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ID:40357818
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页数:20页
时间:2019-07-31
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1、DimensionTheoryofGraphsandNetworksThomasNowotnyManfredRequardtInstitutf¨urTheoretischePhysikUniversit¨atG¨ottingenBunsenstrasse937073G¨ottingenGermanyAbstractStartingfromtheworkinghypothesisthatbothphysicsandthecorre-spondingmathematicshavetobedescribedbymea
2、nsofdiscreteconceptsonthePlanck-scale,oneofthemanyproblemsonehastofaceinthisenter-priseistofindthediscreteprotoformsofthebuildingblocksofcontinuumphysicsandmathematics.Acoreconceptisthenotionofdimension.Inthefollowingwedevelopsuchanotionforirregularstructures
3、like(large)graphsandnetworksandderiveanumberofitsproperties.AmongotherarXiv:hep-th/9707082v19Jul1997thingsweshowitsstabilityunderawideclassofperturbationswhichisim-portantifonehas’dimensionalphasetransitions’inmind.Furthermorewesystematicallyconstructgraphsw
4、ithalmostarbitrary’fractaldimension’whichmaybeofsomeuseinthecontextof’dimensionalrenormalization’orstatisticalmechanicsonirregularsets.11IntroductionIntworecentpapers([1],[2])wedevelopedacertainframeworkinformofaclassof’cellularnetworkdynamics’whicharedesign
5、edtomimicthedynamicsofthephysicalvacuumorspace-timeonthePlanck-scale.Indoingthisourworkingphilosophywasthatbothphysicsandthecorrespondingmathematicsaregenuinelydiscreteonthisprimordiallevel.Thecontinuumconceptsofordinaryspace-timephysicsarethensupposedtoemer
6、gefromcertaindiscretepatternsviaakindof’renormalizationgroupprocess’onthemuchcoarserscaleofresolutiongivenbythecomparativelysmallenergiesofpresentdayhighenergyphysics.Itisoneofouraimstofindthesediscreteprotoforms.Acrucialconceptsinthiscontextisaversionof’intr
7、insicdimension’ofsuchdiscreteirregularnetworkswhichgeometricallyaregraphs.Thisconceptshouldbedefinedinanintrinsicway,withoutmakingopenorimplicitrecoursetocontinuumconceptswhatsoeverorkindofanembeddingdimension,aswewanttounderstand,amongotherthings,whatpropert
8、iesactuallyareencodedinanotionlikedimensiononthemostfundamentalphysicallevel.Ontheotherside,wewanttoknowhowthecontinuumconceptofdimension,whichistoalargeextentofanapriorimathematicalviz.geometri
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