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2 Computer Communications 3 (8) Contents lists available at ScienceDirect Computer Communications journal homepage: Moeling message propagation in ranom graph networks Bin Wu, Ajay D. Kshemkalyani * Computer Science Department, University of Illinois at Chicago, Chicago, IL 667, Unite States article info abstract Article history: Receive 7 February 8 Receive in revise form 3 September 8 Accepte 4 September 8 Available online September 8 Keywors: Message propagation Ranom graphs oe egree oe coverage Peer-to-peer network Message propagation is use in a wie range of applications, such as search in unstructure PP overlays, moeling infection sprea in epiemiology, an moeling the sprea of gossip in social networks. For example, in a PP network that has an unstructure overlay, search for a piece of information is conucte by propagating the query message within the network, usually with the esire that as many noes as possible are covere with as few message forwarings as possible. In this paper, we stuy the behavior of the message propagation process in ranom graph networks an give a simple moel to escribe this process. When applie to a large network with ranom graph topology, the message propagation process can usually be moele as a ranom pick process or the coupon collection problem. We show that these moels are less accurate when the number of covere noes becomes large. We investigate the inaccuracy an then propose refine moels which remey the factors that cause the error. The refine moels have been confirme by our simulations to effectively compensate for the errors, especially uner high coverage conitions. Thus, when a large number of messages is expecte to be use in the message propagation process, the refine moels of higher orers are essential. Ó 8 Elsevier B.V. All rights reserve.. Introuction Message propagation is use in a wie range of applications. For example, message propagation is use in unstructure PP networks to perform keywor an range query searches [4,7]. In such PP overlay networks, search for a piece of information is conucte by propagating the query message within the network, usually with the esire that as many noes as possible are covere with as few message forwarings as possible. The overlay of the unstructure PP network may take various topologies ue to various ynamic characteristics such as the arrival an eparture of noes that form the network. Typical topologies inclue the Gnutella graph [7,5], ranom graph [5,8], power law ranom graph [], or gri [5]. Watt an Strogatz stuie the formation an characteristics of a category of ranom graph networks: the small worl network [9]. A small worl network is at some intermeiary position between a regular network an a ranom graph network in terms of ranomness. Since a small worl network has short network iameter (a property of ranom graphs) an is highly clustere as well, message propagation can be fast. Much research has been performe on constructing PP overlays with the small worl property to balance the search efficiency an the maintenance cost over a partially structure overlay [,3]. In any of these overlays, to fin a specific object without previous knowlege of the object istribution, blin search such as ranom walk [3,4,9], flooing [7,5,8], an their variations * Corresponing author. aresses: bwu@cs.uic.eu (B. Wu), ajayk@cs.uic.eu (A.D. Kshemkalyani). [,6,,,6, 3] has to be applie to the topology. o matter what approach is chosen, the goal is always to propagate the query message to as many noes as possible, an the constraint inclues the response time or the message overhea. Flooing is the approach that spreas message with the maximum spee an also has the maximum message overhea. Ranom walk has high message efficiency in terms of the ratio of the noe coverage to the message overhea, but the spee can be very slow. Variations that benefit from both of these approaches inclue flooing with TTL, expaning ring, an multiple ranom walkers [5]. Some specific phenomena in social activities an epiemiology stuies can also be moele by the message propagation process. For example, isease infection among the population can be cause by contact among people, an the behavior of virus spreaing is heavily epenent on the pattern of contacts. Once infecte by a isease, a person who has a wier range of social links is more likely to sprea the isease to uninfecte people, when the number of allowe contacts is fixe. As another example, consier gossip or the spreaing of information among the population via using one s social contacts. For such examples, the goal may be to stuy how many messages it takes to reach a certain proportion or all of the population. Or alternately, given a certain number of message hops (that etermines the time perio) to be use, ientify the proportion of the population that is infecte by the isease or that receives the news. A real worl social networking example is as follows. In a small town of population,, a new Chinese restaurant is opene. Uner normal behavior, the first few customers woul ranomly /$ - see front matter Ó 8 Elsevier B.V. All rights reserve. oi:.6/j.comcom.8.9.6

3 B. Wu, A.D. Kshemkalyani / Computer Communications 3 (8) tell this information to one or more of their acquaintances. After a while, when a significant portion of the people (say, 6%) alreay know about this restaurant, is it still worthwhile for the restaurant owner to encourage his customers to further sprea this information, if the incentive for avertising is the same? What is the possibility that the next effort introuces a new customer, or, ens up being a repetition to a person who alreay knows it? Using the refine moel gives a more accurate estimate of the probability of reaching a new customer. This will help the owner to make a ecision. The ranom walk approach, with possibly multiple walkers, is of special interest to most research in this area because of its scalable message overhea [3,4,9]. Precisely moeling the ranom walk is not easy because it is har to obtain a precise mathematical moel for the actual network topologies. In this paper, we stuy the behavior of the message propagation process in ranom graph networks an give a moel to escribe this process. We give our quantitative analysis moels base on the Gðn; pþ ranom graph network moel [8] an we focus our analytical target on the noe coverage an message efficiency. If the number of noes in the network is large, then at the steps when the noe coverage is small, the message propagation process can be moele by some other ranom processes, such as the ranom pick [3,4] or the coupon collector s problem [9]. For example, the stuy by Gkantsiis et al. [9] shows that the effect of a k-step ranom walk is statistically similar to that of taking k inepenent samples (ranom pick) in a well-connecte graph. Our moels explain the rationale of such an analogy. We then show that these an similar moels are less accurate when the number of covere noes becomes large. We investigate the inaccuracy an ientify the reasons causing it. We then propose refine moels which remey the factors that cause the error. The refine moels have been confirme by our simulations to effectively compensate for the errors, especially uner high coverage conitions. Thus, when a large number of messages is expecte to be use in the message propagation process, the refine moels of higher orers are essential. ote that a large number of messages can be expecte to be use in the examples of social contacts an epiemiology stuies given above... Contributions. We stuy the message propagation process on the ranom graph topology an formulate a simple mathematical moel in terms of noe coverage an message overhea. We then raw an analogy to other ranom processes, i.e., the ranom pick an the coupon collection problems.. More importantly, we investigate the inheritant ifference between message propagation an other ranom processes an iscover that the noe egree (average number of links per noe) plays an important role for the ifference: the effect of the limite number of links has a negative effect on the probability of forwaring a message to a new noe. 3. By quantitative analysis of such effects, we give a refine moel that increases the accuracy, especially when the number of covere noes has become large. The refinement is introuce by accounting for the irty links of the current noe when forwaring a message. 4. As per the moel, the probability that a next step message reaches a new noe is a function of both the message overhea, x, an the average noe egree,. The orer of refinement is expresse in terms of the maximum number of times ðkþ a noe may have been visite before the current arrival. We thus give a family of enhance moels, the k-orer refine moels. The higher the value k, the more accurate is the refine moel. We verify this by simulations. 5. The estimation of irty links is base on the value of k we specify, which is a ranom variable, whose value may be istribute between an x. We propose two approaches of computing irty links : unconitional an conitional estimations, an we also show that these two approaches actually prouce the same results but they have ifferent computation complexity. The aresse analysis moels are of special importance in stuying the behavior of the querying process in PP networks. The noe coverage an message efficiency are quantitatively moele via the network an process parameters which can be moulate by an aministrator. The n-orer ðn P Þ refine moel shoul be applie for better accuracy in the cases where the noe coverage is high. In these cases an arbitrary noe in the network may have been visite multiple times from its ifferent neighbors. In a typical search in a PP network, a single message may never be propagate very long, an it is also har to trace a single message in the propagation process. We use the term hop count to follow the means of message spreaing using ranom walkers. We exten the hop count to large numbers in orer to reach a higher noe coverage, where our refine moel will give significant ifference from the simple moel. High noe coverage situations are useful when the purpose is to sprea a message to as many noes as possible rather than just fining the esire object. ote that the Gnutella graph is not a ranom network. Hence, Gnutella-like PP networks nee a more specific means of analysis, which must also take into account the istribution of noe egree within the network. This is fairly complicate an outsie the scope of this article. Also, the flooing an partial flooing methos are not moele ifferently from ranom walkers in our investigation of noe coverage. However, the clustering of more realistic networks makes the flooing an partial flooing methos even more sophisticate than the ranom walker metho, an the message efficiency coul be worse than that for ranom walk even at low noe coverage. Section gives a simple algebraic moel for the noe coverage analysis of the message propagation process. In Section 3, we give the analogy between the ranom walk process an ranom sampling. In Section 4, we refine our algebraic moel to factor in the impact of noe egree. In Section 5, we perform a comparison an analysis of our refine moels. Section 6 gives the conclusions.. The algebraic moel When searching an unstructure network using ranom approaches such as ranom walk or flooing, noe coverage is a key concept use to escribe the behavior of the search process. It can be efine as the total number of noes in the network that have alreay been explore; sometimes we can also use the percentage of such noes as well []. In this paper, we use the first form of efinition of noe coverage for convenience. We propose a simple algebraic moel that performs a noe coverage analysis for the message propagation process but makes no istinction so as to whether the messages are forware by flooing or ranom walkers. Each query message is treate as an inepenent sample. This moel expresses the expecte noe coverage in terms of the message overhea, x. In this simple moel, an Erõs Rényi moel ranom graph network is assume, in which there are totally noes in the network. The probability that there exists a link (unirecte) between any two noes is p ð < p < Þ, such that the average noe egree is pð Þ. For simplicity, we assume that each noe has a egree

4 44 B. Wu, A.D. Kshemkalyani / Computer Communications 3 (8) of pð Þ. ext, we assume that the graph is well-connecte an when a noe relays a message, it sens out the message through a ifferent link from the one through which it receives the message, an this outgoing link is selecte ranomly from all links except the incoming one. Finally, this moel makes no istinction between whether or not multiple ranom walkers are use to sprea the message. In the case of multiple ðmþ ranom walkers, we assume they are initiate from m ranomly chosen noes an thereafter are forware inepenently. Suppose at a snapshot of the message propagation process, the current noe coverage is u an a specifie noe is forwaring a message to one of its neighbors, excluing the neighbor who has sent in the message. The probability that a new noe receives this message is u. Thus, the expecte value of noe coverage woul be u þ u after this message is forware. As is large enough, we can use to replace the term for the sake of convenience. Let x enote the number of messages so far. Then uðxþ uðx þ Þ ¼uðxÞþ which can be approximate as: u ðxþ ¼ uðxþ This equation can be solve as uðxþ ¼Ce x þ Here C is a constant etermine by the initial conition. This constant has a minor influence on the results of uðxþ an it can be compute epening on how we configure our initial conition, for example:. If we assume the noe coverage is before any message is forware, uðþ ¼, then C ¼.. If we assume the noe coverage is after a single message has been forware, uðþ ¼, then C ¼ð Þe. 3. For the special case of multiple ranom walkers, suppose there are m walkers sent out from a single initiator (which is specific in the cases of PP search), an also that the first message is require to cover the initiator itself before sening out any walker. Then the initial conition may be uðm þ Þ ¼m þ An the constant C turns out to be C ¼ðmþ Þe mþ Then the noe coverage in this case can be expresse as ( ð m Þemþ x uðxþ ¼ if x > m þ x if x 6 m þ The choice of the constant C has only a trivial influence on the computation of uðxþ an we can etermine this constant base on the special case of interest or just convenience. One of the uses of noe coverage is to estimate the success rate in a PP search: we assume that r copies of a esire object are ranomly istribute in the network. The probability of fining a copy by the effort of spreaing x messages can be expresse as p s ðxþ ¼ uðxþ r ð7þ The probability p s ðxþ is usually referre to as the success rate (of fining a esire copy). ðþ ðþ ð3þ ð4þ ð5þ ð6þ 3. Ranom walk moeling an ranom sampling 3.. Analogy to ranom picking The algebraic moel simulates a process of ranom pick: there is a bag of balls, at each step, we pick a ball an put it back. Then what is the expecte number of istinct balls we will fin after x attempts? Our algebraic moel approximates this process with a ifferential equation an the solution to this equation gives the formula for computing the noe coverage. The rationale of this formula can be valiate by comparing the formula for success rate using the noe coverage above an that given by Bisnik an Abouzei [3,4]. Recall that the algebraic moel gives the noe coverage uðxþ in Eq. (3). The constant C is etermine by the initial conition of a specific search case. If we take the following initial conition: uðxþ ¼ j x ¼ then we have uðxþ ¼ e x The success rate of fining an object with r copies is then compute using noe coverage: p succ ¼ uðxþ r ¼ e x r ðþ On the other han, [3,4] suggest to compute the success rate using the following formula: p s ¼ ð pþ kt ð8þ ð9þ ðþ where p ¼ r an kt ¼ x, the message overhea when k, T enote the number of walkers an the hop number, respectively. This is the result of treating each walk step as a ranom pick. To use the notations in our moel, we rephrase this formula: p s ¼ r x ðþ Also, Eq. () can be rephrase as p succ ¼ e r x ð3þ lim!þ ote that as is large enough, we have r ¼ e r an thus lim r!þ ¼ lim!þ e r ð4þ ð5þ an this equates the computation of p succ in Eq. (3) an that of p s in Eq. (). These two formulas are equivalent as goes to infinity. ote that these two approaches use ifferent formulas an result in the same outcome, because they both simulate the same process of inepenent ranom picks, an the proper computation for the success rate will always give the same result. However, the ranom walk in ranom graphs is not actually inepenent ranom picks, a message forwaring is always associate with the status of the current noe. The probability of fining a new noe by the next message forwaring is ecie not only by the current proportion of unetecte noes in the network, but also by the current status of the sening noe. This impact oes not exist in the inepenent ranom pick process an it contributes to the ifferences between our algebraic moel an the simulation results. This impact will be stuie in etail in Section 4.

5 B. Wu, A.D. Kshemkalyani / Computer Communications 3 (8) In Section 4, we also show how to enhance the algebraic moel to account for the noe egree in the ranom graph overlay, so as to improve accuracy. This consieration allows us to account for the fact that the current noe may have been explore before. 3.. Analogy to coupon collector s problem The coupon collector s problem is another typical ranom sampling process: given types of coupons, a customer buys one coupon at a time. For each time, the probability of getting any specific type is equal: =. Then what is the expecte number of purchases to obtain all the types? If this problem is aske this way: in orer to obtain u types among types of coupons, what is the expecte number of purchases x? then obviously this becomes the reverse problem of the ranom pick we escribe in the previous section. In this section, we show that the analysis metho also reveals the equivalence of the memoryless ranom walk an the coupon collector s problem. It is well known that the expecte times to collect all types is xðþ ¼ X k¼ k ¼ H ð6þ where H is the nth harmonic number an its analytical expression is H ¼ ln þ c þ O ð7þ Here c is the Euler Mascheroni constant an its value is : Similarly, we have xðuþ ¼ þ þ þþ ðu Þ! ¼ X X u k k k¼ k¼ ¼ H ð H u Þ ¼ ln ln ð uþþo O u As an u are large enough, we can omit the term O O u an have xðuþ ¼ ln ð8þ u If we express u in terms of x, we are answering this question: what is the expecte types of coupons we can collect, u, after x purchases? : uðxþ ¼ e x ð9þ which is ientical to Eq. (9), when the initial conition uðxþ ¼ is applie to Eq. (3) in the algebraic moel. This ientity illustrates that the simplifie ranom walk process on a regular ranom graph which is escribe by our algebraic moel, is equivalent to the coupon collector s problem when the sampling space,, is large. 4. Refinements of the algebraic moel The algebraic moel is intene to capture the ranom process of message forwaring in normal ranom graph overlays. The metho we applie is somehow over-simplifie as we ignore the influence of noe egree in our moels an this ignorance leas to overestimate of noe coverage. We now refine the algebraic moel by stuying an factoring in the impact of noe egree. 4.. Impact of noe egree As assume in Section, at each step of message forwaring, our analytical moel assumes the probability of visiting a new noe to be the same as the ratio of the number of unvisite noes to the total number of noes: uðxþ. This seems reasonable at first glance since the next message forwaring is a ranom visit. The above assumption is true if the current noe itself is first visite by the current message. In this case, all the remaining links of this noe are fresh (not probe yet) an whether one such link leas to a visite noe or a new noe is ranom. Hence, the probability of reaching a new noe via that link is thus reasonably etermine by the proportion of new noes currently in the network. However, if the current noe has been visite before, then this probability shoul be lowere because the current noe may forwar the current message via a link that has been traverse before an thus to a visite noe. Consier an extreme conition: if every link of the current noe has been visite before, then there is no way that the next forwaring visits a new noe. The impact of noe egree thus can be expresse in this way: consiering the scenario that the current noe has been visite before, the higher the noe egree, the higher the possibility that the next forwaring traverses a fresh link, an thus the closer the probability of fining a new noe approaches the proportion of unvisite noes within the network. This is confirme in our simulation, see Section Conitional estimate of irty links Recall that in our algebraic moel, the iterative relationship of noe coverage is expresse in Eq. (). We now moify this formula to take into account the possibility that the current noe has been visite before uðx þ Þ ¼uðxÞþ uðxþ pðxþ ðþ Hence, pðxþ is the probability that the next forwaring takes a fresh link. As the network topology is fixe, this probability is etermine only by the proportion of fresh links of the current noe, which in turn is a ranom function of the message overhea x. Hence, we can also express this probability as a function of x. The core challenge of refining the algebraic analysis moel is to compute the probability pðxþ. Let us branch into ifferent preconitions in terms of the number of previous visits to the current noe. ote that this number exclues the arrival of the current message. The notation use is summarize in Table. previous visits: As explaine in the previous section, p ¼. previous visit: p ¼ 3 þ : The number of irty links after the first visit, DL, is one incoming, one outgoing. Hence, enotes the probability that the current message has come from a link other than the two links mae irty by the previous visit. In this case, the probability of probing a fresh link is 3. On the other han, if the current message has come though a irty link, the probability of traversing a fresh link is. previous visits: After visits have been pai to this noe, the probabilities that, 3, or 4 links have been contaminate, enote w ðþ, w ð3þ, w ð4þ, respectively, are: w ðþ ¼ w ð3þ ¼ max½ ;Š þ max½ ;Š w ð4þ ¼ max½ ;Š max½ 3;Š

6 44 B. Wu, A.D. Kshemkalyani / Computer Communications 3 (8) Table otations use in the analysis DL i CDL i UDL i w i ðjþ wða j bþ p i q i ðxþ pðxþ After two visits to a noe, the expecte number of irty links can be expresse as DL ¼ w ðþþ3 w ð3þþ4 w ð4þ ðþ Similarly, the probability that the next message forwaring takes a fresh link, p, can be compute as follows: p ¼ DL DL þ DL DL ðþ 3 previous visits: For simplicity, we assume that DL links have been irty before the thir visit to the current noe. In other wors, we use the conitional probabilities with the assumption that exactly DL (usually a non-integer number as the result of estimation) of the links are known to be irty. Thus the resulting number of irty links after the thir visit, can be DL,DL þ, or DL þ, respectively; the corresponing conitional probabilities, with the given DL as the pre-conition, are enote as wðdl j DL Þ, wðdl þ j DL Þ, an wðdl þ j DL Þ, respectively. These can be compute as follows: wðdl jdl Þ¼ DL DL wðdl þ jdl Þ¼ DL max½ DL ;Š wðdl þ jdl Þ¼ max½ DL ;Š max½ DL ;Š ow DL 3 can be compute as CDL 3 ¼ DL wðdl jdl ÞþðDL þ ÞwðDL þ jdl Þ þðdl þ ÞwðDL þ jdl Þ p 3 can be compute as p 3 ¼ DL 3 DL 3 þ DL 3 DL 3 ð3þ ð4þ i previous visits: Similarly, for i P 3 previous visits, we obtain DL i an p i as follows: wðdl i jdl i Þ¼ DL i DL i wðdl i þ jdl i Þ¼ DL i max½ DL i ;Š wðdl i þ jdl i Þ¼ max½ DL i ;Š max½ DL i ;Š ow, with the preconition that DL i irty links are present before the ith visit, DL i is compute in the following formula: CDL i ¼ DL i wðdl i jdl i ÞþðDL i þ ÞwðDL i þ jdl i ÞþðDL i þ ÞwðDL i þ jdl i Þ ð5þ an p i ¼ DL i Expecte number of irty links after i visits Expecte number of irty links after i visits (conitional estimate) Expecte number of irty links after i visits (unconitional estimate) Probability that after i visits, j links are irty Probability of a irty links after current forwaring, assuming the noe has b irty links before the current visit Probability that the next forwaring after i previous visits takes a fresh link Probability that a noe is visite i times after x messages have been use Probability that the next forwaring takes a fresh link DL i þ DL i DL i ð6þ ote that in the computation of DL, the notations w ðþ, w ð3þ, an w ð4þ enote the probabilities that after two visits,, 3, an 4 links are irty, respectively. These terms escribe the unconitional probabilities associate with those values after visits. Since DL ¼ is a fact known to us, we have w ðþ ¼wð j Þ, w ð3þ ¼ wð3 j Þ, an w ð4þ ¼wð4 j Þ. The conitional an unconitional probabilities are equal when i ¼. In the conitional estimates of irty links, we use the terms DL an DL, while we use CDL i for i P 3 because the preconitions for estimating DL an DL,DL ¼ an DL ¼, are eterministic. Also note that the w i ðjþ is a unconitional probability that is associate with j, an integer number of irty links an i, an integer number of times of visits; whereas wðp j qþ, without subscript, enotes a conitional probability that is inepenent of the times of visits, an the values p an q are usually non-integer numbers Complexity analysis Observe from Eq. (5) that CDL i can be compute in OðiÞ steps because each step to compute CDL j ðj ¼ ;...; iþ takes constant time. Hence, the conitional estimate of the number of irty links can be compute in linear time Unconitional estimate of irty links Since the expecte number of irty links after i visits, DL i,isa ranom variable when i >, the proper estimate of DL i shoul have liste all the possible number of irty links after i visits an then sum them up weighte by their corresponing probabilities. For example, for DL 3, the possible number of irty links after three visits is, 3, 4, 5, or 6, an the formula for computing DL 3 shoul be DL 3 ¼ w 3 ðþþ3 w 3 ð3þþ4 w 3 ð4þþ5 w 3 ð5þþ6 w 3 ð6þ The probabilities are given as follows: w 3 ðþ ¼w ðþwð j Þ w 3 ð3þ ¼w ðþwð3 j Þþw ð3þwð3 j 3Þ w 3 ð4þ ¼w ðþwð4 j Þþw ð3þwð4 j 3Þþw ð4þwð4 j 4Þ w 3 ð5þ ¼w ð3þwð5 j 3Þþw ð4þwð5 j 4Þ w 3 ð6þ ¼w ð4þwð6 j 4Þ ð7þ The computation of these w i ðjþ that are require to compute the unconitional estimate of DL 3 is illustrate in Fig.. w( ) w () 3 w( ) w (3) 3 w () w(4 ) w(3 ) w(3 3) w(4 ) w(3 ) w (3) w(5 3) w(4 3) w(4 4) w (4) 3 w () 4 w(5 4) 5 w (5) 3 Fig.. Unconitional estimate of DL 3. w (4) w(6 4) 6 w (6) 3

7 B. Wu, A.D. Kshemkalyani / Computer Communications 3 (8) Generalizing the above erivation, we can show that 8 w i ðjþwðjjjþ if j ¼ >< w i ðjþwðjjjþþw i ðj Þwðjjj Þ if j ¼ 3 w i ðjþ ¼ w i ðjþwðjjjþþw i ðj Þwðjjj Þþw i ðj Þwðjjj Þ if 3 < j < i w i ðj Þwðjjj Þþw i ðj Þwðjjj Þ if j ¼ i >: w i ðj Þwðjjj Þ if j ¼ i ð8þ As mentione in Section 4., the term with the subscript is calle unconitional probability. For example, w i ðjþ means the probability that exactly j links are irty after the noe has been visite i times. The term without subscript is conitional probability or single step probability. For example, wða j bþ enotes the conitional probability that a links are irty after the next visit, given that the current number of irty links is b ðb þ P a P bþ. The computation of DL i in Section 4. is an estimation that uses only the single step probabilities with the assumption that the number of irty links is given by DL i before the ith visit. A seemingly more accurate formula is given by the following equation: DL i ¼ Xi j¼ j w i ðjþ; where w i ðjþ is given in Eq: ð8þ ð9þ j ¼ ;...; i enumerates all possible number of irty links after i visits. For now, we assume that the noe egree is sufficiently large so that we can reasonably ignore the case that all links become irty. j, j, an j are the only possible number of irty links prior to the ith visit that may result in j irty links after the ith visit. Fig. illustrates on the bottom layer all the possibilities (of the number of irty links) after three visits. The numbers in the circles represent the possible numbers of irty links. The term attache to a circle represents the unconitional probability associate with the circle number (irty links) an the layer inex (number of previous visits). The terms labele on the eges are conitional probabilities which inicate the transition from one state to another after one more visit. When the number of visits to the current noe is, the number of irty links can only be, so we efine w ðþ ¼, an at layer there is only one circle. At layer, which escribes the resulting possibilities after only one visit, we also have only one circle because the number of irty links can only be. The algorithm for computing w i ðjþ is to fin all paths from the top circle to the circle enclosing j at layer i. For each of the paths, the probability corresponing to this path is compute by multiplying all the conitional probabilities wða j bþ along that path. By summing up the path probabilities for all such paths leaing to that circle from the top circle, we obtain the unconitional probability w i ðjþ Complexity analysis To compute DL i ði P Þ, we nee to compute w i ðjþ for j ¼ ;...; i. The computation of w i ðjþ in turn, epens on w i ðkþ for k ¼ j, j, an j. We use ynamic programming to compute an recor this unconitional probability w i ðjþ for all the layers of circles as illustrate in Fig.. Observe that we have i such values for any given i. The single step probabilities wða j bþ along the eges are etermine by a, b, an the noe egree. ote that the value of a can only be b, b þ, or b þ. The probabilities are wðbjbþ ¼ b b wðb þ jbþ ¼ b max½ b;š wðb þ jbþ ¼ max½ b;š max½ b ;Š 9 >= >; ð3þ These are compute the same way we compute the conitional probabilities in Section 4.. It can be seen from Fig. that, to compute DL i, we must compute the values for all eges that lea to layer i from layer (or layer ), the number of which is 3 ði Þ.We must also compute the i unconitional probability values. The time complexity for evaluating w i ðjþ is constant when the unconitional probability values of the upper layer an all the associate ege values are known, as given in Eq. (8). Thus, the complexity for computing DL i is Oði Þ an the space requirement is also Oði Þ Comparison of estimation approaches We claim that the unconitional estimate of the number of irty links UDL i is an accurate estimate because it takes into account all the possible values of the number of irty links after i visits. However, this metho takes time complexity an storage requirement both of Oði Þ. In Section 4., we introuce the conitional estimate that bases the next step estimation on the result of the previous step, which has time complexity of OðiÞ for computing DL i. The ifferences between these two methos may serve as a justification for the simplifie estimation of irty links using the conitional approach. For simplicity, assume that is large enough so that we o not have to bother with the marginal cases of having all links irty. We also enote the conitional estimate of DL i as CDL i an the unconitional estimate as UDL i. It is obvious that CDL ¼ UDL. Assume that at step i, the number of irty links is m. We then compute DL iþ using the unconitional metho: UDL iþ ¼ mw iþ ðmþþðmþþw iþ ðm þ ÞþðmþÞw iþ ðm þ Þ þðmþ3þw iþ ðm þ 3Þþðmþ4Þw iþ ðm þ 4Þ ð3þ We then compare this result with that from the conitional estimation: CDL iþ ¼ DL i wðdl i jdl i ÞþðDL i þ ÞwðDL i þ jdl i Þ þðdl i þ ÞwðDL i þ jdl i Þ ð3þ The value of UDL iþ can be compute as follows. There are only three possible values for DL i : if DL i ¼ m, what is the expecte value for DL iþ? if DL i ¼ m þ, what is the expecte value for DL iþ? if DL i ¼ m þ, what is the expecte value for DL iþ? Then we average these three values weighte by wðm j mþ, wðm þ j mþ, an wðm þ j mþ (compute using Eq. (3)), which are the possibilities that DL i shoul be m, m þ, or m þ, respectively. This is functionally equivalent to the computation escribe in Section 4.3. if DL i ¼ m, then at the next step: DL iþ ¼ mwðmjmþþðmþþwðm þ jmþþðmþþwðm þ jmþ ð33þ

8 444 B. Wu, A.D. Kshemkalyani / Computer Communications 3 (8) mðm Þ mð mþ ¼ m þðmþþ ð Þ ð Þ ð mþð m Þ þðmþþ ð34þ ð Þ ¼ m þ ð35þ We can see that DL iþ epens only on an m. Similarly, we have the following: if DL i ¼ m þ, then DL iþ ¼ðmþÞ þ if DL i ¼ m þ, then DL iþ ¼ðmþÞ þ ð36þ ð37þ ow, to compute DL iþ concerning all possible values of DL i ðm; m þ ; m þ Þ, we make the weighte average: UDL iþ ¼ wðmjmþ m þ wðm þ jmþ ðm þ Þ þ wðm þ jmþ ðm þ Þ ð38þ ote that wðmjmþ ¼ mðm Þ, wðm þ jmþ ¼mð mþ, an wðmþ ð Þ ð Þ jmþ ¼ ð mþð m Þ accoring to Eq. (3). Finally, it turns out: ð Þ UDL iþ ¼ m þ 4 ð39þ ow compute the conitional estimate of DL iþ, accoring to Eq. (35): CDL iþ ¼ DL i þ ¼ m þ 4 ð4þ Comparing Eqs. (39) an (4), we have UDL iþ ¼ CDL iþ ð4þ Base on the above analysis, we have the following conclusion: Conitional estimate an unconitional estimate of irty links give the same result. This result seems counter-intuitive at first glance. However, the conitional metho uses a generalization at each step, an this is a linear generalization of the state at the previous visit. The final result of the unconitional metho is also a linear computation from the previous steps, which implies that the conitional metho oes not lose any precision against the unconitional metho. The unification of the two approaches of estimation gives us the convenience of computing the number of irty links with linear time an space complexity. As per Eq. (35), the expecte numbers of irty links, DL i ði P Þ, can be expresse in terms of the noe egree only. For example, DL ¼ DL þ ¼ 4ð Þ= an DL 3 ¼ DL þ ¼ ð Þ þ Influence of previous visits In the last section, we formulate the concept DL i, which enotes the expecte number of irty links after i visits have been pai to the current noe. Using DL i, we can then compute the corresponing p i, which enotes the probability that the next message forwaring takes a fresh link. In this case, however, the current noe may have been visite,,, or more times before, which means that the value of pðxþ in Eq. () is also a ranom function of the message overhea, x. So, we nee to figure out the expecte value of pðxþ, which epens on the probabilities of how many times the current noe has been visite before. Given that after x messages have been forware, uðxþ noes are covere, we now focus on the computation of such probabilities. Let us enote the probability that an arbitrary noe in the network was visite i times after x messages as q i ðxþ. q ðxþ: After x messages were arbitrarily forware, the noe was not visite. This probability can be expresse as [9]: q ðxþ ¼ x ð4þ q i ðxþ : The probability that the noe was visite exactly i times ði ¼ ;...; xþ is: q i ðxþ ¼ x i i x ð43þ i ow we come back to the computation of pðxþ in Eq. (). ote that pðxþ is the probability that the next message takes a fresh link out, an this probability epens on how many times the current noe has been visite before, which is also a ranom variable epening on x. The expecte value of pðxþ thus can be compute by taking the average of the probabilities of fresh links out, weighte by the probabilities of the corresponing number of previous visits: pðxþ ¼ Xx i¼ p i ðxþq i ðxþ 5. Analysis an simulations ð44þ Eq. () propose a refine moel to compute noe coverage, uðxþ. The refinement introuces the influence of the (potential) previous visits to the current noe on the probability of visiting a new noe at each forwaring step. This consieration was prompte by the eviation between the moel results an the simulation results. The eviation also reflects the ifference between the message forwaring process an a ranom sampling. In a ranom sampling process, a sampling is inepenent of previous steps. Message forwaring, on the other han, is epenent on the status of the current noe. Consier an extreme conition: if all links of the current noe have been explore, then the next forwaring will not fin a new noe at all, while with ranom sampling, this probability is equal to the current proportion of new noes in the network. ote from Eq. (43) that the probability q i ðxþ ecreases ramatically with i when, the total number of noes, is reasonably large an x is not overly large. As an example, let ¼ ;, x ¼ ;, we have: q ðxþ ¼:665 q ðxþ ¼:333 q ðxþ ¼:76 q 3 ðxþ ¼:63 q 4 ðxþ ¼:58 From this example, it is observe that for x values that are not too large, say, less than, the probabilities of multiple visits more than two times are so small that we can reasonably ignore them. When we use k terms for the summation in Eq. (44), the refine moel is terme as the k-orer refine moel. k-orer refinement consiers the influence of,,..., k previous visits. The higher the orer of refinement, the more precise results the moel will prouce.

9 B. Wu, A.D. Kshemkalyani / Computer Communications 3 (8) oe (=, =4 m=3) Simple Moel -Orer Refine Moel oe (=, =6 m=3) Simple Moel -Orer Refine Moel 6 8 Fig.. -Orer refinement on noe coverage. (a) Average noe egree = 4. (b) Average noe egree = oe (=, = m=3) Simple Moel -Orer Refine Moel 4 oe (=, = m=3) Simple Moel -Orer Refine Moel Fig. 3. -Orer refinement on noe coverage. (a) Average noe egree =. (b) Average noe egree =. In Section 5., we apply the -orer refinement of the algebraic moel (Section ) to compute the noe coverage with various average noe egrees. A comparison of the results from the simple algebraic moel, from the -orer refine algebraic moel, an from the simulations is presente in Figs. an 3. In Section 5., we apply the -orer refinement an perform a similar comparison. The istinction between these two refine moels is also analyze in that section. ote that in the graphs, there is an abrupt change in the shape of the curves at a certain number of hops because we change the scale of hops (X-axis) to accommoate both small hop count an large hop count in the same graphs Orer refine moel is still large an unstable. This can be illustrate from the fluctuations (though not visibly prominent) on the curves for the simulation results in Fig.. Possible graph separation at low noe egree: When the average noe egree is small, it has been shown by Erõs an Rényi [8] that is likely that the graph is not connecte: some parts of the network can never be reache from the starting noe. Specifically, consier the Gðn; pþ moel, where p is the probability of a link. One of the properties of the ranom graph is as follows: If pn < ln n, then a graph in Gðn; pþ will almost surely not be connecte. If pn > ln n, then a graph in Gðn; pþ will almost surely be connecte It is observe from Figs. an 3 that the results yiele by the refine moel fall below those for the simple moel, which is expecte since the refinement term, pðxþ, ecreases the probability of visiting a new noe at each step. We also notice that in the cases where average noe egree equals 4 or 6, the -orer refinement still results in noe coverage values greater than those in the simulation results. This is somehow unexpecte at first glance, since the -orer refinement unerestimates the probability. (The -orer term an higher orer terms are neglecte from the computation of pðxþ). Two reasons are possible for this scenario: oe (=,) Simple Moel -Orer Refine Moel (=4) -Orer Refine Moel (=6) -Orer Refine Moel (=) -Orer Refine Moel (=) Precision of simulation: The precision of simulation is significantly affecte by average noe egree. For smaller noe egree setups, the results oscillate willy among multiple runs. Even when we use the average of results from multiple runs, the error Fig. 4. Impact of noe egree on -orer refine moel.

10 446 B. Wu, A.D. Kshemkalyani / Computer Communications 3 (8) In our simulation where ranom graphs of, noes are generate, for the cases where the average noe egree is 4 an 6, the generate graphs are almost surely not connecte, an in the cases where the noe egrees are an, the graphs are almost surely connecte. This phenomena significantly reuces the resulting noe coverage generate from the simulation in which a isconnecte overlay may exist, i.e., for ¼ 4 an 6. ote that in our analysis moels, we assume the network is connecte. Fig. 4 illustrates how the refine moel is affecte by the noe egree. ote that the simple moel takes no account for this effect, thus resulting in over-estimate noe coverage, which is plotte as the top curve. Compare to the simple moel, the refine moel lowers the noe coverage in all cases with respect to the ifferent noe egree values. It is observe from the figure that the smaller the noe egree, the more the eviation of the refine moel from the simple moel. In other wors, the simple moel prouces larger errors for a lower noe egree topology than for higher noe egree topologies. This is because for a smaller noe egree, the effect of noe revisits becomes more important. If the current noe has been visite before, then the chance that the next message forware from this noe visits a new noe is smaller for a lower egree noe than for a higher egree noe (as there are more fresh links for higher egree noes). It is also worth noting that for all noe egree values, the influence of the noe revisits becomes significant only when the hop 4 oe (=, =4 m=3) Simple Moel -Orer Refine Moel 4 oe (=, =6 m=3) Simple Moel -Orer Refine Moel Fig. 5. -Orer refinement on noe coverage. (a) Average noe egree = 4. (b) Average noe egree = oe (=, = m=3) Simple Moel -Orer Refine Moel oe (=, = m=3) Simple Moel -Orer Refine Moel Fig. 6. -Orer refinement on noe coverage. (a) Average noe egree =. (b) Average noe egree = oe (=, =4 m=3) -Orer Refine Moel -Orer Refine Moel oe (=, =6 m=3) -Orer Refine Moel -Orer Refine Moel Fig. 7. Comparing -orer an -orer refine moels. (a) Average noe egree = 4. (b) Average noe egree =

11 B. Wu, A.D. Kshemkalyani / Computer Communications 3 (8) oe (=, = m=3) -Orer Refine Moel -Orer Refine Moel oe (=, = m=3) -Orer Refine Moel -Orer Refine Moel Fig. 8. Comparing -orer an -orer Refine Moels. (a) Average noe egree =. (b) Average noe egree =. number, or, message overhea, becomes consierably large. The eviation of the refine moel from the simple moel reflects the impact of noe revisits. As the hop number increases, the number of noes that have been visite before an the expecte number of visits also increase, which in turn increase the impact of noe links Orer refine moel Observe from Fig. 3 that although the simple moel over-estimates the noe coverage, the -orer moel tens to uner-estimate noe coverage as the hop number becomes large. The curve for our refine moel falls below that for the simulation results, which is more obvious as the hop number grows beyon in this figure. This is ue to the orer of refinement we use for the moel. The impact factor of noe revisits, pðxþ, as efine in Eq. (44), neglects the secon an higher orer terms in the -orer refine moel. These terms represent the scenario of multiple previous visits to the noe. The probability for revisits more than once with small hop number is very small an negligible; this is the reason for using -orer moel for simplicity. However, when the hop number is large enough, the probability will become significant an the corresponing higher orer terms are no longer negligible. The truncation of those higher orer terms results in a reuce value of noe coverage. The greater the hop number, the more significant is this effect, as shown at the right en in Fig. 3. To account for this effect, we inclue the secon orer term in computing pðxþ an the -orer refine moel. We perform the same comparison as in Section 5., however, using this moel to replace the -orer moel. The results are isplaye in Figs. 5 an 6. We base our analysis primarily on Fig. 6 because the simulation results in Fig. 5 have unreasonably large variations an instability ue to small noe egrees, as explaine in Section 5.. Comparing Figs. 6 an 3, the curves for the -orer refine moel turn out to be much closer to the simulation curve than those for the -orer moel when the hop number is larger. This confirms our analysis. The slight ifference between this moel an the simulation is still ue to term truncation in computing pðxþ: even higher orer terms will begin to take effect as hop number grows even larger. The comparison between Figs. 6 an 3 further confirms the valiity of our refine moels an provies a hint for moel esign: if a large number of messages is expecte in the message propagation process, we shoul use the refine moels with higher orer for more accuracy. Finally, Figs. 7 an 8 compare the -orer an the -orer refine moels uner ifferent noe egrees. As preicte from the earlier analysis in this section, the curves for the -orer refine moel always lie above that for the -orer refine moel an their ifference becomes more significant as the hop number grows. 6. Discussion an conclusions Message propagation on ranom graphs is use in a wie range of applications such as search in unstructure PP networks, moeling the sprea of infection in epiemiology, an propagation of gossip in social networks. Until now, message propagation processes were generally moele only as ranom sampling processes. As the noe coverage an the number of messages increase, we showe that ranom sampling no longer serves as an accurate moel. In this paper, we stuie the istinction between the message propagation process an ranom sampling over the Gð; pþ ranom graphs. We investigate the effect of noe egree in the message propagation upon the efficiency of covering istinct noes. Compare to the pure ranom sampling moel, this factor has a negative effect: the actual noe coverage is less than that given by the simplifie moels using ranom sampling. This influence is more significant when the noe coverage becomes high an when the average noe egree is small. The ifference was also quantitatively stuie. We then introuce refine moels that account for irty links, which is the reason for the reuce probability of forwaring a message to a new noe. The number of irty links is a ranom variable an the expectation of this number is epenent upon the number of times that the current noe has been visite before. The number of (previous) visits to a noe is also a ranom variable whose istribution epens on x, the message overhea. Uner normal conitions, the probability that a noe has been visite many times is usually small an these cases can be omitte epening on how precise we want our refine moel to be. 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