network structure - florida state university
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NetworkStructure
WhatisaNetwork?
vertexedge
TechnologicalNetworksTheInternet
Vertices arecomputersorcomputerroutersEdges arecablesoropticalfiberlineslinkingthecomputers
CAIDAAnnualReport(1998)
TechnologicalNetworksAflightpathnetwork
Vertices indicateairportsEdges indicateconnectingflights
BiologicalNetworksA(directed)foodchainnetwork
Vertices indicatespeciesDirectededgeindicatesthatthetargetspeciesdependsontheotherspecies
BiochemicalNetworks
Vertices indicateproteinsDirectededgeindicatesthatoneproteinregulatesthetranscriptionofthetargetprotein
A(directed)genetranscriptionnetwork
Louis(2011)
C.Elegans Connectome
FromMitya Chklovskii.Vertices=neuronEdges=synapticconnectionbetweentwoneurons
SocialNetworks
Patternoffriendshipsamongmembersofakarateclub
Vertices indicateindividualsEdges indicatefriendships
Zachery(1977)
BuildingaSmall-WorldNetwork
Ateachnode,withprobabilitypaddanedgefromanodetosomeothernon-neighboringnode.ThisiscalledtheWatts-Strogatz algorithm.
NetworkClustering
Clusteringinwirelesssensornetworks
Erdos-Renyi RandomNetwork
130nodes215edges
Red =fivenodeswithhighestdegreeGreen =theirneighbors
DegreeDistributionFollowsaPoissonDistribution
degree
prob
ability Pr 𝑋 = 𝑘 =
𝜆'𝑒)*
𝑘!
𝜆 > 0 Isaparameter
Exponentialdecay
Scale-FreeNetwork
130nodes215edges
Red =fivenodeswithhighestdegreeGreen =theirneighbors
Scale-freenetworksarecharacterizedbyafewhubswithveryhighdegree
Scale-FreeDegreeDistributionFollowsaPowerLaw
𝛾 > 0 Isaparameter
Pr 𝑋 = 𝑘 = 𝐶𝑘)0
Forksufficientlylarge
DegreeDistributionFollowsaPoissonDistribution
“Homogeneous” “Heterogeneous”
AlgorithmForBuildingaScale-FreeNetwork
1. Startwithm0 nodesrandomlyconnected
AlgorithmForBuildingaScale-FreeNetwork
1. Startwithm0 nodesrandomlyconnected
2. Addadditionalnodeswhereprobabilityofconnectingtoeachexistingnodeislargerifthatnodehashighdegree
AlgorithmForBuildingaScale-FreeNetwork
1. Startwithm0 nodesrandomlyconnected
2. Addadditionalnodeswhereprobabilityofconnectingtoeachexistingnodeislargerifthatnodehashighdegree
”TheRichGetRicher”
Scale-FreeMetabolicNetworks
Scale-FreeSignalingandTranscriptionNetworks
ShortPathLengthsinScale-FreeMetabolicNetworks
Jeong etal.,Nature,407:651,2000
Citationnumber:8149
Publicationdate:2000
TheirDefinitionofNetworkDiameter
Shortestpathfromnode1tonode6is(1,5,4,6),pathlength=3
Dothiscalculationforallnodepairsandsum
Dividebythenumberofnodepairstogetdiameterd
TheScale-FreeNetworkisMoreResilienttoRandomFailureThantheExponentialNetwork
SF
E
Failurefractionf
Diam
eterd Linearincreaseind
Almostnochange
ButtheScale-FreeNetworkisMoreVulnerabletoAttackThantheExponentialNetwork
SF
E
Failurefractionf
Diam
eterd Samelinearincreaseind
Largerlinearincreaseind
Attackmeanssequentiallytakingoutthenodeswithhighestdegree
GiantComponents
S=Fractionofnodesinthegiantcomponent<s>=meansizeofacomponent
InanExponentialNetworkThereisNoDifferenceBetweenFailureandAttackonGiantComponentSize
Failurefractionf
Sand<s>
<s>
S
fc isthethresholdfornetworkdisintegration
InanExponentialNetworktheNetworkDisintegrationisRoughlyHomogenous
Componentsizedistribution
f=0.05 f=0.18 f=0.45
<s>
S
InaScale-FreeNetworkThereisaBigDifferenceBetweenFailureandAttackonGiantComponentSize
Failurefractionf
Sand<s>
fc isthethresholdfornetworkdisintegration
failureattack
Inset:widerrangeoff
InaScale-FreeNetworktheNetworkDisintegrationinResponsetoFailureisVeryHeterogeneous
Homogenousdisintegration
Heterogeneousdisintegration
InternetNetworkStructure
Endusersarethecomputersandotherdeviceswealluse.
Interiorverticesarerouters,whicharespecial-purposecomputersatthejunctionsbetweendatalines.
DegreeDistributionoftheInternetObeysaPowerLawfork>1
Pr 𝑋 = 𝑘 = 𝐶𝑘)1.34
intheyear2000
In-DegreeandOut-DegreeDistributionsoftheWWWObeyPowerLaws
𝛾56 = 2.1 𝛾9:; = 2.45
Exponentdatafromtheyear2000
Diam
eterd
TheInternetandWWWareResilienttoRandomFailure,butSensitivetoAttack
Failurefractionf
Sand<s>
Failurefractionf
failureattack
IntheInternetandWWWThereisaBigDifferenceBetweenFailureandAttackonGiantComponentSize
fc isthethresholdfornetworkdisintegration
SubwaySystemsareUndirectedNetworks
Nodes=stationsEdges=raillines
Washington,DCmetro
HowManyImportantStationsareThere?
Hub?
DegreeCentralityforWashington,DCSystem
hubs
Notahub
EigenvectorCentralityforWashington,DCSystem
hubs
?
Nowahub
HowAboutAnotherSubwaySystem?
DegreeCentralityfortheClevelandSystem
hubs
Notahub
EigenvectorCentralityfortheClevelandSystem
hubs
Nowahub
? ?
GlycolyticPathway
TheCitricAcidCycle
BondPercolation
Lightedges:thefullnetworkDarkedges:thefilled-inoroccupiededgesp:probabilitythatanedgeisoccupied
ThePercolationTransition
φ =connectivitythreshold
percolationtransition