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Dynamic Phenomena on Complex Networks Luca Dall’Asta Dynamic Phenomena on Complex Networks Luca Dall’Asta To cite this version: Luca Dall’Asta. Dynamic Phenomena on Complex Networks. Data Analysis, Statistics and Probabil- ity [physics.data-an]. Université Paris Sud - Paris XI, 2006. English. tel-00093102 HAL Id: tel-00093102 https://tel.archives-ouvertes.fr/tel-00093102 Submitted on 12 Sep 2006 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE` DE PARIS 11 - U.F.R. DES SCIENCES D’ORSAY Laboratoire de Physique Theorique´ d’Orsay THESE` DE DOCTORAT DE L’UNIVERSITE´ PARIS 11 Sp´ecialit´e: PHYSIQUE THEORIQUE´ pr´esent´epar Luca DALL’ASTA pour obtenir le grade de DOCTEUR DE L’UNIVERSITE´ PARIS 11 Sujet: Ph´enom`enes dynamiques sur des r´eseaux complexes Jury compos´ede: M. Alain Barrat (Directeur de th`ese) M. Olivier Martin M. R´emi Monasson M. Romualdo Pastor-Satorras (Rapporteur) M. Cl´ement Sire (Rapporteur) M. Alessandro Vespignani Juin, 2006 Contents List of Publications iv 1 Introduction 1 1.1 A networked description of Nature and Society . ............. 1 1.2 RelationbetweenTopologyandDynamics . ........... 3 1.3 Summaryofthethesis .............................. ...... 4 2 Structure of complex networks: an Overview 7 2.1 Introduction.................................... ...... 7 2.2 Statistical Measures of Networks Topology . .............. 8 2.2.1 BasicnotionsofGraphTheory . ...... 8 2.2.2 Degreedistribution. ...... 9 2.2.3 Twoand threepoints degreecorrelations . .......... 10 2.2.4 Shortestpathlengthanddistance . ........ 11 2.2.5 Subgraphstructures . ..... 14 2.2.6 Further metrics for weighted networks . .......... 15 2.3 Examplesofrealnetworks . ........ 19 2.3.1 TheInternet ................................... .. 19 2.3.2 The World-wide Air-transportation Network . ........... 22 2.3.3 OtherExamples ................................. .. 23 2.4 Networksmodeling................................ ...... 26 2.4.1 HomogeneousNetworks . .... 26 2.4.2 HeterogeneousNetworks. ...... 29 2.4.3 WeightedNetworks. .. .. .. .. .. .. .. .. .. .. .. .. .... 31 3 Exploration of complex networks 35 3.1 Introduction.................................... ...... 35 3.1.1 Motivations................................... ... 35 3.1.2 Networks sampling methods and their biases . .......... 36 3.2 Statistical physics approach to traceroute explorations.................. 41 3.2.1 Themodel ...................................... 41 3.2.2 Mean-fieldanalysis. .. .. .. .. .. .. .. .. .. .. .. .. ..... 42 3.2.3 Numerical simulations on computer generated networks ............. 46 3.2.4 Accuracyofthemappingprocess . ....... 49 3.2.5 Optimization .................................. ... 54 3.2.6 k-corestructureundersampling . .. 57 i 3.3 NetworkSpeciesProblem . ....... 61 3.3.1 TheSpeciesProbleminNetworks . ...... 61 3.3.2 Inferring N:EstimatorsofNetworksSize . 62 3.3.3 NumericalResults .............................. .... 65 3.4 Conclusions ..................................... ..... 68 4 Spreading and Vulnerability 71 4.1 Introduction.................................... ...... 71 4.1.1 Motivations................................... ... 71 4.1.2 Relation between percolation, vulnerability and spreading ............ 72 4.2 Vulnerability of Weighted Networks: a case study . ............... 75 4.2.1 Weightedcentralitymeasures . ........ 75 4.2.2 VulnerabilityoftheWAN . ..... 78 4.2.3 ComparisonwiththespatialBBVmodel . ....... 81 4.2.4 Conclusions................................... ... 82 4.3 Spreading processes on Weighted Random Graphs . ............ 84 4.3.1 Generalized Molloy-Reed Criterion in Weighted RandomGraphs ........ 85 4.3.2 Examples and Numerical Simulations . ........ 88 4.3.3 DiscussionandConclusions . ....... 92 5 The Naming Game 93 5.1 Introduction.................................... ...... 93 5.2 NamingGame:Generalfeatures . ........ 95 5.2.1 TheModel ...................................... 95 5.2.2 Mean-fieldcase................................. ... 97 5.3 TheRoleoftheTopology ............................ ...... 102 5.3.1 Coarsening dynamics on low-dimensional lattices . .............. 102 5.3.2 Crossover to a fast-converging process in small-worldnetworks . 105 5.3.3 NamingGameoncomplexnetworks . ..... 109 5.4 Agents activity in heterogeneous populations . ............... 122 5.4.1 Numericalresultsonagentsactivity . .......... 123 5.4.2 Theoretical interpretation and future work . ............ 125 5.5 Conclusions ..................................... ..... 129 6 General Conclusions and Outlook 133 Acknowledgements 135 A Generating Functions in Percolation Problems 137 B A General Percolation Theory for Spreading Processes 141 B.1 Markovian Networks with Multi-Type Nodes . ........... 141 B.2 Degree-basedMulti-typeSolution . ............ 145 B.3 LocalHomogeneityApproximation . .......... 146 C Naming Game solution in d =1 149 D Master equation approach to agents internal dynamics 153 Bibliography 157 iv List of Publications The results exposed in this thesis have been published in a series of papers and preprints that we report here divided by argument: Exploration of networks (Chapter 3): • - Dall’Asta, L., Alvarez-Hamelin, J. I., Barrat, A., V´azquez, A., and Vespignani, A. ‘Traceroute-like exploration of unknown networks: A statistical analysis,’ Lect. Notes Comp. Sci. 3405, 140-153 (2005). - Dall’Asta, L., Alvarez-Hamelin, J. I., Barrat, A., V´azquez, A., and Vespignani, A., ‘Exploring networks with traceroute-like probes: theory and simulations, Theor. Comp. Sci. 355, 6-24 (2006). - Dall’Asta, L., Alvarez-Hamelin, J. I., Barrat, A., V´azquez, A., and Vespignani, A., ‘Statistical theory of Internet exploration,’ Phys. Rev. E 71 036135 (2005). - Dall’Asta, L., Alvarez-Hamelin, J. I., Barrat, A., V´azquez, A., and Vespignani, A., ‘How accurate are traceroute-like Internet mappings?,’ Comference Proc. AlgoTel ’05, INRIA, 31-34 (2005). - Viger, F., Barrat, A., Dall’Asta, L., Zhang, C.-H., and Kolaczyk, E., ‘Network Inference from Traceroute Measurements: Internet Topology ‘Species’,’ preprint arxiv:cs/0510007 (2005). k-core analysis of networks (Chapter 3): • - Alvarez-Hamelin, J. I., Dall’Asta, L., Barrat, A., and Vespignani, A., ‘k-core decomposition: a tool for the visualization of large scale networks,’ preprint arxiv:cs/0504107 (2005). - Alvarez-Hamelin, J. I., Dall’Asta, L., Barrat, A., and Vespignani, A., ‘k-core decomposition: a tool for the analysis of large scale Internet graphs,’ preprint arxiv:cs/0511007 (2005). - Alvarez-Hamelin, J. I., Dall’Asta, L., Barrat, A., and Vespignani, A., ‘Large scale networks fingerprinting and visualization using the k-core decomposition,’ in Advances in Neural Information Processing Systems, NIPS ’05 18 (2005). Functional properties of weighted networks (Chapter 4): • - Dall’Asta, L., ‘Inhomogeneous percolation models for spreading phenomena in random graphs,’ J. Stat. Mech. P08011 (2005). v vi - Dall’Asta, L., Barrat, A., Barth´elemy, M., and Vespignani, A., ‘Vulnerability of weighted networks,’ J. Stat. Mech. in press, preprint arxiv:physics/0603163, (2006). Naming Game Model (Chapter 5): • - Baronchelli, A., Dall’Asta, L., Barrat, A., and Loreto, V., ‘Topology Induced Coarsening in Language Games,’ Phys. Rev. E 73, 015102(R) (2006). - Baronchelli, A., Dall’Asta, L., Barrat, A., and Loreto, V., ‘Strategies for fast convergence in semiotic dynamics,’ ALIFE X, Bloomington Indiana (2006), (preprint arxiv:physics/0511201). - Dall’Asta, L., Baronchelli, A., Barrat, A., and Loreto, V., ‘Agreement dynamics on small-world networks,’ Europhys. Lett. 73(6), 969-975 (2006). - Dall’Asta, L., Baronchelli, A., Barrat, A., and Loreto, V., ‘Non-equilibrium dynamics of language games on complex networks,’ submitted to Phys. Rev. E (2006). Other works published during the PhD: • - B¨orner, K., Dall’Asta, L., Ke, W., and Vespignani, A., ‘Studying The Emerging Global Brain: Analyzing And Visualizing The Impact Of Co-Authorship Teams,’ Complexity 10(4), 57-67 (2005). - L. Dall’Asta, ‘Exact Solution of the One-Dimensional Deterministic Fixed-Energy Sandpile,’ Phys. Rev. Lett. 96, 058003 (2006). - M. Casartelli, L. Dall’Asta, A. Vezzani, and P. Vivo, ‘Dynamical Invariants in the Deterministic Fixed-Energy Samdpile,’ submitted to Eur. Phys. J. B, preprint arxiv:cond-mat/0502208 (2006). Chapter 1 Introduction 1.1 A networked description of Nature and Society The recent interest of a wide interdisciplinary scientific community for the study of complex networks is justified primary by the fact that a network description of complex systems allows to get relevant information by means of purely statistical coarse-grained analyses, without taking into account the detailed characterization of the system. Moreover, using an abstract networked representation, it is possible to compare, in the same framework, systems that are originally very different, so that the identification of some universal properties becomes much easier. Simplicity and universality are two fundamental principles of the physical research, in particular of statistical physics, that is traditionally interested in the study
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