Lecture 12: Introduction to Network Modeling ApproachesLecture 12 overviewThe Materials Science MetaphorThe Materials Science Metaphor IIThe Materials Science Metaphor IIINetwork metrics; structural characteristicsThe Materials Science Metaphor IVSchematic of Engineering System Model PurposesSchematic of Complex System ArchitectingModel typesThe Iterative Learning ProcessLecture 12 overviewPoisson Random GraphPoisson Random Graph IIPoisson Random Graph IISmall World Problem as seen by WattsSmall World Network Model (1D)K is the number of nearest neighbors originally with links (=3 below)Small-world networksSmall World Clustering EstimationSmall World Model Path LengthsUbiquity of small-world networksSmall World ModelsGeneralized Random Graphs IGeneralized Random Graphs IISystem Formation (Network Growth) ModelsSchematic of Engineering System Model PurposesBarabasi and Albert ModelD. J. de S. Price’s work ID. J. de S. Price’s work IIPrice’s ModelGeneralizations of Price/Barabasi-Albert Models IClasses of small-world networks:Truncation due to Costs and ConstraintsGeneralizations of Barabasi-Albert Models IIGeneralized Growth ModelsNode Copying ModelsLecture 12 overviewStructure-Property Models for Networked SystemsNetwork Resilience and RobustnessFunction in a network where connection is essential: function is connectivity and/or path lengthConnectivity and path length upon node failure: random networkConnectivity and path length upon node failure: scale free networkRobustness summary #1CascadesThe Watts Model for Global Cascades ILocal DependencyThe Watts Model for Global Cascades IIHetereogeneity of ResistanceThe Watts Model for Global Cascades IIIThe Watts Model for Global Cascades IV“Phase Diagram” for CascadesThe Watts Model for Global Cascades VSize of vulnerable cluster and cascadeThe Watts Model for Global Cascades VIStructural Effects on Cascade Phase DiagramsThe Watts Model for Global Cascades VIIEpidemics and virusesSIR Model for EpidemicsEpidemics and virusesApplying network theory to epidemics: Healthcare Institution NetworkSimulated (1000runs) Outbreak SizesComparison of simulation and analytical predictionComparison to actual outbreak at Evansville Indiana HospitalEvansville Case StudySIS ModelSIS Network Structural EffectsRobustness ObservationsReferences for lecture 12Professor C. Magee, 2006Page 1Lecture 12: Introduction to Network Modeling ApproachesChristopher L. MageeMarch 21, 2006Professor C. Magee, 2006Page 2Lecture 12 overview• Models, metrics and architecture • Understanding• Practice• Overview of model types• “Poisson Random graphs• “Small Worlds”• Random graphs “generalized” for degree sequences• System formation models• Cumulative advantage (aka preferential attachment)• Node copying and others • Structure-Property models• Cascades, epidemics and other initial “applications”Professor C. Magee, 2006Page 3The Materials Science Metaphor• PROCESSING> STRUCTURE> PROPERTIES• Structure determines/affects properties• Structure is a multi-dimensional term that includes many scales and concepts simultaneously (and thus is not a “simple invisible”)• Properties include attributes that encompass dynamics, behavior and “ilities”. • Relationships between Structure and Properties are plentiful andbecame strongest as material classes under detailed study increased• Solid Mechanics, dislocation theory, atomic theory are some of the key enablers for deriving mechanisms to propose structure/property relationships in materials.• In materials, properties of interest (almost always) simultaneously depend on several structural parameters. There is every reason to believe that engineering systems will similarly require numerousstructural parameters to make real progress.Professor C. Magee, 2006Page 4The Materials Science Metaphor II • Processing determines Structure• Different Processing Modes ( e-beam deposition, casting, forging, crystal growth, etc.) have different control parameters(Temperature gradient, stresses, pressure, magnetic and electrical fields, composition, etc.) that affect/determine properties.• Design is thus modifying the processing modes and control parameters to obtain the desired combination of properties. Understanding structure is the chief enabler of effective design• Thermodynamics, phase transformations, thermal and fluid sciences, solid mechanics are useful fundamentals underlying Process/structure relationship• Linking the framework to Engineering Systems requires discussing the structure and properties analogues in such systems.Professor C. Magee, 2006Page 5The Materials Science Metaphor III• Structure Characterization• Materials-Multiple Dimensional and very broadly construed• Engineering Systems Possibilities for Architecture Characterization as Networks.. are also very broadProfessor C. Magee, 2006Page 6Network metrics; structural characteristics• size, sparseness, degree, average degree, degree sequence • degree distribution, power laws, exponents, truncation• geodesic, path length, graph diameter • transitivity (clustering)• connectivity, reciprocity • centrality (degree, closeness, betweenness, information, eigenvector)• prestige, acquaintance• hierarchy• community structure, cliques• homophily, assortative mixing, degree correlation coefficient• motifs, coarse- graining• self-similarity, scale-free, scale-rich• dendograms, cladograms and relationship strength• modularity vs. integralityProfessor C. Magee, 2006Page 7The Materials Science Metaphor IV• Structure Characterization• Materials-Multiple Dimensional and very broadly construed• Engineering Systems Possibilities for Architecture Characterization.. are also very broad (but nonetheless almost surely needs to grow)• Engineering System Properties are also numerous (but some of the most important are not yet adequately quantified)• Robustness (congestion, failure of nodes and links etc.)• Flexibility• Rates of propagation (disease, ideas etc.)• Performance efficiency• The Processing > Structure > Properties “Mantra” from materials becomes for engineering systems• Formation mechanisms + constraints > architecture (structure) > Properties (ilities +)Professor C. Magee, 2006Page 8Schematic of Engineering System Model PurposesSystem StructureQuantified by aRich set of metricsSystem Propertiesunderstood quantitatively in terms of
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