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Inadequacy of SIR Model to Reproduce Key Propertiesof Real-world Spreading Phenomena: Experiments on a

Large-scale P2P SystemDaniel Bernardes, Matthieu Latapy, Fabien Tarissan

To cite this version:Daniel Bernardes, Matthieu Latapy, Fabien Tarissan. Inadequacy of SIR Model to Reproduce KeyProperties of Real-world Spreading Phenomena: Experiments on a Large-scale P2P System. SocialNetwork Analysis and Mining, Springer, 2013, 3 (4), pp.1195-1208. �10.1007/s13278-013-0121-0�. �hal-00857518�

Inadequacy of SIR Model to Reproduce KeyProperties of Real-world Spreading Cascades:

Experiments on a Large-scale P2P SystemDaniel F. Bernardes, Matthieu Latapy, Fabien TarissanLIP6 – CNRS and Universite Pierre et Marie Curie / Paris 6

4 Place Jussieu, 75252 Paris cedex 05 – FranceEmail: firstname.lastname@lip6.fr

Abstract—Understanding the spread of information on com-plex networks is a key issue from a theoretical and appliedperspective. Despite the effort in developing theoretical modelsfor this phenomenon, gauging them with large-scale real-worlddata remains an important challenge due to the scarcity ofopen, extensive and detailed data. In this paper, we explainhow traces of peer-to-peer file sharing may be used to this goal.We reconstruct the underlying social network of peers sharingcontent and perform simulations on it to assess the relevance ofthe standard SIR model to mimic key properties of real spreadingcascades. First we examine the impact of the network topologyon observed properties. Then we turn to the evaluation of twoheterogeneous extensions of the SIR model. Finally we improvethe social network reconstruction, introducing an affinity indexbetween peers, and simulate a SIR model which integrates thisnew feature. We conclude that the simple, homogeneous modelis insufficient to mimic real spreading cascades. Moreover, noneof the natural extensions of the model we considered, whichtake into account extra topological properties, yielded satisfyingresults in our context. This raises an alert against the careless,widespread use of this model.

I. I NTRODUCTION

Diffusion phenomena in complex networks – such asthe spread of virus on contact networks, gossip on socialnetworks and files in peer-to-peer (P2P) networks – havespawned an increasing interest in recent years. The boostof computer networks and online social network platformsoffers data and new insights on these phenomena in largescale networks; the possibility to validate and refine currentmodels might lead to breakthroughs in the field.

Although large scale diffusion phenomena have alwaysattracted considerable interest, it has been historicallychallenging to obtain open, extensive and detailed real-worlddata at this level. Despite this obstacle, diffusion modelsemerged, notably in epidemiology. The early models, bothdiscrete and continuous (see [1], [2] for a survey), focusedprimarily on macroscopicaspects of diffusion – such as theevolution of the number of infected individuals in a population– overlooking themicroscopicdynamic of the epidemic – i.e.,how (by whom) individuals become infected. The advent ofnetwork analysis in various contexts has pushed for a moredetailed description of the diffusion process. Indeed, models

based on the precise interactions of individuals on a networkhave blossomed in sociology [3], computer science [4] andeconomics [5], among others. New epidemic models inspiredby the classical approaches featuring a detailed dynamicdescription in the context of networks also appeared (see [6]for a survey). In particular the network version of the SIRmodel (henceforth called simply SIR model) and derivateshave established themselves as reference models in the studyof information diffusion [7].

In this context, we have seen theoretical developments ofthese models [8], [9], focusing particularly in their asymptoticbehavior. A number of applications of such models werealso explored [10], [11], including works investigatingrelevant properties of epidemic models on real networks [12].However, as pointed out in [13], assessing the pertinenceof such models to describe real-world phenomena is criticaland empirical studies featuringreal spreading dataarekey. Since network-based epidemic models are based onlocal rules of transmission which take into account thenetwork topology, in order to validate these models oneneeds a comprehensive empirical spreading trace, consistingof (1) detailed chronological data of who transmitted theinformation to whom and (2) data describing the underlyingnetwork on which the diffusion process takes place. In largeepidemic bursts the available data often provides the evolutionof an aggregate quantity (such as the number of touchedindividuals) but rarely uncover the local trail of the epidemicat an individual level. Data mining in computer networkscan help providing detailed information at a large scale [14],[15], [16], [17]. In this framework, works typically featurerecords of diffusion events at an individual level but lack thecomplete information of the underlying network on which thediffusion takes place – see discussion in [18]. The presentpaper analyses the relevance of the SIR model for real-worlddiffusions, using data obtained measuring file sharing activityon a peer-to-peer network. Our framework allows one to takeadvantage of this rich dataset to obtain both the real spreadingdata (the detailed diffusion trail) and the underlying network.

This paper extends the results in [19]. It begins with adescription of our dataset and framework in section II, in

which we calculate statistics concerning peer activity andfile sharing and in which we define spreading cascades.In section III we construct the underlying social networkof peers from the diffusion trace. In the following twosections, we confront the SIR model and extensions to thereal data. More precisely, in section IV we simulate thespreading of files as a standard SIR process and compareit with the observed spreading; we also investigate theinterplay between this process and structural properties ofthe underlying network where the spreading takes place. Insection V we examine the spreading pattern when we modifythe SIR model to account for heterogeneity in the behaviorof the peers and in the popularity of files. In section VIwe present a novel approach which consists in simulatingan SIR derived models on an enhanced reconstruction ofthe underlying social network. This reconstructed networkis made possible using an affinity index for each coupleof peers. We conclude the paper with future work perspectives.

II. DATASET AND FRAMEWORK

The data used in this study comes from file sharing inan eDonkey server, obtained from a measurement of eighthours of activity (akin to [20]). In this setting, peers querythe eDonkey server indexing files and for each file they get alist of available peers in the network possessing the requestedfile. Next, peers contact potential providers directly andtransmission between them ensues. Our dataset is a collectionof answers to these queries, encoded as 4-tuples of integersin the following format: (t, P, C, F ), where capital lettersrepresent unique ids (e.g. in Fig. 1). Each tuple accountsfor a query made at timet of the file F by the peerC,satisfied by the peerP – that is,P providedF to the peerC at time t. Let D be the set of all recorded tuples,Pthe set of all peers in these tuples andF the set of allfiles exchanged. In our dataset we have|P| = 1 908 500peers,|F| = 801 280 files and|D| = 22 944 800 file transfers.

Fig. 1. Trace log example with corresponding spreading cascade in blackand underlying network in light gray.

A. File sharing

The traceD naturally induces a relationship between filesand peers (who request or provide them), which we explore de-scriptively in this section. We begin encoding this relationshipbetween the disjoint sets of peersP andF files in a bipartitegraphB = (P,F ,A) as follows. Let(t, P,X, F ) ∈ D bea recorded transmission of the fileF by the peerP to somepeerX at some timet, which we denote simply by(·, P, ·, F ).Likewise, let (·, ·, P, F ) ∈ D be a recorded transmission ofthe fileF to the peerP , provided by some peer at some timeinstant. Then:

A = {(P, F ) ∈ P × F : (·, P, ·, F ) ∈ D ∨ (·, ·, P, F ) ∈ D}

In other words,B is the bipartite graph in which peers arelinked to the files which they have provided or sought. Thedegree of peers and files in this bipartite graph representsthe number of files transfered by a peer and the number ofpeers who shared a file, respectively. The degree distributionof these sets inB (constructed from the P2P trace) areplotted in Fig. 2a. In order to estimate the typical numberof interested peers per file we have computed the mediandegree of files in the bipartite graph, 5, and the averagedegree, 14.73, with standard deviation 34.74. Likewise, wehave calculated the same statistics for peers, to estimate thenumber of files commonly shared by peers: its median degreein the bipartite graph is 3 and the average degree is 6.19,with corresponding standard deviation 12.66. The degreedistribution of both peers and files is however heterogeneousand mostly concentrated on small values; all degree valuesfor peers and files remain below104.

Another important aspect of our P2P trace in terms ofsharing is the abundance offree-riders – that is, peers whobenefit of shared files in the system, but who do not shareback. This characteristic is well known in the P2P literatureand has been observed elsewhere [21]. In our dataset, whilemost peers are clients (i.e., have requested a least one file)only 4.33% of them have supplied files.

B. Spreading cascades

In this work we analyze thespreading cascaderepresentingthe diffusion of each file in the P2P network. For a fileF , thespreading cascade is a directed graph featuring the setPF

of peers who have participated in the spread ofF (as clientsand/or providers) and linksP → C, connecting each clientC with the first peer(s) who providedF to it. More formally,let τF (C) = inf{t : (t, ·, C, F ) ∈ D} be the first instantCobtainedF and let the directed graphKF = (PF ,LF ) be thespreading cascade ofF , with

PF = {P ∈ P : (P, F ) ∈ A}

LF = ∪C∈PF{(P,C) ∈ PF × PF : (τF (C), P, C, F ) ∈ D}

A client requesting a file may receive a response frompotentially several providers simultaneously, which implies

(a) Complementary cumulative degree distributionsof peers and files on the bipartite graphB.

100 101 102 103 104 105Cascade attribute value

10-6

10-5

10-4

10-3

10-2

10-1

100

Fractio

n of cascades with

attribute > x

DepthSizeNumber of links

(b) Complementary cumulative distribution of cas-cade attributes (depth, size, number of links).

(c) Complementary cumulative distribution of thenumber of initial providers in the spreading cas-cades.

Fig. 2. Spreading statistics from the observed diffusion trace.

that nodes in the cascade graph not only have multipleoutgoing links, but also multiple incoming links in general.The causality induced by the fact that we only considerthe links corresponding to the first time a node receivedFprevents the appearance of cycles. Hence the cascade is infact a directed acyclic graph (DAG).

The first key property encoded in the spreading cascadeof a given fileF is the number of nodes who possess it atthe end of the observed period, which is given by thesizeofthe cascade|PF |. We also explore two other key topologicalproperties of the cascade, namely itsdepth and number oflinks. The former is defined as the length of the longest pathon the cascade and captures the maximum number of hopsfrom peer to peer that the file has undergone before it wasrelayed from a provider to a client. The number of links,given by |LF |, combined with the size of the cascade givesinformation on the sharing pattern of the network. An exampleof observed trace and constructed spreading cascade is givenin Fig. 1: the spreading cascade has size7, depth3 and6 links.

From the P2P trace log we have constructed the spreadingcascades for each observed file and calculated the abovementioned features. The distribution of these cascade featuresis presented in Fig. 2b. First, we observe that the cascadedepth distribution is well fitted by a power-law. Examiningindividual cascades with high depth we realize that they arenot typically big in terms of size. Second, most spreadingcascades are quite small, featuring one or few nodes and links– these cascades are essentially trivial trees. The cascadeswith higher number of links, however, display a richerstructure. In fact, the ones with the highest number of linkscannot be tree-like, since their number of links exceeds (byfar) the maximum number of nodes observed in our dataset.

C. Initial providers

Another relevant spreading data concerns theinitialproviders for each file F , namely the set of peers that

possessed it prior to any transfer activity on the observed trace.These nodes are the origin of the spreading cascade, triggeringthe diffusion of the fileF . This information can also be in-ferred from the request log and be determined in the followingway. Let CF (t) = {C ∈ P : (t′, ·, C, F ) ∈ D, t′ < t} be theset of peers who requestedF prior to t. We define the set ofinitial providers ofF as the set of peersP who have providedF at some timet, without having obtained it beforet fromanother peer in the network:

IF = {P ∈ P : (t, P, ·, F ) ∈ D, P /∈ CF (t)}

Plotting the complementary cumulative distribution ofthe number of initial providers for the spreading cascades(Fig. 2c) we obtain an interesting curve, revealing a scale-free distribution. This means that although most spreadingcascades in our observation have few initial providers, thereis a non negligible fraction of cascades with a large numberof initial providers.

III. SOCIAL NETWORK STRUCTURE

As discussed in the introduction, our goal is to investigateand model spreading cascades on the social network of peersparticipating in the P2P system in question. In order toanalyze the empirical spread of files among peers in the lightof detailed network diffusion models mentioned, we neednot only the detailed chronological data of who transmittedthe information to whom (observable in the trace) but alsothe social network on which the diffusion takes place. Aspointed out in [18] it is challenging to reconstruct the networkon which the diffusion takes place. One strategy to unfoldthis network is to explore relations among peers and theircommon shared files. Such strategy was hinted in [22] anddeveloped more substantially in [23], [24], [19], [25]. Wefollow this approach to reconstruct the underlying socialnetwork as well.

Focusing on information content diffusion among peers,it is natural to consider theinterest graph in which each

(a) Degree distributions on the interest graph. Superposedcurves: all peers and clients, providers and initial providers

0.0 0.2 0.4 0.6 0.8 1.0Clustering coefficient

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frac

tion of nod

es with

CC > x

(b) Complementary cumulative clustering coefficient dis-tribution in the interest graph.

Fig. 3. Interest graph statistics

node represents a peer and each edge joining two peers standfor common interest. Interests connecting peers may includebroad subjects such as open source software, folk rock orFrench literature or narrow ones such as movies by QuentinTarantino, a particular computer game or pictures of Beijing.It is reasonable to suppose that peers store and share contentrelated to their interests and, likewise, peers will searchforcontent matching their interests. Hence the diffusion of filesamong peers takes place on the interest graph and occursfrom neighbor to neighbor. Indeed, if a peerP providesa file F (corresponding to a music album for example) toanother peerP ′ then there is link between them in the interestgraph, since both are interested in the same content, namelyF .

It is beyond doubt extremely difficult in a large scaleinteraction network to know precisely whether any two in-dividuals have a common interest. Nonetheless, it is possibleto approximate this graph using the data inD: the inferredinterest graph is given by the projectionG = (P, E) of B onP, connecting the peers who belong to the neighborhood of acommon file in the bipartite graph, for each file:

E = {(P, P ′) ∈ P×P : ∃F ∈ F , (P, F ) ∈ A ∧ (P ′, F ) ∈ A}

See example in Fig. 4. For the sake of readability the inferredinterest graph will be henceforth called simplyinterest graph.

The interest graph obtained from the observed bipartitegraph (as explained above and in Fig. 4) has a single giantcomponent containing essentially all nodes(99.99%) anddensity2.62 × 10−4. In Fig. 3a we have plotted the degreedistribution for the peers: considering the set of all peers, themedian degree is 118 and the mean value is 500.11, withcorresponding standard deviation of 1271.42. We proceed toa finer analysis of the degree distribution, grouping peers incategories (Fig. 3a). Let us consider first the set ofclientsC ∈ P such that(·, ·, C, ·) ∈ D: i.e., peers having requestedfiles during our measurements. Their degree distributionsuperposes the degree distribution of all nodes. This is due

Fig. 4. Interest graph as a projection of the bipartite graphof peers and filesconstructed from the traceD.

to the fact that99.63% of peers in our observations haverequested at least one file, so the clients degree distributionis essentially the global degree distribution. A much morerestrictive category is the set ofproviders P such that(·, P, ·, ·) ∈ D, i.e., peers having supplied files duringour measurements. Their degree distribution has a similarshape, but it is concentrated on larger values, indicatedby a median of 1821 and an average degree of 2906.54 –with corresponding standard deviation of 3471.80. The lastcurve, superposing the curve corresponding to the providers,represents the degree distribution of the initial providers.We have also calculated the clustering coefficient [26] ofthe peers in the interest graph (Fig. 3b): we observe a widerange of clustering values, each represented by a significantfraction of peers. Also, the distribution shows a relativelyhigh fraction of peers with a high clustering coefficient –which is a feature of real complex networks, in contrast torandom graphs.

We close this section with a brief summary: using theintroduced framework, we were able to infer the interestgraph of peers, on which the spreading of files takes place.This graph connects essentially all peers, which can begrouped in two categories: providers and clients. Most peers

in our observations are clients, but only a small fractionsupply files and there is a sharp distinction between clientsand providers in terms of their degree distribution.

IV. SIMPLE SIR MODEL

As mentioned in the introduction, we have decided toinvestigate the file spreading in the light of the simple SIRmodel. In our setting, each file spreading corresponds to anindependent epidemic in the interest graph, in which eachnode is in one of the following states:susceptible, infectedor non-interacting (sometimes denotedremoved, hence theacronym SIR). Susceptible nodes do not possess the file andmay receive it from an infected node, thus becoming infected.Each infected node, in turn, spreads the file to each of itsneighbors, independently, with probabilityp and becomespromptly non-interacting thereafter. Although non-interactingnodes remain in this state, infected nodes may unsuccessfullytry to infect them sending the file.

Supposing the observed diffusion trace was the result ofsuch a simple SIR epidemic we may estimate the spreadingparameterp. Each neighbor-to-neighbor transmission trial canbe seen as a Bernoulli random variable, whose value is1 incase of success and0 otherwise and whose expected valueis p. Assuming each trial is independent and the parameterp is homogeneous for eachP and F , we may estimate itby the empirical proportion of successes over all trials. Sinceeach tuple inD accounts for a successful neighbor-to-neighbortransmission,|D| is the number of successful trials for alldiffusion cascades. The total number of trials, in turn, isgiven by the sum of the degrees of all nodes involved inthe spreading of each file. Hence, we obtain the followingestimate, with a95% confidence intervalp± 10−6:

p = |D| /∑

F∈F

∑

P∈PF

d(P ) = 1.063× 10−3

Since the simple SIR model depends upon a singleparameter, namely the spreading probabilityp, we have fullycharacterized it with the preceding estimation.

A. The underlying network influence

The goal of simulating the standard SIR model andcomparing the simulated cascades with the observed ones isprimarily to assess how realistic this model would performon the interest graph, in terms of size, depth and number oflinks of the spreading cascades. Note that by realistic, wemean able to reproduce the characteristics of the data. Indeed,although the data used in this study can be partial and/orbiased, the present work is independent from the qualityof the data itself. Indeed, the problem of improving themeasurement process is different from the one of identifyingrelevant models able to exploit the features observed in thedata, which is the focus of this paper. This means that whenwe further show the ability of the models to reproduce (or

not) the characteristics of real traces, it has to be understoodas the ability of reproducing the characteristics asobservedin the data, with their flaws. Another approach is to applydetection techniques (such as [27]) in order to removeabnormal events from the raw data before using the modelingtechniques presented in this paper. Although it could improvethe quality of the data, it would at the same time obfuscateour conclusions as it adds another step which interferes inthe analysis process. Thus, promising as it seems, we leavethis approach for a further study.

Secondly, we wish to compare the results with simulationson random networks to understand the role of the networktopological structure on the shape of the spreading cascadesgenerated with the SIR model. With this aim, we have consid-ered the spreading of files in a sequence of random networksderived from the interest graph, with increasing topologicalcomplexity (Fig. 5). More precisely we begin considering anErdos-Renyi (ER) random graph with the same density as ourinterest graph, the simplest random graph in our sequence.Then we have chosen a random graph with the same densityand degree distribution using the Configuration Model (CM)approach [8]. Next we have generated a random bipartitegraph, with the same density and degree distribution as ouroriginal bipartite graphB of peers and files [28]. Compared tothe interest graph, the projection of this random bipartitegraph(RB) has similar density, degree distribution and clusteringcoefficient. In sum, for each new element of this sequenceof (uniformly chosen) random graphs we introduce a newconstraint to make it more realistic – in the sense that itstopological properties will be closer to the interest graph.

Fig. 5. Increasingly realistic random graphs derived from the interest graph.

B. File spreading simulation

Combining the network topology, the initial conditioninformation (the list of initial providersIF calculated foreach fileF ) and the calibrated spreading parameterp we canproceed to the simulations for each underlying network: foreachF , we begin with the initial providers in an infectedstate and the other nodes in a susceptible state. At eachstep, infected nodes will infect each of its neighbors withprobability p, becoming non-interacting afterwards. Theepidemic continues as long as there are active infected nodes.

The first observation concerning the model simulationis that the observed time (measured in seconds) has nodirect relation with the simulation time (number of steps).Furthermore, our dataset corresponds to an observation ina bounded window of time of eight hours, so that we haveno reason to suppose that the file spreading cascades weobserve correspond to the whole spreading cascade of a file.

In other words, if we had measured a longer time window wewould likely observe bigger cascades (in terms of size anddepth) for the same files – due to, among other reasons, newusers who could eventually request the same files. This isalso true for our SIR model: we observe increasingly biggercascades as time increases. In fact, performing unconstrainedsimulations we have obtained a distribution of significantlybigger cascades than the ones we have observed in the realtrace. Thus, in order to perform a suitable comparison withthe observed cascades, we have decided to hold one propertyfixed and compare the other properties. More precisely, foreach file we generate a simulated cascade with the samesize (resp. depth) as the corresponding observed cascadeand compare the depth (resp. size) and number of links.In practice, for each file we simulate the SIR epidemic asdescribed earlier and halt it when it reaches the size (resp.depth) of the corresponding observed cascade.

We have generated populations of simulated cascades foreach underlying network and constraint (on depth and size).We have performed801 280 file spreading simulations (onefor each file inF) for each network and have selected everysimulated file spreading cascade which attained the depth(resp. size) of the real spreading cascade for the same file –and have rejected the others for purpose of comparison. Withthis procedure, each underlying network yields a differentpopulation of file spreading cascades, since the rejectedcascades may be different in each case. However93.80% ofthe files have generated simulated cascades with the samedepth as the corresponding real cascades, for all networks.Similarly, 85.64% of the files have generated simulatedcascades with the same size as the corresponding realcascades, for all networks – except the ER network. Indeed,only 21.76% of the files have generated the contemplated sizein the ER graph. Furthermore the properties of these simulatedcascades on the ER graph deviated significantly from theproperties of the cascades on the other graphs. Hence, in thefollowing analysis we do not include the simulations for theER graph. Rather, we focus on the properties of the fileswith comparable spreading cascade depth (resp. size) on allnetworks but ER.

In Fig. 6a we plotted the complementary cumulativedistribution of the size of cascades with comparable depth.We observe a divergence of the cascade size from the observedcascades: simulated cascades are typically much bigger insize for a given depth compared to real cascades. The rangeof values in both categories is also striking: the biggest realcascade is at least two orders of magnitude smaller than thebiggest simulated ones. Among the simulated cascades, thereis a remarkable matching in size values for the simulationon the CM and the interest graph (curves are superposed). InFig. 6c we plot the complementary cumulative distribution ofthe depth of cascades with fixed size. Real cascades feature amuch higher depth compared to simulations, holding cascadesize constant. In particular there is a cutoff on the cascade

depth for the simulations: we do not observe any cascadedepth bigger than 11 in the simulations. As for the numberof links, we have two interesting situations. If we fix thedepth (Fig. 6b) the number of links distribution resemblesclosely the size distribution (Fig. 6a). This is not completelysurprising, since the two quantities are related. In this case weobserve a larger number of links for all simulations comparedto the number of links in the real cascades since the simulatedcascades themselves are bigger. If, in contrast, we fix thecascade size to fit the observed cascades size (Fig. 6d), weobserve a typically smaller number of links. Combining theseobservations on both plots we conclude that real spreadingcascades are denser than simulated ones, a clear qualitativefeature not captured by the simple SIR model. Finally wenote that most cascades are simple, featuring depth equal toone and correspondingly small size.

To sum up, we have compared simple topological propertiesof real spreading cascades and simulated cascades from acalibrated SIR model, with comparable depth and size. Wehave observed that simulated cascades are relatively “wider”whereas real cascades are relatively “elongated”, that is,real cascades have a smaller size per depth ratio. Moreover,real cascades are typically denser than simulated ones. Interms of interplay between underlying network structure andthe simple SIR spreading cascades, we have observed thatrespecting the interest graph degree distribution was the onlyproperty that caused a striking change in simulations behavioron the considered random networks. Indeed we have observedsharp qualitative dissimilarities between the simulations onthe ER graph (different degree distribution) and no sensibledissimilarities between the simulations on the CM, RB andthe interest graphs.

V. HETEROGENEOUSSIR MODELS

In the previous section we have examined the adequacy ofthe simple SIR model to generate verisimilar file spreadingcascades. We have also inspected the interplay between theunderlying network and the model simulating file spreadingin different networks. Given the generality and simplicityofthe homogeneous model it is not entirely surprising that itdoes not capture key properties of real spreading cascadesin our data. In order to fairly assess the relevance of theSIR model in our context, in this section we consider naturalextensions of the SIR model considered previously, whichtake into account heterogeneous aspects found in the observeddata. More precisely, we perform a complementary analysis,focusing on a single underlying network (the interest graph)and examining two heterogeneous versions of the SIR model,characterized by a distribution of spreading probabilities,instead of a single homogeneous parameter. These modelstake into account the file popularity and peer behaviorheterogeneity and are, thus, presumably better equipped tomimic real spreading cascades.

(a) Size of cascades with fixed depth. Curves corresponding tothe interest graph and CM superposed.

(b) Number of links of cascades with fixed depth. Curvescorresponding to the interest graph and CM superposed.

(c) Depth of cascades with fixed size. (d) Number of links of cascades with fixed size. Curves corre-sponding to the interest graph, RB and CM superposed.

Fig. 6. Simulation of file spreading on different underlying networks: complementary cumulative distribution of cascade properties

A. File popularity

A first refinement of the simple SIR model consists inintroducing different spreading probabilities accordingto thefile being spread. The rationale in this case is to account fordifferent levels of popularity depending on the file. Exogenousreasons – such as a movie release or the death of an artist– can change the supply and demand of a given file andconsequently alter its spreading probability. If we know thespreading probabilities for each file, i.e.,{p(F ) : F ∈ F}, theknowledge of the actual reasons that explain the heterogeneityin file popularity are irrelevant to the characterization ofthismodel. An estimate of these probabilities, in turn, can beobtained from the traceD if we suppose it was generated bya process following this extended SIR model. Indeed, sinceeach file spreading is independent of the others, it is possibleto estimatep(F ) for eachF separately, with the same methodused to derive the homogeneous parameter. Restricting thecalculations to the spreading cascade ofF , p(F ) will be givenby the empirical proportion of successful transmissions ofF

over all possible transmissions ofF :

p(F ) = |{(·, ·, ·, F ) ∈ D}| /∑

P∈PF

d(P )

In Fig. 7a we plot the distribution of the heterogeneousspreading parameters depending on the files. The values ofp are concentrated on the range10−5 to 10−2, indicatingthat there is a considerable fraction of cascades with asignificantly different spreading regime (bigger than oneorder of magnitude). This distribution characterizes theextended SIR model we use in the following simulations.

B. Peer behavior

A second possible refinement is motivated by the fact thatpeers might have intrinsically distinct levels of “generosity”regarding file sharing. Under this hypothesis we extend thestandard SIR model assigning an heterogeneous spreadingprobability to each peer, regardless of which file it is sharing.Thus, we do not need any other information but the spreadingprobability distribution to characterize the model. In this

context altruistic peers, who typically spread files to a largeproportion of their neighbors, would feature a bigger spreadingprobability compared to the homogeneous spreading probabil-ity corresponding to the diffusion aggregates of all peers.Bythe same token, the extreme case of free-riders would havetheir spreading probability assigned to zero. Again we canstudy transmissions as outcomes of Bernoulli trials to estimatethe spreading probabilities. LetFP = {F ∈ F : (P, F ) ∈ A}be the files carried by the peerP ; for each such file thenumber of transmission trialsP could perform corresponds toits degree in the interest graph, namelyd(P ). Hence, to obtainp(P ) for each peerP we divide the number of successfultransmissions ofP to other peers (of any file carried byP )over the total number of potential trials:

p(P ) =|{(·, P, ·, ·) ∈ D}|

|FP | × d(P )

We have plotted the distribution of the positive spreadingprobabilities estimates in this case (Fig. 7b). They accountfor small fraction of all the peers, since the only peers whohave a positive spreading probability are those who provideda file at least once – namely4.33% cf. observations made insection II. Conversely, a large fraction of the peers do notshare the file in this model. We observe a marked range ofvalues, which is significantly greater than the one calculatedfor the homogeneous SIR.

(a) Depending on the files (b) Depending on the peers

Fig. 7. Heterogeneous spreading parameter distributions

Our aim is to generate simulated cascades following bothextensions of the SIR model presented – with heterogeneousspreading probability depending on the files and on the peers– and compare their properties with simulated cascades ofthe simple SIR model and the real observed cascades. Inthis sense, we apply the same methodology as in previoussimulations: we fix the depth (resp. size) for the simulatedcascades and examine the other two properties – the idea isto compare similar spreading cascades in terms of the chosenproperty. As discussed previously, the great majority of thecascades is simple, with depth equal to one and a small size.Hence the simulated cascades corresponding to the simpleobserved cascades will likely correspond in terms of depth,size and number of links. For this reason, we have decidedin this section to focus on the spreading cascades with depth

greater than one.

The simulation results are plotted in Fig. 8: we have plottedthe complementary cumulative distributions of the spreadingcascade depth, size and number of links. Imposing a constrainon the depth for the simulated cascades and comparingtheir size (Fig. 8a) we observe the contrast between thesimulated and the real observed cascades with the samedepth: the former have a typically bigger size compared tolatter. What is remarkable, however, is the agreement amongall the simulated cascade distributions – curves superposedin Fig. 8a. Next, if we fix the size for the simulated cascadesand examine their depth (Fig. 8c), we are faced with the samequalitative similarity among simulated curves. Indeed, thecurves corresponding to the heterogeneous SIR models alsofeature a cutoff in depth, failing to reproduce the scale-freecurve representing the depth of the observed real cascades.Finally, the cascade links distribution plotted in Fig. 8b andFig. 8d reveals the pattern observed previously, namely thatthe observed spreading cascades are typically denser thancorresponding simulated cascades.

In spite of the improvements in the SIR model, introducingan heterogeneous spreading parameter to account for differentprofile of files (respectively peers), simulations indicatethat this refinement does not change qualitatively the basicproperties of simulated spreading cascades. Indeed weobserve a surprising similarity between the three SIR modelscompared, notwithstanding the particularities of each model.

VI. W EIGHTED INTEREST GRAPH

In the previous section we have examined SIR modelextensions that take into account heterogeneous aspects ofpeers and files with the goal of generating more realisticspreading cascades. Another approach is to keep the simpleSIR model and enrich the social network inference. In thissection we will address this question, proposing a way torefine the interest graph taking into account thedegree ofinterest among peers. In other words, we propose a methodto quantify the interest affinity among peers. The rationaleis that peers will be more likely to interact with other peerswith whom they have greater affinity.

A. File spreading simulation

In concrete terms, our affinity score between two peers willbe defined by the number of common files peers shared orprovided. Indeed, instead of approximating the interest graphby the simple projection ofB on P, we consider a richerinferred interest graphG = (P, E ,W), given by theweightedprojection ofB on P such that

E = {(P, P ′) ∈ P ×P : ∃F ∈ F , (P, F ) ∈ A∧ (P ′, F ) ∈ A}

W(P, P ′) = |{F ∈ F : (P, F ) ∈ A ∧ (P ′, F ) ∈ A}|

(a) Size of cascades with fixed depth. Curves corresponding tothe simulations are superposed.

(b) Number of links of cascades with fixed depth. Curvescorresponding to the simulations are superposed.

(c) Depth of cascades with fixed size. (d) Number of links of cascades with fixed size. Curves corre-sponding to the simulations are superposed.

Fig. 8. Simulation of file spreading on the interest graph withdifferent SIR processes: complementary cumulative distribution of cascade properties

In other words, peers belonging to the neighborhood of acommon file inB are connected inG. If a peerP providesa file F (corresponding to a music album for example) toanother peerP ′, then there is a link between them in theinterest graph since both are interested in the same content,namely F . Furthermore, each edge(P, P ′) ∈ E has aninteger weight given by the number of common files theyhave manifested interest in. In Fig. 9a we have plottedthe distribution of weight values in the interest graph: it isheterogeneous, with weights ranging from 1 to 303 and suchthat the vast majority of edges feature small weights. Finally,note that the weight scheme we have introduced is by nomeans the only way to assign an affinity index to each edgeof the interest graph. One could assign a greater affinityto two peers who are both interested in “rarer” files thantwo peers interested to “common” files for instance; anotherpossibility is the Jaccard index of similarity. That said, ourchoice is quite natural and is motivated by the hypothesisthat peers will likely spread files to the neighbors with whomthey have greater affinity, as we explain below.

B. Diffusion models

The diffusion models we have used so far requireadaptation to take into account the enhanced networktopology. We keep the main hypotheses of the SIR model,that is, that each individual is in one of the followingstates: susceptible, infected or non-interacting (sometimesdenotedremoved). Susceptible nodes do not possess the fileand may receive it from an infected node, thus becominginfected. Infected nodes, in turn, try to spread the file toeach of its neighbors, independently, and become promptlynon-interacting thereafter. Each infection attempt from aninfected nodeP to the nodeP ′ is successful with probabilityσ(w) ∈ [0, 1], depending on the weightw of the edgeconnectingP andP ′.

It is reasonable to suppose that a peerP will be moresuccessful in spreading a file to the neighbors with whomhe or she has a greater common interest. In terms of thespreading probabilityσ, this assumption translates itself assupposingσ(w) is increasing withw. Indeed, the weight

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Fig. 9. The interest graph connects peers who share common interests and attributes a weight between this connection proportionally to the the overlapamong their interests. Some peers have several common interestswith others, but most peers have few shared interests. Contagion spreads best among peerswith stronger connection.

connectingP and its neighbors is a measure of how similarare their interests. Hence the more similar two peers are interms of interest, the greater the weight of the edge connectingthem and, in turn, the greater the spreading probability. Toverify this hypothesis we have estimated the value ofσ(w)for each value ofw, adapting estimation methods used insections IV, V. Each observed spreading cascade of a fileF in the trace provides a set of estimated values{σF (w)}:as expected, we have found that the median values ofσare increasing withw up to w = 25 (with the exception oftwo values), after which they essentially reach a plateau atσ(w) = 0.5. In Fig. 9 (right) we have plotted the estimatorvalues for all weights from 1 to 25 in terms of box plots.

Following the approach in [29], we have used a linearfunction to model the spreading probability on the weightedgraph, namelyσ1(w) = a1w+ b1, with a1 = 3.07×10−3 andb1 = 1.54 × 10−3 obtained with a least squares calibration.The number of edges with small weights is much greaterthan the number of edges with big weights in this graph – cf.Fig. 9a. Indeed we observe a greater number of transmissionsbetween peers connected by edges with smaller weight.Hence, the quality of the estimators is greater for small valuesof w and we have taken into account primarily these values inthis model. We have also examined an alternative model forσ, which captures qualitatively the stagnation ofσ for largevalues ofw. In this case we haveσ2(w) = a2 log(w) + b2with a2 = 14.10× 10−3 andb2 = 0.58× 10−3 obtained withthe same calibration method.

C. File spreading simulation

Equipped with the reconstructed social network of peers(the weighted interest graph) and models for the diffusion offiles (described above) we have simulated the spreading of allthe files and compared the corresponding spreading cascadeswith the real, observed, spreading cascades. Simulated traces

corresponding to the spreading of each fileF ∈ F containsthe same number of transfers as the real observed trace ofF .

In Fig. 10 we have plotted the complementary cumulativedistributions of cascade properties from real cascades,compared to the simulated cascades using the diffusionmodels described above. The first general remark is thatsimulated cascades generated by both models are quitesimilar in terms of these metrics. Indeed, the curves of bothsimulations are superposed for the three plots. Comparedto the distribution of real cascades, the sharpest contrastisin terms of depth: the distribution for simulated cascadesfeatures only small values of depth, whereas the depthdistribution for real cascades is remarkably scale-free. Wealso find a discrepancy between simulated and real cascadesin terms of size and number of links: in the former the gapis sharper and in the latter both distributions follow globallythe same trend. Considered together the curves make clearthat these models face a challenge to capture key topologicalproperties simultaneously. Indeed, real cascades have a shapecloser to chain-email cascades [30], in the sense that they arerelatively elongated compared to simulated cascades obtainedwith these contagion models.

VII. C ONCLUSION AND PERSPECTIVES

We have presented a large-scale dataset from a real-worldpeer-to-peer network, featuring diffusion of files among peers.We have proposed a framework to study this dataset whichallows us to obtain, simultaneously, the interest graph ofpeers – where the diffusion of content takes place – and thespreading cascade. Guided by simulations we have examinedspreading cascades generated by the simple SIR model andhave analyzed the interplay between this model and thenetwork topology. We concluded that simulated file diffusionsdo not capture key qualitative properties of the observedspreading cascades. Furthermore, in terms of the studied

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Fig. 10. Spreading cascades profile in terms of: depth, size and number of links respectively. Both models yielded the same cascades profile (simulationcurves superposed), contrasting with real spreading cascades in terms of depth.

properties, the simple SIR model generates similar cascadeson random networks having the same degree distribution asthe interest graph. We have also found that the addition of aclustering coefficient constraint on the random graph did notchange the properties of the spreading cascade qualitatively.

The SIR model is an attractive choice to model theinformation spreading in complex networks: it was inspiredby classical epidemiological models, it is based upon fewassumptions and it can be characterized with few parameters.This flexibility and simplicity explain its popularity asa contagion model, but these characteristics are also itsweakness when used in specific contexts. In this sense, theresults of section IV, mentioned above, are not entirelysurprising. What is surprising, though, is that simulatedcascades from extensions of the SIR model (which takeinto account the heterogeneity in file popularity and peerbehavior) show similar properties as the simple homogeneousSIR model. In addition to these extensions, we have enrichedthe reconstruction of the interest graph, introducing a measureof affinity among peers. Again, simulations reveal anotherunexpected point: despite the enhanced social networktopology, the model simulations did not reproduce qualitativefeatures of real spreading cascades.

In sum, these results suggest that this model is not suitedto describe information spreading in our context. That is,not even the natural extensions of this model, related to keyobserved features of real spreading cascades, offer a betteralternative in terms of the properties we have investigated.It is evidently hard to demonstrate that there is no possiblemodification of this model capable of describing file spreadingcascade profiles in P2P systems, but our results show that thismodel is unlikely to describe spreading cascades generally,as it is commonly taken for granted. In this sense, our workraises a cautionary message against the careless, widespreaduse of this model.

Although the spreading cascade modelling seems to be

more context-dependent than currently thought, the preciserole of context in the choice of model and its parametersremains open. In our case, we have focused primarily onthe interplay between the diffusion process and the networkstructure and have neglected other potentially importantaspects in this context. Two aspects in particular couldexplain why SIR-based models fail to reproduce the profileof spreading cascades.

The first one concerns the time, which is not directlyaddressed in such models except from a logical point ofview. This aspect is however strongly related to the order inwhich the underlying graph is explored during a contagion.The logical nature of the time adopted here is similar toa breadth-first-search exploration which yields short depthstructures. This explains particularly the inadequacy observedwhen the depth is involved in the evaluation of the model,such as in Fig. 6a, Fig. 6c, Fig. 8a, Fig. 8c, and Fig. 10a. Incontrast, figures corresponding to size and number of linksshow a clear improvement of the model efficiency. Thus, itseems very promising to exploit more deeply the temporalinformation in our dataset. One possible way would be totake into account the dynamic aspect of the social network byfiltering the interest graph with pairs of peers that have beenpresent at the same time in the network. Another way wouldbe to incorporate time-related behavior to the contagiondynamics, that is to add a new features in the model itselfthat account for such temporal patterns. Though promising,we leave such approaches for further studies.

The second aspect, which is by far more fundamental asit questions the nature of the model itself, relies on the factthat epidemic models are based on“push” dynamics whereaspeers in P2P systems tend to“pull” content from each others.This might call for a fundamental perspective change on thedynamics of the process. In particular, adoption/thresholdmodels [13], [3] could be more pertinent in this case: we alsoplan to evaluate this possibility in the future.

ACKNOWLEDGMENT

This work is partly funded by the European Commissionthrough the FP7-FIRE project EULER (Grant No.258307) andby the City of ParisEmergenceprogram through the DiReproject.

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