Networks become navigable as nodes move and forget

Net w orks Become Na vigable as No des Mo v e and F orget Augustin Chain treau ∗ Pierre F raigniaud † Emman uelle Lebhar ‡ Abstract W e prop o se a dynamical process for netw ork evolution, aiming at explaining the emergence of the small world phenomenon, i.e., the sta- tistical observ ation that an y pair of individuals ar e link ed b y a short chain o f acquaintances computable by a simple decentralized routing algorithm, k nown a s greedy routing. Previous ly propos ed dynamical pro cesses enabled to demonstra te experimentally (b y sim ula tions) that the small w o rld phenomenon c an emerge from local dynamics. How- ever, the analys is of greedy ro uting us ing the probability distributio ns arising from these dynamics is q uite complex b ecause of mutual de- pendenc ie s . In con trast, our pro cess enables complete formal ana lysis. It is based on the combination of tw o simple pro cesses : a random walk pro cess, and an harmonic forgetting pro cess. Both pro ce sses reflect natural b ehaviors of the individuals, vie wed as no des in the netw ork of inter-individual acqua intances. W e prove that, in k -dimensional lat- tices, the combination of these t wo pro ces ses genera tes lo ng-range links m utually indep endently distributed as a k -harmonic distribution. W e analyze the p erformance s o f gre edy routing at the sta tionary regime of our pro ces s , a nd prov e that the exp ected num b er of s teps for rout- ing fro m any so urce to any target in any multidimensional lattice is a p olylog arithmic function of the distance b etw een the tw o no des in the lattice. Up to our knowledge, these results ar e the first formal pro of that na vig ability in small w orlds can emerge f rom a dynamical pro cess for net work ev o lution. Our dynamical process can find prac- tical applications to the design of spatial goss ip a nd resour c e lo cation proto cols. Keyw ords : Small world phenomenon, dynamical pro ce s s, routing, spatial gossip, resource lo ca tio n, r andom w alk s. ∗ Thomson, Paris. E mail: Augustin.Chaintreau@thomson.net . † CNRS and Univ ersity Paris 7. Email: Pierre.F raigniaud@liafa.juss ieu.fr . Ad- ditional supp orts from the ANR pro jects ALADDIN and ALP AGE, and from the COST Action 2 95 DYNAMO. ‡ CNRS and Univ ers it y Paris 7. Email: E mmanuelle.Lebhar@liafa.jussieu.fr . Ad- ditional supp orts from the ANR pro ject ALADDIN, and fr om the COST Actio n 295 D YNAMO. 1 1 In tro duction Mo dels relating geograph y and so cial-net work friendship enable a go o d un- derstanding of the small w orld phen omenon, a.k.a., six degrees of separation b et ween in dividuals [11, 29 ]. In these mo d els, the p robabilit y of b efriend ing a p articular p erson is assumed to b e in versely p rop ortional to the num b er of closer p eople, fitting with what w as observed e xp erimental ly (cf. [28]). Under this assump tion, it wa s pro ved that, usin g ad ho c probabilit y distri- butions, many classes of graphs are naviga b le, that is, a simple d ecen tralized routing p ro cedure enables efficien t routing from a n y source to an y ta r get. (By efficien t, w e mean, as it is standard in this framework, that routin g from any source s to any target t tak es a p olyloga rithmic expected n u mb er of steps). F or instance, suc h a na vigabilit y prop ert y is satisfied in m ulti- dimensional meshes [24], in graphs of b ound ed ball gro wth [13], and more generally in graphs of b ounded doubling dimension [34]. In all these cases, a graph G , that ma y not only represent geography but also other pro ximit y measures like p rofessional activities, religio us b eliefs, etc., is enh an ced with additional links c hosen at random. More precisely , ev ery no d e is giv en s ome long-r ange links p oin ting at other no des in the graph. F or eac h long-range link ad d ed at a no d e u , the p robabilit y th at the head of this lin k is v is in versely prop ortional to the size of the ball of radius dist G ( u, v ) cen tered at u in G , hence dep end ing on the densit y of G around u . Th is setting applies to w eighte d graphs to o [26], and to infin ite graphs as w ell [13]. F or instance, in the k -d im en sional lattice Z k , the probabilit y th at u has a long-range link p oint ing at v is essen tially prop ortional to 1 /d k where d is the distance b e- t wee n u an d v in the lattice . This setting of the long-range links enables greedy r outing 1 to p erform in p olylogarithmic exp ected num b er of steps (as a function of the d istance in the lattice b et wee n th e source and the target). 1.1 Na vigabilit y as an emerging prop erty In [25] (Problem 7), Jon Kleinber g asks ab out ”what kinds of gro w th p r o- cesses or selectiv e pressures migh t exist to cause n etw orks to b ecome more efficien tly searc h able”. Many at tempts ha ve been made to explain ho w the densit y-b ased d istr ibution of the long-range links can emerge with time fr om the ev olution of a net w ork. Inspired b y the w orld wide web or b y P2P file- sharing systems, all the mo d els we are aw are of hav e considered the augmen- tation pro cess (or rewiring) of a static graph used by its no des for searc h ing information. Ou r work uses a different app r oac h, starting from the follo win g 1 Greedy routin g [24] aims at mo deling the rout in g strategy p erformed by the indi- viduals in Milgram exp eriment. In a graph G enhanced with long-range links, a node u handling a message of destination t selects among all its neigh b ors, including its long-range conta ct(s), the one that is the closest to the target t according to the d istance in the base graph G , and forwards the message t o that node. 1 observ ations. One the one hand, an y on e of us can call or email an y p erson in the w orld. On the other hand, to do so, it is frequently the case that w e hav e met this p erson b efore. W e th u s s tart from the assumption that long-range connections are b et we en remote p eople who hav e met once in the past. In other words, long-range links are emerging f r om no des mobil- it y , that we mo del by random walks in this pap er. Another observ ation is that p eople forget some of their former acquainta nces along with time. Th is forgetting mec h anism repr esen ts the we ll un dersto o d fact that one cannot main tain close relationships with an exp losiv e num b er of p eople. Thus w e couple the random wal k pr o cess w ith a forgetting pr o cess, and prov e that this idealistic setting is s u fficien t to insure p olyloga rithmic na vigabilit y with simply o ne long-range connection per nod e. 1.2 Rewiring pro cesses Clauset a nd Moore [9] prop osed the follo wing rewiring pro cess for the m u l- tidimensional lattice , insp ired by the actions of s urfers on th e we b . While routing from a source s to a target t , if the target is n ot reac h ed after τ steps, then the long-range link of s is rewired t o p oin t at the cu rrent no de x . The threshold τ is set based on th e distance (in the lattice) b et ween s and t , and on the exp ected time of greedy routing from s to t wh en the k -d imensional lattice is augmen ted u s ing the k -harmonic distribution [24]. The sim ulation results presented in [9] sho w that the d istribution f of the link lengths con- v erges to the p o wer la w h ( d ) ∝ 1 /d k . Sandb erg and Clark e [32] prop osed a differen t rewiring pro cess, based o n F r eenet feedbac k mechanisms [8 ]. Th is iterativ e process s elects, at eac h p hase, t wo no des s and t u niformly at ran- dom, and constructs the greedy path s = x 0 , x 1 , . . . , x k − 1 , x k = t from s to t . F or every i ∈ { 0 , 1 , . . . , k } , the long-range link of x i is rewired with p robabil- it y p , to p oin t at t . The k + 1 decisions (rewiring or not) are tak en m utu ally indep end en tly . This p ro cess is analyzed in [31]. It is p ro ved that, under some hyp otheses, the pro cess con v erges. Moreo ver, the stationary distribu- tion f of the lin k lengths can b e fully c haracterized. In the k -dimensional lattice , it is close to the p ow er la w h ( d ) ∝ 1 /d k for an appr opriate p , and sim u lations show th at greedy r outing in r ings and mesh es enhanced using the stationary distribu tion f p erf orm s as efficien tly as when these net wo rks are en hanced u sing the 1- and 2- harmonic distributions, resp ectiv ely . F or b oth [9] and [32], the complete formal analysis of the pro cess remains op en (e v en the formal c h aracterizat ion of the stationary distribution o f the pro cesses describ ed in [9] remains op en). The difficu lty of the analysis is due to the d ep endencies b et ween the long-range lin ks generated b y the pro- cesses. In p articular, the computation of the greedy routing p erformances is a c hallenge when the long-rank links are not mutually ind ep endent. S o, building fu rther theory up on th ese tw o models lo oks quite difficult. In this p ap er, we prop ose a dynamical net work mo d el based on the com- 2 bination of t wo simple pro cesses: a random wa lk pro cess, and a harmonic forgetting pro cess.W e pr o ve th at the com bination of these t w o pro cesses gen- erates long-range links m u tually indep endently distributed as a distribution that resembles the densit y -b ased distribution, and from wh ic h n a vigabilit y pro v ably emerges. 1.3 Sk etch of our netw ork ev olution pro cess In our net w ork evo lution pro cess, called m o ve -and -forget, or m & f for short, individuals are mo deled by tok ens m o ving fr om no de to no d e in the k - dimensional lattice Z k , for some fixed inte ger k ≥ 1 (the dimension of the lattice ma y b e related to th e n umber of pr o ximit y criteria used by the indi- viduals f or routing). Initially , eac h no d e is o ccup ied by exactly one tok en . These tok ens are mo ved m utu ally indep end en tly du ring the execution of the dynamical process, ac cording to a random wa lk. T ok ens are attac hed to the heads of the long-range links, whose tails are the no d es from where the tok en s in itially started their random wa lks. Using the analogy of individuals m oving in the geographical w orld, eac h long- range link indicates an acquainta n ce b et ween an individual lo cated at a fixed geographical p oin t (where the token initia lly stoo d) and some individual lo cated at some geographical co ordinates (wher e the tok en currently s tands). The random walk p r o cess is coupled with anot her dynamic: no des m ay forget their con tacts through their long-range links. T he motiv ation for our forgetting pro cess is that individuals ma y lo ose con tact with former go o d friends, but they m eet new p eople among which some ma y b ecome close friends. Since older acquain tances in dicate stronger relationships, we assu m e that they ha ve less p r obabilit y to b e forgotten th an r ecen t ones. More precisely , a long-range link of age a , that is a long-range link that surviv ed a steps of th e forgetting pro cess, is forgotten with probabilit y φ ( a ) ∝ 1 /a . When a long-rang link is forgotten b y a node, it is rewired to p oin t at this no de (hen ce creating a self-loop ). The tok en at th e head of the forgotten link is remov ed, and a new tok en is launched at the no de. (A new lo cal relationship replace s an old remote relationship). Note that m & f is defined in dep end ently fr om the dimension k of th e lat- tice: tok ens execute rand om walks, and they are forgotten with a probabilit y that d ep ends only of their ages. 1.4 Our results W e p ro ve that, for any fixed integer k ≥ 1, the m & f rewirin g pr o cess sk etc hed abov e conv erges in the k -dimensional lattice to a distribution f of the link lengths that resembles the k -harmonic distrib ution. Precisely , w e prov e that th ere exists d 0 ≥ 0 and t wo p ositiv e constan ts c and c ′ , suc h that, for any d = ( d 1 , . . . , d k ) ∈ Z k with | d i | ≥ d 0 for all i ∈ { 1 , . . . , k } , we 3 Con verg ence Naviga bilit y A. Clauset an d C. Mo ore (2003 ) Simulati ons Simulati ons O. Sandb erg and I. Clarke (2007) Proof Simulati ons Mo ve-and-forget ( m & f ) Proof Proof T able 1: Prop erties of kno wn net w ork ev olution pro cesses compared to m & f ha ve c k d k k · ln 1+ ǫ k d k ≤ f ( d ) ≤ c ′ ln k / 2 k d k k d k k · ln 1+ ǫ k d k where ǫ > 0 is a fixed (arbitrary small) parameter of m & f , and k · k denotes the ℓ ∞ norm. Moreo v er, m & f guaran tees the m u tual indep endence of th e long-range links. As a consequence, the p erformances of greedy routing in the lattice enhanced usin g th e distrib u tion f can b e analyzed formally . W e p ro ve that the exp ected num b er of s teps of greedy routing f rom any source s to an y target t at distance d in the k -dimensional lattic e satisfies E [ X s,t ] ≤ O (ln 2+ ǫ d ) . Therefore, greedy routing p erforms p olylogarithmica lly as a fu nction of the distance b et wee n the sour ce and the target. I n particular, the p erformances of greedy routing are essentia lly the same as the ones obtained by Klein- b erg [2 4] u sing the ad ho c k -harm onic d istribution [2 4]. Up to our knowledge, these results are the first f ormal p ro of that n a v- igabilit y in small worlds can emerge from a dynamical pro cess for net work ev olution (see T able 1). Moreo v er, m & f is simple (b y just coupling tw o simple dynamics), natur ally distribu ted (eac h no d e tak es care of ju s t its to- k en), r ob u st (the loss of one tok en simp ly requires to launc h a new tok en), and scalable (by dir ect adaptations of the infinite lattice setting to square toroidal meshes of arb itrary sizes). Last bu t not least, m & f can find p ractical applications, including the design of distributed s p atial g ossip and resource location protocols. 1.5 Related w orks The searc h for a netw ork ev olution pr o cess that could explain the emergence of the small wo r ld phenomenon in so cial netw orks started with the pioneer- ing w ork of W atts and Strogatz [35] who prop osed a rewiring pro cess in the cycle, generating net works p ossessing small diameter and large clustering co efficien t, simulta neously . Ad ding rand om matc hin gs to cycles, as in [5], yields graph s with small diameter, but non necessarily with small clus tering co efficien t. As f ar as naviga b ilit y is concerned, these net works do not s up- p ort efficien t decen tralized routing mechanisms [24]. Alb ert and Barab´ asi [2] pro du ced a thorough inv estigatio n of the preferentia l attac hment mo del [33]. Although the pr eferen tial attac hment mo del enables the design of efficien t 4 searc h pro cedur es under sp ecific circumstances (see [16] and the references therein), the recen t lo wer b ounds in [1 2] sho w that p olylogarithmic routing cannot b e ac h ieved in general in net w orks generated according to th is mo del. Recen tly , Lib en-No w ell and Klein b erg [27] tried to infer wh ic h inte r actions in so cial net w ork s are lik ely to occur in the near fu ture from the observ ati on of the existing ones, b ut navig abilit y is not of their concern. Actually , as far as we kno w, th e only netw ork evolutio n mo dels fr om which p olylogarith- mic routing emerges are the aforemen tioned ones [9, 32], wh ic h we a lready discussed. F ollo wing up the semin al w ork of Klein b erg [24], a large literature has b een d edicated to the analysis of greedy routing in graph s enhanced b y long- range links set according to v arious kinds of probabilit y distribu tions (see, e.g., [1, 13, 14, 15 , 34]). These p ap ers prov ed that seve ral large classes of graphs can b e enhanced by long-range links so that greedy routing p erforms in p olylogarithmic exp ected num b er of steps. A lo wer b oun d of Ω ( n 1 / √ log n ) exp ected num b er of steps for greedy routing in arbitrary graph s has b een pro ved in [18], and an upp er b ound of O ( n 1 / 3 ) has b een pro v ed in [17]. Lo wer b ounds for the cycle can b e found in [3, 4, 19]. 2 The Mo ve -and-F orget ( m & f ) Rewiring P ro cess 2.1 Pro cess description 2.1.1 Random w alks Let k ≥ 1 b e an in teger. Th e rewiring pr o cess mo ve-a nd-forget ( m & f for short) assu mes th at eac h no d e in the k -dimensional lattice Z k is in itially o ccupied by exactly one tok en . These tok ens mo v e m utually indep endentl y according to r andom w alks. That is, eac h toke n is giv en a set of k fair coins c i , i = 1 , . . . , k . At eac h step of its walk, eac h token flips its k coins, and mov es in the i th dimens ion of the lattice in the p ositiv e direction if c i is head, and in the nega tiv e d irection if it is tail . More precisely , let X ( t ) ∈ Z k denotes the p osition of a tok en in the lattice after t steps of m & f , assuming that th e tok en initially started at no de ( u 1 , . . . , u k ) ∈ Z k . W e ha v e X (0) = ( u 1 , . . . , u k ), and , for t ≥ 1, X ( t ) = ( X 1 ( t ) , . . . , X k ( t )) satisfies X i ( t ) =  X i ( t − 1) + 1 with p robabilit y 1 / 2; X i ( t − 1) − 1 with p robabilit y 1 / 2. (1) 2.1.2 Setting of the long-range links T ok ens are attac hed to the heads of the long-range links, whose tails are the no des from where the tok en s in itially started their r andom w alks (see Figure 1 (a)). The head of a long-range link is called the long-range c ontact 5 Token at node v Long-range link of node u Token trajectory Node u Forgotten token Rewired long- range link (a) (b) New token Figure 1 : Dynamic of the long-range links in m & f . of the tail of this link. Hence the long-range conta ct of a no de u is the no de v cur r en tly occupied b y the tok en launc h ed b y node u . 2.1.3 F orgetting pro cess No des ma y forget th eir cont acts thr ough their long-range links. More pr e- cisely , a long- range link of age a ≥ 0, that is a long-range link that surviv ed a steps of the forgetting pro cess, is f orgotten with prob ab ility φ ( a ). When a long-range link is forgo tten b y a no de, it is rewired to point at this no d e (see Figure 1(b)). The tok en at the h ead of the forgotten link is remo v ed, and a new tok en is launched at th e no de. This new tok en starts another random wa lk in Z k . Hence, if A ( t ) ∈ N denotes the age of the long-range link of some n o de u , that is the n u m b er of steps b et ween time t and the last time this lin k w as rewired during th e executio n of m & f , and if C ( t ) denotes the long-range con tact of no d e u at step t , then w e h a ve C ( t ) = X ( A ( t )). The forgetting function φ has a h u ge impact on the d istribution of the long-range link lengths. In this pap er, we w ill consider φ ( a ) ∝ 1 /a . The precise setting of φ will app ear more complex for tec hnical reasons only 2 (series conv ergence f or infin ite lattice s, normalization, etc.). In fact, its b ehavi or essent ially r eflects a decreasing of th e forgetting p robabilit y that is in v ersely prop ortional to the age of the relationships. The precise setting of φ is d escrib ed in the n ext section whic h explains the connections b et ween the random w alk X , the forgetting function φ , and th e distribution f of the long-range link lengths. 2.2 Setting of the forgett ing function W e fir st pr ov e that the age of the long-range link resulting fr om the execution of m & f at a n o de has a stati onary distribution (the p ro of of this lemma can 2 F or instance, one needs P a ≥ 0 φ ( a ) t o div erge since otherwise the Mark ov chain A ( t ) w ou ld b e transien t, and link s could surviv e infinitely with p ositive probability . How ever, on th e one han d , j ust setting φ ( a ) = 1 /a w ould make A ( t ) recurrent null (and th u s for any a we w ould hav e Pr { A ( t ) = a } con verg ing to 0 as t go es to infinity), but, on the other hand, setting φ ( a ) = 1 /a α with α < 1 w ould not yield navigabili ty . 6 b e found in Ap p endix A). Lemma 1 F or any function φ in [0 , 1) such that the series o f gener al term Π j i =1 (1 − φ ( i )) is finite, ( A ( t )) t ≥ 0 is a M arkov chain which is irr e ducible, ap e rio dic, and p ositive r e curr ent, with stationary pr ob ability distribution π wher e π ( a ) = Π a i =1 (1 − φ ( i )) P j ≥ 0 Π j i =1 (1 − φ ( i )) , for al l a ≥ 0 . Definition 1 We define the for getting pr ob ability φ as the fol lowing func- tion: φ ( a ) = ( 0 if a = 0 , 1 , or 2 ; 1 − a − 1 a  ln( a − 1) ln a  1+ ǫ if a ≥ 3 ; (2) wher e ǫ > 0 is arbitr ary smal l. Note t hat φ ( a ) = 1 a + o  1 a  . Ind eed,  ln( a − 1) ln a  1+ ǫ =  1 + ln(1 − 1 /a ) ln a  1+ ǫ = 1 − 1 + ǫ a ln a + o  1 a ln a  If φ is defined according to Eq. (2), then L emm a 1 enables to giv e a close form u la for π (the proof of this le mma ca n b e found in App endix B). Lemma 2 If φ is define d ac c or ding to Eq. 2, then ther e exists a c onstant c > 0 such that π (0) = π (1) = π (2) = c and for any a ≥ 3 , π ( a ) = c a ln 1+ ǫ a . Finally , the r elationship b et wee n th e statio n ary distribution of th e long- range link ages and the stationary distribution of the long-range link lengths is made explicit in the follo w ing lemma (see pro of in App endix C). Lemma 3 The distribution of the long-r ange links c onver ges to the distri- bution f satisfying, for any d ∈ Z k , f ( d ) = X a ≥ 0 π ( a ) · Pr { X ( a ) = d } . 3 Analysis of the dynamical pro cess m & f In this section, we analyze the stationary distribu tion of the long-range link lengths in the k -dimensional lattice, and prov e that this distrib ution resem b les th e k -h arm onic d istribution. 7 Theorem 1 Ther e exist d 0 ≥ 0 and two p ositive c onstants c and c ′ such that, for any d = ( d 1 , . . . , d k ) ∈ Z k with | d i | ≥ d 0 for al l i ∈ { 1 , . . . , k } , we have c k d k k · ln 1+ ǫ k d k ≤ f ( d ) ≤ c ′ ln k / 2 k d k k d k k · ln 1+ ǫ k d k wher e ǫ > 0 is the fixe d p ar ameter of m & f , and k · k denotes the ℓ ∞ norm. T o p r o ve the theorem, w e first prov e that, for large distances d , a r andom w alk of age a cannot b e of length d unless a ≥ Ω( d 2 ). More pr ecisely , we establish an exp onent ially small upp er b ound for the p robabilit y for a long- range link to b e of length d at a ge a = o ( d 2 ). Second, we pro v e th at if the age a is suffi ciently large, then th e chance for a random walk to reac h a given distance d at age a is prop ortional to 1 √ a . Summing th is probability o ve r all v alues of a larger than d 2 allo ws us to conclude that the transform of the age d istribution π describ ed in Lemma 3 is approac hing the k -h arm onic distribution. Let u s establish some b asic prop erties satisfied b y rand om w alks in di- mension 1. W e will extensiv ely use the follo wing Chernoff b ound . Let T b e a s u m of Bernouilli v ariables, with exp ectation µ . T hen [30]: Pr {| T − µ | > t } ≤ 2 exp( − t 2 4 µ ) for any t ≤ µ . (3) The follo w in g lemma sp ecifies what m u st b e the min im um order of mag- nitude for a in order to contribute significan tly to the sum defi n ing f in Lemma 3. Lemma 4 L et X b e a r andom wa lk in Z . Then, for any age a > 0 a nd any distanc e d ∈ Z , we have Pr { X ( a ) = d } ≤ 2 · exp  − d 2 32 · a  . Due to lac k of space, the pr o of of th e lemma is omitted (it can b e found in Ap p endix D ). W e no w compute an estimation of Pr { X ( a ) = d } when a is sufficien tly large. W e will use the follo wing asymp totic equiv alent of the binomial co- efficien t, th at can b e deriv ed b y application of the Stirling formula. Let n i and m i b e t wo sequences of p ositiv e in tegers such that n i → ∞ , m i → ∞ , and n i − m i → ∞ wh en i gro ws to infinity . Then  n i m i  ∼ 1 √ 2 π · r n i m i · ( n i − m i ) · n n i i m m i i · ( n i − m i ) n i − m i . (4) Lemma 5 L et X b e a r andom walk in Z . F or any ζ > 0 , ther e e xi sts d 0 > 1 such that, for any | d | ≥ d 0 and a ≥ d 2 64 · ln | d | , we have (1 − ζ ) · r 2 π · a exp  − 3 d 2 4 a  ≤ Pr { X ( a ) = d } ≤ (1+ ζ ) · r 2 π · a exp  − d 2 4 a  . 8 Due to lac k of space, the pr o of of the lemma is omitted (it can b e found in App end ix E). W e are no w ready to pro ve of the lo wer b ound of Th eorem 1. F or the sake of simplicit y , let us first assume that the d imension of the lattice is 1. In this ca se, one can apply th e results from the p revious sect ion directly . F or an y a ≥ 3 4 d 2 w e ha ve exp  − 3 d 2 4 a  ≥ 1 /e . Therefore, for any ζ > 0, there exists d 0 large enough and a ≥ 3 4 d 2 suc h that Lemma 5 yields: Pr { X ( a ) = d } ≥ 1 − ζ e r 2 π 1 √ a . Th us : f ( d ) = X a ≥ 0 Pr { X ( a ) = d } π ( a ) ≥ 1 − ζ e r 2 π X a ≥ 3 4 d 2 1 a 3 / 2 · ln 1+ ǫ ( a ) . More generally , in the k -dimensional lattice, let us denote the p osition of the random w alk by X ( a ) = ( X 1 ( a ) , · · · , X k ( a )). F rom the setting of m & f , eac h X i is an unbiased r an d om w alk in dimens ion 1, and the X i s are m utually indep end en t. W e can thus apply all the results from the previous section indep end en tly f or eac h coord inate of d = ( d 1 , . . . , d k ). Assumin g that | d i | ≥ d 0 for all i ∈ { 1 , . . . , k } , w e can apply Lemma 5 to ev ery dimension. W e get: a ≥ 3 4 k d k 2 = ⇒ ∀ i ∈ { 1 , . . . , k } , Pr { X i ( a ) = d i } ≥ 1 − ζ e r 2 π 1 √ a exp  − 3 d 2 i 4 a  . F or a ≥ 3 4 k d k 2 , we ha v e, 3 4 d 2 i a ≤ 3 4 k d k 2 a ≤ 1 and th us exp  − 3 d 2 i 4 a  ≥ 1 /e. As a consequence, by Lemma 5, Pr { X ( a ) = d } = Pr { X 1 ( a ) = d 1 , . . . , X k ( a ) = d k } ≥ 1 − ζ e r 2 π 1 √ a ! k hence f ( d ) = X a ≥ 0 Pr { X ( a ) = d } π A ( a ) ≥ 1 − ζ e r 2 π ! k X a ≥ 3 4 k d k 2 c a 1+( k/ 2) · ln 1+ ǫ ( a ) . The lo wer b oun d is then a d ir ect consequence of th e follo wing result with N = 3 k d k 2 4 (the p ro of of this lemma can b e foun d in App endix F). 9 Lemma 6 F or any ǫ > 0 , and any N ≥ e 2(1+ ǫ ) , we have 2 / ( k + 1) N k / 2 ln 1+ ǫ N ≤ X a ≥ N 1 a 1+( k/ 2) ln 1+ ǫ a ≤ 2 /k ( A − 1) k / 2 ln 1+ ǫ ( N − 1) . (5) Due to lac k of space, the p r o of of the upp er b ound of Theorem 1 is omitted (it can b e foun d in App endix G). 4 Applications In the previous section, we ha v e sh o wn that the distribution f of the long- range link lengths is pr ov ably con v erging to a distribu tion that resembles the k -harmonic distribution. In this section, we show that greedy routing can b e formally analyzed at the stationary state of this distrib ution. Greedy routing can b e form ally analyzed f or t wo reasons: (1) Th e distribu tion f of the long-range links constru cted by m & f can b e b ounded formally (cf. Theorem 1); (2) T he long-range links resulting fr om m & f are m utually in- dep end ent. Based on these t wo facts, we can establish th e th eorem b elo w (see p r o of in App endix H). Theorem 2 In the k -dimensional lattic e augmente d with the long-r ange links at the statio nary distribution of the dynamic al pr o c ess m & f , the ex- p e cte d numb er of steps of gr e e dy r outing fr om any sour c e no de s to any tar g et no de t at distanc e d is O (ln 2+ ǫ d ) . In the rest of the section, w e discuss ho w m & f can find p ractical appli- cations to the design of sp atial gossip and resource lo cation protocols. Gossip-based p roto cols, a.k.a., epidemic algorithms [10], h a ve b een in- tro duced as a metho dology for designing robust and scalable comm unication sc hemes in distributed systems. Roughly , in eac h step, eac h node u c ho oses some other no de v , and sends a message to it. By applyin g suc h sc heme at eac h node, an informatio n originated at some sou r ce s will ev en tually reac h its target(s). This metho d ology can b e adapted to v arious problems, in- cluding information spreading, resour ce lo cation, etc. In [22], Kemp e et al. in tro duced sp atial g ossip , whic h allo wed them to derive efficient solutions for many communicat ion p roblems. In spatial gossip, n o des are arr anged with unif orm densit y in the k -dimensional Euclidean space, and, at eac h step of the gossip proto col, no de u c ho oses no d e v with probabilit y ∝ 1 /d k where  > 0 is a fixed parameter, and d is the distance b et ween u and v . In particular, it is sh o wn that, for  ∈ (1 , 2), spatial gossip enables to p r opa- gate information at distance d in time p olylogarithmic in d . In [23], Kemp e and Klein b erg sho wed that sp atial gossip enables to solv e larger classes of problems, including MST co nstruction and p erm u tation routin g. In partic- ular, th ey pro ve that p ermutation routing using spatial gossip w ith  = 1 p erforms in p olylogarithmic exp ected num b er o f steps. 10 W e sketc h ho w the m & f p ro cess could facilitate the implemen tation of the protocols in [22, 23] for net w orks that tak e adv an tage of no d e mobilit y , as in, e.g., [7, 20, 21]. F or instance, assu me a net wo r k comp osed of a set X of fixed n o des and a set Y of mo v in g no d es. Every x ∈ X connects to the, sa y , k X closer n eigh b ors in X , and to the, sa y , k Y no des y ∈ Y that are currentl y the closest to x . No de x k eeps all these y ’s as temp orary neigh b ors, and it regularly c h ec ks whether these conn ections must b e preserve d. F or that purp ose, no de x regularly flip s biased coins (one for eac h neigh b or y ), and decides whether it should k eep a neigh b or or n ot according to the resu lt of this trial. (Th e bias of the coin is a function φ of the age of the connection). If x decides to forget some y , then x simply replaces y by the n o de y ′ ∈ Y that is currently the closest to x . An so on. Assuming that the moving no des p erform random w alks, and th at all no des are arranged with uniform densit y in the k -dimensional Eu clidean space, Theorem 1 insu res that the distances betw een a no de and its mo ving neigh b ors are roughly distributed according to a k -harmonic distribution. Hence, ev ery fixed no de can mimic spatial gossip for  = 1 by choosing u.a.r. one if its m oving neigh b ors at eac h step. Measuring the impact of the parameters k X and k Y on the p erformances of spatial gossip for  = 1, as w ell as setting up a forgetti ng function φ enabling to implemen t s p atial gossip protocols for  6 = 1 are b ey ond the scop e of this p ap er, b ut are curren tly under our in v estigati on. References [1] I. Abr ah am and C. Gav oille. 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Distance estimatio n and ob ject lo cation via r ings of neigh- b ors. In 24th Annual ACM Sym p osium on P r inciples of Distributed Computing (PODC), pages 41–50, 2005 . [35] D. W atts, and S . S trogatz. Collectiv e Dynamics of S mall-W orld Net- w orks . Nat ure 393 :440-4 42, 1998. 14 APPENDIX A Pro of of Lemma 1 F or all j ≥ 0, w e h a ve: Pr { A ( t + 1) = j | A ( t ) = i } =    1 − φ ( j ) if j = i + 1 φ ( i + 1) if j = 0 0 otherwise . The Mark ov c hain ( A ( t )) t ≥ 0 is irredu cible b ecause any state i ≥ 0 can b e reac hed from an y state j ≥ 0. Also, A is clearly ap erio dic. Let us define the function π as foll o ws. F or a n y a ≥ 0, π ( a ) = Π a i =1 (1 − φ ( i )) P j ≥ 0 Π j i =1 (1 − φ ( i )) with the con ven tion t hat the pro d uct Π 0 i =1 (1 − φ ( i )) equals 1. The function π is w ell defined for all a ≥ 0 b y hypothesis on φ . C learly , P a ≥ 0 π ( a ) = 1. W e no w c heck that π is a stationary distribution. F or all a > 0, we ha ve X i ≥ 0 π ( i ) Pr { A ( t + 1) = a | A ( t ) = i } = π ( a − 1) · (1 − φ ( a − 1)) = π ( a ) , and X i ≥ 0 π ( i ) Pr { A ( t +1) = 0 | A ( t ) = i } = π (0) φ (1)+ π (0) X i ≥ 1 φ ( i +1) Π i j =1 (1 − φ ( j )) . Let B ( i ) = Π i j =1 (1 − φ ( j )) for i > 0. W e ha ve 1 − φ ( i + 1) = B ( i + 1) /B ( i ), therefore φ ( i + 1) B ( i ) = B ( i ) − B ( i + 1). Hence w e get X i ≥ 0 π ( i ) Pr { A ( t + 1) = 0 | A ( t ) = i } = π (0)  φ (1) + X i ≥ 1 ( B ( i ) − B ( i + 1))  = π (0) ( φ (1) − B (1)) = π (0) . Therefore, π is a stationary d istribution for A , and, since A is irr educible and ap erio dic, it is unique. Therefore, A is r ecurrent p ositiv e (se e Theorem 3.1, p. 104 in [6]). B Pro of of Lemma 2 Let B ( j ) = Π j i =1 (1 − φ ( i )). W e ha ve B ( j ) = 1 for j = 0 , 1 , 2, and B ( j ) = 2 ln 1+ ǫ 2 j ln 1+ ǫ j 15 for j ≥ 3. Therefore, the series of general term B ( j ) is fin ite sin ce ǫ > 0, and φ satisfies the conditions of Lemma 1. Precisely , w e ha v e X j ≥ 0 B ( j ) = 3 + X j ≥ 3 2 ln 1+ ǫ 2 j ln 1+ ǫ j ≤ 3 + 2 ln 2 ǫ < ∞ . Since π ( a ) = B ( a ) / P j ≥ 0 B ( j ), the result foll o ws. C Pro of of Lemma 3 F or an y time t ≥ 0 and an y d , we ha ve: Pr { C ( t ) = d } = P r { X ( A ( t )) = d } = X a ≥ 0 Pr { X ( a ) = d and A ( t ) = a } = X a ≥ 0 Pr { X ( a ) = d } Pr { A ( t ) = a } , since the Mark o v c h ain A is indep end en t of the p osition of the tok en. More- o ve r, since A is recurrent p ositiv e (Lemma 1), A ( t ) conv erges in v ariation to π when t gro ws to infin it y , th at is: P a ≥ 0 | Pr { A ( t ) = a } − π ( a ) | tends to 0 as t gro ws to infi n it y (cf. Theorem 2.1, p. 130 in [6]). Therefore, Pr { A ( t ) = a } can b e replace d b y π ( a ) in the ab o ve equali t y wh en t grows to infin it y . Fi- nally Pr { C ( t ) = d } is indep endent of t and its stationary distribution is f ( d ). D Pro of of Lemma 4 First, note th at the r esult is straig h tforward if a < | d | sin ce th e random w alk cannot b e at d istance d in less than | d | time steps. Thus w e can assume a ≥ | d | in the rest of the pro of. Similarly , w e can assume d 6 = 0 since the lemma trivially h olds for d = 0. Let { Y i , i ≥ 1 } b e a coll ection of i.i.d. Bernouilli v ariables that tak e v alue 1 with probab ility 1 / 2 . Let T b e defined b y T ( a ) = Y 1 + · · · + Y a . (6) Th us we get th at X ( a ) and 2 T ( a ) − a ha ve the same d istribution. No w, E [ X ( a )] = 0 for an y a ≥ 0. Thus, for an y d 6 = 0, Pr { X ( a ) = d } ≤ Pr {| X ( a ) − E [ X ( a )] | > | d | / 2 } ≤ Pr { | T ( a ) − E [ T ( a )] | > | d | / 4 } . The rand om v ariable T ( a ) is the sum of a Bernouilli v ariables with exp ec- tation 1 / 2. Thus it has exp ectation a/ 2, and sin ce | d | / 4 is less than this exp ectation, the Chern off b ound of Eq. (3) implies the result. 16 E Pro of of Lemma 5 Assume, w.l.o.g., that d > 0. Fix ζ > 0. According to th e definition of a random walk in Z we ha ve Pr { X ( a ) = d } = 1 2 a  a ( a + d ) / 2  . Let us rewrite ( a + d ) / 2 = ( a/ 2) · (1 + ρ ) and a − ( a + d ) / 2 = ( a/ 2) · (1 − ρ ), where ρ = d/a . According to Eq. (4) w e get that, for an y ζ ′ > 0 with ζ ′ < ζ , there exists d 0 large enough suc h that for | d | ≥ d 0 and a ≥ d 2 64 · ln | d | , w e ha v e: Pr { X ( a ) = d } ≤ r 2 π · a (1 + ζ ′ ) p (1 − ρ 2 )  1 (1 + ρ ) (1+ ρ ) · (1 − ρ ) (1 − ρ )  a 2 . (7) On the other h and, for an y x ∈ ( − 1 , 1) we ha ve ((1 + x ) (1+ x ) · (1 − x ) (1 − x ) ) − 1 = exp ( − (1 + x ) ln(1 + x ) − (1 − x ) ln(1 − x )) . As x approac hes zero w e ha ve (1 + x ) ln (1 + x ) = x + x 2 2 + o ( x 2 ) , and thus (1 + x ) ln(1 + x ) + (1 − x ) ln(1 − x ) = x 2 + o ( x 2 ) . This la tter expression can b e rewr itten as: for any ν > 0 there exists η > 0 suc h that: | x | < η = ⇒ exp  − (1 + ν ) x 2  ≤ 1 (1 + x ) 1+ x (1 − x ) 1 − x ≤ exp  − (1 − ν ) x 2  . Since ρ = d a ≤ 64 ln d d b ecomes arbitrarily close to zero for large v alues of d , one can c hose d 0 large enou gh so that Eq. (7) holds if one replaces th e v alue inside the brac ket b y the ab o ve upp er b ound, with ν = 1 / 2. Hence we get that, for d ≥ d 0 and a ≥ d 2 64 ln d , Pr { X ( a ) = d } ≤ (1 + ζ ′ ) r 2 π · a 1 p 1 − ρ 2 exp  − ρ 2 a/ 4  . Once again, since ρ is arbitrarily clo se to zero for large d , w e can choose d 0 large enou gh so that (1 + ζ ′ ) / p 1 − ρ 2 ≤ 1 + ζ . The upp er b ound in the statemen t of t he lemma follo ws. Only equiv alen t f orms h a ve b een us ed to establish the upp er b ound in the statemen t of the lemma. Thus w e can pro ve the lo w er b ound b y applying exactly th e s ame argumen ts. 17 F Pro of of Lemma 6 Let g : x 7→ − 1 / ( x k / 2 ln 1+ ǫ x ). The deriv ativ e of this function satisfies g ′ ( x ) = k 2 x − k / 2 − 1 (ln − (1+ ǫ ) x ) + (1 + ǫ ) x − k / 2 − 1 (ln − (2+ ǫ ) x ) = 1 x k / 2+1 ln 1+ ǫ x  k / 2 + 1 + ǫ ln x  . Therefore k 2 1 x k / 2 ln 1+ ǫ x ≤ g ′ ( x ) ≤ k + 1 2 1 x k / 2 ln 1+ ǫ x if x ≥ e 2(1+ ǫ ) . As a consequence, 2 k + 1 g ′ ( x ) ≤ 1 x k / 2 ln 1+ ǫ x ≤ 2 k g ′ ( x ) and − 2 g ( x ) k + 1 ≤ Z ∞ x 1 u k / 2 ln 1+ ǫ u du ≤ − 2 k g ( x ) . Eq. (5) follo ws directly fr om this latt er inequalit y . G Pro of of the u pp er b ound of Theorem 1 Again, let u s first consider the simple case of d imension 1. Let d > 1. In this conte xt, whenev er a < d 2 / (64 ln d ), w e get by Lemma 4 that Pr { X ( a ) = d } ≤ 2 · exp ( − 2 ln ( d )) ≤ 2 d 2 . More generally , let u s denote b y i 0 the d imension th at yields the infinity norm o f d (i.e., suc h th at | d i 0 | = k d k ). By app lyin g Lemma 4, we get that if a ≤ k d k 2 64 l n k d k then Pr { X 1 ( a ) = d 1 , · · · , X k ( a ) = d k } ≤ Pr { X i 0 ( a ) = d i 0 } ≤ 2 /d 2 i 0 = 2 / k d k 2 . F or any ζ > 0, there exists d 0 > 0 suc h that if d i ≥ d 0 for all i = 1 , . . . , k , then we can apply Lemma 5 separately for eac h dimension. If a ≥ k d k 2 / (64 ln k d k ), then ∀ i ∈ { 1 , . . . , k } , Pr { X i ( a ) = d i } ≤ (1 + ζ ) · r 2 π · a . 18 Th us, sin ce, for a fi xed a , the random v ariables X i ( a ) are mutually indep en- den t, we get Pr { X 1 ( a ) = d 1 , . . . , X k ( a ) = d k } ≤ (1 + ζ ) · r 2 π ! k 1 a k / 2 . As a consequence, f ( d ) = X a< k d k 2 (64 ln k d k ) Pr { X ( a ) = d } π ( a ) + X a ≥ k d k 2 (64 ln k d k ) Pr { X ( a ) = d } π ( a ) ≤ 2 k d k 2 + (1 + ζ ) r 2 π ! k X a ≥ k d k 2 (64 ln k d k ) c a 1+( k/ 2) ln 1+ ǫ ( a ) . One can then complete the p r o of by using Eq. (5) with N = k d k 2 64 ln k d k . H Pro of of Theorem 2 Let s ∈ Z k b e a source n o de, and t ∈ Z k b e a target no de. Assu me that the distance b et ween s and t in the lattice Z k is dist( s , t ) = d , where dist denotes the ℓ 1 distance in Z k . W e compute the exp ected num b er of steps greedy routing tak es b efore red ucing the d istance to the target by a f actor 2. Let u = ( u 1 , . . . , u k ) ∈ Z k b e the cu rrent no de reac hed by greedy routing, and let B = { v ∈ Z k : dist( v , t ) ≤ dist( u , t ) / 2 } . The probabilit y Pr( u → B ) that u has its long-range link p ointing to a no de in B satisfies Pr { u → B } = X v ∈ B Pr { u → v } . W e pro ve a lo w er b ound on this prob ab ility . Let δ = d ist( u , t ). Let S = { x = ( x 1 , . . . , x k ) , x i ∈ {− 1 , 0 , +1 } for i = 1 , . . . , k } . F or c ∈ Z k and r ≥ 0, let B ( c , r ) denotes the ball of radius r cen tered at c , that is B ( c , r ) = { v ∈ Z k : dist( c , v ) ≤ r } , and, f or x ∈ S , define B x = B ( t + 2 δ 6 k x , δ 6 k ) . W e ha ve B x ⊆ B = B ( t , δ / 2) for an y x ∈ S . Moreo ve r, one can easily sho w that there exists x ∈ S suc h that for an y v = ( v 1 , . . . , v k ) ∈ B x and an y i ∈ { 1 , . . . , k } , w e ha ve | u i − v i | ≥ δ / (6 k ). Pr { u → B } ≥ X v ∈ B x Pr { u → v } ≥ | B x | · min v ∈ B x Pr { u → v } . 19 If δ ≥ 6 kd 0 then | u i − v i | ≥ d 0 for all i , a nd, b y Theorem 1, w e ge t that for an y v ∈ B x , Pr { u → v } ≥ c k u − v k k · ln 1+ ǫ k u − v k . where k · k denotes the ℓ ∞ norm. Since k u − v k≤ dist( u , v ), w e get that Pr { u → v } ≥ c dist( u , v ) k · ln 1+ ǫ dist( u , v ) . No w, for an y v ∈ B x , we h av e v ∈ B and thus dist( u , v ) ≤ 3 δ/ 2. Therefore, Pr { u → v } ≥ c ( 3 δ 2 ) k ln 1+ ǫ ( 3 δ 2 ) . Since | B x | ≥ Ω   δ k  k  , we get that Pr { u → B } ≥ Pr( u → B x ) ≥ Ω  1 ln 1+ ǫ δ  ≥ Ω  1 ln 1+ ǫ d  . As a consequence, at ev ery intermediate no de u of greedy r outing from s to t , if dist( u , t ) ≥ 6 k d 0 then th e probabilit y of halving the distance to the target at the next step is at least Ω( 1 ln 1+ ǫ d ). S ince all the long-range links resulting from m & f are m utually indep endent, w e get that the exp ected n u m b er of steps for halving the distance to the target is O (ln 1+ ǫ d )). By linearit y of the exp ectation, w e get that the tot al exp ected num b er of steps for routing from s to a no de at distance at most 6 k d 0 from th e target t is at most ⌈ log 2 d ⌉ X i = ⌈ log 2 6 kd 0 ⌉ E [halving the distance δ fr om 2 i +1 to 2 i ] ≤ ⌈ log 2 d ⌉ X i = ⌈ log 2 6 kd 0 ⌉ O (ln 1+ ǫ (2 i +1 )) ≤ O (ln 2+ ǫ d ) . Once at distance less than 6 k d 0 to the target, greedy routing completes in O (1) steps, thus the tota l exp ected num b er of steps of greedy routing from s to t is O (ln 2+ ǫ d ). 20