Scale-Free Networks

Complex networks describe a wide range of systems in nature and society. Frequently cited examples include the cell, a network of chemicals linked by chemical reactions, and the Internet, a network of routers and computers connected by physical links. While traditionally these systems have been modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks are governed by robust organizing principles. This article reviews the recent advances in the field of complex networks, focusing on the statistical mechanics of network topology and dynamics. After reviewing the empirical data that motivated the recent interest in networks, the authors discuss the main models and analytical tools, covering random graphs, small-world and scale-free networks, the emerging theory of evolving networks, and the interplay between topology and the network's robustness against failures and attacks.

We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring 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 large sizes of these networks, the distances between most their vertices are short -a feature known as the "smallworld" effect. We discuss how growing networks self-organize into scale-free structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems of non-equilibrium physics, econophysics, evolutionary biology, etc.

Online social networking sites like Orkut, YouTube, and Flickr are among the most popular sites on the Internet. Users of these sites form a social network, which provides a powerful means of sharing, organizing, and finding content and contacts. The popularity of these sites provides an opportunity to study the characteristics of online social network graphs at large scale. Understanding these graphs is important, both to improve current systems and to design new applications of online social networks.

If magnetic fields are frozen in gravitational collapse, the resulting magnetic tension can prevent the outer part of the subcritical envelope of a molecular cloud from falling in with the supercritical core. However, the implied surface magnetic fields much exceed measured values for young stars. Moreover, it is virtually impossible for Keplerian disks to form in these circumstances. Magnetic reconnection can eliminate the long lever arms of the split monopole formed by the gravitational collapse that contibutes to catastrophic magnetic braking. The natural appearance then of a Keplerian disk adjoining a rotating star with an outer convective envelope will lead to an X-wind driven magnetocentrifugally from the inner edge of the disk. This wind can cut off the continued infall from the envelope and build-up of the central stellar mass. We use these ideas and results to calculate the initial mass function and star formation efficiency for the distributed and clustered modes of star formation.

In apparently scale-free protein–protein interaction networks, or 'interactome' networks 1, 2 , most proteins interact with few partners, whereas a small but significant proportion of proteins, the 'hubs', interact with many partners. Both biological and non-biological scale-...

Functional magnetic resonance imaging (fMRI) is used to extract functional networks connecting correlated human brain sites. Analysis of the resulting networks in different tasks shows that: (a) the distribution of functional connections, and the probability of finding a link vs. distance are both scale-free, (b) the characteristic path length is small and comparable with those of equivalent random networks, and (c) the clustering coefficient is orders of magnitude larger than those of equivalent random networks. All these properties, typical of scale-free small world networks, reflect important functional information about brain states.

Recently, it has been demonstrated that many large complex networks display a scale-free feature, that is, their connectivity distributions have the power-law form. In this paper, we investigate the synchronization phenomena in a scale-free dynamical network. We show that its synchronizability is robust against random removal of nodes, but is fragile to specific removal of the most highly connected nodes. * Supported by the UK Engineering and Physical Sciences Research Council through the EPSRC grant number GR/M97923/01 and the Hong Kong GRC CERG Grant 9040565.

Humans often cooperate in public goods games 1-3 and situations ranging from family issues to global warming 4,5 . However, evolutionary game theory predicts 4,6 that the temptation to forgo the public good mostly wins over collective cooperative action, and this is often also seen in economic experiments 7 . Here we show how social diversity provides an escape from this apparent paradox. Up to now, individuals have been treated as equivalent in all respects 4,8 , in sharp contrast with real-life situations, where diversity is ubiquitous. We introduce social diversity by means of heterogeneous graphs and show that cooperation is promoted by the diversity associated with the number and size of the public goods game in which each individual participates and with the individual contribution to each such game. When social ties follow a scale-free distribution 9 , cooperation is enhanced whenever all individuals are expected to contribute a fixed amount irrespective of the plethora of public goods games in which they engage. Our results may help to explain the emergence of cooperation in the absence of mechanisms based on individual reputation and punishment 10-12 . Combining social diversity with reputation and punishment will provide instrumental clues on the self-organization of social communities and their economical implications.

Recently, it has been demonstrated that many large complex networks display a scale-free feature, that is, their connectivity distributions have the power-law form. In the present work, control of a scale-free dynamical network by applying local feedback injections to a fraction of network nodes is investigated. The speciÿcally and randomly pinning schemes are considered. The specifically pinning of the most highly connected nodes is shown to require a signiÿcantly smaller number of local controllers as compared to the randomly pinning scheme. The method is applied to an array of Chua's oscillators as an example.

We explore the network topology of the interbank payments transferred between commercial banks over the Fedwire ® Funds Service. We find that the network is compact despite low connectivity. The network includes a tightly connected core of money-center banks to which all other banks connect. The degree distribution is scale-free over a substantial range. We find that the properties of the network changed considerably in the immediate aftermath of the attacks of September 11, 2001.

Although the "scale-free" literature is large and growing, it gives neither a precise definition of scale-free graphs nor rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and verifiably false claims. In this paper, we propose a new, mathematically precise, and structural definition of the extent to which a graph is scale-free, and prove a series of results that recover many of the claimed properties while suggesting the potential for a rich and interesting theory. With this definition, scale-free (or its opposite, scale-rich) is closely related to other structural graph properties such as various notions of self-similarity (or respectively, self-dissimilarity). Scale-free graphs are also shown to be the likely outcome of random construction processes, consistent with the heuristic definitions implicit in existing random graph approaches. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the scale-free literature, and offers rigorous and quantitative alternatives.

We introduce a general stochastic model for the spread of rumours, and derive mean-field equations that describe the dynamics of the model on complex social networks (in particular, those mediated by the Internet). We use analytical and numerical solutions of these equations to examine the threshold behaviour and dynamics of the model on several models of such networks: random graphs, uncorrelated scale-free networks and scale-free networks with assortative degree correlations. We show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system. In the case of scale-free networks, on the other hand, this threshold becomes vanishingly small in the limit of infinite system size. We find that the initial rate at which a rumour spreads is much higher in scale-free networks than in random graphs, and that the rate at which the spreading proceeds on scale-free networks is further increased when assortative degree correlations are introduced. The impact of degree correlations on the final fraction of nodes that ever hears a rumour, however, depends on the interplay between network topology and the rumour spreading rate. Our results show that scale-free social networks are prone to the spreading of rumours, just as they are to the spreading of infections. They are relevant to the spreading dynamics of chain emails, viral advertising and large-scale information dissemination algorithms on the Internet. r

We construct complex networks from pseudoperiodic time series, with each cycle represented by a single node in the network. We investigate the statistical properties of these networks for various time series and find that time series with different dynamics exhibit distinct topological structures. Specifically, noisy periodic signals correspond to random networks, and chaotic time series generate networks that exhibit small world and scale free features. We show that this distinction in topological structure results from the hierarchy of unstable periodic orbits embedded in the chaotic attractor. Standard measures of structure in complex networks can therefore be applied to distinguish different dynamic regimes in time series. Application to human electrocardiograms shows that such statistical properties are able to differentiate between the sinus rhythm cardiograms of healthy volunteers and those of coronary care patients.

We describe a method for the automatic identification of communities of practice from email logs within an organization. We use a betweenness centrality algorithm that can rapidly find communities within a graph representing information flows. We apply this algorithm to an email corpus of nearly one million messages collected over a two-month span, and show that the method is effective at identifying true communities, both formal and informal, within these scale-free graphs. This approach also enables the identification of leadership roles within the communities. These studies are complemented by a qualitative evaluation of the results in the field.

Most studies of networks have only looked at small subsets of the true network. Here, we discuss the sampling properties of a network's degree distribution under the most parsimonious sampling scheme. Only if the degree distributions of the network and randomly sampled subnets belong to the same family of probability distributions is it possible to extrapolate from subnet data to properties of the global network. We show that this condition is indeed satisfied for some important classes of networks, notably classical random graphs and exponential random graphs. For scale-free degree distributions, however, this is not the case. Thus, inferences about the scale-free nature of a network may have to be treated with some caution. The work presented here has important implications for the analysis of molecular networks as well as for graph theory and the theory of networks in general.

We give a complete geometric description of conformal anomalies in arbitrary, (necessarily even) dimension. They fall into two distinct classes: the first, based on Weyl invariants that vanish at integer dimensions, arises from finite -and hence scale-free -contributions to the effective gravitational action through a mechanism analogous to that of the (gauge field) chiral anomaly. Like the latter, it is unique and proportional to a topological term, the Euler density of the dimension, thereby preserving scale invariance. The contributions of the second class, requiring introduction of a scale through regularization, are correlated to all local conformal scalar polynomials involving powers of the Weyl tensor and its derivatives; their number increases rapidly with dimension.

We find that scale-free random networks are excellently modeled by a deterministic graph. This graph has a discrete degree distribution (degree is the number of connections of a vertex) which is characterized by a power-law with exponent γ = 1 + ln 3/ ln 2. Properties of this simple structure are surprisingly close to those of growing random scale-free networks with γ in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For the large network (ln N ≫ 1) the distribution tends to a Gaussian of width ∼ √ ln N centered at ℓ ∼ ln N . We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent 2 + γ. 87.18.Sn The essence of the modern situation in network science is the change-over from study of classical random graphs with Poisson degree distributions [1,2] to exploration of complex networks with fat-tailed degree distributions . The prominent particular case of such nets are networks with power-law degree distributions (scalefree networks) . While growing, such nets actually selforganize into scale-free structures. These networks play a great role in Nature . The Internet, the WWW, and many basic biological networks belong to this class. Fat tails of the degree distributions produce a number of intriguing effects .

Many natural and social systems develop complex networks, that are usually modelled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semi-circle law is known to describe the spectral density of uncorrelated random graphs, much less is known about the eigenvalues of real-world graphs, describing such complex systems as the Internet, metabolic pathways, networks of power stations, scientific collaborations or movie actors, which are inherently correlated and usually very sparse. An important limitation in addressing the spectra of these systems is that the numerical determination of the spectra for systems with more than a few thousand nodes is prohibitively time and memory consuming. Making use of recent advances in algorithms for spectral characterization, here we develop new methods to determine the eigenvalues of networks comparable in size to real systems, obtaining several surprising results on the spectra of adjacency matrices corresponding to models of real-world graphs. We find that when the number of links grows as the number of nodes, the spectral density of uncorrelated random graphs does not converge to the semi-circle law. Furthermore, the spectral densities of real-world graphs have specific features depending on the details of the corresponding models. In particular, scale-free graphs develop a triangle-like spectral density with a power law tail, while small-world graphs have a complex spectral density function consisting of several sharp peaks. These and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks.

Structural features found in biomolecular networks that are absent in random networks produced by simple algorithms can provide insight into the function and evolution of cell regulatory networks. Here we analyze "betweenness" of network nodes, a graph theoretical centrality measure, in the yeast protein interaction network. Proteins that have high betweenness, but low connectivity (degree), were found to be abundant in the yeast proteome. This finding is not explained by algorithms proposed to explain the scale-free property of protein interaction networks, where low-connectivity proteins also have low betweenness. These data suggest the existence of some modular organization of the network, and that the high-betweenness, low-connectivity proteins may act as important links between these modules. We found that proteins with high betweenness are more likely to be essential and that evolutionary age of proteins is positively correlated with betweenness. By comparing different models of genome evolution that generate scale-free networks, we show that rewiring of interactions via mutation is an important factor in the production of such proteins. The evolutionary and functional significance of these observations are discussed.

Free traffic flow on a complex network is key to its normal and efficient functioning. Recent works indicate that many realistic networks possess connecting topologies with a scale-free feature: the probability distribution of the number of links at nodes, or the degree distribution, contains a power-law component. A natural question is then how the topology influences the dynamics of traffic flow on a complex network. Here we present two models to address this question, taking into account the network topology, the informationgenerating rate, and the information-processing capacity of individual nodes. For each model, we study four kinds of networks: scale-free, random, and regular networks and Cayley trees. In the first model, the capacity of packet delivery of each node is proportional to its number of links, while in the second model, it is proportional to the number of shortest paths passing through the node. We find, in both models, that there is a critical rate of information generation, below which the network traffic is free but above which traffic congestion occurs. Theoretical estimates are given for the critical point. For the first model, scale-free networks and random networks are found to be more tolerant to congestion. For the second model, the congestion condition is independent of network size and topology, suggesting that this model may be practically useful for designing communication protocols.

In this review, we examine cardiovascular metabolism from three different, but highly complementary, perspectives. First, from the abstract perspective of a metabolite network, composed of nodes and links. We present fundamental concepts in network theory, including emergence, to illustrate how nature has designed metabolism with a hierarchal modular scale-free topology to provide a robust system of energy delivery. Second, from the physical perspective of a modular spatially compartmentalized network. We review evidence that cardiovascular metabolism is functionally compartmentalized, such that oxidative phosphorylation, glycolysis, and glycogenolysis preferentially channel ATP to ATPases in different cellular compartments, using creatine kinase and adenylate kinase to maximize efficient energy delivery. Third, from the dynamics perspective, as a network of dynamically interactive metabolic modules capable of self-oscillation. Whereas normally, cardiac metabolism exists in a regime in which excitation-metabolism coupling closely matches energy supply and demand, we describe how under stressful conditions, the network can be pushed into a qualitatively new dynamic regime, manifested as cell-wide oscillations in ATP levels, in which the coordination between energy supply and demand is lost. We speculate how this state of "metabolic fibrillation" leads to cell death if not corrected and discuss the implications for cardioprotection.-Weiss, J. N., L. Yang, and Z. Qu. Network perspectives of cardiovascular metabolism.

The dynamics of Boolean networks with scale-free topology are studied. The existence of a phase transition from ordered to chaotic dynamics, governed by the value of the scale-free exponent of the network, is shown analytically by analyzing the overlap between two distinct trajectories. The phase diagram shows that the phase transition occurs for values of the scale-free exponent in the open interval (2, 2.5). Since the Boolean networks under study are directed graphs, the scale-free topology of the input connections and that of the output connections are studied separately. Ultimately these two topologies are shown to be equivalent. A numerical study of the attractor structure of the configuration space reveals that this structure is similar in both networks with scale-free topologies and networks with homogeneous random topologies. However, an important result of this work is that the fine-tuning usually required to achieve stability in the dynamics of networks with homogeneous random topologies is no longer necessary when the network topology is scale-free. Finally, based on the results presented in this work, it is hypothesized that the scale-free topology favors the evolution and adaptation of network functioning from a biological perspective.

Effective quantitative land surface analyses in soil science need scale-free land surface attributes (morphometric variables, MVs) to be introduced for making comparable results obtained at different scales. To investigate the problem in more detail, a conceptual scheme and curvatures studied earlier in ) [Math. Geol. 27 (1995 373] are further developed in this paper, formulae for a complete system of 12 curvatures and some other MVs are given, and modified Evans -Young algorithm for curvature calculation is described that does not emphasize grid directions. The conceptual scheme is based on that MVs often describe not the land surface itself, but rather the system ''land surface + vector field'', where vector fields of common interest are gravitational field and solar irradiation. Correspondingly, morphometric variables and concepts may (1) refer to this system description (field-specific), or (2) be invariant with respect to any vector field (fieldinvariant), that is, describing the land surface itself, its geometrical form. From the other side, MVs and concepts may be (1) local, (2) regional, which need extended portions of a restricted part of land surface for their determination at a given point, or (3) global (planetary), when elevations of all the Earth are needed for their determination at a given point. Global MVs do not consist subject of this paper; so, the four classes of MVs are considered here: class A (local field-specific MVs), class B (regional field-specific), class C (local field-invariant), and class D (regional field-invariant). MVs of these classes permit description of geometrical land form, pre-requisits of surface runoff, thermal regime of slopes, and altitude zonality. Class A contains three independent MVs expressed by first derivatives of elevation Z by plan coordinates (slope steepness, slope direction, solar insolation) and seven curvatures expressed by second derivatives of Z; class C contains five curvatures; class B contains two variables (catchment and dispersal areas); MVs of class D are not introduced yet. Also, 0016-7061/02/$ -see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 6 -7 0 6 1 ( 0 1 ) 0 0 1 3 6 -7 * Corresponding

We introduce a new family of networks, the Apollonian networks, that are simultaneously scale-free, small-world, Euclidean, space filling, and with matching graphs. These networks describe force chains in polydisperse granular packings and could also be applied to the geometry of fully fragmented porous media, hierarchical road systems, and area-covering electrical supply networks. Some of the properties of these networks, namely, the connectivity exponent, the clustering coefficient, and the shortest path are calculated and found to be particularly rich. The percolation, the electrical conduction, and the Ising models on such networks are also studied and found to be quite peculiar. Consequences for applications are also discussed.

Relatively recent work has reported that networks of neurons can produce avalanches of activity whose sizes follow a power law distribution. This suggests that these networks may be operating near a critical point, poised between a phase where activity rapidly dies out and a phase where activity is amplified over time. The hypothesis that the electrical activity of neural networks in the brain is critical is potentially important, as many simulations suggest that information processing functions would be optimized at the critical point. This hypothesis, however, is still controversial. Here we will explain the concept of criticality and review the substantial objections to the criticality hypothesis raised by skeptics. Points and counter points are presented in dialog form.

Genes and proteins are organized into functional modular networks in which the network context of a gene or protein has implications for cellular function. Highly connected hub proteins, largely responsible for maintaining network connectivity, have been found to be much more likely to be essential for yeast survival. Here we investigate the properties of weighted gene co-expression networks formed from multiple microarray datasets. The constructed networks approximate scale-free topology, but this is not universal across all datasets. We show strong positive correlations between gene connectivity within the whole network and gene essentiality as well as gene sequence conservation. We demonstrate the preservation of a modular structure of the networks formed, and demonstrate that, within some of these modules, it is possible to observe a strong correlation between connectivity and essentiality or between connectivity and conservation within the modules particularly within modules co...

Gene expression data have become an instrumental resource in describing the molecular state associated with various cellular phenotypes and responses to environmental perturbations. The utility of expression profiling has been demonstrated in partitioning clinical states, predicting the class of unknown samples and in assigning putative functional roles to previously uncharacterized genes based on profile similarity. However, gene expression profiling has had only limited success in identifying therapeutic targets. This is partly due to the fact that current methods based on fold-change focus only on single genes in isolation, and thus cannot convey causal information. In this paper, we present a technique for analysis of expression data in a graph-theoretic framework that relies on associations between genes. We describe the global organization of these networks and biological correlates of their structure. We go on to present a novel technique for the molecular characterization of...

We determine stability and attractor properties of random Boolean genetic network models with canalyzing rules for a variety of architectures. For all power law, exponential, and flat in-degree distributions, we find that the networks are dynamically stable. Furthermore, for architectures with few inputs per node, the dynamics of the networks is close to critical. In addition, the fraction of genes that are active decreases with the number of inputs per node. These results are based upon investigating ensembles of networks using analytical methods. Also, for different in-degree distributions, the numbers of fixed points and cycles are calculated, with results intuitively consistent with stability analysis; fewer inputs per node implies more cycles, and vice versa. There are hints that genetic networks acquire broader degree distributions with evolution, and hence our results indicate that for single cells, the dynamics should become more stable with evolution. However, such an effect is very likely compensated for by multicellular dynamics, because one expects less stability when interactions among cells are included. We verify this by simulations of a simple model for interactions among cells.

The conformation space of a 20 residue antiparallel b-sheet peptide, sampled by molecular dynamics simulations, is mapped to a network. Snapshots saved along the trajectory are grouped according to secondary structure into nodes of the network and the transitions between them are links. The conformation space network describes the significant free energy minima and their dynamic connectivity without requiring arbitrarily chosen reaction coordinates. As previously found for the Internet and the World-Wide Web as well as for social and biological networks, the conformation space network is scale-free and contains highly connected hubs like the native state which is the most populated free energy basin. Furthermore, the native basin exhibits a hierarchical organization, which is not found for a random heteropolymer lacking a predominant free-energy minimum. The network topology is used to identify conformations in the folding transition state (TS) ensemble, and provides a basis for understanding the heterogeneity of the TS and denatured state ensemble as well as the existence of multiple pathways.

We present an analytic model for the fully nonlinear power spectrum P and bispectrum Q of the cosmological mass density field. The model is based on physical properties of dark matter halos, with the three main model inputs being analytic halo density profiles, halo mass functions, and halo-halo spatial correlations, each of which has been well studied in the literature. We demonstrate that this model can reproduce the power spectrum and bispectrum computed from cosmological simulations of both an n = −2 scale-free model and a low-density cold dark matter model. To enhance the dynamic range of these large simulations, we use the synthetic halo replacement technique of Ma & Fry (2000), where the original halos with numerically softened cores are replaced by synthetic halos of realistic density profiles. At high wavenumbers, our model predicts a slope for the nonlinear power spectrum different from the often-used fitting formulas in the literature based on the stable clustering assumption. Our model also predicts a three-point amplitude Q that is scale dependent, in contrast to the popular hierarchical clustering assumption. This model provides a rapid way to compute the mass power spectrum and bispectrum over all length scales where the input halo properties are valid. It also provides a physical interpretation of the clustering properties of matter in the universe.

The Cosmic Microwave Background (CMB) anisotropy power on the largest angular scales observed both by WMAP and COBE DMR appears to be lower than the one predicted by the standard model of cosmology with almost scale free primordial perturbations arising from a period of inflation . One can either interpret this as a manifestation of cosmic variance or as a physical effect that requires an explanation. We discuss various mechanisms that could be responsible for the suppression of such low ℓ multipoles. Features in the late time evolution of metric fluctuations may do this via the integral Sachs-Wolfe effect. Another possibility is a suppression of power at large scales in the primordial spectrum induced by a fast rolling stage in the evolution of the inflaton field at the beginning of the last 65 e-folds of inflation. We illustrate this effect in a simple model of inflation and fit the resulting CMB spectrum to the observed temperature-temperature (TT) power spectrum. We find that the WMAP observations suggest a cutoff at kc = 4.9 +1.3 −1.6 × 10 −4 Mpc −1 at 68% confidence, while only an upper limit of kc < 7.4 × 10 −4 Mpc −1 at 95%. Thus, although it improves the fit of the data, the presence of a cutoff in power spectrum is only required at a level close to 2σ. This is obtained with a prior which corresponds to equal distribution wrt kc. We discuss how other choices (such as an equal distribution wrt ln kc which is natural in the context of inflation) can affect the statistical interpretation.

Fractional Brownian motion (fBm) has been used as a theoretical framework to study real time series appearing in diverse scientific fields. Because its intrinsic non-stationarity and long range dependence, its characterization via the Hurst parameter H requires sophisticated techniques that often yield ambiguous results. In this work we show that fBm series map into a scale free visibility graph whose degree distribution is a function of H. Concretely, it is shown that the exponent of the power law degree distribution depends linearly on H. This also applies to fractional Gaussian noises (fGn) and generic f −β noises. Taking advantage of these facts, we propose a brand new methodology to quantify long range dependence in these series. Its reliability is confirmed with extensive numerical simulations and analytical developments. Finally, we illustrate this method quantifying the persistent behavior of human gait dynamics.