réseaux sociaux - licence 2 et 3 - introduction aux ...verel/teaching/08-09/introsc/cm05... · aux...

Réseaux So iauxLi en e 2 et 3 - Introdu tion aux systèmes omplexesSébastien Verelverel�i3s.uni e.frwww.i3s.uni e.fr/∼verelwww.i3s.uni e.fr/teaÉquipe S oBi - Université de Ni e Sophia Antipolis24 mars 2009

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksPlan1 Introdu tion2 Basi de�nitions and random graph3 Small word and s ale free networks

Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksMotivationsS ien e always resear hes onuniversal obje t : Networks is oneof themNetworks : inter onne ted entities,dynami in timeAppears in himi al, biologi al,urbanism, brain, e osystem,so iology, tele ommuni ation, et .


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksMotivations Computer s ien e helps to explorethe world of network : modelisation,simulation and predi tion toolsDes ription of omplex systems(CS) with networks and by omputersFrom modelisation of CSTo design of arti� ial CS−→ Bio-inspired omputationSébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksThe small world problem, MilgramQuestion : What is the distan e betweenpeople in the so ial network ?Experiment of 1967 :send a letter to 60 people randomlysele ted from Omaha (Nebraska)explain the study to peopleAsk them to send the letter to a person inBoston (Massa husetts) with only thename, the profession and the ity. (1933 - 1984)http://maps.google.fr/maps?f=d&sour e=s_d&saddr=Omaha,+Nebraska&daddr=Sharon,+Massa husetts&hl=fr&geo ode=&mra=ls&sll=47.15984,2.988281&sspn=15.664847,39.375&ie=UTF8&ll=42.182965,-83.56269&spn=34.027051,78.75&z=4Sébastien Verel Réseaux so iaux

http://maps.google.fr/maps?f=d&source=s_d&saddr=Omaha,+Nebraska&daddr=Sharon,+Massachusetts&hl=fr&geocode=&mra=ls&sll=47.15984,2.988281&sspn=15.664847,39.375&ie=UTF8&ll=42.182965,-83.56269&spn=34.027051,78.75&z=4

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksThe small world problem, MilgramSome letters goes to the right person !When it arrives, only 6 persons between sour e and destinationThe diameter of the so ial network is small


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksThe small world problem, MilgramSome letters goes to the right person !When it arrives, only 6 persons between sour e and destinationThe diameter of the so ial network is small−→ six degrees of separation


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksThe small world problem, MilgramSome letters goes to the right person !When it arrives, only 6 persons between sour e and destinationThe diameter of the so ial network is small−→ six degrees of separationAs usual in s ien e, there is some ritisms :The e�e t of motivation in rease the rate of su es :Is the so ial network hanged with money ?The so iology of people has some e�e ts on the su es rate :The so ial network have some small network,but it is not sure all the so ial network is small.Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksSix degrees of separation, Guare, 1990Ameri anplaywright, JohnGuare, 1990.

read somewhere that everybody on this planetis separated by only six other people. Sixdegrees of separation between us and everyoneelse on this planet. The President of the UnitedStates, a gondolier in Veni e, just �ll in thenames. I �nd it A) extremely omforting thatwe're so lose, and B) like Chinese watertorture that we're so lose be ause you have to�nd the right six people to make the right onne tion... I am bound to everyone on thisplanet by a trail of six people.Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksCreate a network !Write on a paper, your 4 or 5 singers or bands that you likevery mu hDo not look on the paper of your neighbors


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksCreate a network !Write on a paper, your 4 or 5 singers or bands that you likevery mu hDo not look on the paper of your neighborsConstru t the network :ea h node is a singerthere is a link between two singers, if there are in the same list


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksCreate a network !Write on a paper, your 4 or 5 singers or bands that you likevery mu hDo not look on the paper of your neighborsConstru t the network :ea h node is a singerthere is a link between two singers, if there are in the same listWhat is the diameter of the network ?


Introdu tionBasi de�nitions and random graphSmall word and s ale free networks o-auteurs graph of my bibliography's thesisHao

Belaidouni

Chakrabarti

Chakraborti Stadler

Renner

Flamm

Cupal

GarciaPelayoFontana

Schuster

Reidys

Shipman

Shackleton

Ebner

Johnson

Garey

Segrest

Goldberg

Rothlauf

Toussaint

Igel

Watson

Knowles

Spiessens

Weger

Manderick

Holland

Forrest

Mitchell

Crutchfield

Das

Hordijk

Zecchina

Mezard

Huynen

NimwegenHansonHraber

Vose

Nix Schumacher

Emmerich

Schonemann

Preuss

Surry

Radcliffe

OShea

Layzell

Husbands

Smith

Philippides

Hofacker

BornbergBauer

Griesmacher

Miller

Lobo Weinberger

Tarazona

Tacker

Perelson

Garcia

Bishop

Frauenfelder

Wiles

Tonkes

Geard

Cairns

Skellett

Clergue

Collard

Verel

Tomassini

Vanneschi Platel

Gaspar

Talbi

Weinberg Preux

Robilliard

Fonlupt

YuVassilev

Laarhoven

Aarts

Koza

Bennett

Andre

Schwefel

Hoffmeister

Back

Michel

Philippe

Vendruscolo

Roman

Porto

Bastolla

Chan

Jenkins

Box

Swanson

Drummond

Bresina

Sipper

Capcarrere

Droz

Chopard

Mehlenbein

Crisan

Polito

Chaudhri

Chaudhary

Khoo

Stanhope

Bertram

Daida

Gordon

Spears

Jong

Popovici

Peliti

Derrida

Gould

Eldredge

Ghozeil

FogelSwart

Sherrington Levin

Kurdyumov

Gacs

Kauffman

Kallel

Garnier

Thompson

Harvey

Chellapilla Kreutz Pollack

Juille

Reeves

Naudts

Belew

Land

Poli

Langdon

Thomson

Engelhardt

Newman

Kipnis

Cohen

Asselmeyer

Ebeling

Rose

Whitley Beer

Seys

Dreyfus

Siarry

Schnabl Macready

Wolpert


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksBasi s de�nitions on networkV5

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Graph, networkA dire ted graph G is de�ned as 2-upleG = (V ,E ) where :V is a (�nite) set of nodesE is a set of dire ted edges :E ⊂ V 2.Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksBasi s de�nitions on network

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OutdegreeLet be G = (V ,E ) a dire tedgraph G .deg+(v) is the number of verti esw it is onne ted to by an edge(v ,w) ∈ E .IndegreeLet be G = (V ,E ) a dire tedgraph G .deg−(v) is the number of verti esw that are onne ted to v by anedge (w , v) ∈ E .Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksPath lengthX10

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PathLet be G = (V ,E ) a dire tedgraph G .A path P(v ,w) is a sequen e ofverti es (v0, v1, . . . , vk ) where :v0 = v and vk = w∀i ∈ [|0, k − 1|],(vi , vi+1) ∈ E



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Path lengthThe path length is the number ofedges of a path P(v ,w).Distan eThe distan e between 2 nodes vand w is the minimal path lengthof any path between v and w .Do you know the Dijkstraalgorithm ?Sébastien Verel Réseaux so iaux


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X16 Average path lengthThe average path length is theaverage of distan es between of ouple of nodes.Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksClustering Coe� ientIntrodu ed by Watts and Strogatz in 1998The Clustering Coe� ient of a graph how lose the vertex ofits neighbors are to being a omplete graph.Number of edges links between the verti es within theneighborhooddivided by the number of edges that ould possibly existbetween themv


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksClustering Coe� ient, formal de�nitionClustering Coe� ientLet be G = (V ,E ) a dire ted graph G . (v) = e(N(v))deg+(v)(deg+(v)−1) where N(v) is the nodes in theneighborhood of v and e(N(v)) is the number of edges in theneighborhood.v


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksClustering Coe� ient : examplesv v v


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksClustering Coe� ient : examplesv v vFrom left to right : 14 , 23 and 1.


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksRandom graphRandom graph : the easiest model for any networkSimple underlying asumptionbut more realisti than regular graphFirst family of network studied (≈ 1950)exa t results possibleBasi idea :Edges are added at random between a �xed number n ofverti esMany models were developped, but the most importantthe Erdos-Rényi random graphthe Gilbert random graphSébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksthe Erdos-Rényi random graphDe�nitionGn,m denotes the set of all graphs with n nodes and m edges.This set an be transform into probability spa e by taking theelements of Gn,m with the same probability.


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksThe Erdos-Rényi random graphExpe ted PropertiesGn,m (an instan e of Gn,m) shows the propeties P with highprobability if : Pr(Gn,m has P) → 1 for n → ∞Conne tedness of random graphm = n2 (log(n) + γ(n)).if lim∞ γ = −∞ then a typi al Gn,m is dis onne ted,if lim∞ γ = ∞ then a typi al Gn,m is onne ted.Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksThe Gilbert random graphDe�nitionGn,p is de�ned as graph in whi hprobability that an edge (v ,w) exist is p.Easy to onstru t iteratively


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksThe Gilbert random graphDe�nitionGn,p is de�ned as graph in whi hprobability that an edge (v ,w) exist is p.Easy to onstru t iterativelyRelation between random graph modelsif m ≡ p.n the two models Gn,m andGn,p are almost inter hangeable.


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksRandom graph propertiesGiant onne ted omponentLet > 0 be a onstant and p = /n.If < 1 every omponent of Gn,p has order O(log(n)) with ahigh probability.If > 1 then there will be one omponent with highprobability that has a size of (f ( ) + O(1)).n, where f ( ) > 0.All other omponent have size O(log(n)).Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksRandom graph propertiesGiant onne ted omponentLet > 0 be a onstant and p = /n.If < 1 every omponent of Gn,p has order O(log(n)) with ahigh probability.If > 1 then there will be one omponent with highprobability that has a size of (f ( ) + O(1)).n, where f ( ) > 0.All other omponent have size O(log(n)).Giant onne ted omponentIf p < 1/n : Gn,p omposed by small sub-network of size≈ log(n)If p > 1/n : Gn,p omposed by 1 big one sub-network and a lotof small of size ≈ log(n)Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksRandom graph propertiesDegree distributionLet Xk be the number of nodes with the degree k in Gn,p.Let be a positive onstant and p = /n.Then for k = 0, 1, 2, . . .Pr((1− ǫ)

ke− k!≤

Xkn ≤ (1 + ǫ) ke− k!

) → 1as n → ∞


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksRandom graph propertiesDiameter of Gn,pIf p nlog(n) → ∞log(n)log(pn) → ∞Then the diameter of Gn,p is asymptoti to log(pn) with highprobability.Clustering Coe� ientThe lustering oe� ient of a random graph is asymptoti ally equalto p with high probability.Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksSmall Word phenomenon ba k to physi iansfrom the experiment ofMilgram, no onvin ingnetwork model generating anetwork with high ClusteringCoe� ient or average shortpath lengthWatts and Strogatts startedthree king of real networkssmall world network : neithergrid-like network neither fullrandom network Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksExamplesa tor ollaboration graph (Tjaden, 1997) :


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksExamplesa tor ollaboration graph (Tjaden, 1997) :2 a tors are onne ted by an undire ted edge if they havea ted in at least one �lm


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksExamplesa tor ollaboration graph (Tjaden, 1997) :2 a tors are onne ted by an undire ted edge if they havea ted in at least one �lmThe power grid of the United States (Phadke and Thorp,1988) :Neural network of the nematode C. elegans (White et al.1992) :


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksExamplesDiameter and Clustering Coe� ient of the 3 real networks ompared to random graphD D random CC CC randoma tor ollaboration 3.65 2.99 0.79 0.00027power grid 18.7 12.4 0.08 0.005Neural network 2.65 2.25 0.28 0.05Small World NetworkA small world network is a network with a dense lo al stru ture anda diameter omparable to a random graph with the same numbersof nodes and edges. Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksFrom random to order networksWatts-Strogatz models Small world NetworkNeither :Grid-like networks : regularity and lo ality, but a high averagepath length and diameter ;Random graph : lustering oe� ient of pWatts and Strogatz proposed a mixture of bothSébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksFrom random to order networksWatts-Strogatz modelsGenerative Watts-Strogatz model :Build a ring of n verti es and onne t ea h vertex with its k lo kwie neighbors on the ringDraw a random number between 0 and 1 for ea h edgeRewire ea h edge with probability p :if the edge's random nimber is smaller than p : keep the sour evertex of the edge �xed, and hoose a new target vertexuniformly at random from all other verti es.Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksFrom random to order networksWatts-Strogatz models


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksFrom random to order networksWatts-Strogatz modelsFor p = 0, network is totaly regular.CC is approa hing 3/4 for large kdiameter in O(n)For p = 1, network is regular random networkCC is approa hing p for large kdiameter in O(log n)


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksWatts-Strogatz models


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksNavigability model : Kleinberg (2000)How do people ould�nd a short ut ?Why these pathsexist ?



NavigabilityBetween two pairs of nodes , thereexists a very short path (of logarithmi length) that an be found e� ien y, inspite of a very knowledge of the globalstru ture of this "spontaneous"networks (no global design).What kind of underlying globalknowledge of the stru ture allowsthis phenomenon to arise ?Sébastien Verel Réseaux so iaux


NavigabilityBetween two pairs of nodes , thereexists a very short path (of logarithmi length) that an be found e� ien y, inspite of a very knowledge of the globalstru ture of this "spontaneous"networks (no global design).What kind of underlying globalknowledge of the stru ture allowsthis phenomenon to arise ?→ So ial informationsSébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksNavigability model : Kleinberg (2000)Tori d -dimensional gridv

w Pr(v → w) =Cstedistα(v ,w)Grid : global geographi knowledgeLong range links/ onta ts :Extra lo al knowledge at ea h noderandom onta t meets some daySébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksNavigability model : Kleinberg (2000)Agent based model : De entralized algorithmsroutes only to known onta ts (lo al or long range)i.e. onta ts of previously visited nodes,Using only the geographi positioni.e. the underlying metri


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksNavigability model : Kleinberg (2000)Greedy routing : send the message to the losest neighborroutes in O(log2(n)) on expe tation if α = dif α 6= d , then any de entralized algorithm takes more thanpoly(n)Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksNavigability model : Kleinberg (2000)Greedy routing : send the message to the losest neighborroutes in O(log2(n)) on expe tation if α = dif α 6= d , then any de entralized algorithm takes more thanpoly(n)Noti e that :α ≤ d : logarithmi diameterd < α < 2d : polylogarithmi diameter2d ≤ α : polynomial diameterSébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksDisease Spreading in Stru tured populationsA lot of works from 1980 and 1990 study the deseasespreading :topology grid, ring, starsometime di�erent of relation between individuals


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksDisease Spreading in Stru tured populationsA lot of works from 1980 and 1990 study the deseasespreading :topology grid, ring, starsometime di�erent of relation between individualsCon lusion : spatial stru tured an a�e t the speading


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksDisease Spreading in Stru tured populationsA lot of works from 1980 and 1990 study the deseasespreading :topology grid, ring, starsometime di�erent of relation between individualsCon lusion : spatial stru tured an a�e t the speadingBUT None of this work : treat the global dynami al propertiesof the system as a fun tion of the network stru ture


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksDisease Spreading in Stru tured populationsSimple model (D. Watts)3 possible states for ea helement :sus eptible : ould beinfe tedinfe ted : is infe tedremoved : wasinfe ted

At ea h time step t :every infe ted element an infe tea h one of is neighbbors with theprobability pa newly infe ted elements remainsinfe ted during 1 time step andbe omes removed prematly fromthe populationA time t = 0, a single element isinfe tedSébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksDisease Spreading in Stru tured populationsMeasureThe disease will have run its ourse and died out,Fra tion Fs of the population is uninfe tedThe whole population will have been infe ted (Fs = 0),in some hara teristi time T


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksDisease Spreading in Stru tured populationsMeasureThe disease will have run its ourse and died out,Fra tion Fs of the population is uninfe tedThe whole population will have been infe ted (Fs = 0),in some hara teristi time TIs Fs and T an be understood in terms of path length or lustering oe� ient ?


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksDisease Spreading in Stru tured populationsResults p ≤ 1/(k − 1) : for alltopologies, disease infe tsonly a negligibe fra tion ofthe population1/(k − 1) < p : di�erenttopologies yield di�erent Fs ,it's depends on th CCp ≥ 0.5, for all topologies,all population infe tedSébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksDegree distribution : experimental �ndingsBoth ases : all nodes have very similar degreeFor small world metwork, the degree will be normalydistributed around 2kFor random graphs, the degree will be normal distributedaround (n − 1)pSébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksDegree distribution : experimental �ndingsBoth ases : all nodes have very similar degreeFor small world metwork, the degree will be normalydistributed around 2kFor random graphs, the degree will be normal distributedaround (n − 1)pIs that also the ase in real worls networks ?Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksInternet tra� map

There exist some nodes with high degree : hubSébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksS ale-Free NetworksS ale-free networks (degree distribution)The number of nodes fk with the same outdegree k is proportionalto the outdegree to the power of a onstant :fk ≡ kαS ale-free networks (equivalent)List of all existing outdegreee was made and sorted.The rank of ri of a node i is de�ned as its pla e in the list Theoutdegree k is proportional to its rank to the power of a onstant :k(i) ≡ rβiSébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksS ale-Free Networks


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksS ale-Free NetworksExamplesAirport tra� S ienti� Collaborationsemanti networkssexual partners in humansProtein protein intera tionQuestionWhat kind of model an generate this more realisti degreedistribution ? Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksBarabàsi-Albert modelPrin iple : "The ri h getri her"network grows intime : one vertex isadded at ea h timestepthe new vertex islinked more often toan old node with highdegree Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksBarabàsi-Albert modelPrin iple : "The ri h getri her"network grows intime : one vertex isadded at ea h timestepthe new vertex islinked more often toan old node with highdegreeThe generative model :1 Start with a network (10 verti es,20 edges at random)2 at ea h time step, add a new vertexv . Add m edges from v to verti eswhi h are already there followingthe preferential atta hment :

Π(x) =deg(x)

∑u∈V deg(u)Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksS ale-Free Networks


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksS ale-Free Networks with NetLogo


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksRobustnessStability of s ale-free networks :Random graph :NOT robust against randomfailureNOT sensitive againstatta ks

S ale-Free network :VERY robust againstrandom failureVERY sensitive againstatta ksSébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksDynami and evolution of so ial networksThe des ription of the network was mostly stati The hierar hi al stru ture of s ale free in so iology : ommunautiesModel to explain the ommunauties formation


Introdu tionBasi de�nitions and random graphSmall word and s ale free networksBibliographyD. J. Watts and S. H. Strogatz, Colle tive dynami s of'small-world' networks, Nature 393 (1998), 440�442.D. J. Watts, Networks, Dynami s, and the Small-WorldPhenomenon, Ameri an Journal of So iology 105 2, (1998),493�527.A-L. Barabasi and R. Albert, Emergen e of s aling in randomnetworks, S ien e 286 (1999), 509�512.D. J. Watts, The "new" s ien e of networks, Annual Review ofSo iology 30 (2004), 243�270.Sébastien Verel Réseaux so iaux

Introdu tionBasi de�nitions and random graphSmall word and s ale free networksBibliographyKatharina Anna Lehmann, Mi hael Kaufmann.Random Graphs, Small-Worlds and S ale-Free Networks.Peer-to-Peer Systems and Appli ations (2005) 57-76F. Vazquez, V.M. Eguiluz, and M. San Miguel.Generi Absorbing Transition in Coevolution Dynami sIn Phys. Rev. Lett. 100, 108702 (2008)J. Kleinberg,"Navigation in a small world."Nature 406 (2000)Sébastien Verel Réseaux so iaux

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