Ad-hoc Limited Scale-Free Models for Unstructured Peer-to-Peer Networks

Transcription

1 Eighth International Conference on Peer-to-Peer Computing (P2P'8) A-hoc Limite Scale-Free Moel for Untructure Peer-to-Peer Networ Haan Guclu Center for Nonlinear Stuie Lo Alamo National Laboratory Lo Alamo, NM Durgeh Kumari an Murat Yuel Computer Science an Engineering Department Univerity of Nevaa - Reno, Reno, NV urgeh.rani@gmail.com yuem@ce.unr.eu Abtract Several protocol efficiency metric (e.g., calability, earch ucce rate, routing reachability an tability) epen on the capability of preerving tructure even over the churn caue by the a-hoc noe joining or leaving the networ. Preerving the tructure become more prohibitive ue to the itribute an potentially uncooperative nature of uch networ, a in the peer-to-peer (P2P) networ. Thu, mot practical olution involve untructure approache while attempting to maintain the tructure at variou level of protocol tac. The primary focu of thi paper i to invetigate contruction an maintenance of cale-free topologie in a itribute manner without requiring global topology information at the time when noe join or leave. We conier the uncooperative behavior of peer by limiting the number of neighbor to a preefine har cutoff value (i.e., no peer i a major hub), an the a-hoc behavior of peer by rewiring the neighbor of noe leaving the networ. We alo invetigate the effect of thee har cutoff an rewiring of a-hoc noe on the P2P earch efficiency. 1. Introuction Stability an calability of highly ynamic networ mainly epen on the capability of preerving tructure even over the churn caue by the a-hoc noe joining or leaving the networ. Several protocol efficiency metric (e.g., earch ucce rate, routing reachability rate) epen on thi capability. Preerving the tructure become more prohibitive ue to the itribute an potentially uncooperative nature of uch networ, a in the peer-to-peer (P2P) networ. Thu, mot practical olution involve untructure approache while attempting to maintain the tructure at variou level of protocol tac. In ecentralize P2P networ, the overlay topology (or connectivity graph) among peer i a crucial component in aition to the peer/ata organization an earch. Topological characteritic have profoun impact on the efficiency of earch on P2P networ a well a other networ. It ha been well-nown that earch on mall-worl topologie can be a efficient a O(ln N) [3], an thi phenomenon ha recently been tuie on P2P networ [43, 35, 28, 29]. The bet earch efficiency in realitic networan be achieve when the topology i cale-free (power-law), which offer earch efficiencie lie O(ln ln N). Key limitation of calefree topologie i the high loa (i.e., high egree) on very few number of hub noe. In a typical untructure P2P networ, peer are not willing to maintain high egree/loa a they may not want to tore large number of entrie for contruction of the overlay topology. So, to achieve fairne an practicality among all peer, har cutoff on the number of entrie are impoe by the iniviual peer, which mae the overall networ a limite one. Effect of uch har cutoff on earch efficiency can be ignificant [27]. Due to the uncooperative nature of peer in a P2P networ, protocol cannot completely rely on metho woring with full cooperation of peer. For example, peer may not want to tore large number of entrie for contruction of the overlay topology, i.e., connectivity graph. Even though characteritic of the overlay topology i crucial in etermining the efficiency of the networ, peer typically o not want to tae the buren of toring exceive amount of control information for other in the networ, thereby impoing har cutoff on the amount of control information to be tore. Yet another ey iue i the contruction of cale-free overlay topologie without global information. There are everal technique to generate a cale-free topology [8, 4], by uing global information about the current networ when a noe join or leave. Such global metho are not practical in P2P networ, an local heuritic in generating uch cale-free overlay topologie mut be employe. In other wor, there mut be local an imple operation when peer are joining or leaving the P2P overlay, an alo cauing a minimal inefficiency to the earch mechanim to be run on the networ. The primary focu of thi paper i to invetigate contruction an maintenance of cale-free topologie in a i /8 $ IEEE DOI 1.119/P2P

2 tribute manner without requiring global topology information at the time when noe join or leave. We conier the uncooperative behavior of peer by limiting the number of neighbor to a pre-efine har cutoff value, an the a-hoc behavior of peer by rewiring the neighbor of noe leaving the networ. We alo invetigate the effect of thee har cutoff an the rewiring of a-hoc noe on the P2P earch efficiency. The ret of the paper i organize a follow: Firt, we provie motivation for thi wor, outline the ey parameter to be coniere, an briefly tate the major contribution an fining of the wor. Then, we urvey the previouworonp2pnetworinsection2. InSection3,we urvey the previou wor on cale-free topology generation an briefly cover the importance of cutoff in the cale-free networ an Preferential Attachment (PA) with Har Cutoff. We introuce our topology generation technique uing local heuritic an briefly ecribe the algorithm howing join an leave proce of a noe in the growing networ, in Section 4. In Section 5, we preent our imulation reult of egree itribution of the noe. We alo icu the efficiency of three earch algorithm i.e Flooing (FL), Normalize Flooing (NF) an Ranom Wal (RW) on the topology generate by our imulation. We conclue by ummarizing our current wor an outlining the irection for the future wor in Section Contribution an Major Reult Our wor uncover the relationhip between the a-hoc behavior of peer (i.e., how frequent they join/leave) an the efficiency of earch over an overlay topology where each peer can (or i willing to) tore a maximum number of lin to other peer. In our moel, we parameterize (i) the a-hoc behavior of noe by the probability that a noe leave μ, (ii) the amount of local information to be ue at the time of join by nowlege raiu from the point the noe attempt to join, (i.e., the noe now about the local topology covering hop away from the point the noe attempt to join the networ), an (iii) the amount of local information to be ue at the time of leave by nowlege raiu from the location of the leaving noe, τ l (i.e., each neighbor of the leaving noe now about the local topology covering τ l hop away from itelf). We alo efine the maximum number of lin to be tore by peer a the har cutoff,, for the egree of a peer in the networ a compare to natural cutoff which occur ue to finite-ize effect. Our contribution inclue: Guieline for generating cale-free topologie over a-hoc noe: We introuce a generic moel that can aign availability of ifferent amount of local topology information at the time when a noe join or leave. Our moel provie a way of tuning ahocne of the networ an tuying how to balance tate information for noe joining or leaving. Search efficiency on a-hoc limite cale-free topologie: Through extenive imulation, we tuie efficiency of Flooing (FL), Normalize Flooing (NF), an Ranom Wal (RW) on the topologie generate by our moel with ifferent μ,, τ l,an value. Rewiring methoologie for eigning peer leave algorithm for untructure P2P networ: Our tuy yiele everal guieline for peer leaving an untructure P2P networ, o that the earch performance of the overall overlay topology remain high. Our tuy reveale everal intereting iue. We foun that having more global information about the topology at the time of leave i ignificantly more helpful than having it at the time of join. We how that the egree itribution can be ept cale-free an the earch efficiency can be ept very high by imply eeping τ l at reaonably high value, e.g., Relate Wor Previou wor on P2P networ protocol can be claifie into centralize an ecentralize one. A centralize P2P protocol (e.g. Napter [3]) prove to be uncalable, the majority of the P2P reearch ha focue on ecentralize cheme. The ecentralize P2P cheme can be further claifie into ub-categorie: tructure, untructure, an hybri. In the tructure P2P networ, ata/file content of peer i organize bae on a eying mechanim that can wor in a itribute manner, e.g. Ditribute Hah Table (DHT) [38]. The eying mechanim typically map the peer (or their content) to a logical earch pace, which i then leverage for performing efficient earche. In contrat to the tructure cheme, untructure P2P networ o not inclue a trict organization of peer or their content. Since there i no particular eying or organization of the content, the earch technique are typically bae on flooing. Thu, the earche may tae very long time for rare item, though popular item can be foun very fat ue to poible leveraging of locality of reference [37] an caching/replication [13]. The main focu of the reearch on untructure P2P networ ha been the traeoff between tate complexity of peer (i.e., number of recor neee to be tore at each peer) an flooing-bae earch efficiency. The minimal tate each peer ha to maintain i the lit of neighbor peer, which contruct the overlay topology. Optionally, peer can maintain forwaring table (alo referre a routing table 161

3 in the literature) for ata item in aition to the lit of neighbor peer. Thu, we can claify untructure P2P networ into two bae on the type() of tate peer maintain: (i) per-ata untructure P2P networ (i.e., peer maintain both the lit of neighbor peer an the per-ata forwaring table), an (ii) non-per-ata untructure P2P networ (i.e., peer maintain only the lit of neighbor peer). Non-per-ata cheme are mainly Gnutella-lie cheme [1], where earch i performe by mean of flooing query pacet. Search performance over uch P2P networ ha been tuie in variou context, which inclue pure ranom wal [22], probabilitic flooing technique [34, 23], an ytematic filtering technique [41]. Per-ata cheme (e.g. Freenet [2]) can achieve better earch performance than non-per-ata cheme, though they impoe aitional torage requirement to peer. By maing the peer maintain a number of <ey,pointer> entrie peer irect the earch querie to more appropriate neighbor, where ey i an ientifier for the ata item being earche an the pointer i the next-bet neighbor to reach that ata item. Thi capability allow peer to leverage aociativity characteritic of earch querie [12]. Stuie range from grouping peer of imilar interet (i.e., peer aociativity) [29, 12] to exploiting locality in earch querie (i.e., query aociativity) [11, 37]. Our wor i applicable to both per-ata an non-per-ata untructure P2P networ, ince we focu on the interaction between earch efficiency an topological characteritic. Previou tuy on noe iolation caue by churn in untructure P2P networ introuce a general moel of reilience [42]. In thi tuy, joining an rewiring procee were bae on age-biae neighbor election, where a formal analyi inclue two age-biae technique of neighbor election. In maximal age-election approach, the joining noe elect uniformly ranomly m alive noe from the networ an connect to the one with maximal age. It follow the ame proce when a ea lin i etecte. However, in age-biae ranom wal election approach, the probability of a noe to be electe by another peer i proportional to it current age. Another tuy introuce elf-organizing uper peer networ architecture [21], where uper peer maintain the cache with pointer to file that are recently requete an on the other han client peer ynamically elect uper peer offering bet earch reult. 3. Scale-Free Networ Topologie Recent reearch how that many natural an artificial ytem uch a the Internet [2], Worl Wie Web [5], cientific collaboration networ [9], an networ [19] have power-law egree (connectivity) itribution. Thee ytem are commonly nown a power-law or cale-free networ ince their egree itribution are free of cale (i.e., not a function of the number of networ noe N)an follow power-law itribution over many orer of magnitue. Thi phenomenon ha been repreente by the probability of having noe with egree a P () γ where γ i uually between 2 an 3 [8]. Scale-free networ have many intereting propertie uch a high tolerance to ranom error an attac (yet low tolerance to attac targete to hub) [6], high ynchronizability [25, 26, 31], an reitance to congetion [39]. The origin of the cale-free behavior can be trace bac to two mechanim that are preent in many ytem, an have a trong impact on the final topology [8]. Firt, networ are evelope by the aition of new noe that are connecte to thoe alreay preent in the ytem. Thi mechanim ignifie continuou expanion in real networ. Secon, there i a higher probability that a new noe i line to a noe that alreay ha a large number of connection. Thee two feature le to the formulation of a growing networ moel firt propoe by Barabái an Albert that generate a cale-free networ for which P () follow a power law with γ. Thi moel i nown a preferential attachment (PA or rich-get-richer mechanim) an the reulting networ i calle Barabái-Albert networ [8, 4]. In thi tuy, we ue a imple verion of the PA moel [8]. The moel evolve by one noe at a time an thi new noe i connecte to m (number of tub) ifferent exiting noe with probability proportional to their egree, i.e., P i = i / j j where i i the egree of the noe i. Theaverage egree per noe in the reulting networ i 2m an the minimum egree i m. Scale-free networ are very robut againt ranom failure an attac ince the probability to hit the hub noe (few noe with very large egree) i very mall an attacing the low-egree atellite noe oe not harm the networ. On the other han, eliberate attac targete to hub through which mot of the traffic go can eaily hatter the networ an everely amage the overall communication in the networ. For the ame reaon the Internet i calle robut yet fragile [18] or Achille heel [6, 7]. Scale-free networ alo have mall-worl [4] propertie. In mall-worl networ the iameter, or the mean hop itance between the noe cale with the ytem ize (or the number of networ noe) N logarithmically, i.e., ln N. The cale-free networ with 2<γ<3 have a much maller iameter an can be name ultra-mall networ [14], behaving a ln ln N. Whenγ an m 2, behave a ln N/ ln ln N. However,whenm an γ the Barabái-Albert moel turn into a tree an ln N i obtaine. Alo when γ>3, the iameter behave logarithmically a ln N. Since the pee/efficiency of earch algorithm trongly epen on the average hortet path, calefree networ have much better performance in earch than other ranom networ. 162

4 3.1. The Cutoff One of the important characteritic of cale-free networ i the natural cutoff on the egree (or the maximum egree) ue to finite-ize effect. Natural cutoff can be efine a [16] the value of the egree above which one expect to fin at mot one vertex, i.e., N P () 1. (1) nc By uing the egree itribution for the cale-free networ an the exact form of probability itribution (i.e., P ()=(γ 1)m γ 1 / γ ), one obtain nc (N) mn 1/(γ 1), (2) which i nown a the natural cutoff of the networ. The caling of the natural cutoff can alo be calculate by uing the extreme-value theory [1]. For the cale-free networ generate by PA moel (γ) the natural cutoff become nc (N) m N. (3) 3.2. Preferential Attachment with Har Cutoff The natural cutoff may not be alway attainable for mot of the cale-free networ ue to technical reaon. One main reaon i that the networ might have limitation on the number of lin the noe can have. Thi i epecially important for P2P networ in which noe can not poibly connect many other noe. Thi require putting an artificial or har cutoff to the number of lin one noe might have. In orer to implement the har cutoff in PA, we imply i not allow noe to have lin more than a fixe har cutoff value uring the attachment proce. Thi moifie metho generate a cale-free networ in which there are many noe with egree fixe to har cutoff intea of a few very high egree hub an the egree itribution till ecay in a power law fahion. The egree itribution of PA moel with cutoff i lightly ifferent than that of PA without a cutoff in term of exponent an an accumulation of noe with egree equal to har cutoff. PA moel, in it original form, ha a egree itribution exponent γ for very large networ. However, when a har cutoff i impoe it i oberve that the abolute value of egree itribution exponent ecreae [27]. One can ue the mater-equation [32] approach to analyze the effect of the har cutoff on the topological characteritic. We grow the networ by introucing new noe one by one for implicity. Each new noe lin to m earlier noe in the networ. The probability that the new noe attache to a previou noe of egree i efine to be A /A, Algorithm 1 Networ growth uing paramaterize join an leave procee //Global Variable an Function m - minimum egree μ - probability of a noe to leave the networ N - the maximum noe ID of the exiting networ (the minimum noe ID i ) G - graph of the exiting networ of M lin an N noe PreferentialAttachment(G 1, G 2 ) - a function that perform Preferential Attachment to G 1 by uing the noe in G 2, return the number of ucceful new lin // Join proce of noe i voi Join(i, ) 1: N++ 2: numoflin 3: while numoflin < m o 4: N ran Ranomize(1,N) {Pic a ranom noe from the exiting networ} 5: myg get ubgraph(n ran, ) {Get the ubgraph incluing neighbor noe of N ran up to hop away} 6: numoflin += PreferentialAttachment(G, myg) 7: en while //Leave proce of noe i voi Leave(i, τ l ) 1: myg get ubgraph(n ran,τ l ) {Get the ubgraph incluing neighbor noe of N ran up to τ l hop away} 2: remove(n ran ) {Delete N ran from the exiting networ} 3: N = N 1 4: PreferentialAttachment(G,myG) // Growth proce of a networ with N target noe, parameterize with an τ l voi Grow(N target,, τ l ) 1: for i=m+1; i<n target ;i++o 2: Join(N, ) 3: num Ranom(,1) 4: if N == N target then 5: brea; 6: en if 7: if num < μ then 8: N el Ranomize(1,N) 9: Leave(N el, τ l ) 1: en if 11: en for 163

5 where A i the rate of attachment to a previou noe an thi rate epen only on the egree of the target noe, while A = 1 =m A N i the total rate for all event, an N i the number of noe of egree in the networ. Thu A /A equal to the probability for the newly-introuce noe to attach to a noe of egree. The new feature that we tuy i the effect of a har cutoff on the egree of each noe. Once the egree of a noe reache, it i efine to become inert o that no further attachment to thi noe can occur. Thu only noe with egree = m, m +1,..., 1 are active. Thi retriction i the ource of the cutoff in the efinition of the total attachment rate. We now tuy the egree itribution, N (N), a a function of the cutoff an the total number of noe in the networ N. The mater equation for the egree itribution can be written by uing the fact that N i proportional to N, an thu N Nn a well a A νn a n = mnm ν +1 = m ( 1)n 1 n ν = m +1,..., 1. (4) ( 1)n c 1 ν = By the nature of thee equation, it i evient that n c i of a ifferent orer than n with <. Starting with the olution n m = ν/(m + ν), we can fin n by ubequent ubtitution. Thi recurive approach give u a chance to write n value a prouct [32] an by converting thee prouct into Euler gamma function we how that n c cale a ν, while for <, n ale a (ν+1). We can obtain the coefficient ν in A = νn elf conitently from A = 1 =m A n νn, or equivalently, ν = 1 =m A n. By rewriting the um above a a ifference between two um with limit from the minimum egree to an from cutoff to an by taing aymptotic limit [24] of large N an we get ν 2 2m. (5) Thi reult how that n (3 2m/c) for < an n c c (2 2m/c) confirming the change in the egree itribution exponent [27]. Thi implie that any finite har cutoff value ecreae the egree itribution exponent, i.e., it mae the egree itribution flatter. A better earch efficiency oberve for a maller cutoff can be explaine by the increae in the egree itribution exponent [27]. A-hoc cale-free networ have recently attracte conierable attention in the literature mainly becaue of it mot-eire property of robutne to ranom attac or failure. For example it wa hown that [6] the iameter of the Internet at the autonomou ytem level, which i the mot famou example of cale-free networ, woul not be change conierably if up to 2.5% of the router were remove ranomly. Thi i an orer of magnitue larger than (a) (b) (c) Figure 1. Search trategie: (a) Flooing (b) Normalize flooing (c) Ranom wal the failure rate. It wa alo hown in [6] that for a cale-free networ of ize 1, an a failure rate of 18%, the bigget connecte component hol 8, noe, wherea uner the ame conition a ranom networan urvive thi failure by the bigget connecte component of ize 1. Many moel for a-hoc cale-free networ in which the ege can appear an iappear [15, 33, 17] or ome noe are remove [36] have been tuie. In the firt et of tuie a the noe are joining to the networ ome lin among the pre-exiting noe are rewire or move ranomly by ome probability parameter. Depening on the parameter uch moel exhibit either exponential or power-law egree itribution. In [36], a the noe are joining by preferential attachment ome ranomly electe noe are elete along with their lin from the networ. If the noe whoe neighbor are elete o not reconnect themelve to other noe, it i oberve that the egree itribution i a power law with an exponent ranging from 3 to infinity epening on the eletion probability. The author propoe a remey for the eletion that the neighbor of the elete noe elect ome other noe in the networ an connect by again uing preferential attachment rule. In thi cae the egree itribution i till a power law but the exponent change from 3 to 2 a the eletion probability goe from to 1. The main iavantage of the a-hoc cale-free moel in the literature i that they lac localize algorithmic olution. All require global information to be available to noe o that they can reconnect to ranomly electe noe in the networ. For thi reaon we grow cale-free networ with local heuritic only to imulate the real-life ituation in untructure peer-to-peer ytem. To parameterize our moel we ue two ifferent time-to-live variable: an τ l to ecribe the number of noe available to a new noe an to a neighbor of a elete noe, repectively. In the next ection we explain our moel an it parameter in etail. 164

6 Degree itribution for a hoc cale-free networ m, μ, Degree itribution for a hoc cale-free networ m, μ, =5 Degree itribution for a hoc cale-free networ m, μ, no P, τ l P, τ l P, τ l (a) m, (b) m, =5 (c) m, no cutoff Degree itribution for a hoc cale-free networ m, μ, Degree itribution for a hoc cale-free networ m, μ, =5 Degree itribution for a hoc cale-free networ m, μ (τ l ), no P, τ l P, τ l P =2 = () m, (e) m, =5 (f) m, no cutoff Figure 2. Degree itribution when there i no a-hocne (i.e., μ): P () for variou networ generate by our framewor for varying. 4. Growing Scale-Free Topologie with Local Heuritic In the PA moel an it a-hoc variant a outline in the previou ection, the new noe or the neighbor of a elete noe ha to mae ranom attempt to connect to the exiting noe with a probability epening on the egree of the exiting noe. To implement thi in a P2P (or any itribute) environment, the new noe ha to have information about the global topology (e.g. the current number of egree each noe ha for the PA moel), which might be very har to maintain in reality. Thu, in orer for a topology contruction mechanim to be practical in P2P networ, it mut allow joining or rewiring of the noe by jut uing locally available information. Of coure, the cot of uing only local information i expecte to be lo of cale-freene (or any other eire characteritic) of the whole overlay topology, which will reult in lo of earch efficiency in return. In thi ection, we preent a practical metho uing local heuritic an no global information about the topology. Thi moel imitate the metho for fining peer in Gnutella-lie untructure P2P networ. In our moel, tarting with ome m +1fully connecte noe, at every time tep a new noe with m poible lin i ae to the networ an one ranomly choen noe i elete with probability μ. Since the noe are hortighte, i.e., they o not have global information about the networ, they can only chooe from a ubet of the networ (horizon) they contruct intea of the whole networ. The parameter an τ l are the value ue by the noe an enote the meaure of locality in joining an leaving, repectively. A newly ae noe, firt, elect a ranom exiting noe an contruct a et of noe reachable in hop or le from that noe. Then, thi new noe ranomly elect a noe from thi et an connect itelf with probability proportional to it egree. Thi probability i normalize by the total egree of the noe in the et. The new noe ranomly elect other noe in the et until it egree reache m. If no noe i left in the et to connect but the egree of the new noe i le than m, it elect another ranom noe from the networ an continue thi proce. In the eletion cae, the neighbor of the elete noe elect a noe ranomly from a et of noe reachable in τ l or le tep from the elete noe an connect by uing the preferential attachment rule. Here, in both cae noe cannot connect to other noe with egree equal to the har cutoff. There are pecial cae in thi moel: i) when, the horizon of the new noe contain only the ranomly electe noe an the preferential attachment rule i invali. In thi cae the new noe connect to thi ingle noe in the horizon if it egree i le than the har cutoff. ii) when 165

7 τ l the neighbor of the elete noe o not have any noe in their horizon o no rewiring occur. Thee noe jut loe one of their lin an they o nothing to compenate it. The moel typically become the preferential attachment with global information when value i large an τ l i zero an a BA networ with γ i obtaine. 5. Simulation In the previou ection, we introuce a framewor to invetigate the effect of join an leave procee in term of cale-freene of the topology being contructe within the context of a-hoc untructure P2P networ. Here, we tuy a number of meage-paing algorithm that can be efficiently ue to earch item in P2P networ utilizing the cale-free egree itribution in ample networ generate by our topology contruction algorithm. Thee earch algorithm are completely ecentralize an o not ue any in of global nowlege about the networ. We conier three ifferent earch algorithm: flooing (FL), normalize flooing (NF), an ranom wal (RW). Goal of our imulation experiment inclue: Effect of a-hocne on the earch efficiency in an uncooperative environment with har cutoff: Ahocne of noe joining or leaving the networ affect the earch efficiency, i.e., number of hit per unit time. Further, applying har cutoff on uch a-hoc calefree topologie reuce the egree itribution exponent. We are interete in oberving the effect of thi a-hocne an har cutoff on the earch efficiency for three earch algorithm, i.e., FL, NF, an RW. Thi exten our previou wor in [27], which focue on the effect of har cutoff only. A-hoc cale-free topology contruction with global v. local information: Though we howe in the previou ection that uing local information when a peer i joining yiel a le cale-free topology, the effect of thi on earch efficiency till nee to be he light on. Our imulation aim to invetigate thi too Search Algorithm We ue three earch technique to evaluate our a-hoc cale-free topologie: Flooing (FL): FL i the mot common earch algorithm in untructure P2P networ. In earch by FL, the ource noe, en a meage to all it nearet neighbor. If the neighbor o not have the requete item, they en on to their nearet neighbor excluing the ource noe [ee Fig. 1(a)]. Thi proce i repeate a certain number of time, which i uually calle time-to-live (). Table 1. Parameter of our topology contruction framewor Symbol Parameter Decription Range μ A-hocne of the noe [,1) Available information at join τ l Available information at leave Har cutoff 1 m Minimum egree (# of tub) 1 Normalize Flooing (NF): In NF, the minimum egree m in the networ i an important factor. NF earch algorithm procee a follow: When a noe of egree m receive a meage, the noe forwar the meage to all of it neighbor excluing the noe forware the meage in the previou tep. When a noe with larger egree receive the meage, it forwar the meage only to ranomly choen m of it neighbor except the one which forware the meage. The NF mechanim i illutrate in Fig. 1(b). In thi imple networ with m =2, the ource noe en a meage to it ranomly choen two neighbor an thee neighbor forwar the meage to their ranomly choen two neighbor. In the thir tep, the meage reache it etination. Ranom Wal (RW): RW or multiple RW have been ue a an alternative earch algorithm to achieve even better granularity than NF. In RW, the meage from the ource noe i ent to a ranomly choen neighbor. Then, thi ranom neighbor tae the meage an en it to ranomly electe one of it ranom neighbor excluing the noe from which it got the meage. Thi continue until the etination noe i reache or the total number of hop i equal to. A chematic of RW can be een in Fig. 1(c). RW can alo be een a a pecial cae of FL where only one neighbor i forware the earch query, proviing the other extreme ituation of the traeoff between elivery time an meaging complexity Reult We imulate the three earch algorithm FL, NF, an RW on the topologie generate by our framewor with three ifferent parameter: (i) a-hocne, <μ<1, (ii) available information uring join,, (iii) available information uring leave, τ l, (iv) har cutoff, > 1, an (v) minimum egree (number of tub), m. Thee parameter are lite in Table 1 a well. By aigning ifferent value to each of thee parameter, we generate topologie with 1 noe. We ue ifferent value from 1 to 1 (or jut a few in thi range), in aition to the natural cutoff, i.e., no har cutoff. We varie an τ l from to 3. Minimum egree value (or m) in our topologie were 1, 2, or 3. We tuie maller value of μ from to.3, reflecting no churn to 3% churn, repectively. We performe 5 166

8 P Number of hit Degree itribution for a hoc cale-free networ m, μ.3,, τ l, τ l, τ l =2, τ l =2 = P Degree itribution for a hoc cale-free networ m, μ.3, =5, τ l, τ l, τ l =2, τ l =2 1-5 τ j = τ j (a) m, (b) m, =5 (c) m, no cutoff P Degree itribution for a hoc cale-free networ m, μ.3, no cutoff Figure 3. Degree itribution over a-hoc noe (i.e., μ.3): P () for variou networ generate by our framewor for varying an τ l Flooing in A hoc SF Networ N 4, m, μ no, no, no, =2 =5, =5, =5, =2 =2, =2, =2, =2,,, =2 Figure 4. Flooing (FL) performance over topologie with m an no a-hocne. realization of our reult. We varie the (i.e., time-to-live) value of earch querie in FL an NF to the point we reach the ytem ize. To compare earch efficiencie of RW an NF fairly, we equate of RW earche to the number of meage incurre by the NF earche in the ame cenario. Thu, for the earch efficiency graph of RW when i equal to a particular value uch a 4, thi mean that the number of hit correponing to that =4 value i obtaine by imulating a RW earch with equal to the number of meage that were caue by an NF earch uing a value of 4. A imilar normalization wa one in [23] Effect on Degree Ditribution Our imulation reult how the effect of a-hocne an har cutoff on the egree itribution of the topologie. Figure 2 how the egree itribution of the topologie when there i no a-hocne, i.e., μ. Similarly, Figure 3 how the egree itribution when the noe are a-hoc with μ.3. It i nown that uing more global information (i.e.,, τ l, τ l, τ l =2, τ l =2 =2 nowing more of the networ topology) help to generate better cale-free topologie with lower power-law exponent. Thi phenomenon i clearly hown in Figure 2, i.e., the egree itribution hift from an Exponential one to a powerlaw one a increae from to 2 (i.e., the joining noe ue the topology information at a larger horizon). Thi i true for both m an m, though larger m mae the hift a little le apparent. Further, the har cutoff,, only affect thi proce by imply bouning the very large hub to the cutoff without affecting the tranition from Exponential to power-law. An intereting reult being reveale in Figure 3 i that τ l ha much more ignificant effect than in hifting the egree itribution from Exponential to power-law. Thi i even more apparent for maller value of the cutoff Effect on Search Efficiency In flooing by far the mot important parameter when there i no eletion in the networ i the cutoff which etermine the number of itinct noe one can reach from a noe, ee Figure 4. In thi cae, i alo an important parameter which change the networ from an exponential to a calefree one an give better efficiency in flooing. Our imulation alo how that thi effect can be relieve by increaing the minimum egree in the networ a it can be een in Figure 5. More interetingly, a-hocne play an important role in the efficiency of earch algorithm. Negative effect of the high a-hocne (high μ) can be eliminate by increaing the available information in rewiring, i.e., by increaing τ l in both flooing an normalize flooing, ee Figure 6. In ome cae in normalize flooing higher ahocne yiel better efficiency for enough high value of τ l. Here, we o not preent reult for ranom wal earch algorithm ince the qualitatively they are not ifferent than normalize flooing except that the ranom wal i more vulnerable to iolate cluter in the networ

9 1 Flooing in A hoc SF Networ N 4,, m, μ.2 1 Flooing in A hoc SF Networ N 4,, m, μ.2 1 Flooing in A hoc SF Networ N 4,, m, μ.3 Number of hit , τ l, τ l, τ l =2 =2 =2 Number of hit , τ l, τ l, τ l =2 =2 =2 Number of hit , τ l, τ l, τ l =2 =2 = (a) μ.1 (b) μ.2 (c) μ.3 Figure 5. Flooing (FL) performance over topologie generate with an m. Number of hit Normalize Flooing in A Hoc SF Networ N 4, m, τ J =2, μ.1 =2, μ.1 =5, μ.1 no, μ.1, μ.2 =2, μ.2 =5, μ.2 no, μ Number of hit Normalize Flooing in A Hoc SF Networ N 4, m, τ J, μ =2, μ =5, μ, μ.2 =2, μ.2 =5, μ (a) m, =2 (b) m, (c) m, Number of hit Normalize Flooing in A Hoc SF Networ N 4, m, τ J, μ.1 =2, μ.1 =5, μ.1, μ.2 =2, μ.2 =5, μ.2 Figure 6. Normalize Flooing (NF) performance over topologie generate with variou m,,an τ l value. 6. Summary an Dicuion In ummary, we wore on an a-hoc limite cale-free networ moel for untructure peer-to-peer networ. We firt evelope localize joining an leaving cheme for the peer an meaure the efficiency of earch algorithm uch a flooing an normalize flooing. By coniering the fact that the peer o want to tore too many lin information we alo impoe a har cutoff on the egree a noe can have an analyze it effect on the earch efficiency. We parameterize the locality of the joining an leaving cheme by two parameter: (for joining) an τ l (for leaving) which are the number of hop noe will ue to contruct et of noe from which they will ranomly chooe other noe an attempt to connect by uing the preferential attachment rule an by oberving the har cutoff. Typically, high value of thee parameter will mae the networ a preferential attachment networ with egree itribution exponent 3. We alo moele the ranom eletion of the noe by a probability parameter μ. Our earch imulation how that the negative effect of the low cutoff an high probability of eletion can be eae by increaing the minimum egree in the networ. Thi alo help one to avoi the pathological cae of m for which the networ will liely to have iolate cluter hinering the efficiency of the earch algorithm. To remey the negative effect of high value of μ which etroy the calefreene in the networ we enlarge the locality of the leaving cheme, i.e., increaing τ l for a fixe an cutoff will increae the efficiency of normalize flooing. Our fining are irectly applicable to current untructure P2P networ in which the peer leave the networ unexpectely an they have an upper limit for egree. Acnowlegment Thi wor wa upporte by the U.S. Department of Energy uner contract DE-AC52-6NA25396 an by the National Science Founation uner awar an Author woul lie to than Si Rener for fruitful icuion. Reference [1] Gnutella home page. [2] The Freenet Project. [3] The Napter home page. [4] R. Albert an A.-L. Barabái. Topology of evolving networ: local event an univerality. Phyical Review Letter, 85:5234,

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