Post on 18-Oct-2021
TechnologicalNetworksTheInternet
Vertices arecomputersorcomputerroutersEdges arecablesoropticalfiberlineslinkingthecomputers
CAIDAAnnualReport(1998)
BiologicalNetworksA(directed)foodchainnetwork
Vertices indicatespeciesDirectededgeindicatesthatthetargetspeciesdependsontheotherspecies
BiochemicalNetworks
Vertices indicateproteinsDirectededgeindicatesthatoneproteinregulatesthetranscriptionofthetargetprotein
A(directed)genetranscriptionnetwork
Louis(2011)
SocialNetworks
Patternoffriendshipsamongmembersofakarateclub
Vertices indicateindividualsEdges indicatefriendships
Zachery(1977)
BuildingaSmall-WorldNetwork
Ateachnode,withprobabilitypaddanedgefromanodetosomeothernon-neighboringnode.ThisiscalledtheWatts-Strogatz algorithm.
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
AlgorithmForBuildingaScale-FreeNetwork
1. Startwithm0 nodesrandomlyconnected
2. Addadditionalnodeswhereprobabilityofconnectingtoeachexistingnodeislargerifthatnodehashighdegree
AlgorithmForBuildingaScale-FreeNetwork
1. Startwithm0 nodesrandomlyconnected
2. Addadditionalnodeswhereprobabilityofconnectingtoeachexistingnodeislargerifthatnodehashighdegree
”TheRichGetRicher”
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
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.
In-DegreeandOut-DegreeDistributionsoftheWWWObeyPowerLaws
𝛾56 = 2.1 𝛾9:; = 2.45
Exponentdatafromtheyear2000
Sand<s>
Failurefractionf
failureattack
IntheInternetandWWWThereisaBigDifferenceBetweenFailureandAttackonGiantComponentSize
fc isthethresholdfornetworkdisintegration
BondPercolation
Lightedges:thefullnetworkDarkedges:thefilled-inoroccupiededgesp:probabilitythatanedgeisoccupied