Multidimensional Analysis of Complex Networks
Total Page:16
File Type:pdf, Size:1020Kb
Alma Mater Studiorum Universita’ di Bologna Computer Science Department Ph.D. Thesis Multidimensional analysis of complex networks Possamai Lino Ph.D. Supervisors: Prof. Massimo Marchiori Prof. Alessandro Sperduti Ph.D. Coordinator: Prof. Maurizio Gabbrielli DOTTORATO DI RICERCA IN INFORMATICA - Ciclo XXV Settore Concorsuale di afferenza: 01/B1 Settore Scientifico disciplinare: INF/01 Esame Finale Anno 2013 Ai miei genitori. Abstract The analysis of Complex Networks turn out to be a very promising field of research, testified by many research projects and works that span different fields. Until recently, those analysis have been usually focused on deeply characterize a single aspect of the system, therefore a study that considers many informative axes along with a network evolve is lacking. In this Thesis, we propose a new multidimensional analysis that is able to inspect networks in the two most important dimensions of a system, namely space and time. In order to achieve this goal, we studied them singularly and investigated how the variation of the constituting parameters drives changes to the network behavior as a whole. By focusing on space dimension, we were able to characterize spatial alteration in terms of abstraction levels. We propose a novel algorithm that, by applying a fuzziness function, can reconstruct networks under different level of details. We call this analysis telescopic as it recalls the magnification and reduction process of the lens. Through this line of research we have successfully verified that statistical indicators, that are frequently used in many complex networks researches, depends strongly on the granularity (i.e., the detail level) with which a system is described and on the class of networks considered. We keep fixed the space axes (that is, nodes’ coordinates) and we isolated the dynamics behind networks evolution process. In particular, our goal is to detect new instincts that trigger online social networks utilization and spread the adoption of novel communities. We formalized this enhanced social network evolution by adopt- ing new special nodes, called “sirens” that, thanks to their ability to attract new links, were also able to construct efficient connection patterns. We both simulate the dynamics of individuals and sirens by considering three most known growth models, namely random, preferential attachment and social. Applying this new framework to real and synthetic social networks, we have shown that the sirens, even when used for a limited period of time, effectively shrink the time needed to get a network in mature state. In order to provide a concrete context of our findings, we formalized the cost of setting up such enhancement and we provided the best combinations of system’s parameters, such as number of sirens, time span of utilization and attractiveness, which minimize this cost. 5 6 Contents Abstract 5 1 Introduction 15 1.1 Outlineofthethesis ............................ 17 2 Background 21 2.1 GraphTheory................................ 21 2.1.1 Statisticalproperties. 24 2.2 NetworkModeling ............................. 27 2.2.1 Erd˝os-R´enyiNetworks . 28 2.2.2 SmallWorldNetworks....................... 29 2.2.3 Scale-FreeNetworks .. .. .. .. .. .. .. .. .. .. 31 2.3 Longitudinalanalysis............................ 34 2.4 Critical analysis of complex networks . 36 2.4.1 Staticrobustness .......................... 36 2.4.2 Dynamicalrobustness . .. .. .. .. .. .. .. .. .. 38 2.5 Spreadingprocesses ............................ 39 2.5.1 Epidemic spreading . 39 2.5.2 Rumorsspreading ......................... 40 3 Multidimensionalanalysisofnetworks 43 4 The space dimension 47 4.1 Relatedwork ................................ 49 4.2 TelescopicAlgorithm............................ 51 4.3 ExperimentalResults............................ 54 4.3.1 Datasets............................... 54 4.3.2 Results ............................... 63 4.4 Networkperturbations .. .. .. .. .. .. .. .. .. .. .. 74 4.4.1 Null models as a function of networks’ size . 83 4.4.2 Nullmodelandrealnetworkscomparison . 91 7 8 CONTENTS 5 The time dimension 95 5.1 Introduction................................. 95 5.2 Networkgrowthmodels .......................... 96 5.2.1 Evolutionary models: serial and parallel . 98 5.3 Sirens .................................... 100 5.3.1 NetworkCost............................ 103 5.4 ExperimentalResults. .. .. .. .. .. .. .. .. .. .. .. 103 5.4.1 Datasets............................... 104 5.4.2 Results ............................... 108 6 Conclusions and Future Directions 127 A Supplementarymaterialforspacedimension 133 List of Figures 2.1 Graph abstraction of the seven bridges of K¨onigsberg problem. .... 22 2.2 Graphtypes. ................................ 23 2.3 Regular to random networks via small-world . 30 2.4 Random and scale-free degree distributions . 32 2.5 Impact of cumulative degree distribution on noise reduction in the data 33 2.6 Randomfailureandattacksonnetworks . 37 3.1 High level overview of multidimensional analysis on complex networks 44 4.1 Example of telescopic analysis . 48 4.2 Exampleofbox-coveringonnetworks. 49 4.3 Spatial coarse graining for brain voxels . 50 4.4 One-stepabstractionprocess . 52 4.5 Sequence of box covering iteration for telescopic analysis . ... 53 4.6 Boxcoveringissue ............................. 53 4.7 Exampleofboxcoveringonrealnetworks . 55 4.8 Subwaynetworkmaps ........................... 56 4.9 Examplesofcity-basednetworks . 57 4.10 Exampleofcity-basednetworks. 58 4.11 Pcum distribution of subways, transportation and social networks. 62 4.12 Number of collapsed nodes and edges as a function of f in log-log axes 64 4.13 Effect of the telescopic abstraction on the diameter as a function of f 65 4.14 Effect of the telescopic analysis on the degree . 66 4.15 Impact of the telescopic analysis on the degree correlations ρ ..... 67 4.16 Effect of the telescopic process on Eglob ................. 68 4.17 Effect of the telescopic analysis on Eloc .................. 70 4.18 Effect of the telescopic analysis on ct and cm .............. 71 4.19 Effect of the telescopic analysis on C/Eglob ............... 72 4.20 Effect of the abstraction process on the degree distribution P (k) ... 73 4.21 Number of collapsed nodes and edges with different perturbations .. 75 4.22 Telescopic analysis on various degree based indexes and perturbations 77 4.23 Effect of the telescopic analysis on ρ with different perturbations . 78 4.24 Effect of the telescopic process on Eglob as a function of perturbations 79 9 10 LIST OF FIGURES 4.25 Effect of the telescopic process on Eloc as a function of perturbations . 80 4.26 How different perturbations influence costs in the telescopic analysis . 81 4.27 How different perturbations influence C/Eglob in the telescopic analysis 82 4.28 Effect of the telescopic analysis on random networks on nodes and edges 84 4.29 Effect of the telescopic abstraction on diameter on null models . ... 85 4.30 Effect of the telescopic abstraction on degree in null models . ... 86 4.31 Effect of the telescopic abstraction on Eglob onrandomnetworks . 87 4.32 Effect of the telescopic abstraction on Eloc ................ 88 4.33 Effect of the telescopic analysis on ct and cm in randomnetworks. 89 4.34 How the telescopic analysis influence C/Eglob .............. 90 4.35 Comparison between original and randomized subway networks .... 92 4.36 Comparison between original and randomized city-based networks .. 93 5.1 Example of social rule in evolving social networks . 98 5.2 Impact of C scale parameter on overall simulated time . 99 5.3 Example of on line social network with sirens . 100 5.4 Examples of sirens’ social rule . 102 5.5 Cumulative degree distributions of CM and VT datasets . 106 5.6 NodedegreecorrelationsinCMandVTdatasets . 106 5.7 Serialnetworkgrowthmodel . 110 5.8 Parallelnetworkgrowthmodel . 111 5.9 Parallel network growth model of random networks . 111 5.10 How attractiveness influence the overall network behavior . ..... 113 5.11 Eglob varying the cost, accelerated growth, word of mouth model (CM) 114 5.12 Eglob varying the cost, accelerated growth, preferential model (CM) . 115 5.13 Effect of multiple runs of simulations on VT . 115 5.14 How cost affects Eglob in accelerated growth, broadcast model (VT) . 116 5.15 How cost affects Eglob in accelerated growth, word of mouth model(VT)117 5.16 How cost affects Eglob in accelerated growth, preferential model (VT) 118 5.17 Eglob as a function of cost keeping fixed the number of sirens (CM) . 120 5.18 Effect of attractiveness and time in CM, word of mouth model . 120 5.19 Effect of attractiveness and time in CM, preferential model . .... 121 5.20 Effect of attractiveness and time in VT, broadcast model . 121 5.21 Effect of attractiveness and time in VT, word of mouth model . 122 5.22 Effect of attractiveness and time in VT, preferential model . .... 122 5.23 Correlation between cost and Tmin inCM................ 124 5.24 Correlation between cost and Tmin inVT ................ 125 A.1 Effect of the telescopic analysis on nodes of perturbed subways .... 134 A.2 Effect of the telescopic analysis on edges of perturbed subways .... 135 A.3 Effect of the telescopic analysis on kmax of perturbed subways. 136 A.4 Effect of the telescopic analysis on kmean of perturbed subways . 137 A.5 Effect of the telescopic analysis on σk of perturbed subways . 138 A.6 Effect of the telescopic analysis on Eglob of perturbed subways. 139 A.7 Effect of the telescopic analysis on Eglob of perturbed subways. 140 LIST OF FIGURES 11 A.8 Effect of the telescopic