Collective-dynamics-of-small-world-networks

Nature©MacmillanPublishersLtd19988typicallyslowerthan,1kms-1)mightdiffersignificantlyfromwhatisassumedbycurrentmodellingefforts27.Theexpectedequation-of-statedifferencesamongsmallbodies(iceversusrock,forinstance)presentsanotherdimensionofstudy;havingrecentlyadaptedourcodeformassivelyparallelarchitectures(K.M.OlsonandE.A,manuscriptinpreparation),wearenowreadytoperformamorecomprehensiveanalysis.Theexploratorysimulationspresentedheresuggestthatwhenayoung,non-porousasteroid(ifsuchexist)suffersextensiveimpactdamage,theresultingfracturepatternlargelydefinestheasteroid’sresponsetofutureimpacts.Thestochasticnatureofcollisionsimpliesthatsmallasteroidinteriorsmaybeasdiverseastheirshapesandspinstates.Detailednumericalsimulationsofimpacts,usingaccurateshapemodelsandrheologies,couldshedlightonhowasteroidcollisionalresponsedependsoninternalconfigurationandshape,andhenceonhowplanetesimalsevolve.DetailedsimulationsarealsorequiredbeforeonecanpredictthequantitativeeffectsofnuclearexplosionsonEarth-crossingcometsandasteroids,eitherforhazardmitigation28throughdisruptionanddeflection,orforresourceexploitation29.Suchpredictionswouldrequiredetailedreconnaissanceconcerningthecompositionandinternalstructureofthetargetedobject.MReceived4February;accepted18March1998.1.Asphaug,E.&Melosh,H.J.TheStickneyimpactofPhobos:Adynamicalmodel.Icarus101,144–164(1993).2.Asphaug,E.etal.MechanicalandgeologicaleffectsofimpactcrateringonIda.Icarus120,158–184(1996).3.Nolan,M.C.,Asphaug,E.,Melosh,H.J.&Greenberg,R.Impactcratersonasteroids:Doesstrengthorgravitycontroltheirsize?Icarus124,359–371(1996).4.Love,S.J.&Ahrens,T.J.Catastrophicimpactsongravitydominatedasteroids.Icarus124,141–155(1996).5.Melosh,H.J.&Ryan,E.V.Asteroids:Shatteredbutnotdispersed.Icarus129,562–564(1997).6.Housen,K.R.,Schmidt,R.M.&Holsapple,K.A.Craterejectascalinglaws:Fundamentalformsbasedondimensionalanalysis.J.Geophys.Res.88,2485–2499(1983).7.Holsapple,K.A.&Schmidt,R.M.Pointsourcesolutionsandcouplingparametersincrateringmechanics.J.Geophys.Res.92,6350–6376(1987).8.Housen,K.R.&Holsapple,K.A.Onthefragmentationofasteroidsandplanetarysatellites.Icarus84,226–253(1990).9.Benz,W.&Asphaug,E.Simulationsofbrittlesolidsusingsmoothparticlehydrodynamics.Comput.Phys.Commun.87,253–265(1995).10.Asphaug,E.etal.MechanicalandgeologicaleffectsofimpactcrateringonIda.Icarus120,158–184(1996).11.Hudson,R.S.&Ostro,S.J.Shapeofasteroid4769Castalia(1989PB)frominversionofradarimages.Science263,940–943(1994).12.Ostro,S.J.etal.Asteroidradarastrometry.Astron.J.102,1490–1502(1991).13.Ahrens,T.J.&O’Keefe,J.D.inImpactandExplosionCratering(edsRoddy,D.J.,Pepin,R.O.&Merrill,R.B.)639–656(Pergamon,NewYork,1977).14.Tillotson,J.H.Metallicequationsofstateforhypervelocityimpact.(GeneralAtomicReportGA-3216,SanDiego,1962).15.Nakamura,A.&Fujiwara,A.Velocitydistributionoffragmentsformedinasimulatedcollisionaldisruption.Icarus92,132–146(1991).16.Benz,W.&Asphaug,E.Simulationsofbrittlesolidsusingsmoothparticlehydrodynamics.Comput.Phys.Commun.87,253–265(1995).17.Bottke,W.F.,Nolan,M.C.,Greenberg,R.&Kolvoord,R.A.Velocitydistributionsamongcollidingasteroids.Icarus107,255–268(1994).18.Belton,M.J.S.etal.Galileoencounterwith951Gaspra—Firstpicturesofanasteroid.Science257,1647–1652(1992).19.Belton,M.J.S.etal.Galileo’sencounterwith243Ida:Anoverviewoftheimagingexperiment.Icarus120,1–19(1996).20.Asphaug,E.&Melosh,H.J.TheStickneyimpactofPhobos:Adynamicalmodel.Icarus101,144–164(1993).21.Asphaug,E.etal.MechanicalandgeologicaleffectsofimpactcrateringonIda.Icarus120,158–184(1996).22.Housen,K.R.,Schmidt,R.M.&Holsapple,K.A.Craterejectascalinglaws:Fundamentalformsbasedondimensionalanalysis.J.Geophys.Res.88,2485–2499(1983).23.Veverka,J.etal.NEAR’sflybyof253Mathilde:ImagesofaCasteroid.Science278,2109–2112(1997).24.Asphaug,E.etal.Impactevolutionoficyregoliths.LunarPlanet.Sci.Conf.(Abstr.)XXVIII,63–64(1997).25.Love,S.G.,Ho¨rz,F.&Brownlee,D.E.Targetporosityeffectsinimpactcrateringandcollisionaldisruption.Icarus105,216–224(1993).26.Fujiwara,A.,Cerroni,P.,Davis,D.R.,Ryan,E.V.&DiMartino,M.inAsteroidsII(edsBinzel,R.P.,Gehrels,T.&Matthews,A.S.)240–265(Univ.ArizonaPress,Tucson,1989).27.Davis,D.R.&Farinella,P.CollisionalevolutionofEdgeworth-KuiperBeltobjects.Icarus125,50–60(1997).28.Ahrens,T.J.&Harris,A.W.Deflectionandfragmentationofnear-Earthasteroids.Nature360,429–433(1992).29.ResourcesofNear-EarthSpace(edsLewis,J.S.,Matthews,M.S.&Guerrieri,M.L.)(Univ.ArizonaPress,Tucson,1993).Acknowledgements.ThisworkwassupportedbyNASA’sPlanetaryGeologyandGeophysicsProgram.CorrespondenceandrequestsformaterialsshouldbeaddressedtoE.A.(e-mail:[e-mail]).letterstonature440NATURE|VOL393|4JUNE1998Collectivedynamicsof‘small-world’networksDuncanJ.Watts*&StevenH.StrogatzDepartmentofTheoreticalandAppliedMechanics,KimballHall,CornellUniversity,Ithaca,NewYork14853,USA.........................................................................................................................Networksofcoupleddynamicalsystemshavebeenusedtomodelbiologicaloscillators1–4,Josephsonjunctionarr