RandomGraphs

SocialandTechnologicalNetworks

RikSarkar

UniversityofEdinburgh,2016.

Graph

•  V:setofnodes•  n=|V|:Numberofnodes

•  E:setofedges•  m=|E|:Numberofedges

•  Ifedgea-bexists,thenaandbarecalledneighbors

Graph

•  Howmanyedgescanagraphhave?

Graph

•  Howmanyedgescanagraphhave?

•  InbigO?

✓n

2

◆OR

n(n� 1)

2

Graph

•  Howmanyedgescanagraphhave?

✓n

2

◆OR

n(n� 1)

2

O(n2)

Randomgraphs

•  Mostbasic,mostunstructuredgraphs•  Formsabaseline– Whathappensinabsenceofanyinfluences

•  Socialandtechnologicalforces

•  Manyrealnetworkshavearandomcomponent– Manythingshappenwithoutclearreason

Erdos–RenyiRandomgraphs

Erdos–RenyiRandomgraphs

•  n:numberofverTces•  p:probabilitythatanyparTcularedgeexists

•  TakeVwithnverTces•  Considereachpossibleedge.AddittoEwithprobabilityp

G(n, p)

Expectednumberofedges

•  Expectedtotalnumberofedges

•  Expectednumberofedgesatanyvertex

Expectednumberofedges

•  Expectedtotalnumberofedges

•  Expectednumberofedgesatanyvertex

�n2

�p

(n� 1)p

Expectednumberofedges

•  For

•  Theexpecteddegreeofanodeis:?

p =c

n� 1

IsolatedverTces

•  HowlikelyisitthatthegraphhasisolatedverTces?

IsolatedverTces

•  HowlikelyisitthatthegraphhasisolatedverTces?

•  WhathappenstothenumberofisolatedverTcesaspincreases

Terminologyofhighprobability

•  Somethinghappenswithhighprobabilityif

•  Wherepoly(n)meansapolynomialinn•  Apolynomialinnisconsideredreasonably‘large’

p �✓1� 1

poly(n)

◆

ProbabilityofIsolatedverTces

•  IsolatedverTcesare

•  Likelywhen:

•  Unlikelywhen:

•  Let’sdeduce

p < lnnn

p > lnnn

UsefulinequaliTes

✓1 +

1

x

◆x

e

✓1� 1

x

◆x

1

e

Unionbound

•  ForeventsA,B,C…

•  Pr[AorBorC...]≤Pr[A]+Pr[B]+Pr[C]+...

•  Theorem1:•  If

•  Thentheprobabilitythatthereexistsanisolatedvertex

•  Thusforlargen,w.h.pthereisnoisolatedvertex•  ExpectednumberofisolatedverTcesisminiscule

p = (1 + ✏)lnn

n� 1

1

n✏

•  Theorem2•  For

•  Probabilitythatvertexvisisolated

p = (1� ✏)lnn

n� 1

� 1

(2n)1�✏

•  Theorem2•  For

•  Probabilitythatvertexvisisolated

•  ExpectednumberofisolatedverTces:

p = (1� ✏)lnn

n� 1

� 1

(2n)1�✏

� n

(2n)1�✏=

n✏

2Polynomialinn

Thresholdphenomenon:ProbabilityornumberofisolatedverTces

•  TheTppingpoint

Clusteringinsocialnetworks•  Peoplewithmutualfriendsareodenfriends

•  IfAandChaveacommonfriendB–  EdgesABandBCexist

•  ThenABCissaidtoformaTriad–  Closedtriad:EdgeACalsoexists– Opentriad:EdgeACdoesnotexist

•  Exercise:Provethatanyconnectedgraphhasatleastntriads(consideringbothopenandclosed).

Clusteringcoefficient(cc)

•  MeasureshowTghtthefriendneighborhoodsare:frequencyofclosedtriads

•  cc(A)fracTonsofpairsofA’sneighborsthatarefriends

•  Averagecc:averageofccofallnodes•  Globalcc:raTo #closedtriads

#alltriads

GlobalCCinERgraphs

•  Whathappenswhenpisverysmall(almost0)?

•  Whathappenswhenpisverylarge(almost1)?

GlobalCCinERgraphs

•  WhathappensattheTppingpoint?

Theorem

•  For

•  GlobalccinERgraphsisvanishinglysmall

p = clnn

n

lim

n!1cc(G) = lim

n!1

# closed triads

# all triads

= 0

AvgCCInrealnetworks

•  Facebook(olddata)~0.6•  hhps://snap.stanford.edu/data/egonets-Facebook.html

•  Googlewebgraph~0.5•  hhps://snap.stanford.edu/data/web-Google.html

•  Ingeneral,ccof~0.2or0.3isconsidered‘high’–  thatthenetworkhassignificantclustering/communitystructure