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Chen LM, Holzer M, Shapiro A. Estimating epidemic arrival times using linear spreading theory. CHAOS (WOODBURY, N.Y.) 2018; 28:013105. [PMID: 29390617 DOI: 10.1063/1.5002009] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We study the dynamics of a spatially structured model of worldwide epidemics and formulate predictions for arrival times of the disease at any city in the network. The model is composed of a system of ordinary differential equations describing a meta-population susceptible-infected-recovered compartmental model defined on a network where each node represents a city and the edges represent the flight paths connecting cities. Making use of the linear determinacy of the system, we consider spreading speeds and arrival times in the system linearized about the unstable disease free state and compare these to arrival times in the nonlinear system. Two predictions are presented. The first is based upon expansion of the heat kernel for the linearized system. The second assumes that the dominant transmission pathway between any two cities can be approximated by a one dimensional lattice or a homogeneous tree and gives a uniform prediction for arrival times independent of the specific network features. We test these predictions on a real network describing worldwide airline traffic.
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Affiliation(s)
- Lawrence M Chen
- Department of Mathematics, University of Kansas, Lawrence, Kansas 66045, USA
| | - Matt Holzer
- Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030, USA
| | - Anne Shapiro
- Department of Mathematics and Statistics, Carleton College, Northfield, Minnesota 55057, USA
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2
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Stahn K, Lehnertz K. Surrogate-assisted identification of influences of network construction on evolving weighted functional networks. CHAOS (WOODBURY, N.Y.) 2017; 27:123106. [PMID: 29289055 DOI: 10.1063/1.4996980] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
We aim at identifying factors that may affect the characteristics of evolving weighted networks derived from empirical observations. To this end, we employ various chains of analysis that are often used in field studies for a data-driven derivation and characterization of such networks. As an example, we consider fully connected, weighted functional brain networks before, during, and after epileptic seizures that we derive from multichannel electroencephalographic data recorded from epilepsy patients. For these evolving networks, we estimate clustering coefficient and average shortest path length in a time-resolved manner. Lastly, we make use of surrogate concepts that we apply at various levels of the chain of analysis to assess to what extent network characteristics are dominated by properties of the electroencephalographic recordings and/or the evolving weighted networks, which may be accessible more easily. We observe that characteristics are differently affected by the unavoidable referencing of the electroencephalographic recording, by the time-series-analysis technique used to derive the properties of network links, and whether or not networks were normalized. Importantly, for the majority of analysis settings, we observe temporal evolutions of network characteristics to merely reflect the temporal evolutions of mean interaction strengths. Such a property of the data may be accessible more easily, which would render the weighted network approach-as used here-as an overly complicated description of simple aspects of the data.
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Affiliation(s)
- Kirsten Stahn
- Department of Epileptology, University of Bonn Medical Centre, Sigmund-Freud-Straße 25, 53105 Bonn, Germany
| | - Klaus Lehnertz
- Department of Epileptology, University of Bonn Medical Centre, Sigmund-Freud-Straße 25, 53105 Bonn, Germany
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3
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Mou J, Liu C, Chen S, Huang G, Lu X. Temporal Characteristics of the Chinese Aviation Network and their Effects on the Spread of Infectious Diseases. Sci Rep 2017; 7:1275. [PMID: 28455531 PMCID: PMC5430792 DOI: 10.1038/s41598-017-01380-5] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/05/2016] [Accepted: 03/27/2017] [Indexed: 11/17/2022] Open
Abstract
Aviation transportation systems have developed rapidly in recent years and have become a focus for research on the modeling of epidemics. However, despite the number of studies on aggregated topological structures and their effects on the spread of disease, the temporal sequence of flights that connect different airports have not been examined. In this study, to analyze the temporal pattern of the Chinese Aviation Network (CAN), we obtain a time series of topological statistics through sliding the temporal CAN with an hourly time window. In addition, we build two types of Susceptible-Infectious (SI) spreading models to study the effects of linking sequence and temporal duration on the spread of diseases. The results reveal that the absence of links formed by flights without alternatives at dawn and night causes a significant decrease in the centralization of the network. The temporal sparsity of linking sequence slows down the spread of disease on CAN, and the duration of flights intensifies the sensitiveness of CAN to targeted infection. The results are of great significance for further understanding of the aviation network and the dynamic process, such as the propagation of delay.
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Affiliation(s)
- Jianhong Mou
- College of Information System and Management, National University of Defense Technology, Changsha, 410073, China
| | - Chuchu Liu
- College of Information System and Management, National University of Defense Technology, Changsha, 410073, China
| | - Saran Chen
- College of Information System and Management, National University of Defense Technology, Changsha, 410073, China
| | - Ge Huang
- College of Information System and Management, National University of Defense Technology, Changsha, 410073, China
| | - Xin Lu
- College of Information System and Management, National University of Defense Technology, Changsha, 410073, China.
- School of Mathematics and Big Data, Foshan University, Foshan, 528000, China.
- Department of Public Health Sciences, Karolinska Institutet, Stockholm, 17 177, Sweden.
- Flowminder Foundation, Stockholm, 11 355, Sweden.
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Sun M, Zhang H, Kang H, Zhu G, Fu X. Epidemic spreading on adaptively weighted scale-free networks. J Math Biol 2016; 74:1263-1298. [PMID: 27639702 DOI: 10.1007/s00285-016-1057-6] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2015] [Indexed: 11/28/2022]
Abstract
We introduce three modified SIS models on scale-free networks that take into account variable population size, nonlinear infectivity, adaptive weights, behavior inertia and time delay, so as to better characterize the actual spread of epidemics. We develop new mathematical methods and techniques to study the dynamics of the models, including the basic reproduction number, and the global asymptotic stability of the disease-free and endemic equilibria. We show the disease-free equilibrium cannot undergo a Hopf bifurcation. We further analyze the effects of local information of diseases and various immunization schemes on epidemic dynamics. We also perform some stochastic network simulations which yield quantitative agreement with the deterministic mean-field approach.
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Affiliation(s)
- Mengfeng Sun
- Department of Mathematics, Shanghai University, Shanghai, 200444, China
| | - Haifeng Zhang
- School of Mathematical Science, Anhui University, Hefei, 230039, China
| | - Huiyan Kang
- School of Mathematics and Physics, Changzhou University, Changzhou, 213016, China
| | - Guanghu Zhu
- School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, 541004, China
| | - Xinchu Fu
- Department of Mathematics, Shanghai University, Shanghai, 200444, China.
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Louzada VHP, Araújo NAM, Verma T, Daolio F, Herrmann HJ, Tomassini M. Critical cooperation range to improve spatial network robustness. PLoS One 2015; 10:e0118635. [PMID: 25793986 PMCID: PMC4391338 DOI: 10.1371/journal.pone.0118635] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2014] [Accepted: 12/24/2014] [Indexed: 11/25/2022] Open
Abstract
A robust worldwide air-transportation network (WAN) is one that minimizes the number of stranded passengers under a sequence of airport closures. Building on top of this realistic example, here we address how spatial network robustness can profit from cooperation between local actors. We swap a series of links within a certain distance, a cooperation range, while following typical constraints of spatially embedded networks. We find that the network robustness is only improved above a critical cooperation range. Such improvement can be described in the framework of a continuum transition, where the critical exponents depend on the spatial correlation of connected nodes. For the WAN we show that, except for Australia, all continental networks fall into the same universality class. Practical implications of this result are also discussed.
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Affiliation(s)
| | - Nuno A M Araújo
- Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Lisboa, Portugal; Centro de Física Teórica e Computacional, Universidade de Lisboa, Lisboa, Portugal
| | - Trivik Verma
- Computational Physics, IfB, ETH Zurich, Zurich, Switzerland
| | - Fabio Daolio
- Faculty of Engineering, Shinshu University, Wakasato, Nagano city, Japan
| | - Hans J Herrmann
- Computational Physics, IfB, ETH Zurich, Zurich, Switzerland; Departamento de Física, Universidade Federal do Ceará, Fortaleza, Ceará, Brazil
| | - Marco Tomassini
- Faculty of Business and Economics, University of Lausanne, Lausanne, Switzerland
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6
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Guez OC, Gozolchiani A, Havlin S. Influence of autocorrelation on the topology of the climate network. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:062814. [PMID: 25615155 DOI: 10.1103/physreve.90.062814] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2014] [Indexed: 05/22/2023]
Abstract
Different definitions of links in climate networks may lead to considerably different network topologies. We construct a network from climate records of surface level atmospheric temperature in different geographical sites around the globe using two commonly used definitions of links. Utilizing detrended fluctuation analysis, shuffled surrogates, and separation analysis of maritime and continental records, we find that one of the major influences on the structure of climate networks is due to the autocorrelation in the records, which may introduce spurious links. This may explain why different methods could lead to different climate network topologies.
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Affiliation(s)
- Oded C Guez
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Avi Gozolchiani
- Department of Solar Energy and Environmental Physics, The Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 84990 Midreshet Ben-Gurion, Israel
| | - Shlomo Havlin
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
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Srivastava A, Mitra B, Ganguly N, Peruani F. Correlations in complex networks under attack. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:036106. [PMID: 23030979 DOI: 10.1103/physreve.86.036106] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2012] [Indexed: 06/01/2023]
Abstract
For any initially correlated network after any kind of attack where either nodes or edges are removed, we obtain general expressions for the degree-degree probability matrix and degree distribution. We show that the proposed analytical approach predicts the correct topological changes after the attack by comparing the evolution of the assortativity coefficient for different attack strategies and intensities in theory and simulations. We find that it is possible to turn an initially assortative network into a disassortative one, and vice versa, by fine-tuning removal of either nodes or edges. For an initially uncorrelated network, on the other hand, we discover that only a targeted edge-removal attack can induce such correlations.
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Affiliation(s)
- Animesh Srivastava
- Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, 721302 Kharagpur, India
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Hooyberghs H, Van Schaeybroeck B, Moreira AA, Andrade JS, Herrmann HJ, Indekeu JO. Biased percolation on scale-free networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:011102. [PMID: 20365318 DOI: 10.1103/physreve.81.011102] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2009] [Indexed: 05/29/2023]
Abstract
Biased (degree-dependent) percolation was recently shown to provide strategies for turning robust networks fragile and vice versa. Here, we present more detailed results for biased edge percolation on scale-free networks. We assume a network in which the probability for an edge between nodes i and j to be retained is proportional to (k(i)k(j)(-alpha) with k(i) and k(j) the degrees of the nodes. We discuss two methods of network reconstruction, sequential and simultaneous, and investigate their properties by analytical and numerical means. The system is examined away from the percolation transition, where the size of the giant cluster is obtained, and close to the transition, where nonuniversal critical exponents are extracted using the generating-functions method. The theory is found to agree quite well with simulations. By presenting an extension of the Fortuin-Kasteleyn construction, we find that biased percolation is well-described by the q-->1 limit of the q -state Potts model with inhomogeneous couplings.
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Affiliation(s)
- Hans Hooyberghs
- Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium
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9
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Abstract
Many complex systems, including networks, are not static but can display strong fluctuations at various time scales. Characterizing the dynamics in complex networks is thus of the utmost importance in the understanding of these networks and of the dynamical processes taking place on them. In this article, we study the example of the US airport network in the time period 1990-2000. We show that even if the statistical distributions of most indicators are stationary, an intense activity takes place at the local ("microscopic") level, with many disappearing/appearing connections (links) between airports. We find that connections have a very broad distribution of lifetimes, and we introduce a set of metrics to characterize the links' dynamics. We observe in particular that the links that disappear have essentially the same properties as the ones that appear, and that links that connect airports with very different traffic are very volatile. Motivated by this empirical study, we propose a model of dynamical networks, inspired from previous studies on firm growth, which reproduces most of the empirical observations both for the stationary statistical distributions and for the dynamical properties.
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Moreira AA, Andrade JS, Herrmann HJ, Indekeu JO. How to make a fragile network robust and vice versa. PHYSICAL REVIEW LETTERS 2009; 102:018701. [PMID: 19257248 DOI: 10.1103/physrevlett.102.018701] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2008] [Indexed: 05/27/2023]
Abstract
We investigate topologically biased failure in scale-free networks with a degree distribution P(k) proportional, variantk;{-gamma}. The probability p that an edge remains intact is assumed to depend on the degree k of adjacent nodes i and j through p_{ij} proportional, variant(k_{i}k_{j});{-alpha}. By varying the exponent alpha, we interpolate between random (alpha=0) and systematic failure. For alpha>0 (<0) the most (least) connected nodes are depreciated first. This topological bias introduces a characteristic scale in P(k) of the depreciated network, marking a crossover between two distinct power laws. The critical percolation threshold, at which global connectivity is lost, depends both on gamma and on alpha. As a consequence, network robustness or fragility can be controlled through fine-tuning of the topological bias in the failure process.
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Affiliation(s)
- André A Moreira
- Departamento de Física, Universidade Federal do Ceará, 60451-970 Fortaleza, Ceará, Brazil
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Ramasco JJ, Colizza V, Panzarasa P. Using the Weighted Rich-Club Coefficient to Explore Traffic Organization in Mobility Networks. LECTURE NOTES OF THE INSTITUTE FOR COMPUTER SCIENCES, SOCIAL INFORMATICS AND TELECOMMUNICATIONS ENGINEERING 2009. [DOI: 10.1007/978-3-642-02466-5_66] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/13/2023]
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12
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Opsahl T, Colizza V, Panzarasa P, Ramasco JJ. Prominence and control: the weighted rich-club effect. PHYSICAL REVIEW LETTERS 2008; 101:168702. [PMID: 18999722 DOI: 10.1103/physrevlett.101.168702] [Citation(s) in RCA: 151] [Impact Index Per Article: 9.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2008] [Indexed: 05/27/2023]
Abstract
Complex systems are often characterized by large-scale hierarchical organizations. Whether the prominent elements, at the top of the hierarchy, share and control resources or avoid one another lies at the heart of a system's global organization and functioning. Inspired by network perspectives, we propose a new general framework for studying the tendency of prominent elements to form clubs with exclusive control over the majority of a system's resources. We explore associations between prominence and control in the fields of transportation, scientific collaboration, and online communication.
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Affiliation(s)
- Tore Opsahl
- School of Business and Management, Queen Mary College, University of London, London, United Kingdom
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La Rocca CE, Braunstein LA, Macri PA. Evolution equation for a model of surface relaxation in complex networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:046120. [PMID: 18517703 DOI: 10.1103/physreve.77.046120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/02/2007] [Revised: 04/03/2008] [Indexed: 05/26/2023]
Abstract
In this paper we derive analytically the evolution equation of the interface for a model of surface growth with relaxation to the minimum (SRM) in complex networks. We were inspired by the disagreement between the scaling results of the steady state of the fluctuations between the discrete SRM model and the Edward-Wilkinson process found in scale-free networks with degree distribution P(k) approximately k(-lambda) for lambda<3 [Pastore y Piontti, Phys. Rev. E 76, 046117 (2007)]. Even though for Euclidean lattices the evolution equation is linear, we find that in complex heterogeneous networks nonlinear terms appear due to the heterogeneity and the lack of symmetry of the network; they produce a logarithmic divergency of the saturation roughness with the system size as found by Pastore y Piontti for lambda<3.
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Affiliation(s)
- C E La Rocca
- Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)-Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata-CONICET, Funes 3350, 7600 Mar del Plata, Argentina
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Gautreau A, Barrat A, Barthélemy M. Global disease spread: statistics and estimation of arrival times. J Theor Biol 2007; 251:509-22. [PMID: 18222486 DOI: 10.1016/j.jtbi.2007.12.001] [Citation(s) in RCA: 47] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2007] [Revised: 12/05/2007] [Accepted: 12/05/2007] [Indexed: 10/22/2022]
Abstract
We study metapopulation models for the spread of epidemics in which different subpopulations (cities) are connected by fluxes of individuals (travelers). This framework allows one to describe the spread of a disease on a large scale and we focus here on the computation of the arrival time of a disease as a function of the properties of the seed of the epidemics and of the characteristics of the network connecting the various subpopulations. Using analytical and numerical arguments, we introduce an easily computable quantity which approximates this average arrival time. We show on the example of a disease spread on the world-wide airport network that this quantity predicts with a good accuracy the order of arrival of the disease in the various subpopulations in each realization of epidemic scenario, and not only for an average over realizations. Finally, this quantity might be useful in the identification of the dominant paths of the disease spread.
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Affiliation(s)
- Aurélien Gautreau
- Univ Paris-Sud, 91405 Orsay, France; CNRS, UMR 8627, 91405 Orsay, France.
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Ramasco JJ, Gonçalves B. Transport on weighted networks: When the correlations are independent of the degree. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:066106. [PMID: 18233897 DOI: 10.1103/physreve.76.066106] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2007] [Revised: 09/22/2007] [Indexed: 05/25/2023]
Abstract
Most real-world networks are weighted graphs with the weight of the edges reflecting the relative importance of the connections. In this work, we study nondegree dependent correlations between edge weights, generalizing thus the correlations beyond the degree dependent case. We propose a simple method to introduce weight-weight correlations in topologically uncorrelated graphs. This allows us to test different measures to discriminate between the different correlation types and to quantify their intensity. We also discuss here the effect of weight correlations on the transport properties of the networks, showing that positive correlations dramatically improve transport. Finally, we give two examples of real-world networks (social and transport graphs) in which weight-weight correlations are present.
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Affiliation(s)
- José J Ramasco
- CNLL, ISI Foundation, Viale S. Severo 65, I-10133 Torino, Italy.
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Wu Z, Lagorio C, Braunstein LA, Cohen R, Havlin S, Stanley HE. Numerical evaluation of the upper critical dimension of percolation in scale-free networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:066110. [PMID: 17677328 DOI: 10.1103/physreve.75.066110] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2007] [Revised: 06/01/2007] [Indexed: 05/16/2023]
Abstract
We propose numerical methods to evaluate the upper critical dimension d(c) of random percolation clusters in Erdös-Rényi networks and in scale-free networks with degree distribution P(k) approximately k(-lambda), where k is the degree of a node and lambda is the broadness of the degree distribution. Our results support the theoretical prediction, d(c) = 2(lambda - 1)(lambda - 3) for scale-free networks with 3 < lambda < 4 and d(c) = 6 for Erdös-Rényi networks and scale-free networks with lambda > 4 . When the removal of nodes is not random but targeted on removing the highest degree nodes we obtain d(c) = 6 for all lambda > 2 . Our method also yields a better numerical evaluation of the critical percolation threshold p(c) for scale-free networks. Our results suggest that the finite size effects increases when lambda approaches 3 from above.
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Affiliation(s)
- Zhenhua Wu
- Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA
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