From Random Graph to Small World by Wandering

apport   de recherche ISSN 0249-6399 ISRN INRIA/RR--6489--FR+ENG Thème COM INSTITUT N A TION AL DE RECHERCHE EN INFORMA TIQUE ET EN A UTOMA TIQUE F rom Random Graph to Small W orld by W andering Bruno Gaume — Fabien Mathieu N° 6489 A v ril 2008 Unité de recherche INRIA Rocquenco urt Domaine de V oluceau, Rocquen court, BP 105, 781 53 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30 F rom Random Graph to Small W orld b y W andering Bruno Gaume ∗ , F abien Mathieu † Thème COM  Systèmes omm unian ts Pro jet GANG Rapp ort de re her he n ° 6489  A vril 2008  11 pages Abstrat: Numerous studies sho w that most kno wn r e al-world omplex net w orks share similar prop erties in their onnetivit y and degree distribution. They are alled smal l worlds . This artile giv es a metho d to turn random graphs in to Small W orld graphs b y the din t of random w alks. Key-w ords: Random graphs, small w orlds, random w alks ∗ IRITUPS, T oulouse F-31062 Cedex 4, F rane † Orange Labs, 3840 rue du Général Leler, 92794 Issy-les-Moulineaux Cedex P etit-mondisation par mar hes aléatoires Résumé : De nom breuses études mon tren t un fait remarquable qui est que la plupart des réseaux dits de terrain p ossèden t des propriétés iden tiques bien partiulières et fon t partie de la lasse des graphes p etit-monde . Un autre fait tout aussi remarquable est que ette lasse des p etits mondes est très p etite au regard de l'ensem ble des graphes p ossibles. Dans et artile, nous prop osons une métho de de pro dution de graphes p etit-monde au mo y en de mar hes aléatoires. Mots-lés : Graphes aléatoires, p etits mondes, mar hes aléatoires F r om R andom Gr aph to Smal l W orld by W andering 3 1 In tro dution In 1998, W atts and Strogatz sho w ed that man y real graphs, oming from dieren t elds, share similar prop er- ties [28 ℄. This has b een onrmed b y man y studies sine this seminal w ork [ 4 , 20 , 9, 1 , 13 , 17 , 6, 25 , 5, 23 , 14 , 3℄. The onerned elds inlude, but are not limited to: epidemiology (on tat graphs, . . . ), eonom y (ex hange graphs, . . . ), so iology (kno wledge graphs,. . . ), linguisti (lexial net w orks, . . . ), psy hology (seman ti asso i- ation graphs,. . . ), biology (neural net w orks, proteini in terations graphs), IT (In ternet, W eb). . . W e all su h graphs real-w orld omplex net w orks, or small-w orld net w orks. The ommon prop erties of real-w orld omplex net w orks are a lo w diameter, a globally sparse but lo ally hea vy edge densit y , and a hea vy-tailed degree distribution. The om bination of these prop ert y is v ery unlik ely in random graphs, explaining the in terest that those net w orks ha v e arisen in dieren t sien ti omm unities. In this artile, w e prop ose a metho d to generate a graph with small-w orld prop erties from random graph. This metho d, whi h is based on random w alks, ma y b e a rst step in order to understand wh y graphs from v arious origins share a ommon struture. In Setion 2 , w e briey state the prop erties used to deide wheter a giv en graph is small w orld or not. In Setion 3 , w e surv ey the dieren t existing metho ds to generate omplex net w orks. In Setion 4 , w e analyse the dynamis or random w alks in a graph, and in Setion 5 w e prop ose a new metho d to onstrut small w orlds b y wandering on random graphs. Setion 6 onludes. 2 Small W orlds Struture let G = ( V , E ) b e a reexiv e, symmetri graph with n = | V | no des and m = | E | edges. G is alled smal l world if the follo wing prop erties are v eried: Edge sparsit y Small w orld graphs are sparse in edges, and the a v erage degree sta y lo w: m = O ( n ) or m = O ( n log( n )) Short paths The a v erage path length (denoted ℓ ) is lose to the a v erage path length ℓ rand in the main onneted omp onen t of G ( n, m ) = G ( n, m − n n ( n − 1) ) Erdös-Rén yi graphs. A ording to [ 12 ℄, for d := m − n n ≥ (1 + ǫ ) log( n ) , G ( n, m − n n ( n − 1) ) is almost surely onneted, and ℓ rand ≈ log( n ) log( d ) . ( l = O (log ( n )) ). High lustering The lustering o eien t, C , that expresses the probabilit y that t w o distint no des adjaen t to a giv en third no de are adjaen t, is an order of magnitude higher than for Erdös-Rén yi graphs: C > > C rand = p = m − n n ( n − 1) . This indiates that the graph is lo ally dense, although it is globally sparse. Hea vy-tailed degree distribution Example: DioSyn.V erb e 1 is a reexiv e symmetri graph with 9043 no des and 11093 9 edges. F or sak e of on v eniene, w e only onsider the main onneted omp onen t G c of DioSyn, whi h admits 8835 no des and 11053 3 edges. With an a v erage degree of 12 . 5 , G c is sparse. Other parameters of G c are ℓ ≈ 4 . 17 (to ompare with ℓ rand = 3 . 71 ) and C ≈ 0 . 3 9 (to ompare with C rand = p = 0 . 001 3 ). The degree distribution is hea vy-tailed, as sho wn b y Figure 1 (a least-square metho d giv es a slop e of − 2 . 01 with a ondene 0 . 96 ). Therefore G c v eries the four prop erties of a small w orld. Note, that the degree distribution for random Erdös-Rén yi graphs is far from b eing hea vy-tailed. It is in fat a kind of P oisson distribution : the probabilit y that a no de of a G ( n, p ) graph has degree k is p ( k ) = p k (1 − p ) n − 1 − k  n − 1 k  . Figure 2, where the degree distribution of a Erdös-Rén yi graph with same n um b er of no des and a v erage degree than G c is plotted. This illustrates ho w a small w orld ompares to a G graph with same n um b er of no des and exp eted degree:  Same sparsit y (b y onstrution),  Similar a v erage path length,  Higher lustering, 1 DioSyn is a fren h synon yms ditionnary built from sev en anonial fren h ditionnaries (Bailly , Bena, Du Chaz- aud, Guizot, Lafa y e, Larousse et Rob ert). The A TILF ( http://www.atilf.fr/ ) extrated the synon yms, and the CRISCO ( http://elsap1.uniaen .fr/ ) onsolidated the results. DioSyn.V erb e is the subgraph indued b y the v erbs of Diosyn: an edge exists b et w een t w o v erbs a and b i DioSyn tells a and b are synon yms. Therefore DioSyn.v erb e is a symmetri graph, made reexiv e for on v eniene. A visual represen tation based on random w alks [15℄ an b e onsulted on http://Prox.irit.fr . RR n ° 6489 4 Bruno Gaume & F abien Mathieu 0 50 100 150 200 250 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Degree Probability (a) linear sales 10 0 10 1 10 2 10 3 10 −4 10 −3 10 −2 10 −1 10 0 Degree Probability (b) logarithmi sales Figure 1: Degree distribution of G c  Hea vy-tailed distribution (instead of P oisson distribution) 0 5 10 15 20 25 0 0.02 0.04 0.06 0.08 0.1 0.12 Degree Probability (a) linear sales 10 0 10 1 10 2 10 −4 10 −3 10 −2 10 −1 10 0 Degree Probability (b) logarithmi sales Figure 2: Degree distribution of a t ypial G ( n, p ) graph In [3℄, Alb ert and Barabasi ha v e made a surv ey on existing omplex net w orks studies, inluding [ 4 , 20 , 9 , 1 , 13 , 17 , 6 , 25 , 5 , 23 , 14 , 28 ℄. Some of their ndings are presen ted in T able 1 along G c 's prop erties. Name n < k > ℓ C γ r 2 DioSyn.V erb es 8835 11 . 51 4 . 17 0 . 39 2 . 0 1 0 . 96 In ternet routers 15000 0 2 . 66 11 2 . 4 Mo vie ators 21225 0 28 . 78 4 . 54 0 . 79 2 . 3 Co-authorship, SPIRES 56627 173 4 . 0 0 . 72 6 1 . 2 Co-authorship, math. 70975 3 . 9 9 . 5 0 . 5 9 2 . 5 Co-authorship, neuro. 20929 3 11 . 5 6 0 . 76 2 . 1 Ythan estuary fo o d w eb 134 8 . 7 2 . 43 0 . 22 1 . 05 Silw o o d P ark fo o d w eb 154 4 . 75 3 . 40 0 . 15 1 . 1 3 W ords, synon yms 22311 1 3 . 48 4 . 5 0 . 7 2 . 8 T able 1: Main prop erties of some omplex net w orks INRIA F r om R andom Gr aph to Smal l W orld by W andering 5 3 Generating Small W orlds: State of Art Small-w orld net w orks ha v e b een studied in tensely sine they w ere rst desrib ed in W atts and Strogatz [28 ℄. Resear hs ha v e b een done in order to b e able to generate random datasets with w ell-kno wn  harateristis shared b y so ial net w orks. Most pap ers fo us on either the lustering and diameter, or on the p o w er-la w. 3.1 Clustering and diameter prop ert y W atts and Strogatz [28 ℄, and Klein b erg [19 ℄ ha v e studied families of random graphs that share the lustering and diameter prop erties of small w orlds. W atts and Strogatz mo del onsist in altering a regular ring lattie b y rewiring randomly some links. In Klein b erg's mo del, a d -dimensional grid is extended b y adding extra-links of whi h the range follo ws a d -harmoni distribution. Note, that b oth mo dels fail to apture the hea vy-tail prop ert y met in real omplex net w orks (they are almost regular). 3.2 Hea vy-tail prop ert y There is a lot of resear h dev oted on the pro dution of random graphs that follo w a giv en degree distribution [ 8, 21 , 22 , 26 ℄. Su h generi mo dels easily pro due hea vy-tailed random graphs if w e giv e them a p o w er la w distribution. On the eld of sp ei hea vy-tailed mo dels, there is Alb ert and Barabasi preferen tial atta hmen t's mo del [3, 6℄, in whi h links are added one b y one, and where the probabilit y that an existing no de reeiv es a new link is prop ortional to its degree. A more exible v ersion of the preferen tial atta hmen t's mo del is the tness mo del [1, 7℄, where a pre-determined tness v alue is used in the pro ess of link reation. Lastly , Aiello et al. prop osed a mo del alled α, β gr aphs [2℄, that enompasses the lass of p o w er la w graphs. 3.3 Others mo dels Other mo dels of graph generation are Guillaume and Latap y's A l l Shortest Paths [ 18 ℄, where one onstrut a graph b y extrating the shortest paths of a random graph, and the Dorogo vtsev-Mendes mo del [ 11 ℄. Note, that the latter aptures all desired prop erties, but is not realisti. 4 Conuene & Random W alk in Net w orks 4.1 Random W alk in Net w orks Just lik e Setion 2 , G = ( V , E ) is a reexiv e, symmetri graph with n = | V | no des and m = | E | edges. W e assume that a partile w anders randomly on the graph:  A t an y time t ∈ N the partile is on a no de u ( t ) ∈ V ;  A t time t + 1 , the partile rea hes a uniformly randomly seleted neigh b or of u ( t ) . This pro ess is an homogeneous Mark o v  hain for on V . A lassial w a y to represen t this  hain is a n × n sto  hasti matrix [ G ] : [ G ] = ( g u,v ) u,v ∈ V , with g u,v =    1 deg( u ) if u → v , 0 else. (1) Beause G is reexiv e, no no de has n ull degree, so the underlying Mark o v  hain [ G ] is w ell dened. F or an y initial probabilit y distribution P 0 on V and an y giv en in teger t , P 0 [ G ] t is the result of the random w alk of length t starting from P 0 whose transitions are dened b y [ G ] . More preisely , for an y u , v in V , the probabilit y P t of b eing in v after a random w alk of length t starting from u is equal to ( δ u [ G ] t ) v = ([ G ] t ) u,v , where δ u is the ertitude of b eing in u . One an demonstrate, b y the din t of the P erron-F rob enius theorem [24 ℄, that if G = ( V , E ) is a onneted, reexiv e and symmetri graph, then: ∀ u, v ∈ V , lim t →∞ ( δ u [ G ] t ) v = lim t →∞ ([ G ] t ) u,v = deg( v ) P x ∈ V deg( x ) (2) RR n ° 6489 6 Bruno Gaume & F abien Mathieu In other w ords, giv en than t is large enough, the probabilit y of b eing on no de v at time t is prop ortional to the degree of V , and no longer dep ends on the departure no de u . 4.2 Conuene in Net w orks Equation (2) tells that the only information retained after an innite random w alk is the degree of the no des. Ho w ev er, some information an b e extrated from transitional states. F or instane, assume the existene of three no des u , v 1 and v 2 su h that  u , v 1 and v 2 b elong to the same onneted omp onen t,  v 1 is lose from u , in the sense that man y short paths exist b et w een u and v 1 ,  v 2 is distant from u ,  v 1 and v 2 ha v e the same degree. F rom (2 ) , w e kno w that the sequenes (([ G ] t ) u,v 1 ) 1 ≤ t and (([ G ] t ) u,v 2 ) 1 ≤ t share the same limit, that is deg( v 1 ) / P x ∈ V deg( x ) = deg( v 2 ) / P x ∈ V deg( x ) . Ho w ev er these t w o sequenes are not iden tial. Starting from u , the dynami of the partile's tra jetory on its random w alk is ompletely determined b y the graph's top ologial struture, and after a limited amoun t of steps t , one should exp et a greater v alue for (([ G ] t ) u,v 1 ) than for (([ G ] t ) u,v 2 ) b eause v 1 is loser from u than v 2 . This an b e v eried on the graph of fren h v erbs G c , with:  u = déshabil ler (to undress),  v 1 = eeuil ler (to thin out),  v 2 = rugir (to roar), In tuitiv ely , eeuil ler should b e loser (in G c ) to déshabil ler than rugir , b eause this is the ase seman tially . Also eeuil ler and rugir ha v e the same degree ( 11 ). The v alues of (([ G ] t ) u,v 1 ) and (([ G ] t ) u,v 2 ) with resp et to t are sho wn in Figure 3(a) , along with the ommon asymptoti v alue 11 P x ∈ V deg( x ) . 0 10 20 30 40 50 10 −6 10 −5 10 −4 10 −3 10 −2 t ([ G ] t u,v 1 ) ( str ong confluenc e) ([ G ] t u,v 2 ) ( weak confluence) Comm on asy mpt oti cal v alue (a) F ren h v erbs graph G c 0 10 20 30 40 50 10 −5 10 −4 10 −3 10 −2 t ([ G ] t u,v 1 ) ( str ong confluenc e) ([ G ] t u,v 2 ) ( weak confluence) Comm on asy mpt oti cal v alue (b) Random graph Figure 3: (([ G ] t ) u,v 1 ) and (([ G ] t ) u,v 1 ) for G c and a random graph One an observ e that, after a few steps, (([ G ] t ) u,v 1 ) is ab o v e the asymptoti v alue. W e laim that this is t ypial of no des that are lose to ea h other, and all this phenomen um str ong  onuen e . On the other hand, (([ G ] t ) u,v 2 ) is alw a ys b elo w the asymptoti v alue ( we ak  onuen e ). INRIA F r om R andom Gr aph to Smal l W orld by W andering 7 One ould think that the existene of strong and w eak onuenes is t ypial of graphs with high lustering, b eause the notion of loseness sounds lik e b elonging to a same omm unit y . Ho w ev er, strong and w eak onu- enes also o ur in graphs with lo w lustering o eien ts, su h as Erdös-Rén yi random graphs. F or example, Figure 3(b) sho ws (([ G ] t ) u,v 1 ) and (([ G ] t ) u,v 2 ) for three no des u , v 1 and v 2 arefully seleted in G an Erdös-Rén yi graph with same n um b er of no des and a v erage degree than G c . Figure 3(b) is v ery similar to Figure 3(a). This p oin ts out that the onept of onuene exists in random graphs lik e it do es in small w orlds. In the follo wing Setion, w e will use this to turn random graphs in to small-w orlds. 5 F rom Random Graph to Small W orld b y W andering No w w e w an t to use the onept of onuene to pro vide a w a y to onstrut small-w orld lik e graphs. In order to do that w e in tro due the m utual onuene conf b et w een t w o no des of a graph G at a time t : conf G ( u, v , t ) = max([ G ] t u,v , [ G ] t v, u ) (3) F or not to o lar ge v alues of t , a strong m utual onuene b et w een t w o no des ma y indiate that those no des are lose. W e laim that a go o d w a y to obtain a small w orld from a random graph is to set edges b et w een the pairs of no des with the highest onuene. 5.1 Extrating the onuene graph Giv en an input graph G in = ( V , E in ) , symmetri and reexiv e, with n no des and m in edges, a time parameter t and a target n um b er of edges m , one an extrat a strong onuene graph G = scg ( G in , t, m ) dened b y:  G a symmetri, reexiv e graph with the same no des than G in and m edges,  ∀ r 6 = s, u 6 = v ∈ V , if ( r , s ) ∈ E and ( u, v ) / ∈ E , then conf G in ( r , s, t ) ≥ conf G in ( u, v , t ) . Algorithm 1 : scg (strong onuene graph), extrat highest onuenes Input : An undireted graph G in = ( V , E in ) , with n no des and m in edges A w alk length t ∈ N ∗ A target n um b er of edges m ∈ [ n, n 2 ] Output : A graph G = ( V , E ) , with n no des and m edges b egin E ← − ∅ for i ← 1 to n do E ← − E ∪ { ( i, i ) } /* Make G reflexive */ end M ← − n while M < m do /* Is there unset edges? */ (a) ( r , s ) ← − arg max ( u,v ) / ∈ E ([ G in ] t u,v ) (b) E ← − E ∪ { ( r, s ) } () E ← − E ∪ { ( s, r ) } /* Stay symmetri */ M ← − M + 2 end end Algorithm 1 prop oses a w a y to onstrut scg( G, t, m ) . Note, that b eause of p ossible onuenes with same v alues, line (a) is not deterministi. F urthermore, there is no guaran tee that the strong onuene graph is unique, but the p ossible graphs an only dier b y their (few) edges of lo w est onuene. In pratie, onuenes are distint most of the time 2 RR n ° 6489 8 Bruno Gaume & F abien Mathieu Algorithm 2 : makesw , Making a small w orld Input : A target n um b er of no des for the output graph n ∈ N A target n um b er of edges for the random graph m in ∈ N A w alk length t ∈ N ∗ A target n um b er of edges m ∈ N Output : A graph G = ( V , E ) , with n no des and m edges b egin G in ← − a symmetri, reexiv e, Erdös-Rén yi Random Graph with n no des and m in edges G ← − scg ( G in , t, m ) G ← − largest onneted omp onen t of G end 5.2 Making Small-W orlds W e prop ose to onstrut graphs with a small-w orld struture b y extrating the onuenes of Erdös-Rén yi graphs, as desrib ed in Algorithm 2. Note, that the onuene extration ma y pro due disonneted graphs. Therefore w e ha v e to selet the main onneted omp onen t if w e w an t to study prop erties lik e diameter. Ho w ev er, our exp erimen ts sho w that the size of the main onneted omp onen t is alw a ys more than 80% , so this is not su h a big issue. 5.3 V alidation In order to obtain go o d small-w orlds, the v alues n , m in , m and t m ust b e arefully seleted. In the follo wing, w e set n = 1 000 , m in = 4000 , and m = 100 00 , and w e fo us on the imp ortane of the parameter t . Lik e stated in Setion 2, there is no strit denition of a small-w orld, but t ypial v alues for diameter, lustering and degree distribution. W e arbitrary prop ose to sa y that G = makesw( n, m in , t, m ) is small-w orld shap ed if it v eries:  m ≤ 10 n log( n ) (v eried for n = 100 0 , m = 100 00 ),  its lustering o eien t C G is greater than 10 m n 2 ,  its diameter is lo w er than 3 log( n ) ,  a least square tting on the degree log-log distribution giv es a negativ e slop e of absolute v alue λ greater than 1 , with a orrelation o eien t r 2 grater than 0 . 8 . Remark The p o w er la w estimation w e giv e is not v ery aurate (see for instane [ 27 ℄). Ho w ev er, giving a orret estimation of the o dds that a giv en disrete distribution is hea vy-tailed is a diult issue ([ 16 , 10 ℄), and rening the p o w er-la w estimation is b ey ond the sop e of this pap er. It is is easy to v erify that with those requiremen ts, a random Erdös-Rén yi graph with 1000 no des and 10000 edges is not a small w orld with high probabilit y (for instane b eause of the lustering o eien t). On the other hand, G = makesw( n, m in , t, m ) v eries small-w orld prop erties for some v alues of t , as sho wn in Figure4 :  The upp er urv e sho ws the diameter L (remem b er that w e only onsider the main onneted omp onen t, therefore the diameter is alw a ys w ell dened). The diameter is alw a ys lo w and onsisten t with a small- w orld struture.  The next urv es indiates the lustering o eien t C . F or 2 ≤ t ≤ 40 , C is v ery high. It drops after 40 , as the onuenes on v erge to the no des' degrees, meaning that most of the edges ome from the highest degree no des of the input graph. This leads to star-lik e strutures, that explain the p o or lustering o eien t.  The t w o next urv es indiates that the degree distribution ma y b e a p o w er-la w, with a relativ ely high ondene, for 28 ≤ t ≤ 50 .  Lastly , the lo w er urv e summarizes the v alues of t that v erify the small-w orld requiremen ts (mainly 28 ≤ t ≤ 40 ). 2 If uniqueness really matters, it sues to use a total order on the pairs of V in order to break ties in line (a) . INRIA F r om R andom Gr aph to Smal l W orld by W andering 9 2 4 L 0 0.2 0.4 C 0 1 2 3 λ 0.2 0.4 0.6 0.8 1 r 2 10 20 30 40 50 60 70 80 90 100 Small World? t Figure 4: Small-w orld prop erties of G = makesw( n, m in , t, m ) with resp et to t . 6 Conlusion W e prop osed in this artile a metho d to turn random graphs in to Small-W orld graphs b y the din t of random w alks. This simple and in tuitiv e metho d allo w to set a target n um b er of no des and edges. The resulting graphs p ossess all desired prop erties: lo w diameter, lo w edge densit y with a high lo al lustering, and a hea vy-tailed degree distribution. This metho d is suitable for generating random small-w orld graphs, but it is only a rst step for answ ering the question: why ar e most of r e al gr aphs smal l-worlds, despite the fat that the smal l-world strutur e is very unlikely among p ossible gr aphs? In order to b e eligible for explaining small-w orld eets, a small-w orld generator should b e based on lo al in terations. Therefore it should b e deen tralized, whi h is not the ase of Algorithm 2 . Ho w ev er, there exists v ariations of Algorithm 2 that an b e deen tralized: for instane, if w e in tro due a onuene b ound s , an algorithm where ea h no de u deide to onnet with an y no de it an nd with a m utual onuene greater than s has the same b eha vior that Algorithm 2 (but the n um b er of edges m is then indiretly set b y the parameter s ). Understanding the relationship b et w een m and s is part of our future w ork. Also note, that the random w alks w e used in this rst algorithm ma y b e to o long: for instane, Figure 4 sho ws that a length b et w een 28 and 40 is needed to a hiev e small-w orld prop erties for a 1000 no des graph, whi h is m u h larger than the exp eted diameter of a small-w orld graphs of that size. W e are urren tly w orking on a w a y to shorten the random w alks b y em b edding a preferen tial atta hmen t s heme [3℄ in to our algorithm. Referenes [1℄ L. A. A dami, B. A. Hub erman;, A. Barab'asi, R. Alb ert, H. Jeong, and G. Bianoni;. P o w er-la w distri- bution of the w orld wide w eb. Sien e , 287(5461):2115a+, Mar h 2000. RR n ° 6489 10 Bruno Gaume & F abien Mathieu [2℄ W. Aiello, F. Ch ung, and L. Lu. A random graph mo del for massiv e graphs. In STOC '00: Pr o  e e dings of the thirty-se  ond annual A CM symp osium on The ory of  omputing , pages 171180, New Y ork, NY, USA, 2000. A CM. [3℄ R. Alb ert and A.-L. Barabási. Statistial me hanis of omplex net w orks. R eviews of Mo dern Physis , 74:4797, 2002. [4℄ R. Alb ert, H. Jeong, and A. L. Barabasi. The diameter of the w orld wide w eb. Natur e , 401:130131, 1999. [5℄ A. Barabasi, H. Jeong, Z. Neda, E. Ra v asz, A. S h ub ert, and T. Visek. Ev olution of the so ial net w ork of sien ti ollab orations, April 2001. [6℄ A.-L. Barabási and R. Alb ert. Emergene of saling in random net w orks. Sien e , 286:509512, Otob er 1999. [7℄ G. Bianoni and A. L. Barabási. Bose-einstein ondensation in omplex net w orks. Phys R ev L ett , 86(24):56325635, June 2001. [8℄ B. Bollobás. R andom Gr aphs . Cam bridge Univ ersit y Press, 2001. [9℄ A. Bro der, R. Kumar, F. Maghoul, P . Ragha v an, S. Ra jagopalan, R. Stata, A. T omkins, and J. Wiener. Graph struture in the w eb. Comput. Networks , 33(1-6):309320, 2000. [10℄ A. Clauset, C. R. Shalizi, and M. E. J. Newman. P o w er-la w distributions in empirial data, Jun 2007. [11℄ S. Dorogo vtsev and J. Mendes. Ev olution of net w orks. A dvan es in Physis , 51 (4):10791187, 2002. [12℄ P . Erdos and A. Rén yi. On random graphs. Publi ationes Mathemti ae (Debr e  en) , 6:290297, 1959. [13℄ M. F aloutsos, P . F aloutsos, and C. F aloutsos. On p o w er-la w relationships of the in ternet top ology . In SIGCOMM , pages 251262, 1999. [14℄ R. F errer-i-Can ho and R. V. Sole. The small w orld of h uman language. Pr o  e e dings of The R oyal So iety of L ondon. Series B, Biolo gi al Sien es , 268(1482):2261226 5, No v em b er 2001. [15℄ B. Gaume. Balades aléatoire dans les p etits mondes lexiaux. Information Engine ering Sien es , 4(2), 2004. [16℄ M. L. Goldstein, S. A. Morris, and G. G. Y en. Problems with tting to the p o w er-la w distribution, August 2004. [17℄ R. Go vindan and H. T angm unarunkit. Heuristis for in ternet map diso v ery . In IEEE INF OCOM 2000 , pages 13711380, T el A viv, Israel, Mar h 2000. IEEE. [18℄ J.-L. Guillaume and M. Latap y . Complex net w ork metrology . Complex Systems , 16:8394, 2005. [19℄ J. Klein b erg. The Small-World Phenomenon: An Algorithmi Persp etiv e. In Pr o  e e dings of the 32nd A CM Symp osium on The ory of Computing , 2000. [20℄ R. Kumar, P . Ragha v an, S. Ra jagopalan, D. Siv akumar, A. T omkins, and E. Upfal. The w eb as a graph. In PODS , pages 110, 2000. [21℄ T. Luzak. Sparse random graphs with a giv en degree sequene. R andom Gr aphs , pages 165182, 1992. [22℄ M. Mollo y and B. Reed. A ritial p oin t for random graphs with a giv en degree sequene. R andom Strutur es and A lgorithms , pages 161179, 1995. [23℄ J. M. Mon to y a and R. V. Solé. Small w orld patterns in fo o d w ebs. J. The or. Biol. , 214:405412, F ebruary 2002. [24℄ S. J. Ne. P erronfrob enius theory: A new pro of of the basis, 1994. [25℄ M. E. J. Newman. Sien ti ollab oration net w orks: I. net w ork onstrution and fundamen tal results. Physi al R eview E , 64, 2001. INRIA F r om R andom Gr aph to Smal l W orld by W andering 11 [26℄ M. E. J. Newman. Assortativ e mixing in net w orks. Physi al R eview L etters , 89:208701, 2002. [27℄ M. E. J. Newman. P o w er la ws, pareto distributions and zipf 's la w, Deem b er 2004. [28℄ D. W atts and S. Strogatz. Colletiv e dynamis of small-w orld net w orks. Natur e , 393:440442, 1998. RR n ° 6489 Unité de recherche INRIA Rocquenco urt Domaine de V oluce au - Rocquencourt - BP 105 - 78153 Le Chesnay C edex (France) Unité de reche rche INRIA Futurs : Parc Club Orsay Uni versit é - ZA C des V ignes 4, rue Jacques Monod - 91893 ORSA Y Cedex (Franc e) Unité de reche rche INRIA Lorraine : LORIA, T echnopôle de Nanc y-Brabois - Campus scientifiq ue 615, rue du Jardin Botani que - BP 101 - 54602 V illers-lè s-Nancy Cedex (France ) Unité de reche rche INRIA Rennes : IRISA, Campus uni versitai re de Beauli eu - 35042 Rennes Cede x (France) Unité de reche rche INRIA Rhône-Alpes : 655, ave nue de l’Europe - 38334 Montbonno t Saint-Ismier (France) Unité de reche rche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France) Éditeur INRIA - Domaine de V oluceau - Rocquencourt , BP 105 - 78153 Le Chesnay Cede x (France) http://www.inria.fr ISSN 0249 -6399