@STRING ( JOSAB = "J. Opt. Soc. Am. B" ) @STRING ( APL = "Appl. Phys. Lett." ) @STRING ( ELET = "Electron. Lett." ) @STRING ( PHTLET = "IEEE Photon. Technol. Lett." ) @STRING ( JSTQE = "IEEE J. Sel. Top. Quantum Electron." ) @STRING ( JAP = "J. Appl. Phys." ) @STRING ( OQE = "Opt. Quant. Electron." ) @STRING ( JQE = "IEEE J. Quantum Electron." ) @STRING ( PR = "Phys. Rev." ) @STRING ( PRA = "Phys. Rev. A" ) @STRING ( PRB = "Phys. Rev. B" ) @STRING ( PRC = "Phys. Rev. C" ) @STRING ( PRD = "Phys. Rev. D" ) @STRING ( PRE = "Phys. Rev. E" ) @STRING ( PRL = "Phys. Rev. Lett." ) @STRING ( RMP = "Rev. Mod. Phys." ) @STRING ( JOP = "J. Physiol. London" ) @STRING ( EUPL = "Europhys. Lett." ) @STRING ( NAT = "Nature (London)" ) @STRING ( PREP = "Phys. Rep." ) @STRING ( OL = "Opt. Lett." ) @STRING ( OPEX = "Opt. Express" ) @STRING ( OC = "Opt. Commun." ) @STRING ( ADP = "Ann. Phys. (Leipzig)") @STRING ( JPA = "J. Phys. A") @STRING ( JPD = "J. Phys. D") @STRING ( QSO = "Quantum Semiclass. Opt.") @STRING ( ZPB = "Z. Phys. B") @STRING ( NJP = "New J. Phys.") @STRING ( PU = "Physica (Utrecht)") @STRING ( JLT = "J. Lightwave Technol.") @STRING ( SJQE= "Sov. J. Quantum Electron.") @STRING ( IJBC= "Int. J. Bif. Chaos") @STRING ( PIEEE= "Proc. IEEE") @STRING ( AO = "Appl. Opt.") @STRING ( APA = "Appl. Phys. A") @STRING ( APB = "Appl. Phys. B") @STRING ( SCI = "Science") @STRING ( PNAS = "Proc. Natl. Acad. Sci. U.S.A.") @STRING ( TPB = "Theor. Pop. Biol.") @STRING ( SIAMR = "SIAM Rev.") @STRING ( EPJB = "Eur. Phys. J. B") @STRING ( ARX = "arXiv") @STRING ( JTB = "J. Theor. Biol.") %\def\ao{ Appl.\ Opt.\ } %\def\ap{ Appl.\ Phys.\ } %\def\apa{ Appl.\ Phys.\ A } %\def\apb{ Appl.\ Phys.\ B } %\def\apl{ Appl.\ Phys.\ Lett.\ } %\def\apj{ Astrophys.\ J.\ } %\def\bell{ Bell Syst.\ Tech.\ J.\ } %\def\jqe{ IEEE J.\ Quantum Electron.\ } %\def\assp{ IEEE Trans.\ Acoust.\ Speech Signal Process.\ } %\def\aprop{ IEEE Trans.\ Antennas Propag.\ } %\def\mtt{ IEEE Trans.\ Microwave Theory Tech.\ } %\def\iovs{ Invest.\ Ophthalmol.\ Vis.\ Sci.\ } %\def\jcp{ J.\ Chem.\ Phys.\ } %\def\jmo{ J.\ Mod.\ Opt.\ } %\def\josa{ J.\ Opt.\ Soc.\ Am.\ } %\def\josaa{ J.\ Opt.\ Soc.\ Am.\ A } %\def\josab{ J.\ Opt.\ Soc.\ Am.\ B } %\def\jpp{ J.\ Phys.\ (Paris) } %\def\pl{ Phys.\ Lett.\ } %\def\pspie{ Proc.\ SPIE\ } %\def\vr{ Vision Res.\ } % \bibliographystyle{style} % \bibliography{bibfile} % % where `style' refers to a file `style.bst', which defines how your %citations will look. The standard styles distributed with BibTeX are: % %`alpha' % Sorted alphabetically. Labels are formed from name of author and % year of publication. % %`plain' % Sorted alphabetically. Labels are numeric. % %`unsrt' % Like `plain', but entries are in order of citation. % %`abbrv' % Like `plain', but more compact labels. @ARTICLE{Callaway00, title = {Network Robustness and Fragility: Percolation on Random Graphs}, author = {Callaway, D. S. and Newman, M. E. J. and Strogatz, S. H. and Watts, D. J.}, journal = PRL, year = {2000}, volume = {85}, number = {25}, pages = {5468--5471}, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {Recent work on the Internet, social networks, and the power grid has addressed the resilience of these networks to either random or targeted deletion of network nodes or links. Such deletions include, for example, the failure of Internet routers or power transmission Lines. Percolation models on random graphs provide a simple representation of this process but have typically been limited to graphs with Poisson degree distribution at their vertices. Such graphs are quite unlike real-world networks, which often possess power-law or other highly skewed degree distributions. In this paper we study percolation on graphs with completely general degree distribution, giving exact solutions for a variety of cases, including site percolation, bond percolation, and models in which occupation probabilities depend on vertex degree. We discuss the application of our theory to the understanding of network resilience.}, keywords = {world-wide-web, degree sequence, internet}, OPTISSN = {0031-9007}, } @ARTICLE{Newman03a, title = {The Structure and Function of Complex Networks}, author = {Newman, M. E. J.}, journal = SIAMR, year = {2003}, volume = {45}, number = {2}, pages = {167--256}, month = jun, publisher = {Siam Publications, 3600 Univ City Science Center, Philadelphia, Pa 19104-2688 USA}, abstract = {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.}, keywords = {small-world networks, scale-free networks, barabasi-albert networks, growing random networks, social networks, random graphs, wide- web, evolving networks, food webs, metabolic networks, networks, graph theory, complex systems, computer networks, social networks, random graphs, percolation theory}, OPTISSN = {0036-1445}, } @ARTICLE{Newman03b, title = {Properties of Highly Clustered Networks}, author = {Newman, M. E. J.}, journal = PRE, year = {2003}, volume = {68}, number = {2}, pages = {026121}, month = aug, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {We propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient. Among other things, our results indicate that increased clustering leads to a decrease in the size of the giant component of the network. We also study susceptible/infective/recovered type epidemic processes within the model and find that clustering decreases the size of epidemics, but also decreases the epidemic threshold, making it easier for diseases to spread. In addition, clustering causes epidemics to saturate sooner, meaning that they infect a near- maximal fraction of the network for quite low transmission rates.}, keywords = {small-world networks, complex networks, degree sequence, internet, percolation}, doi = {10.1103/PhysRevE.68.026121}, OPTISSN = {1063-651X}, } @ARTICLE{Newman04, title = {Detecting Community Structure in Networks}, author = {Newman, M. E. J.}, journal = EPJB, year = {2004}, volume = {38}, number = {2}, pages = {321--330}, month = mar, publisher = {Springer-verlag, 175 Fifth Ave, New York, Ny 10010 USA}, abstract = {There has been considerable recent interest in algorithms for finding communities in networks--groups of vertices within which connections are dense, but between which connections are sparser. Here we review the progress that has been made towards this end. We begin by describing some traditional methods of community detection, such as spectral bisection, the Kernighan- Lin algorithm and hierarchical clustering based on similarity measures. None of these methods, however, is ideal for the types of real-world network data with which current research is concerned, such as Internet and web data and biological and social networks. We describe a number of more recent algorithms that appear to work well with these data, including algorithms based on edge betweenness scores, on counts of short loops in networks and on voltage differences in resistor networks.}, keywords = {connectivity, betweenness, algorithm, blocks, graphs, motifs, set}, doi = {10.1140/epjb/e2004-00124-y}, OPTISSN = {1434-6028}, } @ARTICLE{Newman01a, title = {Random Graphs With Arbitrary Degree Distributions and Their Applications}, author = {Newman, M. E. J. and Strogatz, S. H. and Watts, D. J.}, journal = PRE, year = {2001}, volume = {64}, number = {2}, pages = {026118}, month = aug, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the worldwide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.}, keywords = {small-world networks, degree sequence, wide-web, internet, governance, dynamics}, OPTISSN = {1063-651X}, } @ARTICLE{Newman01b, title = {The Structure of Scientific Collaboration Networks}, author = {Newman, M. E. J.}, journal = PNAS, year = {2001}, volume = {98}, number = {2}, pages = {404--409}, publisher = {Natl Acad Sciences, 2101 Constitution Ave Nw, Washington, Dc 20418 USA}, abstract = {The structure of scientific collaboration networks is investigated. Two scientists are considered connected if they have authored a paper together and explicit networks of such connections are constructed by using data drawn from a number of databases, including MEDLINE (biomedical research), the Los Alamos e-Print Archive (physics), and NCSTRL (computer science). I show that these collaboration networks form "small worlds," in which randomly chosen pairs of scientists are typically separated by only a short path of intermediate acquaintances. I further give results for mean and distribution of numbers of collaborators of authors, demonstrate the presence of clustering in the networks, and highlight a number of apparent differences in the patterns of collaboration between the fields studied.}, keywords = {small-world networks, web}, OPTISSN = {0027-8424}, } @ARTICLE{Dorogovtsev02, title = {Pseudofractal Scale-Free Web}, author = {Dorogovtsev, S. N. and Goltsev, A. V. and Mendes, J. F. F.}, journal = PRE, year = {2002}, volume = {65}, number = {6}, pages = {066122}, month = jun, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {We find that scale-free random networks are excellently modeled by simple deterministic graphs. Our graph has a discrete degree distribution (degree is the number of connections of a vertex), which is characterized by a power law with exponent gamma=1+ln 3/ln 2. Properties of this compact structure are surprisingly close to those of growing random scale-free networks with gamma 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 a large network (ln N>>1) the distribution tends to a Gaussian of width similar torootln N centered at (-)lsimilar toln N. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent 2+gamma.}, keywords = {growing random networks, random graphs, intentional attack, internet, breakdown, spectra, law}, doi = {10.1103/PhysRevE.65.066122}, OPTISSN = {1063-651X}, } @Book{Dorogovtsev03, author = {S.N. Dorogovtsev and J.F.F. Mendes}, title = {Evolution of Networks: From Biological Nets to the Internet and WWW}, publisher = {Oxford University Press, Oxford}, year = {2003}, } @ARTICLE{Goltsev06, title = {k-core (bootstrap) percolation on complex networks: Critical phenomena and nonlocal effects}, author = {Goltsev, A. V. and Dorogovtsev, S. N. and Mendes, J. F. F.}, journal = PRE, year = {2006}, volume = {73}, number = {5}, pages = {056101}, month = may, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {We develop the theory of the k-core (bootstrap) percolation on uncorrelated random networks with arbitrary degree distributions. We show that the k-core percolation is an unusual, hybrid phase transition with a jump emergence of the k-core as at a first order phase transition but also with a critical singularity as at a continuous transition. We describe the properties of the k-core, explain the meaning of the order parameter for the k-core percolation, and reveal the origin of the specific critical phenomena. We demonstrate that a so- called "corona" of the k-core plays a crucial role (corona is a subset of vertices in the k-core which have exactly k neighbors in the k-core). It turns out that the k-core percolation threshold is at the same time the percolation threshold of finite corona clusters. The mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point. We show that a random removal of even one vertex from the k-core may result in the collapse of a vast region of the k-core around the removed vertex. The mean size of this region diverges at the critical point. We find an exact mapping of the k-core percolation to a model of cooperative relaxation. This model undergoes critical relaxation with a divergent rate at some critical moment.}, keywords = {regular graphs, bethe lattice}, doi = {10.1103/PhysRevE.73.056101}, OPTISSN = {1539-3755}, } @ARTICLE{Dorogovtsev06, title = {k-core organization of complex networks}, author = {Dorogovtsev, S. N. and Goltsev, A. V. and Mendes, J. F. F.}, journal = PRL, year = {2006}, volume = {96}, number = {4}, pages = {040601}, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {We analytically describe the architecture of randomly damaged uncorrelated networks as a set of successively enclosed substructures-k-cores. The k-core is the largest subgraph where vertices have at least k interconnections. We find the structure of k-cores, their sizes, and their birthpoints-the bootstrap percolation thresholds. We show that in networks with a finite mean number z(2) of the second-nearest neighbors, the emergence of a k-core is a hybrid phase transition. In contrast, if z(2) diverges, the networks contain an infinite sequence of k-cores which are ultrarobust against random damage.}, keywords = {community structure, percolation}, doi = {10.1103/PhysRevLett.96.040601}, OPTISSN = {0031-9007}, } @ARTICLE{Dorogovtsev08, title = {Critical phenomena in complex networks}, author = {Dorogovtsev, S. N. and Goltsev, A. V. and Mendes, J. F. F.}, journal = RMP, year = {2008}, volume = {80}, number = {1275}, pages = {1275}, abstract = { The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, k-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed. }, doi = {doi:10.1103/RevModPhys.80.1275}, } @ARTICLE{Boccaletti06, title = {Complex Networks: Structure and Dynamics}, author = {Boccaletti, S. and Latora, V. and Moreno, Y. and Chavez, M. and Hwang, D. U.}, journal = PREP, year = {2006}, volume = {424}, number = {4--5}, pages = {175--308}, month = feb, publisher = {Elsevier Science Bv, Po Box 211, 1000 Ae Amsterdam, Netherlands}, abstract = {Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks' dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering. (c) 2005 Elsevier B.V. All rights reserved.}, keywords = {small-world networks, scale-free networks, protein-interaction networks, coupled chaotic systems, sexually-transmitted- diseases, gene-expression networks, barabasi-albert networks, law spatial correlations, monte-carlo simulations, primate cerebral-cortex}, doi = {10.1016/j.physrep.2005.10.009}, OPTISSN = {0370-1573}, } @ARTICLE{Watts98, title = {Collective Dynamics of 'small-world' Networks}, author = {Watts, D. J. and Strogatz, S. H.}, journal = NAT, year = {1998}, volume = {393}, number = {6684}, pages = {440--442}, publisher = {Macmillan Magazines Ltd, Porters South, 4 Crinan St, London, England N1 9xw}, abstract = {Networks of coupled dynamical systems have been used to model biological oscillators(1-4), Josephson junction arrays(5,6), excitable media(7), neural networks(8-10), spatial games(11), genetic control networks(12) and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon(13,14) (popularly known as six degrees of separation(15)). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small- world networks than in regular lattices.}, keywords = {pulse-coupled oscillators, synchronization, disease, spread, chaos}, OPTISSN = {0028-0836}, } @ARTICLE{Vazquez02, title = {Large-Scale Topological and Dynamical Properties of the Internet}, author = {V\'{a}zquez, A. and Pastor-Satorras, R. and Vespignani, A.}, journal = PRE, year = {2002}, volume = {65}, number = {6}, pages = {066130}, month = jun, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {We study the large-scale topological and dynamical properties of real Internet maps at the autonomous system level, collected in a 3-yr time interval. We find that the connectivity structure of the Internet presents statistical distributions settled in a well-defined stationary state. The large-scale properties are characterized by a scale-free topology consistent with previous observations. Correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the Internet. The study of the Internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models. We compare the properties of growing network models with the present real Internet data analysis.}, keywords = {growing random networks, small-world networks, random graphs, evolving networks, complex networks, degree sequence, wide-web, attack, growth}, doi = {10.1103/PhysRevE.65.066130}, OPTISSN = {1063-651X}, } @ARTICLE{Pastor-Satorras01, title = {Epidemic Spreading in Scale-Free Networks}, author = {Pastor-Satorras, R. and Vespignani, A.}, journal = PRL, year = {2001}, volume = {86}, number = {14}, pages = {3200--3203}, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {The Internet has a very complex connectivity recently modeled by the class of scale-free networks. This feature, which appears to be very efficient for a communications network, favors at the same time the spreading of computer viruses. We analyze real data from computer virus infections and find the average lifetime and persistence of viral strains on the Internet. We define a dynamical model for the spreading of infections on scale-free networks. finding the absence of an epidemic threshold and its associated critical behavior. This new epidemiological framework rationalizes data of computer viruses and could help in the understanding of other spreading phenomena on communication and social networks.}, keywords = {small-world networks, internet}, OPTISSN = {0031-9007}, } @ARTICLE{Boguna03, title = {Absence of Epidemic Threshold in Scale-Free Networks With Degree Correlations}, author = {Bogu\~{n}\'{a}, M. and Pastor-Satorras, R. and Vespignani, A.}, journal = PRL, year = {2003}, volume = {90}, number = {2}, pages = {028701}, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {Random scale-free networks have the peculiar property of being prone to the spreading of infections. Here we provide for the susceptible-infected-susceptible model an exact result showing that a scale-free degree distribution with diverging second moment is a sufficient condition to have null epidemic threshold in unstructured networks with either assortative or disassortative mixing. Degree correlations result therefore irrelevant for the epidemic spreading picture in these scale- free networks. The present result is related to the divergence of the average nearest neighbor's degree, enforced by the degree detailed balance condition.}, keywords = {complex networks, dynamics}, doi = {10.1103/PhysRevLett.90.028701}, OPTISSN = {0031-9007}, } @ARTICLE{Boguna04, title = {Models of Social Networks Based on Social Distance Attachment}, author = {Bogu\~{n}\'{a}, M. and Pastor-Satorras, R. and Diaz-Guilera, A. and Arenas, A.}, journal = PRE, year = {2004}, volume = {70}, number = {5}, pages = {056122}, month = nov, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {We propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance. Social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space. We derive analytical results (corroborated by extensive numerical simulations), showing that the model reproduces the main statistical characteristics of real social networks: large clustering coefficient, positive degree correlations, and the emergence of a hierarchy of communities. The model is confronted with the social network formed by people that shares confidential information using the Pretty Good Privacy (PGP) encryption algorithm, the so-called web of trust of PGP.}, keywords = {small-world networks, betweenness}, doi = {10.1103/PhysRevE.70.056122}, OPTISSN = {1063-651X}, } @ARTICLE{Albert00, title = {Error and Attack Tolerance of Complex Networks}, author = {Albert, R. and Jeong, H. and Barab\'{a}si, A. L.}, journal = NAT, year = {2000}, volume = {406}, number = {6794}, pages = {378--382}, publisher = {Macmillan Publishers Ltd, Porters South, 4 Crinan St, London N1 9xw, England}, abstract = {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(1). Complex communication networks(2) 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(3-5), the Internet(6), social networks(7) and cells(8). 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.}, keywords = {small-world networks, wide-web, internet, dynamics}, OPTISSN = {0028-0836}, } @ARTICLE{Albert99, title = {Internet - Diameter of the World-Wide Web}, author = {Albert, R. and Jeong, H. and Barab\'{a}si, A. L.}, journal = NAT, year = {1999}, volume = {401}, number = {6749}, pages = {130--131}, publisher = {Macmillan Magazines Ltd, Porters South, 4 Crinan St, London N1 9xw, England}, OPTISSN = {0028-0836}, } @ARTICLE{Barabasi99, title = {Emergence of Scaling in Random Networks}, author = {Barab\'{a}si, A. L. and Albert, R.}, journal = SCI, year = {1999}, volume = {286}, number = {5439}, pages = {509--512}, publisher = {Amer Assoc Advancement Science, 1200 New York Ave, Nw, Washington, Dc 20005 USA}, abstract = {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 was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of Large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.}, keywords = {world-wide-web, complexity, internet, dynamics}, OPTISSN = {0036-8075}, } @ARTICLE{Ravasz03, title = {Hierarchical Organization in Complex Networks}, author = {Ravasz, E. and Barab\'{a}si, A. L.}, journal = PRE, year = {2003}, volume = {67}, number = {2}, pages = {026112}, month = feb, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {Many real networks in nature and society share two generic properties: they are scale-free and they display a high degree of clustering. We show that these two features are the consequence of a hierarchical organization, implying that small groups of nodes organize in a hierarchical manner into increasingly large groups, while maintaining a scale-free topology. In hierarchical networks, the degree of clustering characterizing the different groups follows a strict scaling law, which can be used to identify the presence of a hierarchical organization in real networks. We find that several real networks, such as the Worldwideweb, actor network, the Internet at the domain level, and the semantic web obey this scaling law, indicating that hierarchy is a fundamental characteristic of many complex systems.}, keywords = {scale-free networks, scientific collaboration networks, small-world networks, metabolic networks, evolving networks, wide-web, internet, evolution, topology, attack}, doi = {10.1103/PhysRevE.67.026112}, OPTISSN = {1063-651X}, } @ARTICLE{Serrano06a, title = {Clustering in Complex Networks. I. General Formalism}, author = {Serrano, M. {\relax \'A}. and Bogu\~{n}\'{a}, M.}, journal = PRE, year = {2006}, volume = {74}, number = {5}, pages = {056114}, month = nov, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {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 different metrics that are able to thoroughly characterize transitive relations. A rigorous analysis of several real networks, which makes use of this formalism and the metrics, is also provided. It is also found that clustered networks can be classified into two main groups: the weak and the 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.}, doi = {10.1103/PhysRevE.74.056114}, OPTISSN = {1539-3755}, } @ARTICLE{Serrano06b, title = {Clustering in Complex Networks. {II}. Percolation Properties}, author = {Serrano, M. {\relax \'A}. and Bogu\~{n}\'{a}, M.}, journal = PRE, year = {2006}, volume = {74}, number = {5}, pages = {056115}, month = nov, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {The percolation properties of clustered networks are analyzed in detail. In the case of weak clustering, we present an analytical approach that allows us 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.}, keywords = {degree sequence, random graphs, epidemics}, doi = {10.1103/PhysRevE.74.056115}, OPTISSN = {1539-3755}, } @ARTICLE{Serrano06c, title = {Percolation and Epidemic Thresholds in Clustered Networks}, author = {Serrano, M. {\relax \'A}. and Bogu\~{n}\'{a}, M.}, journal = PRL, year = {2006}, volume = {97}, number = {8}, pages = {088701}, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {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, thus extending 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.}, keywords = {scale-free networks}, doi = {10.1103/PhysRevLett.97.088701}, OPTISSN = {0031-9007}, } @ARTICLE{Carmi07, title = {A Model of Internet Topology Using K-Shell Decomposition}, author = {Carmi, S. and Havlin, S. and Kirkpatrick, S. and Shavitt, Y. and Shir, E.}, journal = PNAS, year = {2007}, volume = {104}, number = {27}, pages = {11150--11154}, publisher = {Natl Acad Sciences, 2101 Constitution Ave Nw, Washington, Dc 20418 USA}, abstract = {We study a map of the Internet (at the autonomous systems level), by introducing and using the method of k-shell decomposition and the methods of percolation theory and f ractal geometry, to find a model for the structure of the Internet. In particular, our analysis uses information on the connectivity of the network shells to separate, in a unique (no parameters) way, the Internet into three subcomponents: (i) a nucleus that is a small (approximate to 100 nodes), very well connected globally distributed subgraph; (ii) a fractal subcomponent that is able to connect the bulk of the Internet without congesting the nucleus, with self-similar properties and critical exponents predicted from percolation theory; and (iii) dendrite-like structures, usually isolated nodes that are connected to the rest of the network through the nucleus only. We show that our method of decomposition is robust and provides insight into the underlying structure of the Internet and its functional consequences. Our approach of decomposing the network is general and also useful when studying other complex networks.}, keywords = {complex networks, emergence, fractals, networks, percolation}, doi = {10.1073/pnas.0701175104}, OPTISSN = {0027-8424}, } @ARTICLE{Cohen00, title = {Resilience of the Internet to Random Breakdowns}, author = {Cohen, R. and Erez, K. and ben--Avraham, D. and Havlin, S.}, journal = PRL, year = {2000}, volume = {85}, number = {21}, pages = {4626--4628}, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {A common property of many large networks, including the Internet, is that the connectivity of the various nodes follows a scale-free power-law distribution, P(k) = ck(-alpha). We study the stability of such networks with respect to crashes, such as random removal of sites; Our approach, based on percolation theory, leads to a general condition for the critical fraction of nodes, p(c), that needs to be removed before the network disintegrates. We show analytically and numerically that for alpha less than or equal to 3 the transition never takes place, unless the network is finite. In the special case of the physical structure of the Internet (alpha approximate to 2.5), we find that it is impressively robust, with p(c) > 0.99.}, keywords = {world-wide-web, networks, topology, dynamics}, OPTISSN = {0031-9007}, } @ARTICLE{Vazquez03, title = {Resilience to Damage of Graphs With Degree Correlations}, author = {V\'{a}zquez, A. and Moreno, Y.}, journal = PRE, year = {2003}, volume = {67}, number = {1}, pages = {015101(R)}, month = jan, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {The existence or nonexistence of a percolation threshold on power law correlated graphs is a fundamental question for which a general criterion is lacking. In this work we investigate the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena. We obtain some general expressions that allow the computation of the transition thresholds or their bounds. Using these results we study the effects of assortative and disassortative correlations on the resilience to damage of networks.}, keywords = {complex networks, percolation, internet, attack}, doi = {10.1103/PhysRevE.67.015101}, OPTISSN = {1063-651X}, } @ARTICLE{Eames08, title = {Modelling Disease Spread Through Random and Regular Contacts in Clustered Populations}, author = {K. T. D. Eames}, journal = TPB, year = {2008}, volume = {73}, number = {1}, pages = {104--111}, month = feb, publisher = {Academic Press Inc Elsevier Science, 525 B St, Ste 1900, San Diego, Ca 92101-4495 USA}, abstract = {An epidemic spreading through a network of regular, repeated, contacts behaves differently from one that is spread by random interactions: regular contacts serve to reduce the speed and eventual size of an epidemic. This paper uses a mathematical model to explore the difference between regular and random contacts, considering particularly the effect of clustering within the contact network. In a clustered population random contacts have a much greater impact, allowing infection to reach parts of the network that would otherwise be inaccessible. When all contacts are regular, clustering greatly reduces the spread of infection; this effect is negated by a small number of random contacts. (C) 2007 Elsevier Inc. All rights reserved.}, keywords = {social networks, mouth epidemic, transmission, strategies, virulence, gonorrhea, evolution, patterns, foot, epidemic, network, clustering, pair-wise approximation}, doi = {10.1016/j.tpb.2007.09.007}, OPTISSN = {0040-5809}, } @PhdThesis{Trapman06, author = {P. Trapman}, title = {On stochastic models for the spread of infections}, school = {Vrije Univ. Amsterdam}, year = {2006}, OPTurl = {http://www.cs.vu.nl/~ptrapman/proefschrifttrapmanrevisie.pdf}, } @ARTICLE{Trapman07, title = {On Analytical Approaches to Epidemics on Networks}, author = {Trapman, P.}, journal = TPB, year = {2007}, volume = {71}, number = {2}, pages = {160--173}, month = mar, publisher = {Academic Press Inc Elsevier Science, 525 B St, Ste 1900, San Diego, Ca 92101-4495 USA}, abstract = {One way to describe the spread of an infection on a network is by approximating the network by a random graph. However, the usual way of constructing a random graph does not give any control over the number of triangles in the graph, while these triangles will naturally arise in many networks (e.g. in social networks).}, keywords = {complex networks, dynamics, models, spread, random graphs, percolation, epidemics}, doi = {10.1016/j.tpb.2006.11.002}, OPTISSN = {0040-5809}, } @ARTICLE{Gleeson07a, title = {Seed Size Strongly Affects Cascades on Random Networks}, author = {Gleeson, J. P. and Cahalane, D. J.}, journal = PRE, year = {2007}, volume = {75}, number = {5}, pages = {056103}, month = may, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {The average avalanche size in the Watts model of threshold dynamics on random networks of arbitrary degree distribution is determined analytically. Existence criteria for global cascades are shown to depend sensitively on the size of the initial seed disturbance. The dependence of cascade size upon the mean degree z of the network is known to exhibit several transitions-these are typically continuous at low z and discontinuous at high z; here it is demonstrated that the low-z transition may in fact be discontinuous in certain parameter regimes. Connections between these results and the zero- temperature random-field Ising model on random graphs are discussed.}, keywords = {scale-free networks, field ising-model, complex networks, dynamics, hysteresis}, doi = {10.1103/PhysRevE.75.056103}, OPTISSN = {1539-3755}, } @ARTICLE{Gleeson07b, title = {An Analytical Approach to Cascades on Random Networks}, author = {Gleeson, J. P. and Cahalane, D. J.}, journal = {Noise and Stochastics in Complex Systems and Finance Book Series: Proceedings of the Society of Photo-optical Instrumentation Engineers (spie)}, year = {2007}, volume = {6601}, pages = {U235--U246}, publisher = {Spie-int Soc Optical Engineering, 1000 20th St, Po Box 10, Bellingham, Wa 98227-0010 USA}, abstract = {The expected steady-state fraction of active nodes in Watts' model of threshold dynamics on random networks is determined analytically. The analysis applies to random graphs with arbitrary degree distributions, and includes the effect of finite seed fractions. The seed fraction is shown to have a strong impact upon the existence of global cascades and Watts' cascade condition is extended to include these effects.}, keywords = {scale-free networks, complex networks, dynamics, hysteresis, model, complex networks, cascades, threshold dynamics}, doi = {10.1117/12.724525}, OPTISSN = {0277-786X}, } @ARTICLE{Gleeson08a, title = {Cascades on Correlated and Modular Random Networks}, author = {Gleeson, J. P.}, journal = PRE, year = {2008}, volume = {77}, number = {4}, pages = {046117}, month = apr, publisher = {Amer Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {An analytical approach to determining the mean avalanche size in a broad class of dynamical models on random networks is introduced. Previous results on percolation transitions and epidemic sizes are shown to be special cases of the method. The time-dependence of cascades and extensions to networks with community structure or degree-degree correlations are discussed. Analytical results for the rate of spread of innovations in a modular network and for the size of k cores in networks with degree-degree correlations are confirmed with numerical simulations.}, keywords = {scale-free networks, field ising-model, complex networks, random graphs, dynamics, hysteresis, internet}, doi = {10.1103/PhysRevE.77.046117}, OPTISSN = {1539-3755}, } @ARTICLE{Gleeson08b, title = {Mean Size of Avalanches on Directed Random Networks With Arbitrary Degree Distributions}, author = {Gleeson, J. P.}, journal = PRE, year = {2008}, volume = {77}, number = {5}, pages = {057101}, month = may, publisher = {Amer Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {The mean size of unordered binary avalanches on infinite directed random networks may be determined using the damage propagation function introduced by [B. Samuelsson and J. E. S. Socolar, Phys. Rev. E 74, 036113 (2006)]. The derivation of Samuelsson and Socolar explicitly assumes a Poisson distribution of out- degrees. It is shown here that the damage propagation function method may be used whenever the in-degree and out-degree of network nodes are independently distributed; in particular, it is not necessary that the out-degree distribution be Poisson. The general case of correlated in- and out-degrees is discussed and numerical simulations (on large finite networks) are compared with the theoretical predictions (for infinite networks).}, doi = {10.1103/PhysRevE.77.057101}, OPTISSN = {1539-3755}, } @ARTICLE{Warren02, title = {Geography in a Scale-Free Network Model}, author = {Warren, C. P. and Sander, L. M. and Sokolov, I. M.}, journal = PRE, year = {2002}, volume = {66}, number = {5}, pages = {056105}, month = nov, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {We offer an example of a network model with a power-law degree distribution, P(k)similar tok(-alpha), for nodes, but which nevertheless has a well-defined geography and a nonzero threshold percolation probability for alpha>2, the range of real-world contact networks. This is different from p(c)=0 for alpha<3 results for the original well-mixed scale-free networks. In our lattice-based scale-free network, individuals link to nearby neighbors on a lattice. Even considerable additional small-world links do not change our conclusion of nonzero thresholds. When applied to disease propagation, these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering.}, keywords = {sexual life-styles, percolation}, doi = {10.1103/PhysRevE.66.056105}, OPTISSN = {1063-651X}, } @ARTICLE{Eguiluz02, title = {Epidemic Threshold in Structured Scale-Free Networks}, author = {Eguiluz, V. M. and Klemm, K.}, journal = PRL, year = {2002}, volume = {89}, number = {10}, pages = {108701}, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {We analyze the spreading of viruses in scale-free networks with high clustering and degree correlations, as found in the Internet graph. For the susceptible-infected-susceptible model of epidemics the prevalence undergoes a phase transition at a finite threshold of the transmission probability. Comparing with the absence of a finite threshold in networks with purely random wiring, our result suggests that high clustering (modularity) and degree correlations protect scale-free networks against the spreading of viruses. We introduce and verify a quantitative description of the epidemic threshold based on the connectivity of the neighborhoods of the hubs.}, keywords = {small-world networks, transmission dynamics, complex networks, hiv}, doi = {10.1103/PhysRevLett.89.108701}, OPTISSN = {0031-9007}, } @ARTICLE{Radicchi04, title = {Defining and Identifying Communities in Networks}, author = {Radicchi, F. and Castellano, C. and Cecconi, F. and Loreto, V. and Parisi, D.}, journal = PNAS, year = {2004}, volume = {101}, number = {9}, pages = {2658--2663}, publisher = {Natl Acad Sciences, 2101 Constitution Ave Nw, Washington, Dc 20418 USA}, abstract = {The investigation of community structures in networks is an important issue in many domains and disciplines. This problem is relevant for social tasks (objective analysis of relationships on the web), biological inquiries (functional studies in metabolic and protein networks), or technological problems (optimization of large infrastructures). Several types of algorithms exist for revealing the community structure in networks, but a general and quantitative definition of community is not implemented in the algorithms, leading to an intrinsic difficulty in the interpretation of the results without any additional nontopological information. In this article we deal with this problem by showing how quantitative definitions of community are implemented in practice in the existing algorithms. In this way the algorithms for the identification of the community structure become fully self-contained. Furthermore, we propose a local algorithm to detect communities which outperforms the existing algorithms with respect to computational cost, keeping the same level of reliability. The algorithm is tested on artificial and real-world graphs. In particular, we show how the algorithm applies to a network of scientific collaborations, which, for its size, cannot be attacked with the usual methods. This type of local algorithm could open the way to applications to large-scale technological and biological systems.}, keywords = {metabolic networks, small-world, complex networks, food webs, organization, betweenness}, doi = {10.1073/pnas.0400054101}, OPTISSN = {0027-8424}, } @ARTICLE{Kim05, title = {Cyclic Topology in Complex Networks}, author = {Kim, H. J. and Kim, J. M.}, journal = PRE, year = {2005}, volume = {72}, number = {3}, pages = {036109}, month = sep, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {We introduce a cyclic coefficient R which characterizes the degree of circulation in complex networks. If a network has a perfect treelike structure, then R becomes zero. The larger value of R represents that the network has more cyclic structure. We measure both the cyclic coefficients and the distributions of local cyclic coefficients for various networks and discuss the cyclic structures of them.}, keywords = {organization, internet, web}, doi = {10.1103/PhysRevE.72.036109}, OPTISSN = {1539-3755}, } @Article{Britton07, author = {T. Britton and M. Deijfen and A. N. Lager\.{a}s and M. Lindholm}, journal = ARX, title = {Epidemics on random graphs with tunable clustering}, year = {2007}, eprint = {0708.3939}, archivePrefix = {arXiv}, primaryClass = {}, version = {v1}, url = {http://arxiv.org/abs/0708.3939v1}, SLACcitation = {%%CITATION = 0803.0586;%%}, abstract = {Cite as: arXiv:0708.3939v1 [math.PR] In this paper, a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. It is investigated how these quantities varies with the clustering in the graph and it turns out for instance that, as the clustering increases, the epidemic threshold decreases. The network is modelled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if they share at least one group.} } @Article{Miller08, author = {J. C. Miller}, journal = ARX, title = {The spread of infectious diseases through clustered populations}, year = {2008}, eprint = {0806.2888}, archivePrefix = {arXiv}, primaryClass = {}, version = {v1}, url = {http://arxiv.org/abs/0806.2888v1}, SLACcitation = {%%CITATION = 0803.0586;%%}, abstract = {Cite as: arXiv:0806.2888v1 [q-bio.QM] Contact networks form the substrate along which infectious diseases spread. Most network-based studies of the spread have only considered the impact of variations in node degrees. However, a number of other effects such as clustering, variations in infectiousness or susceptibility, or variations in closeness of contacts are expected to play a significant role. We find analytic techniques to predict how these effects alter the growth rate, probability, and size of epidemics which we validate for a realistic social network. We find that (for given degree distribution and average transmissibility) clustering is the dominant factor controlling the growth rate, heterogeneity in infectiousness is the dominant factor controlling the probability of an epidemic, and heterogeneity in susceptibility is the dominant factor controlling the size of an epidemic. Edge weights (measuring closeness or duration of contacts) have impact only if correlations exist between different edges. Combined, these effects can play a minor role in reinforcing one another, with the impact of clustering largest when the population is maximally heterogeneous or if the closer contacts are also strongly clustered. Our results have a number of implications for design of interventions.}, } @Article{Guardiola02, author = {X. Guardiola and R. Guimera and A. Arenas and A. Diaz-Guilera and D. Streib and L. A. N. Amaral}, journal = ARX, title = {Macro- and micro-structure of trust networks}, year = {2002}, eprint = {0206240}, archivePrefix = {arXiv}, primaryClass = {cond-mat}, version = {v1}, url = {http://arxiv.org/abs/cond-mat/0206240v1}, abstract = { Cite as: arXiv:cond-mat/0206240v1 [cond-mat.dis-nn] A challenging problem in the social sciences is the characterization of the formation of ``social capital''. It is believed that societies with more social capital are more democratic and economically developed than societies with little social capital. However, social capital is a concept hard to quantify and measure in a ``real-word'' context. Here, we take advantage of an existing ``web of trust'' between users of the ``Pretty-Good-Privacy'' (PGP) encryption algorithm, which is used in digital communication to ``sign'' documents so that the recipient knows for sure who the author is. Our analysis reveals the coexistence in the web of trust of a macro scale-free structure with micro strongly-connected cells in a complex network. We also show that when this network is intentionally attacked the scale-free structure rapidly disintegrates and that the resulting network is partitioned into a large number of small strongly-connected cells that are resilient to the intentional attack. } } @Misc{PGP_url, title = {Giant component of the network of users of the {P}retty-{G}ood-{P}rivacy algorithm for secure information interchange}, url = {http://deim.urv.cat/~aarenas/data/xarxes/PGP.zip}, } @Misc{WWW_url, title = {World {W}ide {W}eb data for webpages within \emph{nd.edu} domain}, url = {http://www.nd.edu/~networks/resources/www/www.dat.gz; www.barabasilab.com/resources/www/www.dat.gz}, } @Misc{Actor_url, url = {www.barabasilab.com/resources/actor/actor.dat.gz} } @Misc{asrel_url, title = {The {CAIDA} {A}utonomous {S}ystem {R}elationships {D}ataset, 30-{J}un-2008}, url = {http://www.caida.org/data/active/as-relationships; http://as-rank.caida.org/data/2008/as-rel.20080630.a0.01000.txt}, } @Misc{ITDK_url, title = {Internet router-level graph computed from {ITDK0304} skitter and iffinder measurements. ``{CAIDA}'s {I}nternet {T}opology {D}ata {K}it \#0304.'' {S}an {D}iego {S}upercomputer {C}enter, {U}niversity of {C}alifornia, {S}an {D}iego}, year = {2003}, url = {www.caida.org/tools/measurement/skitter/router_topology/itdk0304_rlinks_undirected.gz}, } @Misc{cond_mat_url, title = {Network of coauthorships between scientists posting preprints on the {C}ondensed {M}atter {E-P}rint {A}rchive, includes all preprints posted between 1-{J}an-1995 and 31-{M}ar-2005}, url = {http://www-personal.umich.edu/~mejn/netdata/cond-mat-2005.zip}, } @Misc{power_url, title = {An undirected, unweighted network representing the topology of the {W}estern {S}tates {P}ower {G}rid of the {U}nited {S}tates}, url = {http://cdg.columbia.edu/uploads/datasets/power_unweighted}, } @ARTICLE{Boguna02, title = {Epidemic Spreading in Correlated Complex Networks}, author = {Bogu\~{n}\'{a}, M. and Pastor-Satorras, R.}, journal = PRE, year = {2002}, volume = {66}, pages = {047104}, number = {4}, month = oct, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {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, which 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.}, keywords = {degree sequence, graphs}, doi = {10.1103/PhysRevE.66.047104}, OPTISSN = {1063-651X}, } @Article{Boguna03b, author = {Bogu\~{n}\'{a}, M. and Pastor-Satorras, R. and Vespignani, A.}, journal = ARX, title = {Epidemic spreading in complex networks with degree correlations}, year = {2003}, eprint = {0301149v1}, archivePrefix = {arXiv}, primaryClass = {}, version = {v1}, url = {http://arxiv.org/abs/cond-mat/0301149v1}, SLACcitation = {%%CITATION = %%}, abstract = {Cite as: arXiv:cond-mat/0301149v1 [cond-mat.stat-mech] We review the behavior of epidemic spreading on complex networks in which there are explicit correlations among the degrees of connected vertices.} } @ARTICLE{Goltsev08, title = {Percolation on Correlated Networks}, author = {Goltsev, A. V. and Dorogovtsev, S. N. and Mendes, J. F. F.}, journal = PRE, year = {2008}, volume = {78}, pages = {051105}, number = {5}, month = nov, publisher = {Amer Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {We reconsider the problem of percolation on an equilibrium random network with degree-degree correlations between nearest- neighboring vertices focusing on critical singularities at a percolation threshold. We obtain criteria for degree-degree correlations to be irrelevant for critical singularities. We present examples of networks in which assortative and disassortative mixing leads to unusual percolation properties and new critical exponents.}, keywords = {complex networks, 2-dimensional systems, internet, growth}, doi = {10.1103/PhysRevE.78.051105}, OPTISSN = {1539-3755}, } @ARTICLE{Burda04, title = {Perturbing General Uncorrelated Networks}, author = {Burda, Z. and Jurkiewicz, J. and Krzywicki, A.}, journal = PRE, year = {2004}, volume = {70}, pages = {026106}, number = {2}, month = aug, publisher = {American Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {This paper is a direct continuation of an earlier work, where we studied Erdos-Renyi random graphs perturbed by an interaction Hamiltonian favoring the formation of short cycles. Here, we generalize these results. We keep the same interaction Hamiltonian but let it act on general graphs with uncorrelated nodes and an arbitrary given degree distribution. It is shown that the results obtained for Erdos-Renyi graphs are generic, at the qualitative level. However, scale-free graphs are an exception to this general rule and exhibit a singular behavior, studied thoroughly in this paper, both analytically and numerically.}, keywords = {complex networks}, doi = {10.1103/PhysRevE.70.026106}, OPTISSN = {1063-651X}, } @ARTICLE{Bianconi08, title = {Local Structure of Directed Networks}, author = {Bianconi, G. and Gulbahce, N. and Motter, A. E.}, journal = PRL, year = {2008}, volume = {100}, pages = {118701}, number = {11}, publisher = {Amer Physical Soc, One Physics Ellipse, College Pk, Md 20740-3844 USA}, abstract = {Previous work on undirected small-world networks established the paradigm that locally structured networks tend to have a high density of short loops. On the other hand, many realistic networks are directed. Here we investigate the local organization of directed networks and find, surprisingly, that real networks often have very few short loops as compared to random models. We develop a theory and derive conditions for determining if a given network has more or less loops than its randomized counterparts. These findings carry broad implications for structural and dynamical processes sustained by directed networks.}, doi = {10.1103/PhysRevLett.100.118701}, OPTISSN = {0031-9007}, } @InCollection{Bollobas84, author = {B. Bollob\'{a}s}, OPTtitle = {Graph Theory and Combinatorics}, booktitle = {Graph Theory and Combinatorics: Proc. Cambridge Combinatorial Conf. in honour of Paul Erd\H{o}s}, publisher = {Academic Press, New York}, pages = {35}, year = {1984}, editor = {B. Bollob\'{a}s}, OPTchapter = {}, } @article{Newman03c, author = {Newman, M. E. J. }, citeulike-article-id = {3061406}, doi = {http://dx.doi.org/10.1103/PhysRevE.67.026126}, journal = PRE, keywords = {analysis, methodology, network, reciprocity, social}, month = {February}, number = {2}, pages = {026126}, posted-at = {2009-01-19 05:54:29}, priority = {0}, publisher = {American Physical Society}, title = {Mixing patterns in networks}, volume = {67}, year = {2003} } @article{Newman02, abstract = {A network is said to show assortative mixing if the nodes in the network that have many connections tend to be connected to other nodes with many connections. Here we measure mixing patterns in a variety of networks and find that social networks are mostly assortatively mixed; but that technological and biological networks tend to be disassortative. We propose a model of an assortatively mixed network; which we study both analytically and numerically. Within this model we find that networks percolate more easily if they are assortative and that they are also more robust to vertex removal.}, author = {Newman, M. E. J. }, citeulike-article-id = {693937}, doi = {http://dx.doi.org/10.1103/PhysRevLett.89.208701}, journal = PRL, keywords = {assortativity, complex\_network, degree\_correlation, social\_network}, month = {October}, number = {20}, pages = {208701}, posted-at = {2008-12-11 06:28:40}, priority = {2}, publisher = {American Physical Society}, title = {Assortative Mixing in Networks}, volume = {89}, year = {2002} } @article{Kossinets06, abstract = {}, author = {Kossinets, G. and Watts, D. J.}, journal = SCI, month = {January}, pages = {88--90}, title = {Empirical Analysis of an Evolving Social Network}, volume = {311}, year = {2006} } @article{Watts02, abstract = {}, author = {Watts, D. J.}, journal = PNAS, month = {April}, pages = {5766--5771}, title = {A simple model for global cascades on random networks}, volume = {99}, year = {2002} } @Article{Gleeson08c, author = {Gleeson, J. P. and Melnik, S.}, journal = ARX, title = {Analytical results for bond percolation and k-core sizes on clustered networks}, year = {2008}, eprint = {0811.4511}, archivePrefix = {arXiv}, primaryClass = {}, version = {v1}, url = {http://arxiv.org/abs/0811.4511v1} } @Article{Traud08, author = {Traud, A. L. and Kelsic, E. D. and Mucha, P. J. and Porter, M. A.}, journal = ARX, title = {Community structure in online ccollegiate social networks}, year = {2008}, eprint = {0809.0690}, archivePrefix = {arXiv}, primaryClass = {}, version = {v1}, url = {http://arxiv.org/abs/0809.0690v1} } @Article{Barthelemy05, author = {Barth\'{e}lemy, M. and Barrat, A. and Pastor-Satorras, R. and Vespignani, A.}, journal = JTB, title = {Dynamical patterns of epidemic outbreaks in complex heterogeneous networks}, year = {2005}, pages = {275--288}, volume = {235} }