Growing Networks Through Random Walks Without Restarts

Transcription

1 Growing Network Through Random Walk Without Retart Bernardo Amorim, Daniel Figueiredo, Giulio Iacobelli, Giovanni Neglia To cite thi verion: Bernardo Amorim, Daniel Figueiredo, Giulio Iacobelli, Giovanni Neglia. Growing Network Through Random Walk Without Retart. Proceeding of the 7th Workhop on Complex Network (CompleNet 2016), Mar 2016, Dijon, France. <hal > HAL Id: hal Submitted on 13 Dec 2016 HAL i a multi-diciplinary open acce archive for the depoit and diemination of cientific reearch document, whether they are publihed or not. The document may come from teaching and reearch intitution in France or abroad, or from public or private reearch center. L archive ouverte pluridiciplinaire HAL, et detinée au dépôt et à la diffuion de document cientifique de niveau recherche, publié ou non, émanant de établiement d eneignement et de recherche françai ou étranger, de laboratoire public ou privé.

2 Growing Network Through Random Walk Without Retart Bernardo Amorim Daniel Figueiredo Giulio Iacobelli Giovanni Neglia March 2016 Abtract Network growth and evolution i a fundamental theme that ha puzzled cientit for the pat decade. A number of model have been propoed to capture important propertie of real network. In an attempt to better decribe reality, more recent growth model embody local rule of attachment, however they till require a primitive to randomly elect an exiting network node and then ome kind of global knowledge about the network (at leat the et of node and how to reach them). We propoe a purely local network growth model that make no ue of global ampling acro the node. The model i baed on a continuouly moving random walk that after tep connect a new node to it current location, but never retart. Through extenive imulation and theoretical argument, we analyze the behavior of the model finding a fundamental dependency on the parity of, where network with either exponential or a conditional power law degree ditribution can emerge. A increae parity dependency diminihe and the model recover the degree ditribution of Barabái-Albert preferential attachment model. The propoed purely local model indicate that network can grow to exhibit intereting propertie even in the abence of any global rule, uch a global node ampling. 1 Introduction The growth and evolution of network i a fundamental problem in Network Science pecially in the light that network are contantly changing over time. Explaining how and why different real network grow and evolve the way they do ha kept reearcher buy for the pat decade. Not urpriing, variou mathematical model for network growth and evolution have been propoed in the literature, either ad-hoc model tailored to pecific domain, or Publihed in the Proceeding of the 7th Workhop on Complex Network (CompleNet 2016), March 23-25, 2016, Dijon, France. Department of Computer and Sytem Engineering (PESC), Federal Univerity of Rio de Janeiro (UFRJ), Brazil, [bamorim,daniel,giulio]@land.ufrj.br MAESTRO Team, INRIA, Sophia-Antipoli, France, giovanni.neglia@inria.fr 1

3 general model aiming to capture general principle. A celebrated general network growth model i the Barabái-Albert (BA) model [1] which embodie the principle of preferential attachment found in variou real network. A recognized drawback of mot propoed network growth and evolution model i the aumption of global information about the network [2, 4, 5, 8]. For example, the BA model ha a primitive to randomly elect a node from the exiting network according to the degree ditribution. To relax uch aumption, model that attach new node and edge to the exiting network uing local attachment rule, uch a the Random Walk Model [9, 10], have been propoed. Clearly, random walk require knowledge of the current node degree and it neighbor, a much more localized information. Moreover, it eem more reaonable that new node connect to nearby node (through ome local proce) rather than electing new neighbor from the entire population (through ome global proce). However, the Random Walk Model tudied in [9, 10] and other [5, 8] till require a primitive to randomly elect a node from the network (for the purpoe of retarting the walker, for example) and are thu not purely local, becaue they need to know the number and the identity of all network node a well a a way to reach them. Such model have local attachment rule, but global entry point election. More recently, model that have no global primitive have tarted to be explored [6, 7]. A drawback of thee other model i that they rely on an initial network already containing all node uch a a lattice or a regular tree, that i then modified according to local rule, and thu are technically not growth model. In thi work we propoe and explore a network growth model that i purely local, requiring no global election over the node or any initial network. The model work a follow: 0. Start a network with a ingle node with a elf-loop and place a random walk on thi node. 1. Let the random walk take exactly tep. 2. Connect a new node to the node where the walker reide. 3. Stop if the number of node in the network i n, otherwie go to Step 1. Intuitively, the random walk move around continuouly and after every tep a new node i added and connected to it current location. The new node immediately become part of the network and the walker ee no difference between it and any other node. Note that the model ha two parameter and n and grow an undirected tree (apart from the elf-loop at the initial node) ince every new node tart with degree one. Moreover, the random walk i uniform on the neighbor and i never retarted, thu it name NRRW (No Retart Random Walk) model. Figure 1 illutrate a ample path of network growth with = 1 and = 2. Can uch purely local model give rie to intereting network tructure uch a network that exhibit a power-law degree ditribution? Interetingly, we uncover variou non-trivial feature of thi model uch a the fundamental dependency on the parity and magnitude of and it relationhip to the degree ditribution. If i odd and mall we find that network generated by the model tend to have very hort-tailed degree ditribution and 2

4 Figure 1: Example of ample path for network growth for = 1 (top) and = 2 (bottom). The red quare denote the walker poition. The naphot repreent the growing network jut after the new node i connected. very long ditance. On the contrary, if i even and mall, network exhibit a pecial kind of power law degree ditribution (to be formalized later) and very hort ditance! A increae the effect of parity decreae and network exhibit a heavy-tailed degree ditribution. Interetingly, with large enough, the oberved degree ditribution follow a power law with exponent identical to the network generated by the BA model, recovering the effect of preferential attachment. We alo rigorouly prove that for = 1 the random walk i tranient and the degree of every node i bounded from above by a geometric ditribution. Other intereting feature will be highlighted in what follow. The model here propoed i very related to the Random Walk Model [9] which alo allow a random walk to take tep before connecting a new node. The key difference i that in [9], after a new node i attached to the network, the random walk i retarted uniformly at random acro all exiting node in the network. Our random walk never retart, and i therefore a purely local model. Interetingly, the author of [9] how (through imulation and approximation) that their model i cloely related to the BA model and yield a power law degree ditribution independently of. However, recently thi finding ha been quetioned and for = 1 it wa mathematically proven that thi i not the cae [3]. Our model and finding contribute to thi debate and poibly hed light on how both reult could be reconciled (more on thi on Section 6). The remainder of thi paper i organized a follow. Section 2 dicue the model and it intuitive behavior, a well a the connection with prior work. Section 3 preent the evolution of node degree induced by the model. Section 4 analyze the depth of the tree generated by the model. Section 5 preent our theoretical finding for the cae = 1, howing the tranient nature of the model in thi cae. Finally, we ummarize our finding and preent a brief dicuion in Section 6. 3

5 2 Network Growth Model A preented in Section 1, NRRW (No Retart Random Walk) model can be interpreted a a imple random walk that attache a new node to it current location every tep. Similar propoed random walk model for growing network aume that the random walk retart either after connecting a new node or adding ome number of edge to the new node [9, 10]. A retart conit of randomly electing a node from the exiting network (uually uniformly) and placing the random walk on that node. Depite the imilaritie, the lack of retart make NRRW fundamentally different from model with retart. In particular, the retart ignificantly reduce the correlation between conecutive node addition ince it i very unlikely that the random walk will viit the previou new node when walking to add a new node. Intuitively, the random walk loe memory at every retart. Moreover, retart have the drawback of auming knowledge of all network node and random acce to any uch node, and i thu not a purely local growth model. What i intuitively the behavior of NRRW? In a ene, when i large the random walk will have little memory between node addition. However, thi behavior i different from retart ince the random walk will not find itelf on a node choen uniformly at random but on a node choen randomly proportional to it degree. 1 Thu, when i large the NRRW eem imilar to the BA model ince new node connect to random node choen proportionally to their degree. However, ince i fixed and the network grow, will NRRW indeed exhibit a behavior imilar to BA model when << n and then will finally become mall in comparion to the network ize? What about mall value for? Intuitively, the random walk will frequently tumble over the newly created node. Interetingly, thi local behavior depend fundamentally on the parity of. If = 1 then the random walk can alway walk to the newly created node and add a new node to it. Such behavior i jut not poible if = 2 and the walker i not on the root. Thi qualitative difference i not limited to = 1 and = 2. When i odd the walker can alway land to the mot recently added node after tep and then add a new node. For even, thi i impoible unle the walker doe not travere the loop at the initial node. The above obervation jutifie why in the NRRW model we conider a ingle node with a elf-loop a a tarting point. If thi wa not the cae, for any even the random walk would only add node to the original node, trivially contructing a tar ince it can never tep on a newly created node. The loop allow a change of parity with repect to the level of the tree where new node can be added. In fact if the random walk i at level k of the tree and i even, the random walk can only add new edge at the level k +2h for h = 2 k,..., 1,0,1,2,... until it doe not travere the loop. For odd thi i not neceary a the walker can tep on a newly created node to be able to add to node in any level of the tree without returning to the root. Thu, yet another fundamental difference between even and odd. Will thee difference between mall and large and even and odd manifet themelve in tructural propertie of the tree generated by the model? 1 Recall that the teady tate ditribution of a random walk on a fixed network i given by d i / j d j, where d i i the degree of node i. 4

6 In particular, will the degree ditribution fundamentally depend on? In what follow we explore the degree and other propertie of the tree generated by the model howing in fact, that play a key role. 2.1 Simulation In order to tudy the model we deigned and implemented an efficient imulator (in C++) for the NRRW model which ha a parameter, n and r, with r denoting the number of independent run. For each run, we tart with a ingle node with a elf loop, move the random walk tep, connect a new node to it current location, and repeat. We collect tatitic for the variou propertie merging the reult acro the r imulation run, uch a degree ditribution (fraction of node with degree k acro all run). The wort cae time complexity of a imulation run i O(nlogn) but the amortized time complexity i O(n), a we ue a growing vector to repreent the neighbor of a node that double it capacity when needed. Thu, a walker tep require Θ(1) time and a node addition take O(1) amortized time. 3 Degree Behavior In thi ection we tudy the degree ditribution of NRRW through extenive imulation illutrating it behavior and dependencie. Figure 2 how the Complementary Cumulative Ditribution Function (CCDF) of node for variou value for. Surpriingly, when i mall (between 1 and 8) the repective degree ditribution are fundamentally different, exhibiting a kind of power law for even and an exponential tail for odd. Note that when = 1 we do not oberve node with degree larger than 40 while for = 2 a nonnegligible fraction of node have degree greater than We alo oberve oppoite trend in the degree ditribution a increae. For odd, increaing yield a ditribution with heavier tail, while for even increaing yield a ditribution with a lighter tail. A increae into a medium range (between 15 and 64) the trend continue and the two ditribution approach each other. For even larger (between 127 and 256) the degree ditribution become very cloe, being almot inditinguihable. Interetingly, with large the degree follow a power law ditribution, uggeting that the effect of even value dominate the dynamic. Moreover, for large the CCDF exhibit a power law with exponent approximately 2 a it i alo the cae for the BA model which i baed on linear preferential attachment. 2 Thi upport our initial intuition that when i large, the random walk ample node (adding a new node and connecting to it) with probability proportional to their degree, behaving imilarly to the BA model. Figure 3 how the degree ditribution for = 2 but over different value for n. Interetingly, note that independent of n the degree ditribution exhibit the ame power law exponent. However, a n increae the fraction of node greater than k become maller for any fixed k > 0 (with the exception of the 2 Recall that if D follow a power law ditribution, then P(D = k) k α where α > 1 i the power law exponent, and it follow that P(D k) k (α 1). Thu, the CCDF ha an exponent that i one unit le than the PDF. 5

7 Degree ditribution for mall, n=10^6, r=10^ P(D>d) P(D>d) 10 2 Degree ditribution for medium, n=10^6, r=10^ d d Degree ditribution for large, n=10^6, r=10^ P(D>d) d Figure 2: Empirical degree Complementary Cumulative Ditribution Function (CCDF) for variou value of in log log cale (n = 106, r = 103 ). cut-off regime which occur when k i near n). Thi implie that the fraction of node with k = 1, the minimum degree, i increaing with n. Thi i clear by oberving d = 1 (leftmot point in x-axi) and noting that the fraction of node with degree greater than 1 i decreaing with n. Note that uch behavior doe not occur for = 1 which maintain it degree ditribution a n increae (the dot for different n value are barely ditinguihable in plot). If for = 2 the fraction of node with degree 1 increae and converge to 1 a n goe to infinity, then we cannot claim that the degree ditribution follow a power law. However, we can conider the degree ditribution of the node that do not have degree 1. In particular, the conditional degree ditribution, conditioned on D > 1, i hown in Figure 3. Note that the conditional degree ditribution doe not how dependance on n and moreover eem to follow a power law. Thi finding i quite intereting ince the fraction of node with degree 1 can converge to 1 (a n ) while the remainder of node can till follow a power law. Thi may hed new light on the contrating reult in [9] and [3]. We return to thi dicuion in Section 6. Figure 3 alo how the fraction of node with degree greater than 1, P(D > 1), a a function of n for different even value (for odd, it doe not depend ignificantly on n a hown in the top right part of Figure 3 for = 1). Note that for mall the fraction goe to zero reaonably fat (and thu, the fraction of node with degree 1 goe to one). A increae the rate at which P(D > 1) decreae alo decreae. Note that for large (128 or 256) thi decreae i barely noticeable, depite being preent. Interetingly, a odd increae, P(D > 1) decreae but without howing any dependency on n. When = 257, P(D > 1) approache the value hown in Figure 3 for = 256 (reult not hown due to pace contraint). 6

8 P(D>d) Degree ditribution for =2, r=10^9/n n P(D>d) Degree ditribution for =1, r=10^9/n n d d Conditional degree ditribution for =2, r=10^9/n 10 0 Denity of non leave for each N; even P(D>d D>1) n P(D>1) D N Figure 3: Empirical degree CCDF for variou value of n in log log cale (top plot). Empirical degree CCDF conditioned on the degree being greater than 1; fraction of node with degree greater than 1 (bottom plot). A hown, the NRRW model ha a very particular behavior with repect to the degree ditribution. In Section 6 we provide a further dicuion with a few conjecture for it aymptotic behavior with n. 4 Level Behavior We now invetigate the level of the node on the tree generated by the NRRW model. 3 A we have hown above, the model dynamic ha a fundamental dependence on the parity of, pecially when i mall. Indeed, thi dependence alo manifet itelf on the level of the node. Figure 4 how the level ditribution (fraction of node at level larger than l) for a few mall value of eparated into odd and even, repectively. The level ditribution for even decreae very fat. Note that although n = 10 6, when = 2, 90% of node are at level 4 or le and no node i at level 7 or higher. A even increae the level ditribution decreae relatively lower, with 90% of node found at level greater than 4 when = 8. Still, no node i found at level greater than 10. The behavior i completely different for odd, and the level ditribution eem to be uniform (traight line on a linear-linear plot). For = 1 the ditribution ha the heaviet tail with about 4 node per level, giving rie to different level. For = 7 there are about 40 node per level, giving rie to different level. Interetingly, a even increae 3 Recall that the level of node on a tree i given by it ditance to the root, and thu the root i at level zero. 7

9 the level ditribution become heavier while a odd increae the level ditribution become lighter. Figure 4 alo how the level ditribution for large. Indeed, a increae the level ditribution for even and odd become more imilar and the dependency on the parity diminihe. Thi behavior i imilar to what oberved for the degree ditribution, illutrated in Section 3. Note that from the level ditribution we can infer the kind of tree that NRRW generate. When i mall and even, the tree generated are fat and hort, with mot node near the root and a few with very large degree. When i mall and odd, the tree are thin and long with few node pread acro many level and no node with large degree. A increae, the two kind of tree move in each other direction, becoming more and more imilar Depth level ditribution for even, n=10^ Depth level ditribution for odd, n=10^ P(L>l) P(L>l) l l 1.00 Depth level ditribution for large, n=10^6 P(L>l) l Figure 4: Empirical CCDF of the node level for different value (n = 10 6, r = 10 3 ). 5 Theoretical finding for = 1 The numerical imulation with odd and in particular with = 1 ugget that tree grow in depth a the number of node increae. In particular, the growth in depth eem linear on the number of node. Thi i an indication that the random walk i continuouly puhing the tree to lower depth jut never to return to it origin. In a nuthell, the random walk i tranient and viit each node in the tree only a relative mall number of time, with high probability. The following Theorem rigorouly formalize thi intuition. Theorem 1. In the NRRW model with = 1, the number of viit to a node i tochatically dominated by 1 plu a geometric random variable with upport on Z >0. 8

10 Proof. We conider here that the initial network conit of a ingle node with no elf-loop. Thi implifie the notation and doe not compromie the main argument. Let r denote the initial node of the growing network hereafter referred to a the root of the tree (at any tep the growing network i a tree) where the walker reide at time zero. Note that r i the only node at level zero. Let X n be the level (i.e. the ditance from the root) of the node viited by the NRRW at tep n. We call the proce {X n,n Z 0 } the level proce. Note that the random walk viit r the ame number of time that the level proce viit level zero. At tep n > 0 the NRRW i in a node v n with at leat two edge: the one the NRRW ha arrived from and the new one added a a conequence of the NRRW arrival. Let d n 2 denote the degree of node v n. If v n r, the NRRW jump from v n to a node with larger level with probability d n 1 d n 1 2 and with the complementary probability d 1 n 1 2 to a node with maller level. If v n = r, then the level can obviouly only increae. Note that, due to the fact that degree keep changing becaue of the arrival of new edge, the level proce i non-homogeneou (both in time and in pace). We now tudy the evolution of the level proce every two tep, i.e. we conider the proce Y n X 2n. Given that the network i a tree and X 0 = 0, the two-tep level proce can be een a a non-homogeneou reflecting lazy random walk on 2Z 0 = {0,2,4,...}. We denote by p k,h (n) the probability that the level at tep n + 1 i h conditioned on the fact that it i k at tep n. Although the notation hide it, we oberve that the probabilitie p k,h (n) depend on the whole hitory of the NRRW until tep n. The reaon to conider the two-tep level proce i that we can get bound on the tranition probabilitie p k,h (n) that allow a imple comparion with a (biaed) homogeneou random walk. The bound derived above for X n lead immediately to conclude that p k,k+2 (n) = 4 1 for any level k 0 and p k,k 2(n) = 1 4 for k 2, but we can get a tighter bound for p k,k 2 (n). If the NRRW i at level k, all the node on the path between it current poition and the root r have degree at leat 2. If it then move to node v at level k 1, a new edge i attached to v, whoe degree i now at leat 3. The probability to move from v further cloer to the root to a node with level k 2, i then at mot 3 1. It follow then that p k,k 2 (n) = 1 6 for k 2. We conider now a homogeneou biaed lazy random walk (Yn ) n 0 on 2Z 0 tarting from 0 with tranition probabilitie p k,k+2 = 1 4 for all k 2Z 0 and p k,k 2 = 1 6 for k 2Z 0 and k 0. We how that if (Yn ) n 0 alo tart in 0 (Y0 = 0), it i tochatically dominated by (Y n) n 0. We prove it by coupling the two procee a follow. Let (ω n ) n 0 be a equence of independent uniform random variable over [0, 1]. We ue them to generate ample path for both procee (Y n ) n 0 and (Yn ) n 0 a follow: Y n 2, if ω n [0, p k,k 2 (n)) Yn 2, if ω n [0, p Y n+1 = Y n + 2, if ω n [1 p k,k+2 (n),1] Yn+1 k,k 2 ) = Yn + 2, if ω n [1 p k,k+2,1] otherwie otherwie Y n where p k,k 2 (n) and p k,k 2 are 0 if k = 2. We tart oberving that if Y n and Y n have the ame value k, then every time Y n increae alo Y n increae becaue p k,k+2 = 1 4 p k,k 2(n). On the contrary if Y n decreae (a it can happen only Y n 9

11 for k 2), then Y n may decreae or not becaue p k,k 2 (n) 1 6 = p k,k 2. It follow that if Y n and Yn are at the ame level, then Yn+1 Y n+1. We are going to prove by induction on n that Yn+1 Y n+1 for every n. With a light abue of terminology we ay that Y n increae (rep. decreae) if Y n+1 > Y n (rep. Y n+1 < Y n ). We tart oberving that indeed Y0 Y 0, becaue both procee tart in 0. Let u aume that Yn = h k = Y n. For all value of h, every time Yn increae alo Y n increae becaue p k,k+2 = 1 4 p k,k+2(n) and then Yn+1 = h + 1 k + 1 = Y n+1. If h 2, then p h,h 2 = 1 6 p k,k 2 (n) and if Y n decreae then Yn mut alo decreae (Yn+1 = h 1 k 1 = Y n+1 ). It follow that for h 2 then Yn+1 Y n+1. The only cae when Y n may decreae without Yn decreaing i when h = 0 and k 0, but in thi cae Yn+1 = 0 and Y n+1 0. Thi prove that Yn+1 Y n+1 for every n. Given that Yn Y n and both procee tart at level zero, the number of viit of (Y n ) n 0 to level zero i bounded by the number of viit of (Yn ) n 0 to level zero. The homogeneou biaed lazy random walk (Yn ) n 0 i tranient ince p k,k+2 = 1/4 > p k,k 2 = 1/6. Thu, the probability of the firt return time to level 0 i f 0 < 1. By the trong Markov property, the number of viit to level 0 i geometrically ditributed on the et Z >0 with parameter equal to 1 f 0. Since a viit to level zero in (X n ) n 0 (one level proce) implie a viit to level zero in (Y n ) n 0 (two level proce), then it follow that the number of viit of (X n ) n 0 to level zero i bounded by a geometric random variable and then even more o by 1 plu the ame geometric random variable. Now let u conider any node v in the growing network. If the NRWW never viit v, then the degree of b i 1 and the thei follow immediately. Otherwie, let conider the firt time the NRWW viit v to be time t = 0 and let conider v to be the root of the current tree. We can retrace the ame reaoning and conclude that the number of viit to v for t > 0 i bounded by a geometric random variable on Z >0 with parameter equal to 1 f 0. Then the total number of viit to v i bounded by 1 plu uch random variable. Thi conclude the proof. Corollary 1. In the NRRW model with = 1, the degree ditribution of any node i bounded by a geometric ditribution. Thi follow ince the degree of every node equal the number of viit of the random walk to the node plu 1 (the plu 1 account for the fact that any node joining the network, although not yet viited by the walker, ha degree 1). 6 Dicuion and Concluion A we have hown, the NRRW model exhibit intereting feature that fundamentally impact the network it generate. For = 1 the random walk i tranient and node degree i bounded by a geometric ditribution (Theorem 1). For = 2, the fraction of node with degree 1 eem to converge to 1 a n. However, the conditional degree ditribution eem to follow a power law. Can uch reult be made mathematically rigorou? Other intereting quetion alo emerge from our analyi of the NRRW model. In particular, our numerical imulation eem to indicate that for any even, the 10

12 fraction of node with degree 1 will converge to 1 a n. On the other hand, our imulation alo indicate that thi i not the cae for any odd. So will there be a fundamental difference between a fixed but arbitrarily large even and odd? It i hard to imagine that = 2 10 and = would have fundamentally different behavior, ince in both cae the random walk move quite a lot before adding a new node. Of coure, any fixed will be mall a n. Thu, we make the following conjecture: Conjecture 1. For any fixed even, the fraction of node with degree one converge to 1, a n. For any fixed odd, the fraction of node with degree one converge to a number trictly le than 1, a n. If true, uch conjecture would imply that the degree ditribution are alo never identical, for any fixed even or odd. However, our numerical reult indicate that the conditional degree ditribution (conditioned on degree being greater than one) for even, eem to converge to a power law a n. On the other hand, for = 1 we have proved that the random walk i tranient and degree ditribution i bounded by a geometric ditribution (Theorem 1). Can fixed odd value really generate power law? If thi i the cae, then there would be a phae tranition on, from inducing a network with degree ditribution with an exponential tail ( = 1) to a power law tail. Depite the numerical reult indicating the heavy tail degree for = {127, 255}, we make the following conjecture: Conjecture 2. For any fixed even, the conditional degree ditribution i bounded from below by a power law, a n. For any fixed odd, degree ditribution i bounded from above by an exponential, a n. Such conjecture conider that n diverge. In practice n mut be finite when generating a network with NRRW model. Thu, for a fixed n, the difference induced by an even or odd may diminih a increae. In particular, the degree ditribution generated by even and odd value may become arbitrarily cloe a increae, a we have oberved in numerical imulation for a fixed n (Figure 2). Lat, we return to the recent dipute if the Random Walk model with retart generate a power law degree ditribution, independently of [3, 9]. It ha been mathematically proved that when = 1 the fraction of node with degree one converge to 1, a n [3]. At the ame time, imulation reult ugget that the degree ditribution follow a kind of power law [9]. We can attempt to reconcile uch finding by leveraging our own finding on NRRW model. When = 1 the new node i connected to a given exiting node u if i) a neighbor of u i elected at the retart and then ii) the random walk move to u. A node whoe neighbor are all leave would be elected with a probability proportional to it degree. Now it ha been hown that when n the fraction of node that are leave converge to 1, then mot of the neighbor of a non-leaf node are leave and thi node i eentially elected proportionally to it degree, imilarly to the BA model embodying preferential attachment. Thu, the conditional degree ditribution, leaving out degree 1 node, will follow a power law ditribution with the ame exponent a in the BA model. In ome ene, thi reconcile the finding of the two prior work [3, 9]. To conclude, a exemplified above, a fundamental undertanding of NRRW model add to our undertanding of purely local network growth model. In 11

13 particular, beide requiring a le trict aumption to operate, model that do not rely on any global primitive can alo generate network with rich and divere tructural propertie. 7 Acknowledgement Reearch conducted within the context of the THANES Aociate Team, jointly upported by Inria (France) and FAPERJ (Brazil). Thi work ha alo been partially funded through reearch grant from CNPq and CAPES (Brazil). Reference [1] Barabái, A.L., Albert, R.: Emergence of caling in random network. cience 286(5439), (1999) [2] Blanchard, P., Krueger, T., Ruchhaupt, A.: Small world graph by iterated local edge formation. Phy. Rev. E 71, 046,139 (2005) [3] Canning, C., Jordan, J.: Random walk attachment graph. Electronic Communication in Probability 18, 1 5 (2013) [4] Colman, E.R., Rodger, G.J.: Local rewiring rule for evolving complex network. Phyica A: Statitical Mechanic and it Application 416, (2014) [5] Gabel, A., Redner, S.: Sublinear but never uperlinear preferential attachment by local network growth. Journal of Statitical Mechanic: Theory and Experiment 2013(02), P02,043 (2013) [6] Ikeda, N.: Network formation determined by the diffuion proce of random walker. Journal of Phyic A: Mathematical and Theoretical 41(23), 235,005 (2008) [7] Ikeda, N.: Network formed by movement of random walker on a bethe lattice. In: Journal of Phyic: Conference Serie, vol. 490, p IOP Publihing (2014) [8] Li, M., Gao, L., Fan, Y., Wu, J., Di, Z.: Emergence of global preferential attachment from local interaction. New Journal of Phyic 12(4), 043,029 (2010) [9] Saramäki, J., Kaki, K.: Scale-free network generated by random walker. Phyica A: Statitical Mechanic and it Application 341, (2004) [10] Vázquez, A.: Growing network with local rule: Preferential attachment, clutering hierarchy, and degree correlation. Phyical Review E 67(5), 056,104 (2003) 12