. Numerical simulations on a correlated growing network model provide support for our conclusions.

The phenomenon of resonant activation of a Brownian particle over a fluctuating barrier is revisited. We discuss the important distinctions between barriers that can fluctuate among "up" and "down" configurations, and barriers that are always "up" but that can fluctuate among different heights. A resonance as a function of the barrier fluctuation rate is found in both cases, but the nature and physical description of these resonances is quite distinct. The nature of the resonances, the physical basis for the resonant behavior, and the importance of boundary conditions are discussed in some detail. We obtain analytic expressions for the escape time over the barrier that explicitly capture the minima as a function of the barrier fluctuation rate, and show that our analytic results are in excellent agreement with numerical results., Comment: 33 pages of text, 9 figures

We present a continuous time random walk model for the scale-invariant transport found in a self-organized critical rice pile [Christensen et al., Phys. Rev. Lett. 77, 107 (1996)]. From our analytical results it is shown that the dynamics of the experiment can be explained in terms of L\'evy flights for the grains and a long-tailed distribution of trapping times. Scaling relations for the exponents of these distributions are obtained. The predicted microscopic behavior is confirmed by means of a cellular automaton model., Comment: 4 pages, RevTex, includes 3 PostScript figures, submitted to Phys. Rev. Lett