Dimension Theory of Graphs

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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