Future Generation Computer Systems

Transcription

1 Future Generaton Computer Systems 29 (2013) Contents lsts avalable at ScVerse ScenceDrect Future Generaton Computer Systems journal homepage: Gosspng for resource dscoverng: An analyss based on complex network theory Stefano Ferrett Department of Computer Scence, Unversty of Bologna, Mura Anteo Zambon 7, Bologna, Italy a r t c l e n f o a b s t r a c t Artcle hstory: Receved 11 January 2012 Receved n revsed form 31 May 2012 Accepted 4 June 2012 Avalable onlne 23 June 2012 Keywords: Resource dscovery Unstructured overlays Complex networks Gossp protocols Modelng Ths paper analyzes the adopton of unstructured P2P overlay networks to buld resource dscovery servces. We consder a smple dstrbuted communcaton protocol, whch s based on gossp and on the local knowledge each node has about resources hold by ts neghbors. In partcular, upon recepton (or generaton) of a novel query, a node relays the message to those neghbors that have resources whose profle matches the query. Moreover, the node gossps the query to other remanng neghbors, so that the query can be dssemnated through the overlay. A mathematcal analyss s provded to estmate the number of nodes recevng the query (and consequently, the porton of query hts), based on the network topology, resource avalablty and gossp probablty. Results show that the use of unstructured networks, coupled wth smple dssemnaton protocols, represent a vable approach to buld P2P resource dscovery systems Elsever B.V. All rghts reserved. 1. Introducton The amount of servces and resources avalable n current computer networks has become so hgh that ther dscovery results as one of the more dffcult ssues to cope wth. Ths remans true whatever the type of the consdered resource mght be, e.g. some data content rather than a computng faclty. Dfferent ways are possble to dstrbute (and locate) resources on a network. The employed system archtecture, ts topology (.e. the graph resultng from the nteracton map among nodes of the network) and the consequent node organzaton mpose constrants on the typology of searchng methods to dscover these resources and nfluence ther performance. In partcular, resource dscovery mechansms can be bult on top of ether structured or unstructured archtectures. Structured networks are those where lnks are created based on the contents hold by nodes. Examples of structured archtectures are the tradtonal clent/server, tree-based and herarchcal content-dependent structures [1], Peer-to-Peer (P2P) systems bult usng Dstrbuted Hash Tables (DHTs) [2 5]. Conversely, n an unstructured P2P overlay network, lnks among nodes are establshed arbtrarly,.e. they do not depend on the contents beng dssemnated through the overlay [6]. These solutons are partcularly smple to buld. Thus, unstructured overlays may be useful n very dynamc contexts. Peers locally manage E-mal address: sferrett@cs.unbo.t. URL: ther connectons based on some general desred topology. Such a selected topology may vary dependng on the characterstcs the system should have. For nstance, a unform graph where nodes have all the same degree (.e. number of connected nodes) mght be useful to balance the communcaton load at peers. Conversely, a scale-free topology mght be preferred to create a robust overlay, wth a reduced network dameter (ths s obtaned at the cost of havng some nodes subject to hgher workloads) [7]. A man queston here s how queres can be dssemnated to locate resources effectvely. In ths paper, we show that a smple gossp protocol (augmented wth some addtonal local nformaton about the network) can be employed to buld effectve resource dscovery mechansms on top of unstructured P2P overlay networks. The proposed analytcal model s based on complex network theory. In partcular, accordng to the protocol, peers mantan local knowledge about ther neghbors,.e. each node s aware of ts resources and those of nodes whch are drectly lnked to t (ths requres very lttle management costs). The dscovery process passes through the lnks that compose the overlay. Hence, a node generatng a query can ask ts neghbors only. In turn, each tme a node receves a query, t relays the related message to those neghbors holdng resource tems matchng the query (f there s any). In addton, t gossps the message to other neghbors (whch do not own any matchng resource), so that the query can be dssemnated through the overlay. The proposed model s able to estmate the amount of nodes recevng the query and the amount of query hts,.e. how many resource tems can be found usng ths smple resource dscovery protocol. The approach s qute general; the network topology X/$ see front matter 2012 Elsever B.V. All rghts reserved. do: /j.future

2 1632 S. Ferrett / Future Generaton Computer Systems 29 (2013) can be set by defnng the node degree dstrbuton probablty. Dependng on the network topology, the resource avalablty and the gossp probablty, t s possble to understand f the query reaches a lmted amount of nodes, or f t s spread through the whole network,.e. t mght reach an nfnte amount of nodes. In substance, ths study demonstrates that the use of unstructured networks employng gosspng dssemnaton strateges, based on local decson processes, guarantees that queres can percolate through the network. Ths outcome promotes the development of vable resource dscovery mechansms over unstructured P2P archtectures. In order to valdate the effectveness of the mathematcal model, we have compared numercal outcomes wth those obtaned va smulaton. A dscrete event smulator has been bult, whch s able to mmc the dstrbuted communcaton protocol on top of a randomly generated unstructured overlay whose topology can be specfed usng a gven node degree probablty dstrbuton. We smulated a wde number of overlay networks, varyng the network topology and degree dstrbuton parameters, the network sze, the resource avalablty. We also vared the parameters characterzng the communcaton protocol,.e. the gossp probablty. Results obtaned va smulaton are comparable wth those comng from the analytcal model. As a fnal remark, t s worth pontng out that we are not suggestng here to replace completely structured and relable dstrbuted schemes, whch are usually employed to buld resource dscovery approaches, wth unstructured overlays usng gossp. Rather, our clam s that ths soluton represents an nterestng alternatve when dealng wth large scale and hghly dynamc systems. In ths case, n fact, the costs for managng and mantanng a structured (or centralzed) dstrbuted system s qute hgh. The proposed approach s partcularly effectve when dscovery mechansms are needed whch allow to locate a set of (one or several) resource tems deployed on a network, rather than all resources (n ths case a structured approach mght provde better results). In fact, the protocol allows to span the unstructured network wthout floodng t. Hence, through the descrbed protocol a requestng node obtans a set of avalable resources deployed on the unstructured system. Then, dependng on ther characterstcs, t can decde whch one to ask/utlze. The remander of ths paper s organzed as follows. Secton 2 dscusses some background and the related work. Secton 3 presents the system model. Secton 4 states the local protocol executed at each node. Secton 5 presents the mathematcal model. Secton 6 outlnes results comng from a numercal analyss and smulaton. Fnally, Secton 7 provdes some concludng remarks. 2. Background and related work Resource dscovery n dstrbuted systems can be mplemented usng dfferent technques [8]. Usually, the proposed solutons rely ether on centralzed archtectures (e.g. [9 11]) or P2P structuredoverlays (e.g. [12 15]) to store and dssemnate contents. In essence, these are archtectural solutons where lnks among nodes are created based on the contents hold by nodes. These solutons are wdely employed n several real use cases, but they mght have sgnfcant drawbacks. For nstance, accordng to the clent/server archtecture, all nodes connect to the sngle node that has the whole knowledge about resources present n the system,.e. the server. Ths soluton s really smple and can be qute effectve to look for a resource, but the server, whch acts as the ndex system, consttutes the bottleneck and the sngle pont of falure. Other classc structured searchng approaches, such as DNS, rely on tree based or herarchcal structures [1]. Whle more scalable, these solutons are typcally subject to hgh costs for the management of the overlay structure. Another usual structured approach conssts on the use of Dstrbuted Hash Tables (DHTs) n P2P systems [2 5]. In ths case, the dea s to characterze resources through a specfc key-word. Such nformaton s dstrbuted across the network, so that each node s responsble for a gven key. A hashng algorthm s used to dentfy the node who stores the searched resource. Hence, each query based on some partcular key passes through the correspondng node n the DHT. Dscovery mechansms on top of DHTs guarantee that a gven key can be found n the network (f present) n a lmted number of hops. However, DHT-based approaches need ntensve mantenance on hash table updates. Moreover, the use of key-words for searchng a resource strongly lmts the expressveness of the queres. On the other hand, the use of unstructured overlays enables scalable and effcent solutons that obvate the need for a structure [16 19]. In an unstructured P2P overlay, lnks among nodes are establshed arbtrarly. These archtectural solutons are partcularly smple to buld and manage, wth lttle mantenance costs, yet at the prce of a non-optmal organzaton of the overlay. Peers locally manage ther connectons to buld some general desred topology and lnks do not depend on the contents beng dssemnated [6]. Unstructured overlays are qute useful when the number of nodes s very hgh, wth very frequent topology changes and churns,.e. hgh number of nodes jonng and leavng the system. They are employed n several communcaton scenaros, rangng from classc dstrbuted P2P systems to wreless delay tolerant and opportunstc networks. As concerns the dscovery of resources, whle many mentoned P2P structured approaches (e.g. those employng DHTs) lmt the expressveness of a query by forcng nodes to search resources based on a lmted number of key-words, systems bult over unstructured overlays may support partal-match and complex queres. Ths s because resources can be characterzed n many dfferent forms. A profle can be assocated to a resource that descrbes t. Hence, queres can be made that explot such resource profles. A dsadvantage of such a knd of unstructured P2P solutons s that t mght be more dffcult to dscover rare tems, when compared wth more popular ones. Thus, ther performance depends heavly on the search mechansms mplemented on top of the communcaton overlay. To dssemnate queres several alternatves exst. These range from smple floodng or uncastng to the use of more sophstcated methods based, for example, on machne learnng [20]. For nstance, dscovery n Gnutella s based on a floodng protocol; dscovery requests are relayed to all neghbors untl a matchng resource s found or a tmeout occurs. It has been recognzed that such a floodng-based algorthm used n Gnutella does not scale, snce each query generates a huge amount of traffc and large systems quckly become overwhelmed by the query-nduced load. When we look at other possble approaches, a man dstncton s whether the search method s nformed or unnformed. Informed schemes rely on the partal knowledge of the network that nodes have dscovered prevously; and thus, these mechansms are based on some heurstcs to effectvely explot the knowledge each node has on the network. In contrast, n unnformed schemes nodes know nothng of the surroundng network; the network must be explored untl some resource s found. Ths can be done n a systematc way, e.g. breadth-frst search, depthfrst search, floodng. Alternatvely, random search protocols can be exploted, e.g. random walk, gossp-based probablstc forwardng, probablstc floodng [21 23]. Data replcaton strateges, coupled wth random walk search, are another opton [24 26,18]. Accordng to a random walk protocol, when a node receves a query, t relays t to a randomly chosen neghbor. A sequence of random walks can be executed f the number of results of a query s lower than a certan

3 S. Ferrett / Future Generaton Computer Systems 29 (2013) threshold. Wth respect to floodng, such protocol strongly reduces the number of propagated messages. Furthermore, the bottleneck rsk s lmted for the recever because durng a random walk, the maxmum number of responses that can be obtaned n a short tme s the depth of the random walk tself, whereas wth floodng the number of receved responses can be exponental [27]. Accordng to best-neghbor-based propagaton, queres are not flooded over the network, but drected to only some selected nodes,.e. to those for whch a query ht s more probable [28, 29]. A smlarty forwardng algorthm s employed that caches some nformaton on prevous requests and dscovery results. Informaton mantaned on the cache expres after some deadlne. Just a few works employ gossp solely to dscover contents (not general resources) [30,17,31]. These schemes are nspred by the theory of epdemcs, snce a node that gossps a content may be seen as t nfects ts neghbor. Weaker relablty guarantees are provded for better scalablty. The nterestng dea s that by employng gossp, dssemnaton strateges can be devsed whch ensure a small probablty of delverng contents just to a lttle porton of nodes, and a very hgh probablty of delverng contents to (almost) all destnatons [32 34]. The dea underlyng gossp protocols dates back to the orgnal USENET news protocol, NNTP, developed n the early 1980 s. Semnal works that employ gossp protocols to buld effectve dstrbuted systems have been presented n [32,35]. Gossp protocols can explot push or pull approaches. In a push protocol, each node gossps perodcally to dssemnate ts contents. Conversely, n a pull protocol, a node solcts the transmsson of nformaton from other nodes to compensate for loss of nformaton. Gossp algorthms employed n an unstructured network may represent an nterestng opton to dscover resources. As mentoned, despte the fact that these solutons are very smple and do not have nformaton on how to route queres, they provde statstcal guarantees on the number of nodes beng nvolved n a data dssemnaton process. Moreover, these solutons are very reslent to network changes and churns, snce they do not rely on the exstence of one or more nodes. A hgh level of scalablty s ensured; n fact, ther propertes are preserved as the sze of the system ncreases. Not only, nodes have a load that depends on ther degree (.e. the hgher the number of neghbors of a node, the hgher the number of messages the node wll probably send). The topology of the overlay has thus a strong nfluence on the performances of the content dssemnaton, and t can be selected based on the preferred characterstcs that the overlay network must guarantee. For nstance, scale-free networks have a very low dameter (.e. rangng from log log N to log N, beng N the number of nodes). Ths means that a message may requre very few hops to reach the other part of the network (f correct lnks are exploted). However, n these nets hubs (.e. peers wth hgher degrees) wll lkely sustan a hgher workload than the other low-degree nodes. Conversely, f a network has unform degree dstrbuton (where nodes have approxmately the same number of neghbor nodes), then the workload s equally shared among all peers. However, the dameter of the network ncreases, and so does the number of hops needed to cover the whole network wth a broadcast [36]. The queston we are dealng wth n ths paper s f gossp-based approaches (augmented wth local nformaton about neghbors) can be effectve to dscover resources n hghly dstrbuted and dynamc P2P scenaros. In partcular, the communcaton protocol we consder s a mx of a push gossp-based scheme and a bestneghbor-based propagaton, snce each node relays messages to those neghbors that ht the query and gossps the message to others. The proposed theoretcal framework allows to evaluate the performance of the presented communcaton protocol n dscoverng resources, gven a certan network topology wth gven statstcal characterstcs. Moreover, t allows to understand how the gossp probablty can be tuned to make the protocol effectve, gven the network topology. 3. System model We consder a P2P system bult on top of an unstructured overlay network. (Note that n the followng we use the terms peer and node as synonyms.) Peers are organzed n a way that does not depend on the dstrbuton of resource tems n the system [36]. Rather, a pseudo-random attachment process s employed that allows to shape the overlay based on a specfc network topology. Ths s a smple way to buld a network, snce no sophstcated overlay management technques need to be exploted. Moreover, there s no central component that controls the dssemnaton of generated queres. Notce that the overlay s unstructured,.e. there s no assocaton between nodes and the resources they hold. Hence, the system does not employ any routng tables or mechansms employng some knowledge base storng a lst of relatonshps between resources and nodes, as done n certan P2P systems lke NeuroGrd or Freenet. Each tme a node generates a novel query for a resource (dentfed by a profle), t dssemnates a message contanng t to ts neghbors (the algorthm s explaned n the next secton). Each node recevng such a message acts as a relay and forwards the query to other (neghbor) nodes. The dssemnaton s based on pure local decsons; n fact, peers employ a mxed strategy that combnes gossp together wth a local knowledge of (profles of) resources mantaned by ther neghbors. As a fnal remark, the model s thought to characterze the executon of the protocol at a gven nstant durng the evoluton of the network, assumng that network changes are slower than a gven executon of the communcaton protocol Overlay network We consder a set of nodes organzed as a P2P overlay network. Each node n s connected to a gven subset of nodes, whose number s specfed usng some probablty dstrbuton. 1 We do not mpose any restrcton on the overlay, whch can be generated usng any knd of algorthm and attachment protocol executed when peers jon the network. We denote wth p the probablty that a peer n has neghbors (the number of nodes connected to a node n s usually referred as ts degree). We assume that the overlay has a hgh number of nodes. Ths assumpton comes from the fact that the soluton we are studyng s thought for very large and hghly dynamcal systems. If the number of nodes s low, or n presence of a relatvely stable network, probably the use of an unstructured soluton mght be avoded, snce other approaches can be profcently employed, such as centralzed solutons or structured dstrbuted systems [37 39,20,40]. The hgh number of nodes, together wth the random nature of contacts among peers n the overlay, augments the probablty of havng a low clusterng n the network [7]. Queres are ncluded wthn messages spread through the overlay. The message contans addtonal nformaton also, such as the d of the node orgnatng the query (e.g. ts IP address, wth other addtonal applcaton dependent nformaton). When a query s submtted, we assume that drect communcaton occurs among neghbor nodes only. Hence, to dssemnate nformaton through the overlay, peers must act as relays and forward messages to ther neghbors. Once a query ht occurs,.e. the node has some resource tems matchng the query, the node can contact the orgnatng node drectly by sendng a message. 1 We use bold fonts to dentfy real enttes n the dstrbuted system, e.g. host nodes or message queres; all ths n order to dstngush them from mathematcal elements of the model, durng the dscusson.

4 1634 S. Ferrett / Future Generaton Computer Systems 29 (2013) It s clear that the topology of the overlay has a strong nfluence on the performance of the dscovery process [33]. For nstance, f a scale-free network s employed, then the network has a low dameter [41]. However, a scale-free net contans a nonneglgble fracton of peers whch mantan a hgh number of actve connectons, and hence they sustan a workload hgher than the other low-degree nodes [42 44]. Conversely, f a network has a more unform degree dstrbuton, then the workload s equally shared among all peers. However, the dameter of the network ncreases, and so does the number of hops needed to cover the whole network wth a broadcast [36]. The framework employed n ths work allows to assess how the topology of the overlay mpacts the effectveness of the dstrbuted protocol by specfyng the node degree probablty dstrbuton. We focus on the network coverage and on the ablty of the dssemnaton scheme to spread a query, dependng on the topology of the overlay Resource dscovery In ths paper, we focus on the searchng technque,.e. how to dstrbute the query n the unstructured network. We are not nterested n the query matchng process,.e. how to make proper queres and how these are matched aganst the exstng resources. Indeed, the dssemnaton protocol s ndependent to ths method. We can model the system as a set of resources representng whatever knd of data or faclty to be exploted by nodes. Each resource class s defned by a set of attrbutes whch specfy ts characterstcs. There can be several resource nstances (tems), descrbed by one or a set of key attrbutes wth an assocated value. Queres are made to locate resources matchng certan constrants. In general, we mght thnk that each resource tem has a profle descrbng t. A varety of descrpton languages exst to descrbe resources, dependng on the resource type. For nstance, f the resource s a computng faclty, a possble soluton s to use the Composte Capabltes/Preference Profles (CC/PP), or some dervaton of the protocol [45]. Conversely, f the resource s some data content, then t mght be descrbed by ts name combned wth addtonal metadata assocated to t. It s responsblty of the query matchng process to understand f, gven a query, a resource tem wth a gven profle matches that query. In any case, the dssemnaton of the query does not depend on how resources are characterzed and dentfed. 4. The protocol Two man actvtes are accomplshed by peers. The frst one s concerned wth the management of resources hold by the peer. The other one refers to the dstrbuton of a receved/generated query The resource management protocol The resource management protocol s very smple (see Algorthm 1). When a peer n holds a novel resource tem, t nforms ts neghbors (lnes 3 6 n the algorthm). In partcular, a message s sent specfyng that a novel resource s avalable wth a gven profle. In turn, when a node m receves a message contanng a declaraton that ts neghbor n holds a novel resource tem, m adds a related entry n ts neghbor table to store the resource profle (lne 14 n the algorthm). Ths way, each tme m receves a query that hts that resource tem, m forwards the query to n. When a resource tem becomes unavalable at a node, t nforms ts neghbors that wll remove the related entry (lnes 8 11 n the algorthm). In turn, upon recepton at n of a control message from a node m, statng that such resource tem s no longer avalable at m, the related entry s removed from n s cache (lne 16). Algorthm 1 Resource management protocol executed at node n 1: N n s neghbors 2: Requre: Novel resource tem avalable wth profle rp 3: msg = avalable, rp 4: for all m N do {send the resource profle to all neghbors} 5: send(msg, m) 6: end for 7: Requre: Resource tem wth profle rp no more avalable 8: msg = unavalable, rp 9: for all m N do {remove the profle} 10: send(msg, m) 11: end for 12: Requre: Recepton of a control message from a peer m for a resource profle rp 13: f avalablty of a resource tem rp then 14: addincache(m, rp) {new resource at m} 15: else 16: removefromcache(m, rp) {unavalable resource at m} 17: end f 4.2. The query dstrbuton protocol The dstrbuton of a query s based on a push protocol [33, 46]. Algorthm 2 shows the pseudo-code of the algorthm executed at each peer n when a query must be dssemnated. The used notaton s summarzed n Table 1. It s worth mentonng that such code descrbes only the dstrbuton of the query. We mplctly assume that another software module s n charge of analyzng the query, comparng t wth the profles of resource tems hold by the consdered node, and those cached n the neghbor table, and n case managng the query ht. As concerns the query dstrbuton, once a node n makes a novel query, or upon recepton of a novel query from a neghbor m, n checks f t has handled t already (we assume that each query can be unvocally dentfed through the message d of the orgnatng node). In such a case, n drops the query (lnes 1 3). Ths avods that multple copes of the same query are processed and dssemnated, thus lmtng the amount of messages n the network. Then, the query s processed and f there s a query ht the sender of the query s contacted (lnes 4 9). In partcular, the node that has the resource contacts the node that created the query. Ths happens through a drect communcaton between the two peers. Then, n forwards the query to ts neghbors, unless the (TTL) assocated to the message has reached a 0 value (lnes 10 13). (In ths last case, n fact, the message does not need to be forwarded elsewhere, snce the maxmum number of hops has been reached.) In partcular, n forwards the query to the subset of neghbors holdng some resource tems whose profle matches the query (lnes 14 18). Then, n consders the remanng set of ts neghbors,.e. those nodes that do not hold resource tems matchng the query. For each node n ths subset, n gossps the message wth a probablty γ 1 (lnes 19 23). An mportant aspect s concerned wth the (TTL) value, employed to avod that messages are forwarded forever n the net. In partcular, such (TTL) must be suffcently large to guarantee that the message can be spread through the whole network. Fndng the optmal value for the TTL s a very mportant ssue. In fact, several works employ very dfferent values to set such value. For nstance, the works presented n [33,46 49] employ n ther smulatons values of the TTL rangng from 6 up to 100. Floodng protocols usually set the value of the TTL equal to 7, as done n Gnutella. Ths s because, by floodng a message, the number of reached peers ncreases almost exponentally, and n many networks (e.g. scale free ones and small worlds) such value s wdely suffcent to cover the whole network [50,51].

5 S. Ferrett / Future Generaton Computer Systems 29 (2013) Table 1 Notaton. γ := gossp probablty f := probablty that a node forwards a query to neghbors f := probablty that followng a lnk, a node s reached that forwards a query to neghbors F := generatng functon of f F := generatng functon of f p := probablty that a peer has degree equal to q := excess degree probablty,.e. probablty that followng a lnk a node s reached whch has lnks other that the consdered one r := average number of nodes that receve a query r := probablty that peers receve a query, startng from a gven node r := probablty that peers receve a query, startng from a gven lnk R := generatng functon of r R := generatng functon of r σ := probablty that a node has a resource tem matchng the consdered query s := average number of owners of a matchng resource tem that receve the query durng the search process Algorthm 2 Query dstrbuton protocol executed at node n Requre: Query Q generated at n Q receved n a message relayed by a neghbor peer m 1: f Q already handled then 2: Return 3: end f 4: f QueryHt(Q) then 5: s = Sender(Q) 6: rf = ProfleMatchngResource(Q) 7: msg = avalable, rp 8: send(msg, s) 9: end f 10: decreasettl(q) 11: f TTL(Q) = 0 then 12: Return 13: end f 14: N n s neghbors \ m {m = NULL f Q orgnated at n} 15: I { N has a resource tem matchng e} 16: for all I do 17: send(q, ) 18: end for 19: for all N \ I do {gossp to the remanng neghbors} 20: f random() < γ then 21: send(q, ) 22: end f 23: end for In essence, the TTL mght be set based on an estmaton of the network dameter (.e. the largest value among the shortest paths between any two nodes n the network). Ths estmaton can be obtaned from the degree probablty dstrbuton, and n most knds of nets t s usually a low number. However, answerng the queston f such a value s suffcent to spread contents on the network (and n general to dentfy the optmal TTL value), s however not trval. In fact, when we consder the dameter of a network, we are focusng on the shortest paths among nodes, and n general ths does not mean that the average path length among nodes scales wth the dameter value. Ths happens n several complex networks. For nstance, t s recognzed that gven two Web pages, they are just a few clcks away f the shortest path s traced (small world phenomenon). However, very long alternatve paths exst between two Web pages; and n general, t mght be dffcult to select the shortest path only wth local nformaton. Moreover, n the matter of ths ssue, a related aspect s that the proposed protocol mposes that each node handles a gven message only once (nodes cache the ds of handled queres); hence, queres do not make cycles. Furthermore, f the network has a low clusterng, then locally t appears as f t has a tree-lke structure. Therefore, each tme a node selects a neghbor to gossp, then t lkely chooses to follow a path that cannot be taken otherwse. All ths (.e. the possblty of havng long paths and the avodance of redundant transmssons for a query) mght suggest that a hgh value of the TTL should be preferred, snce a too low value of the TTL mght cause that the query ends along a path, wthout beng spread through the network. On the other hand, f a too hgh value of the TTL s selected, t mght happen that whle a query s travelng through the network, a node, that has handled t already, overwrtes/cancels the nformaton on that query t has on ts cache; and thus, f that node receves that query once agan, t reprocesses the query. In substance, the value of the TTL should be tuned based on several factors, such as the network topology, the cache sze of nodes and, probably most mportant, the gossp probablty. In fact, the hgher the gossp probablty (.e. the protocol resembles a floodng) the lower the TTL needed to spread a message through the network. 5. System analyss In ths secton, we analyze the performance of the decentralzed P2P protocol presented n the prevous secton. We specfcally focus on the coverage of the overlay. Ths allows to estmate the amount of query hts of a resource search s. We denote wth σ the probablty that a node has a resource tem,.e. σ represents the porton of nodes n the overlay havng some resources matchng the needs of the peer makng the request. We model each sngle query dssemnaton as a standalone actvty. In other words, the model treats the dstrbuton of generated queres as ndependent tasks. Ths s a correct assumpton f peers have a buffer cache whose sze s suffcently large to handle smultaneous queres passng through t. Otherwse, the model should be extended to consder possble buffer overflows. Of course, upon a query ht, the two peers can drectly communcate to exchange/provde use of the resource. In partcular, the node that has the resource contacts the node that created the query. Ths happens through a drect communcaton between the two peers, that does not nfluence the rest of the network. For ths reason, there s no need to consder such part of the protocol n the model. We consder networks wth a large number of nodes. Followng the approach presented n [41,7], we assume that lnks among nodes are randomly generated, based on a gven node degree dstrbuton [52]. Ths does not represent a problem, snce the overlays we are consderng here are synthetc communcaton networks, whch can be bult usng whatever algorthm chosen durng the network desgn phase. A consequence of the random nature of the attachment process s that, regardless of the node degree dstrbuton, the probablty that one of the second neghbors (.e. nodes at two hops from the consdered node) s also a frst neghbor of the same node, goes as N 1, beng N the number of nodes n the overlay. Hence, ths stuaton can be gnored snce the number of nodes s hgh.

6 1636 S. Ferrett / Future Generaton Computer Systems 29 (2013) Degree and excess degree dstrbutons We denote wth p the probablty that a peer n has degree equal to. Startng from n, another measure of nterest s the number of connectons (lnks) that a node m, whch s a neghbor of n, may provde, other than the one that connects m wth n. In partcular, the probablty that, followng a lnk n the overlay, we arrve to a peer m that has other lnks (hence ts total degree s + 1) s q = ( + 1)p +1. jp j j The probablty q s often referred as the excess degree dstrbuton [41]. Probabltes p and q represent two smlar concepts.e. the number of contacts of a consdered peer (ts degree), and the number of contacts obtaned followng a lnk (ts excess degree), respectvely. In the followng, we ntroduce measures obtaned by consderng the degree p of a node, and consderng the excess degree q of a lnk. In ths last case, wth a slght abuse of notaton we denote all the probabltes/functons related to the excess degree wth the same letter used for the degree, wth an arrow on top of t, just to recall that the quantty refers to a lnk Probablty of dssemnaton Gven a peer n n charge of relayng a query, the probablty that n forwards t to of ts neghbors s f = [σ + (1 σ )γ ] j p j j [(1 σ )(1 γ )] j, (1) whch s obtaned by consderng all the possble cases of n, havng a degree hgher than, whch forwards the query to neghbors ether because there s a query ht (wth probablty σ ), ether because there s no query ht but n decdes to gossp the query to the node nevertheless (wth probablty (1 σ )γ ). Moreover, n does not gossp the query to ts remanng j neghbors, that would not generate a query ht (wth probablty (1 σ )(1 γ )). In the rest of the dscusson, for the sake of a more readable presentaton, we denote Γ = σ + (1 σ )γ and 1 Γ = (1 σ )(1 γ ). A smlar reasonng can be made to measure the probablty that, followng a lnk we arrve to a node that forwards the query to other nodes. Ths probablty s readly obtaned by substtutng, n (1), p j wth q j,.e. f = j Γ q j (1 Γ ) j. (2) j To proceed wth the reasonng, we need to ntroduce the generatng functons for f, f, as well as for p, q,.e. G(x) = p x G (x) = q x, (3) F(x) = f x F (x) = f x. (4) If we consder the F generatng functon, we have F(x) = f x = j Γ x p j (1 Γ ) j j = j j p j Γ x (1 Γ ) j j = j =0 p j (Γ x + 1 Γ ) j = G Γ x + 1 Γ. (5) One mght notce that all the coeffcents of the ntroduced generatng functons are probabltes. In fact, G(1) = p = 1, as well as F(1) = f = 1, and so on. Now, t s also possble to evaluate the average of the values f, by calculatng the dervatve of F measured at x = 1, snce F (1) = f [53]. We have F (x) x=1 = dg Γ x + 1 Γ dx x=1 = Γ G (1) = Γ p, (6) where p s the mean node degree. From a smlar reasonng, F (x) x=1 = Γ G (1) = Γ q, (7) where q s the mean value of the excess degree, that s [7] q = ( + 1)p +1 q = jp j = ( 1)p jp j j j = p2 p. (8) p 5.3. Number of recevers and query hts We can now consder the whole number of nodes reached by a message startng from a gven node, regardless of the number of hops. Let denote wth r the probablty that peers receve a query, startng from a gven node. Smlarly, denote wth r the probablty that peers are reached by the query dssemnaton, startng from a lnk. In general, r can be defned usng the followng recurrence, r 0 = 0, r +1 = f j r a1 r a2 r aj. (9) j 0 a 1 +a 2 + +a j = Eq. (9) can be explaned as follows. It measures the probablty that followng a lnk we dssemnate the query to + 1 peers. (The case r 0 s mpossble, snce at the end of a lnk there must be a node.) In general, one peer s that reached at the end of the lnk tself. Then, we consder the probablty that the peer has other j lnks (varyng the value of j). Each lnk k allows to dssemnate the query to a k peers, and the sum of all these reached peers equals to. Smlarly, we can calculate r as follows r 0 = 0, r +1 = j 0 f j a 1 +a 2 + +a j = r a1 r a2 r aj. (10) In ths case, we start from the peer tself, consderng t has a degree equal to j; and as before, from ts j lnks we can reach other peers, globally. The use of generatng functons may be of help to handle these two equatons [53]. In fact, f we consder the generatng functons for r and r, R(x) = r x, R (x) = r x (11)

7 S. Ferrett / Future Generaton Computer Systems 29 (2013) then, after some manpulaton typcal for generatng functons (e.g. [7]) we arrve to the followng result R (x) = x f j [ R (x)] j = x F ( R (x)) (12) j 0 and, smlarly, R(x) = x f j [ R (x)] j = xf( R (x)). (13) j 0 From the generatng functons, we mght recover the elements r, r composng them. Unfortunately, Eqs. (12) and (13) may be dffcult to solve, dependng on the degree probablty dstrbuton p whch controls the whole ntroduced measures [7]. But actually, we are not nterested that much n the sngle values of r, r. In fact, t s easer and more useful to measure the average number r of peers that receve a gven query through the dssemnaton protocol. To ths am, we can employ the typcal formula for generatng functons r = R (1) [53]. In fact, takng the frst equaton of (11), dfferentatng and evaluatng the result for x = 1, and snce r 0 = 0, we have R (x) x=1 = r, whch s the mean value related to the dstrbuton of r coeffcents. We already observed that the coeffcents of the ntroduced generatng functons are probabltes, and thus F(1) = f = 1, and smlarly F (1) = 1, R(1) = 1, R (1) = 1. Hence, takng (13) and dfferentatng r = R (1) = F( R (x)) + xf ( R (x)) R (x) x=1 = 1 + F (1) R (1). (14) Smlarly, from (12), R (1) = F ( R (x)) + x F ( R (x)) R (x) Thus, x=1 = 1 + F (1) R (1). (15) 1 R (1) = 1 F. (16) (1) Ths last equaton allows to fnd the fnal formula for r, r = 1 + F (1) 1 F (1) Γ p 2 = 1 + (1 + Γ ) p Γ p 2. (17) Now, r s the number of peers that receve the query, regardless f these nodes have a resource tem matchng that query. To obtan the average number of query hts s, t suffces to multply r by the probablty that a peer has a resource tem matchng that query σ, hence obtanng s = σ r. From ths last formula, t s clear that r s the man measure to consder here, snce t s the one on whch the effectveness of ths resource dscovery protocol depends. For ths reason, n the expermental evaluaton we wll focus on ths value, based on the system characterstcs. Then, the presence of resources whose profle matches the query σ s qute mportant as well. Indeed, ths equaton confrms the clam that n an unstructured overlay t s more dffcult to dscover rare tems Dscusson on percolaton As t s qute typcal n complex network theory, t s actually easer to examne nfnte networks rather than just large ones. The analyss of nfnte networks, under condtons smlar to those of large scale networks, allows to understand mportant peculartes of the real networks and on protocols executed by ther nodes. For nstance, t s possble to understand f a message can percolate through the network. Ths assumpton s perfectly reasonable n our scenaro, snce we consder very large dynamcal systems (wth a number of nodes that tends to nfnty) where peers know only ther neghbors and manage contents based on local knowledge about ther neghbors. Eq. (17) has a dvergence when (1 + Γ ) p = Γ p 2, whch sgnfes that the query reaches an nfnte number of nodes,.e. the query percolates through the network. By lookng at the parameters, ths stuaton depends, frst, on the nodes connectvty,.e. the node degree probablty dstrbuton p. In fact, the degree probablty dstrbuton determnes f the overlay has a gant component (.e. the largest subset of connected nodes whch scales wth the network sze, and thus has a number of nodes whose lmt tends to ), rather than beng parttoned nto a set of components of lmted sze [7]. The query can be spread to a large (nfnte) number of nodes only when there s such a gant component; otherwse,.e. when the network s parttoned nto a hgh number of components of lmted sze, the query can be sent to a lmted number of nodes only. Studes exst that allow to understand how to buld networks wth a gant component [7,54]. Second, the value of σ has nfluence on both the number of nodes to be reached holdng some resource tem matchng the query and on the dssemnaton of queres. In fact, the hgher σ the hgher the probablty that a node has some of ts neghbors whch generate a query ht; these nodes wll be recevers of the query and subsequently they wll act as relays for such query. Thrd and fnal, the gossp probablty γ determnes f the message query s spread through the network even when the amount of nodes holdng a resource tem n the overlay for a gven query s small,.e. when σ has a very low value. Of course, settng γ = 1 allows to flood the query to the whole component (from whch the query has been orgnated). Ths mght represent a far choce when the network has a tree-lke structure, or when the network has a very low clusterng. Conversely, a low value for γ should be employed when there are loops n the overlay. A completely dfferent scenaro s concerned wth the stuaton when the network s formed by lmted clusters only (there s no gant component). In such a case, n fact, the number of reached nodes does not grow proportonally wth the network sze, and a fnte number of nodes mght receve a query. 6. Expermental results Ths secton presents an assessment performed to valdate the model dscussed n the prevous secton and evaluate whether effectve resource dscovery mechansms can be bult on top of an unstructured P2P system. The evaluaton s performed by consderng the analytcal model and results obtaned through a smulaton of the dstrbuted protocol. The two approaches provde smlar outcomes. In partcular, when the theoretcal model estmates that an nfnte amount of nodes s reached through the dssemnaton, smulatons show that a sgnfcant porton of the smulated network receves the queres, as expected. For the reasons explaned above, the focus here s on network coverage. In fact, snce we employ a probablstc model, from r we can recover the amount of query hts, represented by a gven percentage of the recevers.

8 1638 S. Ferrett / Future Generaton Computer Systems 29 (2013) Fg. 1. Number of recevers and query hts: topology based on a Posson degree dstrbuton wth mean λ = 5. Another mportant metrc to consder s the number of messages sent n the network. In ths sense, the protocol ensures that peers dssemnate a gven query at most once. Moreover, the tree-lke structure of the overlay lmts that multple copes of the same query are receved by a peer. We consder two knds of networks,.e. random graphs and scale-free networks. Accordng to a random graph, peers have the same probablty to attach to other lnks. In substance, when the overlay s generated, a lnk between each par of peers s created wth a certan probablty p. Hence, based on ths model the average degree s k = pn, beng N the network sze. It s well known that when the number of peers N s large, nodes degrees may be well characterzed usng a Posson dstrbuton k e k. Several works! employ ths constructon tool for generatng random graphs [7]. On the other hand, scale-free networks ganed a lot of nterest n recent years. These networks are characterzed by a degree dstrbuton followng a power law p λ. They are characterzed by the presence of hubs,.e. nodes wth degrees hgher than the average, that have an mportant mpact on the connectvty of the net. The nterest on scale-free networks n ths work relates to the fact that several P2P systems are ndeed scale-free networks [43,41] Theoretcal model We employed the framework presented n Secton 5 to assess the performance of the dssemnaton protocol, based on the overlay network topology,.e. node degree dstrbuton, the probablty σ that a node has a resource tem matchng the query, and the gossp probablty γ. Fg. 1 shows the number of nodes recevng a query when the unstructured overlay has a topology based on a Posson node degree dstrbuton wth mean value λ = 5 (we tested the framework wth other λ values, obtanng smlar results). Lnes n the chart correspond to the whole number of recevers (.e. relay nodes and query hts), whle ponts correspond to the number of query hts. Results are obtaned varyng the value of σ (on the x-axs),.e. the porton of nodes n the overlay that have resource tems that would correspond to a query ht. From these two fgures t s easy to see that, for each specfc γ value, there s a phase transton,.e. as σ s vared there s an abrupt ncrement on the number of recevers (and query hts), passng from a lmted value to,.e. the query percolates through the network. Ths phase transton depends on the parameters used to set the dstrbuted system. In fact, the value of σ not only represents the resource avalable probablty, but t nfluences the query dstrbuton n the overlay also (a node forwards wth Fg. 2. Number of recevers and query hts: scale-free topology wth exponent λ = 3.3. probablty 1 the query to each of ts neghbors that have matchng resource tems). Fnally, the value of γ does not change the trend of the curves; bascally, the hgher γ the smaller the value of σ to have a transton. Smlar consderatons can be made for Fg. 2, where the estmated amount of recevers and query hts s reported for a scale-free network wth a degree dstrbuton p λ, wth λ = 3.3. Also n ths case, each curve corresponds to a specfc γ value, whle varyng σ. The chart shows that for each curve there s a phase transton, where the number of recevng nodes passes from a lmted (low) value to an nfnte number. As mentoned, we are employng a probablstc model and the nterest s on the trend of the results, based on a gven network topology and on the behavor of the protocol. Results show that, gven a number of recevers of a gven query, the amount of query hts s proportonal to ths value Fndng one resource, at least In many P2P resource dscovery systems, nodes look for a sngle resource to retreve/acqure/explot. In other words, n certan stuatons we are not nterested n fndng as many resources as possble; rather, a sngle query ht s suffcent. It s clear that the lower the resource avalablty σ the more dffcult s to fnd a gven resource and thus the hgher the number of nodes to ask n order to retreve such a resource. It s worth notcng that, as shown n the prevous results, above the phase transton the number of query hts dverges. Hence, t s suffcent to employ a smple gossp communcaton protocol to fnd a sngle resource. In any case, gven a network topology and a specfc settng for the σ parameter, the level of message dssemnaton on the network s controlled by the gossp probablty γ. In ths secton, we show the mnmum value of γ to guarantee that at least one resource s found, gven a certan overlay topology and a resource avalablty. Fg. 3 shows such a value for dfferent settngs of σ, when the network topology s a Posson degree dstrbuton wth mean λ = 5. Outcomes demonstrate that t s suffcent to employ very low values of the gossp probablty to fnd a resource, even when the resource avalablty s low. Ths confrms the effectveness of the proposed approach. As concerns scale-free networks, we recall that wth hgher values of λ the moments of the degree dstrbuton dverge,.e. when λ > 2 the mean dverges, when 2 < λ < 3, the network has a gant component and the mean s fnte but the varance and hgher moments dverge [55]. Hence, n these cases the query easly percolates through the network and resources are

9 S. Ferrett / Future Generaton Computer Systems 29 (2013) Fg. 3. Mnmum γ to fnd at least one resource: topology based on a Posson degree dstrbuton wth mean λ = 5. For hgher values of σ an nfnte number of resources s found. Fg. 4. Mnmum γ to fnd at least one resource: scale-free topology wth exponent λ = 3.3. For hgher values of σ an nfnte number of resources s found. found wth hgh probablty. For ths reason, we put the focus on overlays wth a lower value of λ, wth respect to those cted here above. Fg. 4 shows the mnmum value of γ when the overlay topology s a scale-free network wth an exponent λ = 3.3. Based on ths consdered value of λ, t s possble to see that when we are dealng wth rare resource tems, a hgh gossp probablty γ s needed to ensure that a resource s found. However, as σ augments, the needed value of γ decreases Smulaton In order to assess the theoretcal model proposed n the paper, we have bult a dscrete-event smulator mmckng the presented protocol. The smulator was wrtten n C code. Pseudorandom number generaton was performed by employng the GNU Scentfc Lbrary, a lbrary that provdes mplementaton of several mathematcal routnes for numercal and statstcal analyss [56]. The smulator allows to test the behavor of a gven amount of nodes employng the protocol explaned n Secton 4. The smulator generates a random network based on a chosen degree dstrbuton. In partcular, once havng (randomly) assgned a specfc target degree to each node, usng the selected degree dstrbuton, a random mappng s made so that lnks are created untl each node has reached ts own target degree. Durng the ntalzaton phase, for each node a random choce was made to Fg. 5. Model vs. smulaton: topology based on a Posson degree dstrbuton wth mean λ = 5. dstrbute resources; the resource avalablty was set based on a probablty σ,.e. for each network node, the resource was present wth probablty σ. We vared the network topology, the number of nodes and statstcal parameters characterzng the network degree dstrbuton. For each network settng, we repeated the smulaton usng a corpus of 20 dfferent randomly generated networks. For each network, we analyzed the dssemnaton of 400 queres sent by random nodes. In the results that follow, for each generated network we show the average number of recevng nodes,.e. query hts and relays; ths number allows to understand f the dstrbuted protocol s able to dssemnate the query through the unstructured network, usng the presented protocol Posson degree dstrbuton Here, we show results for networks generated through a Posson degree dstrbuton. Fg. 5 shows results obtaned from smulaton and the theoretcal model. We smulated dfferent corpuses of networks, varyng the number of nodes and the value of the gossp probablty γ. Each pont n the chart corresponds to the average number of recevers for a smulated network. The lne corresponds to the theoretcal value measured usng Eq. (17). It s possble to observe that all results from the smulatons lye near the theoretcal value, regardless on the consdered number of smulated network nodes. Hence, the model s able to capture the behavor of the dstrbuted protocol. Fgs. 6 and 7 show results obtaned n our smulatons when γ = 0.1 (resp. σ = 0.1), whle varyng σ (resp. γ ), above the phase transton. Accordng to the model, the system s above the phase transton. Hence, assumng an nfnte number of nodes n the network, an nfnte number of nodes receve the message query, and due to the σ settng, an nfnte number of query hts s obtaned. As concerns smulatons, nstead, we expect that a nonneglgble porton of nodes s reached durng the dssemnaton of a query. Of course, snce the dssemnaton s based on rather low values of γ, σ probabltes, and snce the network clusterng of these consdered networks s qute low (a random attachment process was employed to buld lnks n the network [41,7]), t s unlkely that all network nodes receve the query beng dssemnated. In fact, because of the tree-lke structure of the network, every tme a lnk s not employed to dssemnate the message, t s lkely that some branch (and consequently some sub-graph) of the overlay s cut away. Indeed, results confrm our outlook. A non-neglgble porton of nodes s reached n each confguraton (wth respect to the network sze). Yet, the whole overlay s not covered completely. The amount of the reached nodes ncreases wth the vared parameter σ (resp. γ ).

10 1640 S. Ferrett / Future Generaton Computer Systems 29 (2013) a=6, b=0.4 a=4, b=0.4 a=2.5, b=0.5 Number of Peers Fg. 6. Model vs. smulaton: Posson degree dstrbuton, λ = 5, varyng σ above the phase transton. The chart reports the number of recevng nodes through smulaton. The theoretcal model returns an nfnte number of nodes (beng the modeled overlay an nfnte graph), not shown here Degree Fg. 8. Degree dstrbuton of some scale-free networks usng the constructon method proposed n [57]. Fg. 7. Model vs. smulaton: Posson degree dstrbuton, λ = 5, varyng γ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). Smlar results were obtaned for dfferent networks bult varyng the statstcal parameters of the random graph (not shown here). All ths confrms that the protocol s able to spread a gven query n the network n random graphs wth Posson degree dstrbutons Scale-free networks To buld scale-free networks, our smulator mplements a constructon method whch has been proposed n [57]. The nterestng aspect of ths algorthm s that t dffers from other proposals, whch buld networks wth a power law dstrbuton by contnuously addng novel nodes and edges, hence havng networks that grow n tme [42]. Conversely, the method n [57] bulds a network of fxed sze, characterzed by two parameters a, b. More specfcally, the number of nodes y whch have a degree x satsfes log y = a b log x,.e. y = ea. Thus, the total number x of nodes of the generated network s b a e b e a N = x, b x=1 beng e a b the maxmum possble degree of the network, snce t must be that 0 log y = a b log x. Once the number of nodes and Fg. 9. Model vs. smulaton: scale-free network, a = 6, b = 1, varyng γ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). ther degrees have been determned, edges are randomly created among nodes untl nodes reach ther desred degrees. Fg. 8 shows some examples of networks bult wth our smulator, mplementng the constructon method proposed n [57]. In partcular, the chart reports, for three dfferent settngs of a, b, the number of nodes whch have a gven degree, n a log log scale. It s possble to apprecate how such dstrbutons are almost lnear n a log log scale, hence confrmng they all follow a power law functon. As made above for random graphs, Fgs. 9 and 10 show results obtaned n our smulatons when we employ a scale-free network topology, wth γ = 0.1 (resp. σ = 0.1), whle varyng σ (resp. γ ), above the phase transton. Agan, based on the model an nfnte number of recevers s reached (assumng a network of nfnte sze). From the smulatons, a non-neglgble porton of nodes s reached durng the dssemnaton of queres, that ncreases together wth the γ (resp. σ ) parameter. Indeed, t s nterestng to observe that when γ = 0.6, σ = 0.1 almost all network peers receve the query durng the dssemnaton, and thus, almost all owners of some matchng resource tem receve the queres. In the scenaros reported n the pctures, n fact, we employed scale-free networks generated through a = 6, b = 1, resultng n networks composed of 2482 nodes. In ths case, smulaton results provde average results above 2200 nodes. A smlar behavor s obtaned

11 S. Ferrett / Future Generaton Computer Systems 29 (2013) Fg. 10. Model vs. smulaton: scale-free network, a = 6, b = 1, varyng σ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). Fg. 12. Model vs. smulaton: scale-free network, a = 6, b = 1.1, varyng σ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). Fg. 11. Model vs. smulaton: scale-free network, a = 6, b = 1.1, varyng γ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). Fg. 13. Model vs. smulaton: scale-free network, a = 6, b = 1.2, varyng γ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). when σ = 0.6, γ = 0.1. Agan, ths result s n accordance wth the outcomes from the model, statng that an nfnte number of nodes s reached wth these settngs. Fgs show smlar results for dfferent networks settngs. A sgnfcant porton of network nodes s reached, whose sze ncreases together wth the γ, σ values. Agan, all ths confrms that the theoretcal model s able to predct that a gven query, dstrbuted n an unstructured P2P system, can percolate through the whole overlay. 7. Conclusons Ths paper analyzed the performance of an unstructured P2P overlay network that explots a very smple gossp-based dssemnaton strategy to perform a P2P resource dscovery. The mathematcal analyss has been performed by resortng to the complex network theory. Results show that by tunng the gossp probablty, t s possble to spread queres through the overlay, wthout the need to resortng to sophstcated dssemnaton strateges bult on top of costly structured dstrbuted systems. Ths s true when networks are large n sze and the number of owners of some resource tems matchng the query s not neglgble. Of course, the use of more sophstcated approaches, such as adaptve dssemnaton schemes bult on top of gossp-based strateges [33], Fg. 14. Model vs. smulaton: scale-free network, a = 6, b = 1.2, varyng σ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). can be useful as well, yet at the cost of addtonal nteractons and communcaton among nodes. In ths work we focused on the network coverage. As concerns the communcaton overhead, t s evdent that the use of more

12 1642 S. Ferrett / Future Generaton Computer Systems 29 (2013) Fg. 15. Model vs. smulaton results: scale-free network, a = 6, b = 1.3, varyng γ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). Fg. 18. Model vs. smulaton: scale-free network, a = 6, b = 1.4, varyng σ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). Fg. 16. Model vs. smulaton: scale-free network, a = 6, b = 1.3, varyng σ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). Fg. 19. Model vs. smulaton: scale-free network, a = 6, b = 1.5, varyng γ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). Fg. 17. Model vs. smulaton: scale-free network, a = 6, b = 1.4, varyng γ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). Fg. 20. Model vs. smulaton: scale-free network, a = 6, b = 1.5, varyng σ above the phase transton. Number of recevng nodes obtaned through smulaton (the model returns an nfnte sub-graph). costly solutons (n terms of network management and node workload), such as centralzed approaches or structured overlays, would provde better performances. In any case, the protocol consdered n ths work lmts the amount of messages sent n the network, snce each node relays a gven query only once. Hence, no duplcate transmssons occur on a lnk. Moreover, the

13 S. Ferrett / Future Generaton Computer Systems 29 (2013) low clusterng guarantees that tree-lke overlays are obtaned; ths lmts the possblty that a peer receves multple messages contanng the same query. Ths s accomplshed wthout the need (and the costs) of mantanng a structured overlay. Indeed, the proposed approach represents an nterestng alternatve to determnstc algorthms executed on top of structured overlays when dealng wth large scale and hghly dynamc systems. References [1] D.A. Tran, T. Nguyen, Herarchcal multdmensonal search n peer-to-peer networks, Comput. Commun. 31 (2008) [2] S. Basu, S. Banerjee, P. Sharma, S.J. Lee, Nodewz: peer-to-peer resource dscovery for grds, n: Proc. of IEEE/ACM GP2PC 05, pp [3] M. Ca, M. Frank, J. Chen, P. Szekely, Maan: a mult-attrbute addressable network for grd nformaton servces, J. Grd Comput. (2003) 184. IEEE Computer Socety. [4] E. Carln, M. Coppola, P. Dazz, D. Laforenza, S. Martnell, L. 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14 1644 S. Ferrett / Future Generaton Computer Systems 29 (2013) [48] J. Sacha, J. Dowlng, R. Cunnngham, R. Meer, Dscovery of stable peers n a self-organsng peer-to-peer gradent topology, n: F. Elassen, A. Montresor (Eds.), Dstrbuted Applcatons and Interoperable Systems, n: Lecture Notes n Computer Scence, vol. 4025, Sprnger, Berln, Hedelberg, 2006, pp [49] H. Tanta-nga, E.E. Mlos, V. Kešelj, Self-organzng peer-to-peer networks for collaboratve document trackng, n: Proceedngs of the 1st ACM Internatonal Workshop on Complex Networks Meet Informaton & Knowledge Management, CNIKM 09, ACM, New York, NY, USA, 2009, pp [50] S. Jang, L. Guo, X. Zhang, H. Wang, Lghtflood: Mnmzng redundant messages and maxmzng scope of peer-to-peer search, IEEE Trans. Parallel Dstrb. Syst. 19 (2008) [51] M. Rpeanu, A. Iamntch, I. Foster, Mappng the Gnutella network, IEEE Internet Comput. 6 (1) (2002) [52] E.A. Bender, E.R. Canfeld, The asymptotc number of labeled graphs wth gven degree sequences, J. Combn. Theory Ser. A 24 (1978) [53] H.S. Wlf, Generatngfunctonology, second ed., Academc Press, Inc., [54] A.D. Flaxman, D. Gamarnk, G.B. Sorkn, Embracng the gant component, Random Structures Algorthms 27 (2005) [55] R. Cohen, S. Havln, D. Ben-Avraham, Structural Propertes of Scale Free Networks, Wley-VCH, [56] G.P. Contrbutors, GSL GNU scentfc lbrary GNU project free software foundaton, FSF, [57] W. Aello, F. Chung, L. Lu, A random graph model for power law graphs, Experment. Math. 10 (2000) Stefano Ferrett s an assstant professor at the Department of Computer Scence, Unversty of Bologna. He receved the Laurea degree (wth honors) and the Ph.D. n Computer Scence from the Unversty of Bologna respectvely n 2001 and n He s the char of the Workshop on DIstrbuted SImulaton & Onlne gamng (DISIO). Hs current research nterests nclude dstrbuted and multmeda systems, synchronzaton algorthms, complex and computer networks, wreless networks, dstrbuted smulaton.