Peer-to-peer systems have attracted considerable attention

Transcription

1 Reputaton Aggregaton n Peer-to-Peer etwork Usng Dfferental Gossp Algorthm Ruhr Gupta, Yatndra ath Sngh, Senor Member, IEEE, arxv:20.430v4 [s.i] 28 Jan 204 Abstrat Reputaton aggregaton n peer to peer networks s generally a very tme and resoure onsumng proess. Moreover, most of the methods onsder that a node wll have same reputaton wth all the nodes n the network, whh s not true. Ths paper proposes a reputaton aggregaton algorthm that uses a varant of gossp algorthm alled dfferental gossp. In ths paper, estmate of reputaton s onsdered to be havng two parts, one ommon omponent whh s same wth every node, and the other one s nformaton reeved from mmedate neghbours based on the neghbours dret nteraton wth the node. The dfferental gossp s fast and requres less amount of resoures. Ths mehansm allows omputaton of ndependent reputaton value by a node, of every other node n the network, for eah node. The dfferental gossp trust has been nvestgated for a power law network formed usng preferental attahment PA) Model. The reputaton omputed usng dfferental gossp trust shows good amount of mmunty to the olluson. We have verfed the performane of the algorthm on the power law networks of dfferent szes rangng from 00 nodes to 50,000 nodes. Index Terms Trust, Reputaton, Dfferental Gossp, Free Rdng, Colluson. Introduton Peer-to-peer systems have attrated onsderable attenton n reent past as these systems are more salable than the lent-server systems. But, free rdng has emerged as a bg hallenge for peer-to-peer systems [], [2]. Tendeny of nodes to draw resoures from the network and not gvng anythng n return s termed as Free Rdng. As nodes have onfltng nterests so, selfsh behavour of nodes leads to problem of free rdng [3]. Ths behavour of nodes an be explaned by famous prsoners dlemma [4]. In a fle sharng network, f nodes are onsdered as players, ther E wll be the strategy where none of them wll share the resoures [5]. Expermental studes [6], [7] on Gnutella network onfrmed ths fat. Trust or reputaton management systems an be used n peer to peer networks to overome the problem of free rdng as well as to ward off some of the attaks [8], [9]. Suh reputaton management systems have been n e-ommere portals lke e-bay [0], but they have the advantage that they are based on lent server arhteture. In peer-to-peer fle sharng networks, as there s no entral server or repostory, trust has to be estmated and stored by eah node n a dstrbuted fashon, and all suh trust values need to be aggregated to buld an effetve reputaton management system. Aggregaton of trust generally onsumes a lot of tme and memory espeally wth large number of nodes. Apart from t, exstng methods assume that the reputaton of a peer must have a global value,.e. the peers behave unformly wth all other peers, but ths s not true. In ths paper, we propose Ruhr Gupta and Yatndra ath Sngh are wth the Department of Eletral Engneerng, Indan Insttute of Tehnology, Kanpur, Inda. E-mal:{rgupta, ynsngh}@tk.a.n a method whh aggregates the trust estmated by a node dretly, trust reported by neghbours and trust averaged usng a varaton of gossp algorthm to make trust vetors at eah node. Studes show that unstrutured peer-to-peer networks generally follow Preferental Attahment PA) model. In partular Gnutella has a power law degree dstrbuton f d) d α wthα2.3 [9], [23], [24]. Therefore ths algorthm has been smulated for networks havng power law degree dstrbuton,.e. networks formed by Preferental Attahment PA) model [], [2]. Aordng to [2], a graph G m evolves from G m when a new node wth m edges ons the network. Here m s the number of onnetons that the new node wll make at the tme of onng the network. The onng nodes hooses a node wth probablty P gven by degree o f node be f ore ths onneton s made P sum o f degree o f all the nodes be f ore ths onneton s made. Remander of ths paper s organsed as follows. Seton two presents the related work n reputaton management. Seton three desrbes the system model. Seton four, proposes the dfferental gossp trust algorthms for the aggregaton of trust. Seton fve presents the analyss of the algorthms and numeral results. Seton sx onludes the presented work. 2 Related Work Many methods have been proposed n the lterature [3], [4], [5], [6], [7], [8] for the reputaton aggregaton. Egen-Trust [3] depends largely on pre-trusted peers.e. peers that are globally trusted. Ths s salable to a lmted extent. Peer-Trust [4] stores the trust data.e. trust values of all the peers n the network) n a dstrbuted fashon. Ths s performed usng a trust manager at every node. In Peer-Trust, hash value of a node d s alulated to dentfy the peer where the trust

2 2 value of the node wll be stored. Song et.al. [5] used fuzzy nferene to ompute the aggregaton weghts. These weghts are assgned on the bass of global trust value of the opnng peer, transaton amount and date of transaton wth the peers beng evaluated. In Fuzzy- Trust, eah peer mantans a loal trust value and the transaton hstory wth the remote peers. At the tme of aggregaton, system queres for the trust value, from the qualfed peers, for the peer beng evaluated, and ombne the reeved values usng aggregaton weghts to ompute the updated trust values. The trust value alulated n ths way s global n nature. Zhou et.al. [6] dentfed the power law dstrbuton n the users feedbak by observng ebay transaton traes. Power Trust leverage on ths dstrbuton. Every node keeps the reord of loal trust sores of the nodes wth whom t has nterated, as per the qualty of serve obtaned from them. Global reputaton s alulated by weghted aggregaton of loal trust sores. The global reputaton of opnng node s taken as weght. Smlar to fuzzytrust, trust value generated here are global and all peers n the network use the same value for all the transatons. Gossp Trust [7] uses push gossp algorthm as gven n [2] for aggregaton n omplete graph based network. Ths ensures fast reputaton aggregaton wth low message overhead. Ths tehnque also alulates the global reputaton of a node by weghted aggregaton of opnons of dfferent nodes. It also uses the bloom flter arhteture for the effent rankng. [2] analyses gossp algorthm for aggregaton of nformaton n omplete graph based network and also omes up wth a upper bound on onvergene. Ths paper also studes dffuson n the network. As evdent, generally the earler work [3], [4], [6], [7] assumes that the reputaton of a peer must have a global value,.e. peers behave unformly wth all the peers. But ths s not the ase due to nodes beng selfsh n nature. A peer behaves wth dfferent deeny levels wth dfferent peers. The same holds true for the opnon as well,.e. peer gves dfferent weghts to opnons of dfferent peers. These aspets have been taken nto aount n our algorthm alled Dfferental Gossp Trust. The urrent work ombnes loal, reported value from trusted neghbours and global trust usng the new varaton of gossp to defne a novel trust aggregaton algorthm. 3 System Model In ths paper, we are studyng a peer-to-peer network. Typally, there wll be mllons of nodes n a peer-topeer network. These nodes are assumed to be onneted by a network graph G m generated by PA preferental attahment) proess [9] for m 2. Here, all the nodes whose addresses are stored by a node, are onsdered to be neghbours of the node. Generally the nodes wll have small number of neghbours. There s no dedated server n ths network. Peers n ths network are ratonal,.e. they are only nterested n ther own welfare. They are onneted to eah other by an aess lnk followed by a bak bone lnk and then agan by an aess lnk to the seond node. We are assumng that the network s heavly loaded.e. every peer has suffent number of pendng download requests, hene these peers are ontendng for the avalable transmsson apaty. We also assume that every peer s payng the ost of aess lnk as per the use for both download and upload as per the bllng prate of most of the serve provders). Downloaded data s more valuable than the ost of aess lnk. Moreover, data that s of nterest to peer s always avalable. So, every peer wants to maxmses ts downloads and mnmse ts uploads so that t an get maxmum utlty of ts spendng. Ths optmsaton leads to problem of free rdng. If a node s downloadng, some other node has to upload. So the desred ondton s that the download should be equal to upload for a node. Usually ths means that there s no gan. Even n ths senaro, the node gans due to nteraton wth others, as the hanes of survval of any entty s more wth ommunaton apablty. Thus nteraton tself s an nentve. A node wll usually try to get the ontent and avod uploadng to maxmse gan. Thus free rdng beomes optmal strategy. So a reputaton management system need to be enfored to safeguard the nterest of every node by ontrollng the free rdng behavour. In a reputaton management system, every node mantans a reputaton table. In ths table, a node mantans the reputaton of the nodes wth whom t has nterated. Whenever t reeves a resoure from some node, t adusts the reputaton of that node aordngly. When another node asks for the resoure from ths node, t heks the reputaton table and aordng to the reputaton value of the requestng node, t alloates resoure to the other node. Ths ensures that every node s faltated from the network as per ts ontrbuton to the network and onsequently free rdng s dsouraged. For usng suh a reputaton management system, the nodes need to estmate the trust value of the nodes nteratng wth them. There are number of ways to estmate the reputaton [20]. We assume that trust value observed by node a for the node an be defned as t. 4 Aggregaton of Trust Whenever a node needs a resoure, t asks from ts neghbours; f they have the resoure, the node gets the answer of ts query. If neghbours do not have t, they forward the query to ther neghbours and so on. The node that have the resoure, reples bak to the requestng node. The requestng node now asks for the resoure from the node havng the resoure. The answerng node provdes the resoure now dretly aordng to the reputaton of the node. If a node reeves a request from another node that s not ts neghbour, the reputaton of that node needs to

3 3 be estmated some how n order to dede the qualty of serve to be provded. If two nodes are gong to transat for the frst tme they should have reputaton of eah other. Ths an be done by gettng the reputaton of node from neghbours and then usng t to make an ntal estmate. When for a node, multple trust values are reeved, we need an aggregaton mehansm to get the trust value. Trust value should always le n between zero and one. For the whole network, we an defne a trust matrx of dmensons. Here t represents the trust value of as mantaned by based on dret nteraton. Ths matrx s generally sparse n nature as generally a node wll have very small number of neghbours beng dretly transated wth as ompared to total number of nodes n the network. It may be noted that t s estmated based on transaton between nodes and and an be alled as loal trust value. These trust values wll be propagated and aggregated by all the nodes n a network. The trust estmate whh should be atually used wll be based on aggregaton of loal trust values and trust estmates reeved from neghbours. For the reputaton nformaton reeved from dret neghbours, the weghts an be assgned based on neghbours reputaton. 4. Dfferental Gossp Trust We have modfed the gossp based nformaton dffuson algorthm to allow faster dffuson of the trust values enablng faster estmaton of global trust vetors at all the nodes. The algorthm an be dvded nto two parts. In frst part, we wll dsuss about the method of nformaton dffuson whereas n the seond part, we wll dsuss about the nformaton that s to be dffused. 4.. Dfferental Gossp Algorthm Gossp Algorthms are used for spreadng nformaton n large deentralsed networks. These algorthms are random n nature as n these, nodes randomly hoose ther ommunaton partner n eah nformaton dffuson step. These algorthms are generally smple, lght weght and robust for errors. They have less overhead ompared to the determnst algorthms [2], [22]. The gossp algorthms are also used for the dstrbuted omputaton lke takng average of the numbers stored at dfferent nodes. These algorthms are sutable for the omputaton of reputaton vetor n the peer-to-peer networks [7]. There are three types of gossp algorthms: push, pull and push-pull. In push knd of algorthms, n every gossp step, nodes randomly hoose a node from ts neghbours and push a pee of nformaton to t. Whereas n pull algorthms, nodes take the nformaton from one of the randomly seleted neghbourng node. Both these proesses happen smultaneously n the pushpull based algorthms. Cherhett et.al. [25] stated that n a PA model based network, push or pull alone wll take long n spreadng the nformaton n the network. If the push model s mplemented and the nformaton s wth a power node, t wll take many rounds n pushng nformaton to low degree nodes. If pull model s beng used, and the nformaton s wth low degree node, t wll agan take many rounds for a power nodes to pull the nformaton. Ths phenomenon wll be evdent n the average omputaton usng push or pull gossp algorthm as nformaton of every node need to be dstrbuted to every other node. To avod ths problem we propose dfferental push gossp algorthm. In ths algorthm, every node makes dfferent number of pushes dependng upon the rato of ts own degree to the average neghbour degree. n a sngle gosspng step. If every node also pushes ts degree to all the neghbourng nodes, then eah node an estmate the average degree of all ts neghbours. Here, we make three assumptons, ) every node has a unque dentfaton number known to every other node. So, f some node pushes some nformaton about another node, reevng node knows that ths nformaton s about whh partular node; ) tme s dsrete; and ) every node knows about the startng tme of gossp proess. All nodes that have some feedbak about a sngle node, gossp ther feedbak about that sngle node. All the nodes estmate the global reputaton of the sngle node based on the outome of gossp. Let the feedbak about the th node by node be y. If t does not have any feed bak about t keeps the value of y as 0. Every node that has feed bak about node assumes the gossp weght g as and rest of the nodes assume the gossp weght as zero. It s done so that as a result of gossp, every node overages to the rato of summaton of all gossped values as well as summaton of all gossp weghts averaged over all the nodes. The rato of two y onverged values,.e. g gves the sum of gossped values averaged over node who started wth gossp weght of unty. If only one of the nodes assumes gossp weght as unty and remanng as zero, all nodes wll be able to estmate the summaton of gossped values. Every node has y and g as nformaton to be gossped. Let us all ths par as gossp par. The rato of gossp par s traked n every step to dede on onvergene. Every node, frst alulates the rato k of ts own degree and average degree of ts neghbours. As k wll be a real number, t s rounded off to nearest nteger f k. For all other ases, k. The node hooses k nodes randomly n ts neghbourhood and sends k y, k g ) as gossp par to all randomly seleted k nodes and tself. The node s also onsdered as one of the neghbours of tself. After reevng all gossp pars from dfferent nodes nludng tself, the node sums up all the pars. Ths summaton now beomes new gossp par. The rato of ths gossp par s the value that node has evolved n ths step. If at least one gossp par has been reeved from a node other than tself, the ondton of onvergene wll

4 4 be heked between the ratos of ths step and prevous step wth a predefned error onstant. If onvergene ondton s satsfed, t means a node needs not to run the gossp proess any more for ts onvergene. But one onvergene s aheved by a partular node, onvergene of other nodes s not assured. Hene, f a node wll stop gosspng, onvergene of ts neghbours may suffer. To avod ths problem, one a node gets onverged, t wll announe among all ts neghbours that t has aheved onvergene. Every neghbour wll note ths announement. When a node fnds that tself and all ts neghbours have onverged, t wll stop the gossp proess. When a round of gosspng starts t takes some tme to omplete. After gosspng, nodes get a value that s used tll the next new value s onverged upon at the end of next round. After the end of a round, next round of gossp wll start after some tme. The tme dfferene between the two rounds wll depend upon the hange n the behavour of the nodes n the network and the number of new nodes omng n the network per unt tme. For smplty, ths tme dfferene has been taken as a onstant. In realty, ths should be dynamally adusted Dfferental Reputaton Aggregaton When a node requests resoure from another node, the node needs the reputaton of node so that t an dede the qualty of serve to be offered to node. There are two possble ondtons between node and node. Frst, node may have served node earler and hene node has some trust value about node. In ths ase there s no problem for node and node wll serve as per the reputaton avalable wth t. Seond, node and node are unknown to eah other. In ths ase node needs general reputaton about node. For ths, aggregaton of reputaton s needed. Aggregaton of reputaton should not be resoure ntensve and should be mmune to olluson and whtewashng. We an have two knd of optons for ths. Frst, by gosspng all the nodes an reah a onsensus about the reputaton of node [7]. Seond all the nodes exhange ther reputaton tables about the nodes they have nterated wth and ths proess should always be runnng, lke exeutng a routng protool at network layer. Frst proess s more vulnerable to olluson where as seond proess s resoure ntensve. As we an see that unstrutured peer-to-peer network s very smlar to human network, thus we an observe the human behavoural strateges to dentfy the soluton. In human network when we need the reputaton value of some body we rely on personal experene wth hm. If we don t have any personal experene wth hm, we rely on two thngs. Frst the general perepton about hm whh we reeve from gossp flowng around and seond the nformaton gven by our frends f they have any dret nteraton wth hm. We ombne these two and at aordngly. odes follow the same knd of approah n our proposed algorthm. The nodes gather opnon of ther neghbours and ombne t wth the opnon, obtaned from general gossp after weghng the neghbours opnon aordng to the onfdene n the neghbours. In general, t an be sad that a node gves weght to every node n the network. The nodes that have not nterated wth t are gven weght as where as those whh have nterated are gven weght aordng to the onfdene n them always ). Let us onsder that there are nodes n the network. Every node perodally alulates the trust value of the other nodes on the bass of qualty of serve provded by them aganst the requests made. Let us assume that t s the trust value measured by node for node. Here t, ) wll always le between 0 and suh that the t wll represent the omplete trust n node, whereas, t 0 wll represent no trust n the node. If a node A has not transated wth a node B, then the trust value of node B wll also reman 0 wth the node A. Ths ntal value s taken as 0 to avod the whte washng attak. Ths ntal value an also be taken as hgher than zero and an be dynamally adusted thereafter as per the level of whtewashng n the network. In ths paper, we have not studed ths aspet. We wll dsuss the algorthm n four steps to make t smple to understand. In the frst step, global reputaton aggregaton for a sngle node wll be dsussed. In the seond step, we wll dsuss globally albrated loal reputaton aggregaton for ths sngle node. In the thrd step, smultaneous global reputaton aggregaton for all the nodes wll be dsussed, and fnally n the fourth step, smultaneous aggregaton of globally albrated loal reputaton aggregaton for all the nodes wll be dsussed. In the frst step to keep thngs the smple, we assume that weghts of all the nodes for every node be. Ths leads us to alulaton of global reputaton R global ) of a node. Ths an be equvalently represented usng a matrx vetor multplaton as follows, R global n) tt n) ). ) Here n s the tme nstant and R global s the global reputaton vetor ontanng global reputatons of dfferent nodes. Let us assume that R s global reputaton of th node.e. th element n R global n) olumn vetor. s a vetor of s,.e. [...] T. Dfferental Gossp algorthm an be used for dong ths omputaton n dstrbuted fashon. Ths proess s shown n algorthm. Although eah node onsders the average of feedbaks from every other node n the network, t s desrable to assgn dfferent weghts to the dret feedbaks reeved from neghbourng nodes. The dret feedbak from a node s based on the dret nteraton whh t had experened. The weghts an be assgned by a

5 5 degree o f average neghbour degree ) Algorthm Global Reputaton aggregaton for a sngle node Requre: t The reputaton estmated by node for node only on the bass of dret nteraton) for, gossp error tolerane ξ Ensure: Global Reputaton of node R ) f has some reputaton value about then Assume weght g, and y t else Assume weght g 0, and y 0 end f Push self degree to neghbourng node Take the average of neghbours degree Calulate the rato of ts degree and average of neghbour degree k Round off k to nearest nteger for k else take k m {Intalse Gossp Step} u y g for nodes havng g 0; otherwse u 0. repeat Do for node y s, g s ) are all pars of gossp weght and gossp value) reeved by the node n the prevous step y y s ; g g s {update gossp pars}{s s s S s S the set of nodes sendng the gossp to } hoose k random nodes n ts neghbourhood send gossp par k y, k g ) to all k nodes and also to tself m m{nrement the gossp step} f S > then f y g u ξ then Inform all neghbours about self onvergene end f end f u y g for nodes havng g 0; otherwse u 0. untl Self onvergene and all neghbours onvergene has happened output R y g node on the bass of number and qualty of transatons made wth the other node who s provdng the feedbak. The trust value of a node s a good metr of qualty and number of transatons. The weghts for dfferent nodes an be derved on the bass of the trust values of these nodes. Same dea s used n seond varaton of algorthm where nodes estmate globally albrated loal reputaton vetor. So n seond step, we propose the weght w to be of the form: w a b t. 2) Here a and b are two parameters that a node an dede on ts own. Frst parameter an be adusted aordng to the overall qualty of serve reeved by the node from the network, whereas seond parameter an be adusted aordng to the reommendaton of a partular neghbour and qualty of serve from the network. So the seond parameter wll be adusted for every neghbour ndependently. In ths paper, a and b has been taken as the onstants for every node for smplty. Salent features of ths sheme are as follows. Even f a node has no neghbourhood relatonshp wth the estmatng node, ts feed bak wll stll get some onsderaton. If a node has bad reputaton wth the estmatng node, ts feedbak wll have weght lose to the node whh have no neghbourhood relaton wth the estmatng node. odes wth hgher reputaton wll be gven hgher weghts and t wll help n makng better qualty of serve groups. Values of a and b an be dynamally adusted by nodes as per ther requrement. Though n ths work, a and b have been taken as onstants. Colluson wll be sgnfantly redued. A weghted trust matrx, that s dfferent at every node, s formed by multplyng trust values wth weghts.e. for node I, the element n weghted matrx wll be W I suh that, W I w I t. 3) A node gves hgh weght to the feedbak gven by those nodes whh have provded better qualty of serve. Ths leads to the alulaton of globally albrated loal reputaton vetor. It s a olleton of the reputaton of all the nodes n the network aordng to reeved feedbak about the node and the weghts of nodes gvng feedbak to the alulatng node. It means f some node I s alulatng globally albrated loal reputaton vetor, the th element of ths vetor wll be Rep I, W I. 4) w I ow globally albrated loal reputaton at node I, R glr I an be equvalently represented as the matrx vetor multplaton R glr I n) Sum I W T I n) ). 5) Here Sum I w I. It s nterestng to see that f we onsder the weghts of all nodes as n 5), ths equaton degenerates to ). Eah node wll have four dfferent knd of data about other nodes - frst t.e. the trust value as result of dret nteraton, seond y, the ntermedate varable for gosspng and thrd g, gosspng weght. In the start of every gosspng round y assume the value of t whereas g wll be only for one of the values of, and 0 for all the others. Ths wll happen for all values of and. Fourth value Rep s obtaned after gosspng and onsderaton of neghbours opnon. It wll also be mantaned at every node. Apart from these four

6 6 enttes, we also wants to ount the total number of nodes opnng about node. For ths purpose every node that have opned about node wll assume ount and others wll assume ount 0 and hene n the proess of gossp, all these s wll sum up and we wll get the ount. Equaton 4) an be alternatvely represented as, W I W I S I S I Rep I, w I, w I S I S w I t w I t S I S I w I, w I S I S I w I ) t w I ) t t S I S I w I ) w I ). S I S I Here S I s the set of neghbours of node I. As neghbourhood between two nodes s based upon the nteraton between them so for non neghbour nodes the weght wll be. Usng ths fat, w I ) t S I Rep I, w I ) S I t. 6) In order to ompute the globally albrated loal reputaton of node, eah node wll need the reputaton of as estmated by neghbours on the bass of the dret nteraton wth node. Whereas n the gossp algorthm, after every step, the value at the node keeps on hangng as t gets added to the values pushed by other nodes and s dstrbuted after dvson to neghbours. After few steps, the values onverge when nomng and outgong values statstally balane eah other. Hene after the frst step of gosspng t s dffult to get the value of t from a neghbour for the nodes wth whom t has dret nteraton, by gosspng proess. If a node s partpatng n the proess of gossp about node for the frst tme or reputaton of node at ths node has hanged onsderably sne start of prevous round of gossp, ths node wll nform the reputaton of node to all of ts neghbours before the start of next gosspng round fgure ). Ths wll be done by all the nodes. After ths proess, every node has opnon of ts neghbours about node If a node does not nform reputaton of, the already avalable earler value wll be onsdered). If node wll not hear from a node for a long tme, t wll assume that ths node s no longer present and hene t wll drop ts feedbak after some tme. ow these reputatons wll be multpled by W I ) as requred n equaton 6) and summed up as value ŷ I see algorthm 2). ow normal gossp wll be done as n algorthm 2 wth a dfferene that only one node wll be gven gossp weght and rest wll be gven 0 gossp weght The nodes that have reputaton nformaton about node Fg.. Sequene of omputaton for estmatng globally albrated loal reputaton at eah node under onsderaton.e., ) wll also push ount. After stablsaton of gossp, eah node wll have d), where d s number of nodes havng dret nteraton and s total number of nodes, sum of total values/total number of nodes, and as gossp weght. Ths wll lead to the summaton of all reputaton values avalable and total number of nodes gvng these reputaton values. ow reputaton an be alulated usng 6)[algorthm 2]. Algorthm 2 Globally albrated loal Reputaton aggregaton for a sngle node Requre: Feedbak matrx t, gossp error tolerane ξ. Ensure: Globally albrated loal Reputaton of node Rep ) Assume weght g ode do f then g 0 end f f has some reputaton value about then take ount, and y t else take ount 0, and y 0 end f alulate w for all neghbours of by formula w a b t f ode s partpatng frst tme n gosspng proess then Push feedbak due to dret nteraton about the node under onsderaton to all neghbours else f Feedbak about the node under onsderaton has hanged by more than some onstant then Push the new feedbak to all the neghbours end f end f Push self degree d to neghbourng nodes Calulate ŷ d w k ) f eedbak f rom node k about k Algorthm Contnued...

7 7 degree o f average neghbour degree ) Algorthm Algorthm 2 ontnued) Take the average of neghbours degree Calulate the rato of ts degree and average of neghbour degree k Round off k for k else take k End do m {Intalse Gossp Step} u y g for nodes havng g 0; otherwse u 0. repeat Do for node y s, g s, ount s ) are all 3-tuples of gossp value of reputaton, gossp weght and ount reeved by node n the prevous step y g s ; ount ount s {update s S s S y s ; g s S gossp pars} {S s the set of nodes sendng the gossp to } hoose k random nodes n ts neghbourhood send gossp par k y, k g ) to all k nodes and tself m m{nrement the gossp step} f S > then f y g u ξ then Inform all neghbours about self onvergene end f end f u y g for nodes havng g 0, otherwse u 0. untl Self onvergene and all neghbours onvergene has happened outputrep ŷ y g wk ) ount g In thrd varaton we want to aggregate the global reputaton of all nodes smultaneously. Ths algorthm s qute smlar to algorthm exept few hanges. Unlke algorthm, node wll push omplete vetor y whh onssts of feedbak from node about all the other nodes t has transated wth. Smlarly, nstead of sngle gossp weght g, node wll send vetor g. A node d wll also be attahed wth every par of y and g so that reevng node an dstngush among gossp pars. So, n fat, node pushes gossp tro onsstng of y, g and node d. The onvergene of algorthm s heked by the followng ondton: y n) g n) y n ) g n ) ξ 7) Where n s the tme nstant andξs the permssble error bound. In the fourth varaton we want to aggregate the globally albrated loal reputaton of all the nodes smultaneously. Ths algorthm s qute smlar to seond varaton expet that we wll use the thrd varaton for gosspng proess. Moreover n ths varaton nodes wll push full vetor t, n plae of only t for node. Fg. 2. Topology of the example network It an be noted here that the tme omplexty of all four varatons of algorthm wll be of the same order beause reputatons of all the nodes wll be pushed smultaneously as a vetor. Whereas the ommunaton omplexty n thrd and fourth varaton wll nrease proportonally to the sze of vetor, as now the reputaton aggregaton s happenng for these many nodes. 4.2 Example for Dfferental Gossp Algorthm We wll onsder a network of 0 nodes and observe the aggregaton n ths network. Fgure 2 shows the topology of the network. Table 2 shows the aggregated value after every teraton at eah node. 5 Analyss of Algorthm 5. Analyss of onvergene of Gossp Algorthm In ths seton we wll study the tme needed by nodes to onverge to the average of loal dret estmate values at dfferent nodes. Frst, we wll study the spreadng of gossp n power law network. Then, we wll study the dffuson speed of gossp. Then, we wll study the dffuson speed of gossp. We have taken power law network as most atual P2P network tends to follow power law degree dstrbuton. Based on these results, we wll fnd the tme of onvergene. Cherhett et.al. [25] proved that n PA based graph, {G m } for m 2 push or pull alone wll fal n spreadng the gossp. They also proved that push-pull wll sueed n Olog 2 ) 2 ) tme steps where low degree nodes push nformaton to low degree nodes and power nodes and pull nformaton from power node. But n a peer-to-peer networks, t s dffult to dentfy the power nodes. Moreover pullng the nformaton s more expensve than pushng the nformaton. So we have proposed to use dfferental push gossp n plae of push pull gossp. In the followng theorem we have proved that the dfferental push gossp wll take same tme as the push pull gossp. Theorem 5.. Gossp wll spread wth hgh probablty n a PA based graph,{g m } for m 2, wthn Olog 2) 2 ) tme

8 8 ode degree k 3 tr tr tr tr tr tr tr tr TABLE Aggregated value after every teraton at eah node usng dfferental-push by hgh probablty we mean, o), where o) goes to zero as nreases). Proof: Dfferental push means that every node wll push ts data to dfferent number k ) of nodes nstead of one node. Here k s the rato of node s degree and average degree of all ts neghbours. {G m } for m 2 an be thought as the unon of few fnte onneted omponents. These omponents wll onsst of many low degree nodes wth average degree log 2 and few hgh degree power nodes. These omponents wll have dameter as log 2. In a network omponent) wth average degree log 2 and dameter log 2, the gossp wll spread n wthn Olog 2 ) 2 ) steps usng normal push [26]. But, these omponents also have power nodes that may lead to large spreadng tme n that omponent [25]. As power nodes wll have hgh degree and makng one push at a tme wll take longer n transferrng the nformaton to all of ts neghbours. Therefore f some low degree node s onneted only to power nodes, the nformaton transfer to that node wll take muh longer. Dfferental push wll solve ths problem beause now power nodes wll be makng multple pushes as per the rato of degrees of neghbour nodes and degree of the node tself. Hene, n every omponent, gossp spreads wthn Olog 2 ) 2 ) steps usng dfferental push. As there are fnte suh omponents, gossp wll spread n omplete network wthn Olog 2 ) 2 ) steps. In our ase, few nodes have nformaton reputaton of a node) that has to be averaged and ths average has to be spread to all the nodes. So we wll prove the onvergene for the ase where every node has nformaton whh has to be averaged. Let at n0, eah node has a number. For th node ths number s d 0,. So the obetve of gossp s to have d 0, d avg at eah node after some rounds of gossp. The number of rounds needed should be least possble. For n0, the gossp weght at eah node wll be unty. After n steps, let the node have the evolved number as d and evolved gossp weght as g. To study the tme taken n the onvergene of algorthm we assume that eah node mantans a vetor m. The dmenson of ths vetor wll be. Ths vetor wll reord the ontrbuton reeved from every node nludng tself about the node m. So ntally at n 0 eah node wll have a vetor n whh ) elements wll be zero and one element, the one for tself, wll be unty. If n the proess of frst step of gosspng, only the node hooses node for pushng the gossp about m, then the ontrbuton by to.e. m wll be n,, reorded n the th element of ontrbuton vetor of node. So after frst step of gossp the ontrbuton vetor of wll ontan two non-zero elements one reeved from and one pushed to tself. We assume here that only push has been reeved by node. In ase of l pushes beng reeved the vetor wll have l non zero entres. ow node wll hoose some node o. ode wll push the omplete ontrbuton vetor dvded by p f p push gossp s under onsderaton) to node o. ow node o wll do vetor addton of all the reeved vetors nludng the one reeved from tself. The resultant vetor wll be the new ontrbuton vetor. Ths proess wll be repeated at all the nodes. So t an be sad that d m m n,, d m suh that 0, g m m. When a node wll reeve same amount n,, of ontrbuton from all nodes, at that tme the rato of evolved number d m ) and evolved gossp weght gm ) wll be the average of all the numbers. Theorem 5.2. Unform Gossp dffuses wth dfferental push n PA based graph wthn Olog 2 ) 2 log 2 ξ ) tme wth hgh probablty suh that ontrbutons at all nodes wll be ξ unform after ths amount of tme,.e. max m ξ where n,,. On the bass of these two theorems, t an be seen as n [2]), that wth hgh probablty relatve error n average estmaton and sum estmaton f only one node s gven weght one and others are gven zero) wll be bounded byξafter Olog 2 ) 2 log 2 ξ ) gossp steps. 5.2 Analyss of Colluson In our proposed system a node may get trust values about a node by three possble ways, frst by dret nteraton, seond from neghbours and thrd by gosspng. Frst mehansm an not be affeted by olluson. We m n,,

9 9 are assumng that seond mehansm wll also not be affeted by olluson as neghbours have a defnte level of trust for eah other. We are onsderng the olluson beause of thrd mehansm. For analyss of olluson, we wll alulate the dfferene of real reputaton and estmated reputaton of a node by some node o n the presene of olluson usng our proposed method, we wll ompare t wth the method proposed n [7]. Lets us assume that the network s formed by the member nodes of set. There s a subsetcof setsuh that member nodes of set C are nvolved n olluson. The ardnalty of sets and are assumed to be and C respetvely. We also assume that members nodes of setcare olludng n groups wth a group sze of G. By olludng n a group we mean that f some node s the member of that group then group members of olludng group wll report ts reputaton as. Whereas for others nodes they wll report the reputaton value as 0. Let us say that real reputaton of a node s R, and estmated reputaton s real R est, f s a olludng node, R estn, f s not a olludng node. t R real. 8) Here t s normalsed trust value of the node x at node. If s not a olludng node then, R estn If s a olludng node, then R est t \C \C. 9) t G. 0) So the expeted value of reputaton estmate E[R est ]) wll be t G \C t \C E[R est ] C ) C GC t \C 2 ) So dfferene n real reputaton and expeted value of estmated reputaton by node o for node R o old ) wll be, t C R o GC old 2. 2) ow, we norporate the trust based weghted opnon of neghbours. Lets us assume that w o s the weght gven to the opnon of node by node o. It may be noted w o, equaton 2). So the real reputaton of node for node o wll be, R real t w o )t w o ). 3) If x s not a olludng node then, t w o )t R estn \C w o ). 4) And f s a olludng node then, t w o )t G R est \C w o ). 5) So the expeted value of reputaton estmate E[R est ]) wll be t w o )t G E[R est ] \C C w o ) C ) t w o )t \C w o ) GC w o )) t w o )t \C w o ). 6) So dfferene n real reputaton and expeted value of estmated reputaton by node o for node R o new)wll be equaton 6 - equaton 3), R o new GC w o )) C w o ) w o )) GC t C 2 w o )) Ro 7) old 5.3 umeral Results Performane of algorthm for reputaton aggregaton for peer to peer fle sharng system s also evaluated by smulaton as well. The smulaton experments has been onduted for 00 to nodes. A power law network has been bult usng Preferental Attahment model. Performane of dfferental algorthm has been evaluated n terms of number of teratons to assess the rate of onvergene) requred to onverge wthn a ertan aggregaton error. umber of pakets per node per gossp step that are requred to be transmtted for onvergene have also been alulated to assess the network overhead. Algorthm has also been tested aganst olluson. Fgure 3 shows the number of gossp steps requred for dfferent error bounds for dfferent number of nodes. t

10 0 Ths s learly evdent that number of gossp steps s nreasng wth a rate muh less than normal push gossp. Peer to peer networks operate above TCP layer,.e. these knd of networks assume a relable bt ppe between sender and reever. So peer to peer network suffers by paket loss only when some node leaves the network.e. due to hurnng.fgure 4 shows the requred number of gossp steps wth dfferent paket loss probablty for 0000 nodes. Here the assumpton s when a node leaves durng gossp proess, t hands over the gossp par vetors to some other node so mass onservaton stll apples. Whenever a node pushes gossp par to ths absent node, the pushng node doesn t reeve any aknowledgement. In suh ases pushng node pushes the gossp par to tself so that mass onservaton stll apples. We an see a small nrement n the number of gossp steps wth the nrease n the paket loss probablty. Fgure 5 and 6 shows the mmunty of algorthm aganst olluson n terms of RMS error n ase of ndvdual fg 6) and group olluson fg 5). Here average RMS error s defned as follows. Average RMS error r ˆr )/r ) 2 8) Here r s the reputaton of node at node omputed by dfferental gossp n presene of olludng nodes, whereas rˆ s the omputed reputaton f olludng nodes would not have been there. Ths s learly evdent that effet of olluson on reputaton omputaton by dfferental gossp s qute less even wth very hgh perentage of olludng users. The olludng group sze s makng a small dfferene n dfferental gossp reputaton omputaton. Table 2 shows the number of message transfers requred by a node n one gossp step. It an be seen that ths s dereasng slghtly wth the nrease n number of nodes. Ths s happenng beause as number of gossp steps nreases the overhead nurred n the begnnng gets dstrbuted and a node s less burdened as the number of total nodes nreases. Smlar thng happens when a lower value ofξs hosen. Ths s also evdent from Table 2 and fg 3 that n ase of dfferental gossp per step ommunaton ost s more than normal push gossp but total ommunaton ost for onvergene s less for networks bgger than 000 nodes moreover ths dfferenes nreases substantally as network sze nreases. We have not verfed t for normal pull gossp but ntutvely t an be observed that same thng wll be true. 6 Conluson In peer-to-peer networks, free rdng s a maor problem that an be overome by usng reputaton management system. A reputaton management system nludes two Fg. 3. Gossp step ounts wth dfferent number of nodes) and dfferent error boundsξ Fg. 4. Gossp step ounts for 0000 wth dfferent error boundsξ for dfferent paket loss probablty proesses, frst estmaton of reputaton and seond aggregaton of reputaton. In ths paper we have proposed an aggregaton tehnque by modfyng push gossp algorthm to dfferental push gossp algorthm. The proposed aggregaton tehnque effently aggregates the trust values from dfferent nodes n a power law network. Ths tehnque does not requre the dentfaton of power nodes. Ths makes algorthm easly mplementable as dentfaton of power nodes n a dstrbuted settng s hard. Ths algorthm s also robust aganst hurnng as an be seen n fgure 4. Proposed tehnque aggregates the reputaton n a dfferental manner. Ths s done by onsderng the feedbak of trusted nodes wth a hgher weght. Ths leads to robustness aganst olluson as evdent from fgure 5.

11 ξ0.0 ξ0.00 ξ0.000 ξ TABLE 2 umber of messages per node per step transmtted due to gosspng Fg. 5. Average RMS error wth dfferent sze olludng groups for dfferent perentage of olludng peers Fg. 6. Average RMS error wth ndvdual peers for dfferent perentage of olludng peers Proposed algorthm has been presented to avod the problem of free rdng but t an also be used to avod malous users n the network ust by hangng the method of estmaton of a and b. Referenes [] M. Feldman and J. Chuang, Overomng free-rdng behavor n peer-to-peer systems, SIGeom Exh., vol. 5, no. 4, pp. 4 50, Jul [Onlne]. Avalable: [2] M. Karakaya, I. Korpeoglu, and O. Ulusoy, Free rdng n peerto-peer networks, IEEE Internet Computng, vol. 3, no. 2, pp , [3] B. Yang, T. Conde, S. D. Kamvar, and H. Gara-Molna, on-ooperaton n ompettve p2p networks. n ICDCS. IEEE Computer Soety, 2005, pp [Onlne]. Avalable: [4] M. J. Osborne, Introduton to Game Theory: Internatonal Edton. Oxford Unversty Press, [Onlne]. Avalable: [5] Y. Tang, H. Wang, and W. Dou, Trust based nentve n p2p network, n Proeedngs of the E-Commere Tehnology for Dynam E-Busness, IEEE Internatonal Conferene, ser. CEC-EAST 04. Washngton, DC, USA: IEEE Computer Soety, 2004, pp [Onlne]. Avalable: [6] E. Adar and B. A. Huberman, Free rdng on gnutella, Frst Monday, vol. 5, p. 2000, [7] D. Hughes, G. Coulson, and J. Walkerdne, Free rdng on gnutella revsted: The bell tolls? IEEE Dstrbuted Systems Onlne, vol. 6, no. 6, pp., Jun [Onlne]. Avalable: [8] K. Chen, K. Hwang, and G. Chen, Heurst dsovery of rolebased trust hans n peer-to-peer networks, IEEE Transatons on Parallel and Dstrbuted Systems, vol. 20, no., pp , [9] E. Daman, S. De Captan D Vmerat, S. Parabosh, and P. Samarat, Managng and sharng servants reputatons n p2p systems, Knowledge and Data Engneerng, IEEE Transatons on, vol. 5, no. 4, pp , [0] [] A.-L. Barabás and R. Albert, Emergene of Salng n Random etworks, Sene, vol. 286, no. 5439, pp , Ot [Onlne]. Avalable: [2] B. Bollobás, O. Rordan, J. Spener, and G. Tusnády, The degree sequene of a sale-free random graph proess, Random Strut. Algorthms, vol. 8, no. 3, pp , May 200. [Onlne]. Avalable: [3] S. D. Kamvar, M. T. Shlosser, and H. Gara-Molna, The egentrust algorthm for reputaton management n p2p networks, n Proeedngs of the 2th nternatonal onferene on World Wde Web, ser. WWW 03. ew York, Y, USA: ACM, 2003, pp [Onlne]. Avalable: [4] L. Xong and L. Lu, Peertrust: Supportng reputaton-based trust for peer-to-peer eletron ommuntes, IEEE Trans. on Knowl. and Data Eng., vol. 6, no. 7, pp , Jul [Onlne]. Avalable: [5] S. Song, K. Hwang, R. Zhou, and Y.-K. Kwok, Trusted p2p transatons wth fuzzy reputaton aggregaton, IEEE Internet Computng, vol. 9, no. 6, pp , ov [Onlne]. Avalable: [6] R. Zhou and K. Hwang, Powertrust: A robust and salable reputaton system for trusted peer-to-peer omputng, IEEE Trans. Parallel Dstrb. Syst., vol. 8, no. 4, pp , Apr [Onlne]. Avalable: [7] R. Zhou, K. Hwang, and M. Ca, Gossptrust for fast reputaton aggregaton n peer-to-peer networks, IEEE Transatons on Knowledge and Data Engneerng, vol. 20, no. 9, pp , [8] Z. Lang and W. Sh, Pet: A personalzed trust model wth reputaton and rsk evaluaton for p2p resoure sharng, n Proeedngs of the Proeedngs of the 38th Annual Hawa Internatonal Conferene on System Senes - Volume 07, ser. HICSS 05. Washngton, DC, USA: IEEE Computer Soety, 2005, pp [Onlne]. Avalable: [9] S. Sarou, P. Gummad, and S. Grbble, A Measurement Study of Peer-to-Peer Fle Sharng Systems, [Onlne]. Avalable: [20] R. Gupta and Y.. Sngh, Trust estmaton n peer-to-peer network usng blue, CoRR, vol. abs/ , 203. [2] D. Kempe, A. Dobra, and J. Gehrke, Gossp-based

12 2 omputaton of aggregate nformaton, n Proeedngs of the 44th Annual IEEE Symposum on Foundatons of Computer Sene, ser. FOCS 03. Washngton, DC, USA: IEEE Computer Soety, 2003, pp [Onlne]. Avalable: [22] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, Randomzed gossp algorthms, IEEE/ACM Trans. etw., vol. 4, no. SI, pp , Jun [Onlne]. Avalable: [23] D. Group, Gnutella: To the bandwdth barrer and beyond, [24] H. Chen, H. Jn, J. Sun, D. Deng, and X. Lao, Analyss of largesale topologal propertes for peer-to-peer networks, n Cluster Computng and the Grd, CCGrd IEEE Internatonal Symposum on, 2004, pp [25] F. Cherhett, S. Lattanz, and A. Panones, Rumor spreadng n soal networks, Theor. Comput. S., vol. 42, no. 24, pp , 20. [26] U. Fege, D. Peleg, P. Raghavan, and E. Upfal, Randomzed broadast n networks. Random Strut. Algorthms, vol., no. 4, pp , 990. [27] G. Grmmett and D. Welsh, Probablty: An Introduton, 986. Appendx proof of theorem 5.2 We an see the property of mass onservaton proposton A.) [2] holds n ths ase as well. Proposton A.. Under the dfferental Push protool wth Unform Gossp, the sum of all of th node s ontrbutons at all nodes s m and hene the sum of all weghts s n,, g m Proof of theorem 5.2: As we know when a node wll get equal ontrbuton from every node t wll reah the average value. Takng varane around the mean value of the ontrbutons from all the nodes at a partular node wll gve the level of onvergene at one node. If we sum these varanes for all the nodes, we wll get the dea about the onvergene of network. We are ust referrng to reputaton of node m n ths proof, and super srpt m have not been expltly shown. It means m n n and g m n g n. The varane at node wll be E n,, n,, ) 2.e. n,, g )2. As ths quantty s small, we an drop. Ths wll stll gve the dea about the onvergene. Further addng the varane at all the nodes gves us the dea about further onvergene n the whole network. We all ths as potental funtonψ n ψ n n,, g ) 2 9), Let us studyψfor p-push gossp,.e. when every node s makng p pushes to p nodes. Here we are assumng that node hooses every node nludng tself for push ndependently. Here fk) means that a node k hooses a node and pushes gossp par. The d and g are dvded by p); one part s always retaned by the node and remanng are used for p push. n,, p n,, n,,k. 20) p k: f k) g n,, 2) As ψ n p n,, p p g p k: f k) g k: f k) n,,k n,, g ) 2, 22), substtutng the values of n,, and g, we get ψ n n,, g ) n,,k g ) 2 p p, k: f k) p) 2 n,, g ) 2, p) 2 n,,k g ) 2, k: f k) 2 p) 2 n,, g ) n,,k g ), k: f k) p) 2 n,,k g ), ˆk k: f k) f ˆk),k ˆk g ) n,ˆk,ˆk. In seond term of the prevous equaton, we are dong summaton over, and k : f k). Eah node k s ontrbutng p opes to ts p neghbors. If ths ontrbuton s summed over k, t should be equal to ontrbuton reeved by eah node when summed over all nodes. Thus p n,,k g )2 n,,k g )2 23). k ψ n k: f k) p) 2 n,, g ) 2 p p) 2 n,, g,, 2 p) 2 n,, g ) n,,k g ), k: f k) 2 p) 2 n,,k g ) g ) n,ˆk,ˆk, ˆk, k ˆk: f k) f ˆk) p) ψ 2 n p) 2 n,, g ) n,,k g,,k: f k) n,,k g ) k: f k) f ˆk),ˆk k ) 2 p) 2 n,,ˆk g n,ˆk ). ) 2

13 3 We know that node wll hoose a node randomly among ts neghbours. Let us assume that the degree of sad node k s d k wth probablty P dk. Out of remanng nodes, we an form group of d k nodes n ) d k ways. If we fx one node say as one of the neghbour, then there are 2 C dk ) ways of havng other dk neghbours. Thus the probablty of beng a neghbour of a node k havng degree d k wll 2 d k ). Further the probablty that k wll d k ) hoose wll be d k. P[ f k) ] P[ˆk k, f k) f ˆk) ] ) 2 d k d k P dk ) P dk 24) Smlarly 25) P dk P dˆk 26) The P dk s the probablty that a node has degree d k. For networks generated by PA Model, d k Let us assume that the maxmum value of P d s P dmax and mnmum value s P dmn and the dfferene of P dmax and P dmn s K, then E[ψ n ψ n ] p ψ n 2 p) 2 ) n,,k g ) P dk k p) 2 ) 2 k g n,ˆk n,,ˆkp dmax P d mn ˆk ˆk p) 2 ) 2,k n,, g n,,k P dmax n,,k g k ) 2 P 2 d mn g P d mn ˆk Pd k ) d γ k. Here γ s network exponent. Hene, E[ψ n ψ n ] p) ψ 2 n p) 2 n,, g ),k, n,,k g ) P[ f k) ] 2 p) 2 n,,k g ), ˆk k:ˆk k g ) n,ˆk n,,ˆk P[ˆk k, f k) f ˆk) ] p ψ 2 n p) 2 n,, g ),k, n,,k g ) P dk p) 2 n,,k g ) g ) n,ˆk n,,ˆk k P dk P dˆk p) 2 n,,k g ) 2 P 2 d k ) 2 In the last lne we use the fat that p ψ 2 n p) 2 ) n,, g ) n,,k g k p) 2 ) 2 n,,k g k g ) n,ˆk n,,ˆk P dˆk ˆk,k p) 2 ) 2 k, n,,k g ) P dk ) 2 P 2 d k ) P dk Applyng mass onservaton g n,,, n,,k, and k wll be equal to, so the seond term wll beome k g zero and thrd term wll beome K 2. Fourth term s always non negatve so removng ths term wll not affet the bound. So E[ψ n ψ n ] p ψ n p) 2 ) 2 2 K 2 p) 2 ) 2 P2 d k n,,k g K 2 2 k, p ψ n p) 2 ) 2 p) 2 ) 2 P2 d mn ψ n p ψ n K 2 2 p) 2 ) 2 ) 2 p) 2 ) 2 P 2 d mn ψ n wll always reman non negatve. We know that d mn s 2. If we onsder the value ofγto be 2, maxmum value of K 2 wll be 6 onsderng P d max to be zero and the maxmum possble value of 2 ) 2 s 4. Thus, E[ψ n ψ n ] p ψ n 4 p) 2. 27) ow we wll alulate the value ofψ 0. We know that ntally the ontrbuton vetor ontans only sngle non-zero value.e. ontrbuton reeved from t self and

14 4 that value s, rest all elements are 0. So ψ 0 0,, g ) 2 0,, [ 0,, g ) 0, 2 0,2, g ) 0, ,, g ) 0, ,, g ) 0, 2 ] [ 0 ) 2 0 ) 2... ) ) 2 ] [ 2 2 ) ] 2 [ ] 28) ow we wll substtute the value ofψ 0 n 27). Ths wll gve us the bound onψ n E[ψ ψ 0 ] p ψ 0 29) 4 p) 2 E[ψ ] p ) 4 p) 2 Smlarly 30) E[ψ 2 ] p p ) 4 p) ) 2 E[ψ n ] 6 p) 2 p) 2 6p) 3 4p) 2 p) n 6p) n 4p) n... 4p) 2 ) p) n 4pp ) n ) p) n 4 p)p 4 p)p It an be seen that rght hand sde of the above equaton s maxmum when p. It means potental funton s deayng at the slowest rate for p. So tme taken n onvergene for p wll be maxmum. For normal push, algorthm wll at as upper bound for dfferental push algorthm.e. ombnaton of dfferent values of postve nteger p s. So takng p; E[ψ n ] ) 2 n 8 ) 2 n k d 3) Here k d s an nteger onstant that s greater than the rato of maxmum value of ) 2 n and 8. Ths an be seen that k d wll depend on the number of steps requred for onvergene. After gosspng for nlog 2 )log 2 k d log 2 ξ steps, E[ψ n ] ) 2 log 2 )) 2 log 2k d )) 2 log 2/ξ)) k d E[ψ n ] ξ 32) n,, g ) 2 ξ, If summaton of some non-negatve numbers are less thanξthen ndvdually eah number must be less than ξ,.e. n,, g ξ 2 for all nodes. If we onsder weght as an nformaton to be spread among the nodes, aordng to theorem 5., nformaton wll reah to all the nodes n a power law network wth hgh probablty, n n log 2 ) 2 rounds. After these n rounds every node wll reeve at least 2 n weght. So applyng unon bound [27] over weght spreadng and potental deay event We have seen potental s deayng at every step) and dvdng wth g gves n,, g ξ at steps Olog 2) 2 log 2 log 2 k d log 2 ξ ).e. wthn Olog 2 ) 2 log ξ ) steps wth hgh probablty.