Eitan Altman
email: eitan.altman@inria.fr

Philippe Nain
email: philippe.nain@inria.fr

Adam Shwartz

Yuedong Xu
email: yuedong.xu@inria.fr

Predicting the Impact of Measures Against P2P Networks on the Transient Behaviors

The paper has two objectives. The first is to study rigorously the transient behavior of some peer-to-peer (P2P) networks whenever information is replicated and disseminated according to epidemic-like dynamics. The second is to use the insight gained from the previous analysis in order to predict how efficient are measures taken against P2P networks. We first introduce a stochastic model which extends a classical epidemic model, and characterize the P2P swarm behavior in presence of free riding peers. We then study a second model in which a peer initiates a contact with another peer chosen randomly. In both cases the network is shown to exhibit phase transitions: a small change in the parameters causes a large change in the behavior of the network. We show, in particular, how phase transitions affect measures of content providers against P2P networks that distribute non-authorized music or books, and what is the efficiency of counter-measures.

I. INTRODUCTION

Along with the worldwide penetration of the Internet, a huge demand has appeared to copyrighted music and movies that have been accessible for free over the Internet. While benefiting a very large internaut community as well as potentially providing higher benefits for Internet access providers, it seems unclear whether the creators and the copyright owners have gained anything from this unregulated access. Two opposing approaches appeared, both proposing to protect the copyright owners. The first consists of fighting against nonauthorized access whereas the second one, aims at finding cooperative solutions that would benefit both the Internauts as well as all other economic actors. An example of a cooperative solution is a flat taxation that would allow Internauts to pursue downloading freely music and films, and that would distribute the tax money between the copyright owners. This cooperative approach has several difficulties in its implementation; a major one is how to distribute the tax income fairly. A major drawback of the confrontation policy is the huge monitoring effort that it requires and that seems not to provide credible evidence for unauthorized downloads [START_REF] Piatek | Challenges and Directions for Monitoring P2P File Sharing Networks -or -Why My Printer Received a DMCA Takedown Notice[END_REF]. In order to assess the efficiency of non-cooperative measures against unauthorized downloads, the authors of [START_REF] Wong | P2P Networks: The Interplay between Legislation and Information Technology[END_REF] have analyzed the impact of the effort, of the authorities or of content provider companies, invested in (i) reducing file uploading in P2P networks and in (ii) reducing the demand for files, on the availability of files and, more generally, on the operation of the P2P networks. The stationary analysis there is based on a M/G/∞ queuing model.

In this paper we are interested in predicting the impact of measures as described in the previous paragraph, on the transient behavior of torrents. By how much should the request or departure rate in a P2P network be reduced in order to have a significant change in file availability? To achieve that, we consider abstract models of a torrent in simplified P2P networks, where a large number of peers are interested in a file which is initially available at a small fraction of the population.

Our models are formulated as epidemic type processes of file dissemination. We consider both cooperative peers, which are those that make a file available to other peers as soon as they obtain the file, and free riders, who leave the system immediately after obtaining the file. To understand the impact of measures against the cooperative sharing behavior, we parameterize the degree of free-riding in the system as well as the degree of cooperation.

The P2P dynamics is modeled by a Markov chain (Section II) which is approximated in two specific regimes: the first (Section III) is the early stage when a large fraction of the population does not yet have the file. The system is then well approximated by a branching process. In the case that there is a positive probability of not getting extinct in the first regime, the system is shown in Section IV to move with some nonzero probability to a second regime in which, for the case of a sufficiently large population size, its dynamics is close to the solution of a differential equation. A similar fluid limit is studied in VI for the case of limitation on uplink or downlink speed. We briefly state our contributions:

1. Modeling and approximating the transient behavior Our first important contribution is to show in what sense each of the above two models approximates the original Markov chain, and how to use both in order to get the whole transient behavior of the P2P network. This is in contrast with all other models of P2P networks that we know of, which either use only a branching process approach [START_REF] Yang | Service Capacity of Peer to Peer Networks[END_REF] or which use only an epidemic mean-field approximation [START_REF] Qiu | Modeling and Performance Analysis of Bittorrent-like Peer-to-Peer Networks[END_REF]. The latter approach (of using only the mean-field limit) is shown to provide a tight approximation when the initial number of peers with the file scales linearly with the total size N of the population of peers. With a fixed initial number of nodes that does not scale with N , there is a positive probability of early extinction (see Section VII for detail) for any set of system parameters, and this probability cannot be predicted by the mean-field limit alone.

Analysis and identifying phase transitions

We first study a P2P model that corresponds to the epidemic-like file dissemination (Sections II-IV) We then study a second model (Section VI) in which, at random times, each peer contacts another peer randomly chosen within the set of existing peers.

In both cases, we show the existence of phase transitions: a small change in the parameters causes a large change in the network behavior.

A phase transition occurs both in the branching model for the extinction time and in the epidemic model for the file availability. In the branching process, the existence of two phases was not known to Galton and Watson (considered as the founders of branching processes) and was only discovered and proved later in [START_REF] Steffensen | Deux problèmes du calcul des probabilités[END_REF]. In the epidemiology community, the phase transition was already known in [START_REF] Mollison | Epidemic Models: their Structure and Relation to Data[END_REF] for a model equivalent to our first model without the free riders. For the the second model [START_REF] Nunez-Queija | Scaling Laws for File Dissemination in P2P Networks with Random Contacts[END_REF], we show the existence of two phase transitions, one for the file availability and the other one for the maximum torrent size.

Application.

In Section V, we present a counteraction against unauthorized file sharing in the presence of illegal publishers. We evaluate the impact of measures against Internet piracy on the performance of P2P systems in Section VII (see Figure 11).

The accuracy of the various approximations is investigated in Section VII, related studies are discussed in Section VIII, and concluding remarks are given in Section IX.

II. MODEL A. Assumptions

Assume there is a population N of peers interested in a single file. Let Y (t) be the number of peers that possess the file at time t. A peer acquires the file when it encounters another peer that has the file. We will consider two types of peers: cooperative and non-cooperative peers. Once a cooperative peer has acquired the file, it stays in the network for a random time distributed according to an exponential rv with parameter 1/µ ≥ 0 and then leaves the network. During the lingering time of a cooperative peer with the file, it participates in the file dissemination. A non-cooperative peer, also called a freerider, leaves the network at once when it receives the file. Note that "free riders" in our context is an abstract description of noncooperative behaviors, which is different from that in the current BitTorrent system.

Let X c (t) and X f (t) denote the number of cooperative peers without the file and the number of free-riders (necessarily without the file) at time t, respectively. Define the process

Y := {(Y (t), X c (t), X f (t)), t ≥ 0}. Let (Y (0), X c (0), X f (0)) denote the initial state of Y that has Y (0) + X c (0) + X f (0) = N . Let the ratio of various types of peers be (y 0 , x c,0 , x f,0 ) := ( Y (0) N , Xc(0) N , X f (0) N ). For simplicity, we introduce new variables N c = X c (0) and N f = X f (0).
We consider an abstract P2P network in which the file acquisition is via random contact between pair-wise peers. When two such peers meet, the cooperative peer transmits the file to the other peer. It is assumed that it takes an exponential time with rate λ > 0 for a peer without a file to encounter a cooperative peer with the file. The transmission of the file is always supposed to be successful. This model describes a general P2P swarm without a tracker, and even the spreading of a file in current Internet. It is inspired by the contact process in [START_REF] Nunez-Queija | Scaling Laws for File Dissemination in P2P Networks with Random Contacts[END_REF] and [START_REF] Massoulié | Coupon Replication Systems[END_REF]. One of the main difference lies in that a peer contacts all other connected peers in the system, instead of only one random peer periodically. We assume that the file transmission time is negligible compared to the time it takes for two peers to meet and therefore this time is taken to be zero.

All the random variables (rvs) introduced so far are assumed to be mutually independent. As a consequence, if Y (t) = k then any peer without the file will meet a cooperative peer with the file after a time that is distributed according to the minimum of k independent and exponential rvs with rate λ, that is after a time distributed according to an exponential rv with rate λk.

Measures of the authorities or of content provider companies against file sharing systems may have an impact on the decrease in the population N interested in the file and an increase in the fraction of free riders among the population interested in the file. It can however have an impact also on the behavior of cooperative peers that would leave the system sooner (i.e. µ is expected to increase). Our model combines an epidemic type propagation of the file together with a description of the free riding behavior. Define ρ := λN c /µ.

We first consider (Section II-B) the case where all peers are fully cooperative in the sense that µ = 0 and X f (0) = 0 (no free riders). µ = 0 implies that cooperative peers do not leave the network after receiving the file. We then move to the general case where µ > 0 and X f (0) ≥ 0 (Section II-C).

B. Fully cooperative network

When all peers are fully cooperative (i.e. µ = 0 and X f (0) = 0) the population of peers remains constant and equal to N , that is, Y (t) + X c (t) = N at any time t. The network dynamics can be represented by the process {Y (t), t ≥ 0}. This is a finite-state continuous-time Markov process with non-zero transitions given by

Y (t) → Y (t) + 1 with rate λY (t)(N -Y (t)). (1)
In other words the process {Y (t), t ≥ 0} is a pure birth Markov process on the state-space {y 0 , . . . , N }, where state N is an absorbing state which is reached when all peers have the file. Define m(t) := E[Y (t)], the expected number of peers with the file at time t. Standard algebra shows that

dm(t) dt = λE[Y (t)(N -Y (t))], t > 0. (2)
Unfortunately, the right-hand side of (2) does not express as a function of m(t), thereby ruling out the possibility of finding m(t) in closed-form as the solution of an ODE. Assume that λ is written as λ = β/N and that lim N →∞ N -1 Y (0) = y 0 ∈ (0, 1]. Then, for large N , m(t) is well-approximated by N y(t) where y(t) is obtained as the unique solution of the ODE [6, Thm 3.1]

dy(t) dt = f (y(t)), t > 0, (3)
where f (u) := βu(1 -u) and y(0) = y 0 ∈ (0, 1] (conditions (3.2)- [START_REF] Steffensen | Deux problèmes du calcul des probabilités[END_REF][START_REF] Mollison | Epidemic Models: their Structure and Relation to Data[END_REF] in [START_REF] Kurtz | Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes[END_REF]Thm 3.1] are clearly satisfied). It is found that

y(t) = y 0 (y 0 + (1 -y 0 ) exp(-βt)) , t ≥ 0. (4)
This is a well-known instance (see e.g. [START_REF] Zhang | Performance Modeling of Epidemic Routing[END_REF]) of what is known as mean-field approximation, a theory that focuses on the solution of ODEs obtained as limits of jump Markov processes [START_REF] Kurtz | Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes[END_REF]. The ODE (3) has been extensively used in epidemiology studies, where y(t) represents the fraction of infected patients at time t when the population is of size N .

Proposition 1 below, whose proof can be found in [START_REF] Altman | Predicting the Impact of Measures Against File Sharing Networks on the Transient Behaviors[END_REF], states that the mean-field approximation is an upper bound for

E[Y (t)]. Proposition 1: E[Y (t)] ≤ N y 0 /(y 0 +(1-y 0 )e -βt ) ∀t ≥ 0.

C. General network

We consider the general network defined in Section II-A. Define the vector X(t) = Xc(t) X f (t) , where we recall that X c (t) is the number of cooperative nodes in the system who do not have the file at time t and X f (t) is the number of freeriders in the system at time t (by definition, none of these have the file at time t). Let e c = (1, 0) and e f = (0, 1). Under the statistical assumptions made in Section II-A it is seen that the process Y = {(Y (t), X(t)), t ≥ 0} is a finite-state Markov process whose non-zero transitions are given by

Y (t) X(t) → Y (t) + 1 X(t) -ec with rate λY (t)Xc(t), (5)
Y (t) X(t) → Y (t) -1 X(t) with rate µY (t), (6)
Y (t) X(t) → Y (t) X(t) -e f with rate λY (t)X f (t). (7)
Throughout this paper we will assume that λ > 0 and µ > 0.

The process Y takes its values in the set

E := {(i, j, k), 0 ≤ i ≤ y 0 + N c , 0 ≤ j ≤ N c , 0 ≤ j + k ≤ N -y 0 }.
Furthermore, all states in E of the form (0, j, k) are absorbing states since there are no more transitions when the file has disappeared.

An explicit characterization of the transient behavior of the absorbing Markov process Y is a difficult task due both to the presence of non-linear and non-homogeneous transition rates in the state variables and to the dimension of Y. In this paper we will instead develop two approximations of the Markov process Y. The first one, in Section III, will consist in replacing X c (t) by N c = X c (0) in the transition rate (5), which will introduce a birth and death Markov branching process. As expected, this (so-called) branching approximation will loose its accuracy as the ratio X c (t)/N c decreases.

The second approximation, in Section IV, will use an asymptotic argument as N → ∞ based on a mean-field approximation of Y. This approximation is justified if the initial state of Y is of the order of N . Both the branching and the mean-fielf approximations approaches will allow us to approximate key characteristics of Y such as the probability of disappearance of the file, the time before all files disappear, the maximum number of cooperative peers in the network and the fraction of peers that eventually receive the file.

III. BRANCHING APPROXIMATION

Let Y b := {Y b (t), t ≥ 0} be a Markov process on IN := {0, 1, . . .} (the subscript b refers to "branching") with non-zero transition rates given by

Y b (t) → Y b (t) + 1 with rate λY b (t)Nc (8) Y b (t) → Y b (t) -1 with rate µY b (t) (9)
where we recall that N c is the number of cooperative peers without the file at time t = 0. Since X c (t), the number of cooperative peers without the file at time t, is non-increasing in t, a quick comparison between ( 5)-( 7) and ( 8)- [START_REF] Harris | The theory of Branching Processes[END_REF] indicates that the process Y b should dominate the process Y. This bounding result is formalized and proved in the proposition below.

A word on the notation: a real-valued rv Z 1 is stochastically smaller than another real-valued rv Z 2 , denoted as

Z 1 ≤ st Z 2 , if P (Z 1 > x) ≤ P (Z 2 > x) for all x. Proposition 2: If Y (0) ≤ Y b (0) then Y (t) ≤ st Y b (t) for any t > 0.
The Markov process Y b is an absorbing continuous-time birth and death process on IN with absorbing state 0. Because its transition rates are linear functions of the system state, this is also a continuous-time Markov branching process [START_REF] Harris | The theory of Branching Processes[END_REF], namely, a process in which at any time t each member of Y b (t) evolves independently of each other. The next section specializes known results of the theory of branching processes to the process Y b .

A. Extinction probability and extinction time

As previously observed the process Y b is a birth and death branching process [9, Chapter V]. Each object (peer) of this process has a probability of change in the interval (t, t + h) given by bh + o(h) with u = λN c + µ; with probability p 0 = µ/u an object dies (a peer leaves) and with probability p 2 = λN c /u an object is replaced by two objects (a peer receives the file).

Given Y b (0) = k the extinction time T k is defined by

T b (k) = min{t > 0 : Y b (t) = 0} Let G k (t) := P (T b (k) < t) be the CDF of T b (k). Given Y b (0) = k, the extinction probability, q k , is given q k = G k (∞). The CDF of T b (1)
is obtained from [9, Eq. (7.3), p. 104] and is given by

G 1 (t) = 1 -e -µ(1-ρ)t 1 -ρe -µ(1-ρ)t , t ≥ 0, (10)
where we recall that ρ = λN c /µ. From (10) we find

q 1 = min{1, 1/ρ}. (11)
In other words, the extinction will be certain iff ρ ≤ 1. Since all objects behave independently of each other we have

q k = q k 1 = min{1, 1/ρ k } and G k (t) = G 1 (t) k = 1 -e -µ(1-ρ)t 1 -ρe -µ(1-ρ)t k , t ≥ 0. (12)
In particular, if ρ = 1,

G k (t) = µt 1 + µt k , t ≥ 0. (13)

B. Expected time to extinction

Assume that ρ < 1 (extinction is certain). The expected extinction time is equal to

E[T b (k)] = ∞ 0 (1 -G k (t))dt. In particular E[T b (1)] = - log(1 -ρ) µρ . (14)
Let us now come back to the original process Y. Define T (y 0 ) := inf{t : Y (t) = 0}, the first time when the file has disappeared from the network given that Y (0) = y 0 . When

Y (0) = Y b (0) = y 0 , Proposition 2 implies that P (T (y 0 ) > t) = P (Y (t) > 0) ≤ P (Y b (t) > 0) = G y0 (t).
In particular

E[T (y 0 )] ≤ E[T b (y 0 )], so that E[T (1)] ≤ -log(1-ρ)
µρ from ( 14) for ρ < 1 .

IV. MEAN-FIELD APPROXIMATION

In this section we investigate the behavior of the process Y defined in Section II-C as N , the number of peers, gets large. We first show that this behavior (to be made more precise) is well approximated by a deterministic limit solution of an ODE, an approach known as mean-field approximation. See [START_REF] Kurtz | Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes[END_REF] for the theory and [START_REF] Chaintreau | The Age of Gossip: Spatial Mean Field Regime[END_REF], [START_REF] Massoulié | Coupon Replication Systems[END_REF], [START_REF] Zhang | Performance Modeling of Epidemic Routing[END_REF] for recent applications in the area of file sharing systems.

Like in Section II-B we assume that the pairwise contact rate, λ, is of the form λ = β/N with β > 0. We recall that the initial state of Y is given by [START_REF] Fan | On the Performance and Availability of BitTorrent-like P2P File Sharing Systems[END_REF] with y 0 + x c,0 + x f,0 = 1. [The analysis below holds under the weaker condition lim

Y (0) = N y 0 , X c (0) = N x c,0 , X f (0) = N x f,0
N →∞ N -1 (Y (0), X c (0), X f (0)) = (y 0 , x c,0 , x f,0 ).]
Let v 1 = (1, -1, 0), v 2 = (-1, 0, 0) and v 3 = (0, 0, -1). Denote by g(Y, Y + v i ), i = 1, 2, 3, the non-zero transition rates of the process Markov process Y out of state Y = (Y 1 , Y 2 , Y 3 ). We have (cf. ( 5)-( 7))

g(Y, Y +v1) = β N Y1Y2, g(Y, Y +v2) = µY1, g(Y, Y +v3) = β N Y1Y3
which can be rewritten as

g(Y, Y + v i ) = N f Y N , v i , i = 1, 2, 3 (16)
where

f (u, v 1 ) = βu 1 u 2 , f (u, v 2 ) = µu 1 and f (u, v 3 ) = βu 1 u 3 for u = (u 1 , u 2 , u 3 ).
We may therefore use Theorem 3.1 in [START_REF] Kurtz | Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes[END_REF] (it is easily that conditions (3.2)-(3.4) in [START_REF] Kurtz | Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes[END_REF] are satisfied) to obtain that the rescaled process N -1 Y converges in probability as N → ∞, uniformly on all finite intervals [0, T ], to the solution

(y, x c , x f ), 0 ≤ y, x c , x f , y + x c + x f ≤ 1, of the system of ODEs d dt   y x c x f   =   y(βx c -µ) -βyx c -βyx f   (17)
with initial condition (y 0 , x c,0 , x f,0 ). In particular, for any finite t the solution y, x c , x f of (17) will approximate the fraction of peers with the file, the fraction of cooperative peers without the file and the fraction of free-riders, respectively, at time t. The accuracy of this approximation will increase with N , the total number of peers.

A. Peers that never receive the file: a phase transition

The fraction of cooperative peers x c and the fraction of freeriders x f that do not have the file monotonically decrease (this is true also for the original system) to some limit values. They can continue decreasing until there are no copies of the file in the system, namely until y = 0.

The first question we wish to address is whether these limits are close to 0 or are large. In other words, we wish to know whether all (or almost all) peers interested in the file are able to obtain it or not. If the answer is no, then we shall be interested in computing the fraction of peers that never receive the file.

Let θ := β/µ. From the first two equations in [START_REF] Mundinger | Optimal scheduling of peer-topeer file dissemination[END_REF] we obtain x c as

dy dx c = -1 + 1 θx c . ( 18
)
The solution of this differential equation is

x c + y = θ -1 ln x c + φ(θ) (19)
where φ(θ) := x c,0 + y 0 -θ -1 ln x c,0 . Let y max be the maximum ratio of cooperative peers with the file. According to the first equation in ( 17), y max is reached when x c = θ -1 if θ > 1 and is expressed as

y max = -θ -1 (1 + ln θ) + φ(θ). (20)
When θ ≤ 1, y max is reached when x c = x c,0 (i.e. at time t = 0). On the other hand, as t → ∞ y is approaching 0 (since we have assumed that µ > 0) so that, from [START_REF] Altman | P2P Business and Legal Models for Increasing Accessibility to Popular Culture[END_REF], x c (∞) satisfies the equation

x c (∞) -θ -1 ln(x c (∞)) -φ(θ) = 0. (21)
It is easily seen that this equation has a unique solution in (0, x c,0 ) (note that x c (t) ≤ x c,0 for any t since x c is nonincreasing from the second equation in [START_REF] Mundinger | Optimal scheduling of peer-topeer file dissemination[END_REF]). From [START_REF] Mundinger | Optimal scheduling of peer-topeer file dissemination[END_REF] we find that x f (t) =

x f,0 xc,0 x c (t) for all t. As recalled earlier the mean-field approximation only holds for finite t and there is therefore no guarantee that it will hold when t = ∞, namely, that N Y -1 will converge in probability to (0, x c (∞), x f,0 x c (∞)/x c,0 ) as N → ∞. However, due to the particular structure of the infinitesimal generator of Y this convergence takes place as shown in [13, Sec. 5.2] (Hint: consider the rescaled Markov process Ỹ := {( Ỹ (t), Xc (t), Xf (t)), t ≥ 0} with generator g(•, •) = g(•, •)/Y 1 and same state-space as Y, so that starting from the same initial condition the terminal values of X c (t) and Xc (t) (resp. X f (t) and Xf (t)) will have the same distribution. The mean-field approximation for Ỹ shows that the solution of the associated ODE's is given by (0, x c,0 e -βτ , x f,0 e -βτ ) for any t ≥ τ , with τ the unique solution in (0, ∞) of x c,0 + y 0 = x c,0 e -βτ + µτ , from which the result follows).

In summary, as N is large, the fraction of cooperative (resp. free riders) peers which will never receive the file is approximated by x min c := x c (∞) (resp. x min f := x f,0 x min c /x c,0 ) where x c (∞) can be (numerically) calculated from [START_REF] Simatos | A Queueing System for Modeling a File Sharing Principle[END_REF].

We are interested in whether there is an abrupt change in content availability (i.e. x c (∞)) with the parameter θ. Obviously, if θ is 0, all the cooperative peers that do not have the file at time 0 will never receive it. To find a phase transition, we approximate log(x c (∞)) in ( 21) by using its Taylor extension at x c,0 and obtain

x c (∞) ≈ (( 1 θ -x c,0 -y 0 ) + 1 θ ( x c (∞) x c,0 -1) 2 ) ( 1 θx c,0 -1)
.

Since the expression 1 θ ( xc(∞) xc,0 -1) 2 is bounded, the phase transition happens at θ = 1/x c,0 .

Despite the similarity in the definitions of ρ in Section III and of θx c,0 in the present section, the phase transition at ρ = 1 is different in nature from that at θx c,0 = 1. The former indicates whether or not the file will be extinct while the latter will drastically impact the final size of the torrent.

Figure 1 displays the mapping log 10 (θx c,0 ) → x min c for x c,0 ∈ {0.01, 0.1, 0.3, 0.5, 0.9} and y 0 = 0.05. The curves for x min c are monotonically decreasing in x c,0 (the curve that intersects the vertical axis close to 1 is the one corresponding to x c,0 = 0.01, and so on.). For each curve we note the existence of a phase transition at θx c,0 = 1, which is more pronounced as the ratio of free riders increases.

B. Combining the branching and the epidemic model

The mean-field approximation is accurate for large N if the initial state scales with N linearly. In the case that N is very large but the initial condition does not scale with N (e.g. Y (0) = 1, X c (0) + X f (0) = N -1), we can do the following. Fix some N 0 much smaller than N but larger than 1. Use the branching process approximation until the number of peers with the file is N 0 . Then, switch to the epidemic model. (For the branching process, we recall that given that there is no extinction, the population size grows exponentially fast).

V. CONTROL ACTIONS AGAINST P2P NETWORKS

In this section, we first investigate the major findings in the analysis of content availability. A set of control actions are proposed to protect copyrighted files against P2P file sharing.

A. Observations on file availability

Before proposing the counteractions against illegal P2P swarms, we investigate the impact of measures on file availability. The main question is how does a decrease or increase in one of the system parameters affect measures such as

• the size of the torrent: the fraction of those who are interested in the file and are able to get a copy of it. This can be seen as a global availability measure. • the extinction probability or the expected extinction time,

• the maximum availability: the maximum number of copies that can be found simultaneously in the system. This can be viewed as an instantaneous availability measure.

According the analysis in Sections III-IV, all above measures depend on the ratio λ µ (or β µ equivalently). A small ratio λ µ means a poor availability of the file. However, the contact rate λ is an intrinsic parameter of P2P swarms that can hardly be changed technically. An even more challenging problem is that there usually exist several illegal publishers residing in the system for a very long time. They aim to spread the copyright protected file as wide as possible in the P2P swarm. To combat with undesirable file sharing, we present two methods, the cooperation control and the pollution attack. The former is to discourage the degree of cooperation of peers with the file. The latter introduces a number of polluters before the file dissemination begins, which can be found in [START_REF] Altman | Predicting the Impact of Measures Against File Sharing Networks on the Transient Behaviors[END_REF] due to the page limit of this paper.

B. Control of cooperation

We introduce the cooperation control to prevent the dissemination of copyrighted files. We aim to reduce the degree of cooperation (i.e. increasing µ) so that the delay of obtaining the file is increased. To achieve this goal, the content owner can invest a certain amount of money in the very beginning to discourage the cooperation of peers. The cooperation control does not contradict with our opposition of collaborative solutions such as flat tax. In fact, we are focusing on this unilateral action of the content owner against unauthorized file dissemination.

We consider the same model as in Section II-C but we now assume that all peers are cooperative and that there is a number Y * N > 0 of permanent publishers, where the subscript N refers to the total number of peers in the system at time t = 0. The pairwise contact rate is λ = β/N . Denote by α the investment level of the content owner against P2P networks. The departure rate is an increasing function of α, denoted by µ(α). We denote by Y N (t) the number of non-permanent publishers and by X N (t) the number of peers without the file at time t. Observe that Y N (0) + X N (0) = N -Y * N . If lim N →∞ Y N (0)/N = y 0 and lim N →∞ X N (0)/N = x 0 , which implies that lim N →∞ Y * /N = 1 -x 0 -y 0 := y * , then, by Kurtz's result [START_REF] Kurtz | Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes[END_REF], the rescaled process N -1 (Y (t), X(t)) converges in probability as N → ∞, uniformy on all finite intervals [0, T ], to the solution of the ODEs

d dt y x = β(y + y * )x -µ(α)y -β(y + y * )x (22)
with initial state (y 0 , x 0 ). From now on we will assume that y * > 0.

Consider an arbitrary peer without the file at time t = 0 and denote by T N the time that elapses before it receives it. Let P N (t) := P (T n < t). Similarly to [START_REF] Zhang | Performance Modeling of Epidemic Routing[END_REF]Page 6] we find

dP N (t) dt = β(1 -P N (t)) E[Y N (t)] + Y * N N . ( 23
)
Solving for P N (t) gives

P N (t) = 1 -e -β t 0 E[(Y N (s)]+Y * N N ds , t ≥ 0. ( 24
)
Hence

E[T N ] = ∞ 0 (1 -P N (t))dt = ∞ 0 e -β t 0 E[Y N (s)]+Y * N N ds dt.
From the above we know that (E[Y N (t)] + Y * N )/N → y + y * as N → ∞ for every t > 0, so that from (24) lim N →∞

(1 -P N (t)) = e -β t 0 (y(s)+y * )ds (25) for every t > 0. On the other hand, lim N →∞ Y * N /N = y * implies that for 0 < ǫ < y * there exists N ǫ such that Y * N /N > y * -ǫ for all N > N ǫ . Therefore, from (24),

1 -P N (t) ≤ e -β(y * -ǫ)t (26)
for N > N ǫ , t > 0. Since the r.h.s of ( 26) is integrable in [0, ∞), (25) and ( 26) allow us to apply the bounded convergence theorem to conclude that

T (α) := ∞ 0 lim N →∞ (1 -P N (t))dt = ∞ 0 e -β t 0 (y(s)+y * )ds dt.
The objective of the content owner is to choose an investment level α ≥ 0 which will maximize its utility

h(α) := T (α) -α. ( 27
)
To understand the impact of cooperation control on the delay, we present numerical studies in Section VII.

VI. P2P WITH A FIXED REQUEST RATE PER NODE

A. Model

In this section we will consider a slight variation of the model in [START_REF] Nunez-Queija | Scaling Laws for File Dissemination in P2P Networks with Random Contacts[END_REF]: there are N peers at time t = 0, at least one of them having a file. Each peer without the file sends a request for the file to another peer selected at random. These requests are initiated at Poisson rate λ > 0. It is assumed that a peer with the file leaves the system after an exponentially distributed random duration with rate µ > 0. All these rvs are mutually independent. Let Y (t) (resp. X(t)) be the number of peers with the file (resp. without the file) at time t. We have Y (0) + X(0) = N with Y (0) ≥ 1. Under the above assumptions Z := {(Y (t), X(t)), t ≥ 0} is a Markov process on the set E := {(y, x) ∈ {0, 1, . . . , N } 2 : 0 ≤ y + x ≤ N }. Let q(z, z ′ ), z = (y, x), z ′ = (y, x) ∈ E, denote its generator. Non-zero transition rates are given by

Y (t) X(t) → Y (t) + 1 X(t) -1 with rate λY (t)X(t) Y (t) + X(t) , (28)
Y (t) X(t) → Y (t) -1 X(t) with rate µY (t). ( 29
)
This model differs from our previous model in that the rate of increase is normalized by the total number of peers in the system. More precisely, the rate in (28) follows from the fact that with probability Y (t)/(Y (t) + X(t)) a peer without the file will contact a peer with a file at time t (the latter implicitly assumes that a peer may contact itself as otherwise this probability would be Y (t)/(Y (t) + X(t) -1); the reason for doing this will next become apparent. Note that this assumption will have no effect when N gets large) so that the total rate of increase of the number of peers with the file is λY (t)X(t)/(Y (t) + X(t)) at time t. The same model is considered in [START_REF] Nunez-Queija | Scaling Laws for File Dissemination in P2P Networks with Random Contacts[END_REF] with the difference that in [START_REF] Nunez-Queija | Scaling Laws for File Dissemination in P2P Networks with Random Contacts[END_REF] there is one permanent publisher, thereby implying that all peers will receive the file. These authors show that that the mean broacast time is O(N ) if λ < µ and is O(log(N )) if λ > µ. Thus, there is a phase transition at λ = µ.

In this section we will instead focus on (i) the fraction of peers that will receive the file (in the absence of a permanent publisher this fraction is not always equal to 1) and on (ii) the maximum torrent size (maximum number of copies of the file at one time) as N is large. In both cases we will show the existence of phase transitions.

Our analysis will use Kurtz's theorem [6, Thm 3.1] like in Section IV. Note, however, that both metrics (i) and (ii) above require to use the mean-field limit as t → ∞, something that Kurtz's result does not cover.

To overcome this difficulty, we will use the same argument as in [START_REF] Darling | Differential Equation Approximations for Markov Chains[END_REF] (see also Section IV where this argument was already used), taking advantage of the particular structure of the infinitesimal generator of the process Z. More specifically, it is seen that the generator of Z writes in the form q(z, z ′ ) = y q(z, z ′ ) for z = (y, x), z ′ = (y ′ , x ′ ) ∈ E, where non-zero transition rates are given in (28)-(29).

Let Z = {( Ỹ (t), X(t)), t ≥ 0} be a Markov process with generator g(z, z ′ ) and state-space E (same state-space as Z).

Since Z has been obtained by changing the time-scale of Z, the final values of X(t) and of X(t) will have the same distribution (note that the final state of Ỹ (t) and Y (t) is always zero since states (0, •) are all absorbing states) and so will have the maximum torrent size.

Since the generator g(z, z ′ ) can be written as g(z, z ′ ) = N f (z/N, z ′ /N ) (this is where the assumption that a peer may contact itself is useful) and since conditions (3.2)-(3.4) in [START_REF] Kurtz | Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes[END_REF]Thm 3.1] are clearly satisfied, we may apply [6, Thm 3.1] to obtain that, at any finite time t, N -1 ( Ỹ (t), X(t)) converges in probability as N → ∞ to the solution (ỹ, x), 0 ≤ ỹ, x ≤ 1, ỹ + x ≤ 1, of the ODEs

ẏ = -µ + λx/(x + ỹ), ẋ = -λx/(x + ỹ) (30)
given that lim N →∞ N -1 ( Ỹ (0), X(0)) = (ỹ(0), x(0)). Let (y 0 , x 0 ) := (ỹ(0), x(0)). We will assume that 0 < y 0 < 1 and y 0 + x 0 = 1 (the case y 0 = 0 (resp. y 0 = 1) has no interest since it corresponds to a P2P network with no file at any time (resp. where all peers have the file at time t = 0).

B. Phase transitions

Define ξ := λ/µ. Adding both ODEs in (30) yields ỹ(t) + x(t) = -µt + 1. Plugging this value back into (30) gives x(t) = x 0 (1 -µt) ξ for 0 ≤ t < 1/µ and, by continuity,

x(t) = x 0 (1 -µt) ξ for 0 ≤ t ≤ 1/µ with x(1/µ) = 0.
In order to approximate the fraction of peers which will never receive the file as N is large, one needs to find the first time τ > 0 where either x(τ ) = 0 or ỹ(τ ) = 0. This time τ is easy to find as shown below.

We already know that x(t) > 0 for 0 ≤ t < 1/µ and x(1/µ) = 0 so that we only need to focus on the zeros of

ỹ(t) in [0, 1/µ]. By writing ỹ as ỹ(t) = (1 -µt)(1 -x 0 (1 - µt) ξ-1 ) we conclude that the smallest zero of ỹ in [0, 1/µ] is (1 -x 1/(1-ξ) 0 )/µ if ξ < 1 and is 1/µ if ξ ≥ 1. Therefore, τ = (1 -x 1/(1-ξ) 0 )/µ > 0 if ξ < 1 and τ = 1/µ if ξ ≥ 1. Introducing this value of τ in x(t) yields x(τ ) = x 1/(1-ξ) 0 if ξ < 1 and x(τ ) = 0 if ξ ≥ 1.
In other words, as N is large, all peers will get the file if ξ ≥ 1 and a fraction x 1/(1-ξ) 0 of them will not if ξ < 1. In other word we observe a phase transition at ξ = 1: all peers will get the file if ξ ≥ 1 and a fraction ξ → x 1/(1-ξ) 0

will not if ξ < 1.

Let us now turn to the maximum torrent size. As N is large it will be approximated by the maximum of ỹ over the interval [0, τ ]. A straightforward analysis of the mapping t → ỹ(t) in [0, τ ] shows that

• it is decreasing if ξ ≤ 1 or if ξ > 1 and ξx 0 ≤ 1 -these conditions can be merged into the single condition ξ ≤ 1/x 0 -so that its maximum, y max , is given by y max = y 0 = 1 -x 0 ,

• it is unimodular (first increasing then decreasing) if ξ > 1/x 0 , with its maximum reached at t 1 := (1 -(ξx 0 ) 1-ξ )/µ and given by

y max = (x 0 ) 1/(1-ξ) ξ ξ/(1-ξ) (ξ -1) > 0. (31)
In summary, as N is large, the maximum torrent size is approximated by N y max with y max given in (31) if ξ > 1/x 0 and y max = 1 -x 0 if ξ ≤ 1/x 0 . This shows another phase transition (see Fig. 2) at ξ = 1/x 0 (i.e. at λx 0 = µ) in the sense that the torrent is maximum at t = 0 if ξ ≤ 1/x 0 and is maximum at a later time if ξ > 1/x 0 .

VII. NUMERICAL RESULTS

This section has two goals: to investigate the accuracy of the approximations developed in the previous sections (to be made more precise) and to study the impact of measures against non-authorized uploading or downloading on the file availability in P2P swarming systems. Due to lack of space, we will not report any numerical result for the P2P model considered in Section VI; we will instead focus on the P2P model introduced in Section II-C and on its branching and mean-field approximations developed in Sections III and IV, respectively (Fig. 345678910), as well as on the optimization problem set in Section V (Fig. 11).

For each set of parameters, between 200 and 1000 discreteevent simulations of the Markov model in Section II-C have been run. In each figure (except in Fig. 56where only simulation results are displayed and in Fig. 11 where only mean-field results are shown) both simulation and approximation results are reported for the sake of comparison. Let r := (Y (0) + X c (0))/N be the ratio of cooperative peers at time t = 0 and recall that N c = X c (0) (see Section II). The total number of peers, N , at time t = 0 is equal to 400 in Fig. 3456, to 300 in Fig. 7 -8 and to 500 in Fig. 11.

A. File extinction time and the branching approximation

In this section we focus on Fig. 345678. Fig. 3 (resp. Fig 4) compares the CDF of the extinction time obtained by simulation and by the branching approximation in [START_REF] Zhang | Performance Modeling of Epidemic Routing[END_REF] when Y (0) = 1 (resp. Y (0) = 3), λ = 6 • 10 -3 , µ = 1, and for two values of r (r = 0.6 implies that N c = 239 and X f (0) = 160, r = 1 implies that there are no free riders (X f (0) = 0) and N c = 399). Note that ρ = λN c /µ is close to 2.4 when r = 1 and is close to 1.43 when r = 0.6. In all cases, the simulation and the branching approximation are in close agreement up to a certain time (time T B in Fig. 3) which, interestingly, corresponds to the extinction time in the branching model. After this time, the extinction of the file in the Markov model increases sharply (the larger r the larger the increase). In other words, the extinction of the file in the original Markov ml has two modes, an early extinction mode and a late extinction mode. The former occurs when the file disappears before the dissemination has reached its peak value (i.e. most peers do not get the file) and the latter when most peers leave the network with the file. One may also check that the branching approximation provides an upper bound for the CDF of the extinction time, as predicted by Proposition 2. Last, we note that when there are less cooperative peers (r = 0.6) the file lifetime is prolonged (see e.g. point D in Fig. 3 where simulation curves for r = 1 and r = 0.6 cross each other); this can be explained by the fact that there are less contact opportunities between cooperative peers when r = 0.6. The main difference between Fig. 3

and Fig 4 lies

in the increase of the probability of the late extinction that is steeper with three initial seeds (Y (0) = 3) than with one initial seed (Y (0) = 1).

Simulation results in Fig. 5-6 exhibit the same early-late extinction pattern as in Fig. 34; they have been obtained for λ = 25 • 10 -4 and for two different values of µ, r and Y (0). Fig. 78show the expected time to extinction as a function of the pairwise contact rate λ, in the case of an early extinction (i.e. for small values of λ), for µ = 1 and for two values of r. The curves "Model" display the mapping λ → E[T b (k)], with E[T b (k)] the expected extinction time in the branching process given Y (0) = k (see Section III). We observe an excellent match between the simulation and the branching approximation thereby showing that the latter works well for early file extinction. Also note that having three seeds instead of one greatly extends the expected extinction time.

B. File availability and the mean-field approximation

We now look at the fraction of peers that will not acquire the file. We assume that Y (0) = 10 and we recall that N = 300. Fig. 9 (resp. Fig. 10) displays this fraction as a function of λ (resp. µ) for two different values of r (r = 1 corresponding to X c (0) = 290 and r = 0.5 corresponding to X c (0) = 140 and X f (0) = 150). In each figure, both simulation and meanfield approximation results are reported. The fraction of peers without the file is a decreasing function of the pairwise contact rate λ and an increasing function of the cooperation degre µ. The mean-field approximation is obtained as the unique solution x c (∞) in (0, x c (0) of equation [START_REF] Simatos | A Queueing System for Modeling a File Sharing Principle[END_REF] where the initial condition of the ODEs ( 17) is given by (y 0 , x c (0), x f (0)) = (Y (0)/N, X c (0)/N, X f (0)/N ). In both figures we observe a remarkable agreement between the simulation and the meanfield results (relative errors never exceed 2% when all peers are cooperative (r = 1) and never exceed 7% when half of the peers are free riders (r = 0.5)). We also note that the fraction of peers without the file considered as a function of λ (resp. µ) is larger (resp. smaller) when r = 0.5 than when r = 1; this is of course not surprising since, unlike cooperative peers, free riders do not contribute to the file dissemination.

C. Action against unauthorized file downloading

We now evaluate the impact of actions against unauthorized file downloading. For that, we use the framework developed in Section V. Since the simulations in [START_REF] Altman | Predicting the Impact of Measures Against File Sharing Networks on the Transient Behaviors[END_REF] show that, for large N , T (α) in Section V is a good approximation of the expected time, T N , needed for an arbitrary peer to get the file we only consider the utility function h(α) (see ( 27) ). We assume that the cooperation degree µ(α) is given by µ(α) = µα. There are 500 peers (N = 500) at time t = 0 including two persistent publishers (Y * = 2). We assume that Y (0) = 0 so that X(0) = N -Y * = 498. The initial condition of the ODE ( 22) is (y 0 , x 0 ) = (0, X N (0)/N ) with y * = 2/N . Fig. 11 displays the mapping α → h(α), for two different values of β and µ. We observe that a small investment cannot obviously postpone the expected delivery delay of the file, resulting in a decreased utility. As the investment grows, the utility of the content owner increases significantly. The curves in Fig. 11 also show how large an investment has to be to counteract P2P illegal downloading. Note the content owner can still have an increased utility when the ratio β/µ(α) is greater than one, as the utility is maximized across all curves when the ratio β/µ(α) lies between two and three.

Fig. 1 .

 1 Fig.1. Ratio of cooperative (as N is large) that never receive the file as a function of log 10 (θx c,0 ).

Fig. 2 .

 2 Fig. 2. Maximum torrent size over N (as N is large) as a function of ξ for x 0 = 0.95 (left figure) and x 0 = 0.8 (right figure).

Fig. 3 .Fig. 4 .

 34 Fig. 3. CDF of extinction time for Y (0) = 1, λ = 0.006, µ = 1.

Fig. 5 .

 5 Fig. 5. CDF of extinction time for Y (0) = 1 and different µ.

Fig. 6 .

 6 Fig. 6. CDF of extinction time for Y (0) = 3 and different µ.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Early extinction time as a function of λ with Y (0) = 1.

Fig. 9 .

 9 Fig. 9. Fraction of peers without the file as a function of λ.

  Fig. 10. Fraction of peers without the file as a function of µ.

Fig. 11 .

 11 Fig. 11. Investment vs. utility of content owner.

VIII. RELATED WORK

There has been a number of work on the mathematical studies of structured and unstructured P2P-based content distribution. A seminal work can be found in [START_REF] Yang | Service Capacity of Peer to Peer Networks[END_REF]. The authors propose a continuous-time branching process to analyze service capacity (i.e. maximum rate of downloading) and a coarse-grain Markov model to characterize the steady state of downloading rate. In [START_REF] Qiu | Modeling and Performance Analysis of Bittorrent-like Peer-to-Peer Networks[END_REF], Qiu and Srikant propose a fluid model composed of ordinary differential equations to describe the dynamics of BitTorrent systems. Authors in [START_REF] Fan | On the Performance and Availability of BitTorrent-like P2P File Sharing Systems[END_REF] further propose a novel fluid model based on stochastic differential equations. This new model also extends [START_REF] Qiu | Modeling and Performance Analysis of Bittorrent-like Peer-to-Peer Networks[END_REF] to multi-classes system and is able to describe chunk availability. Mundinger et al. [START_REF] Mundinger | Optimal scheduling of peer-topeer file dissemination[END_REF] propose a deterministic scheduling algorithm to achieve the optimal makespan for a structured system which requires global knowledge. A coupon model is put forward in [START_REF] Massoulié | Coupon Replication Systems[END_REF] to investigate the effectiveness of a generic P2P file sharing system.

Recently, the process of file dissemination has attracted a lot of attentions. Clévenot et al. adopt a hybrid approach (fluid and stochastic) to analyze Squirrel, a P2P cooperative web cache in [START_REF] Clévenot | A Simplified Model for the Analysis of the Squirrel Peer-to-Peer Caching System[END_REF]. In [START_REF] Nunez-Queija | Scaling Laws for File Dissemination in P2P Networks with Random Contacts[END_REF] Queija et al. study the scaling law of mean broadcasting time in a closed P2P swarm with constant request rate. Authors in [START_REF] Simatos | A Queueing System for Modeling a File Sharing Principle[END_REF] formulate a ball-and-urn model to characterize the "flash crowd" effect in a closed P2P networks. The content provided by P2P networks such as music, movies and software are usually unauthorized. Content provided are therefore inclined to combat illegal downloading/uploading via technical solutions. Authors of [START_REF] Wong | P2P Networks: The Interplay between Legislation and Information Technology[END_REF] and [START_REF] Altman | P2P Business and Legal Models for Increasing Accessibility to Popular Culture[END_REF] propose a M/G/∞ queueing model to access the efficiency of non-cooperative measures against unauthorized downloading. Authors in [START_REF] Menasche | Bundling Builds Availability in BitTorrent[END_REF] propose a similar but elegant queuing model to study the impact of bundling strategy of file availability and downloading rate. Our general model is inspired by the one in [START_REF] Nunez-Queija | Scaling Laws for File Dissemination in P2P Networks with Random Contacts[END_REF]. However, it differs from [START_REF] Nunez-Queija | Scaling Laws for File Dissemination in P2P Networks with Random Contacts[END_REF], [START_REF] Qiu | Modeling and Performance Analysis of Bittorrent-like Peer-to-Peer Networks[END_REF], [START_REF] Yang | Service Capacity of Peer to Peer Networks[END_REF], [START_REF] Altman | P2P Business and Legal Models for Increasing Accessibility to Popular Culture[END_REF] in four ways: 1) we are studying the transient behavior; 2) a peer can initiate a number of random contacts, instead of one, with other peers; 3) we observe several phase transitions in response to system parameters; 4) we adopt Markov branching process and meanfield approaches to characterize the file dissemination model comprehensively.

IX. CONCLUSION

In this paper we have proposed to use the theory of continuous time branching process as well as of the dynamics of epidemics in order to study the transient behavior of torrents that occur in P2P systems. The use of these tools allowed us to compute the probability of early extinction of the torrent as well as the expected time until that extinction, the availability of a file in the system, the maximum availability and when it occurs, and the size of the torrent. This is used for analyzing the impact of measures to decrease non-authorized Internet access to copyrighted files. We identify regimes in which the performance measures are quite sensitive to such measures and others in which the measures have very limited impact. In particular, we present two counteractions against unauthorized file sharing in the presence of illegal publishers.

Our methodology can be extended to analyze file bundling that serves as a positive action of file dissemination.


MD Venla Lehti
email: venla.lehti@utu.fi

MD Andre Sourander

PhD, Lecturer, Anat Klomek

MD Solja Niemelä

Lauri Sillanmäki

MD Jorma Piha

MD Kirsti Kumpulainen

MD Tuula Tamminen

MD Irma Moilanen

MD Fredrik Almqvist

Childhood bullying as a predictor for becoming a teenage mother in Finland

Keywords: teen pregnancy, bully, victim, risk factor, epidemiology

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Becoming a teenage mother in Western societies is a phenomenon often associated with various psychosocial problems.

The socioeconomic status of teenage mothers is on average lower compared with older mothers [START_REF] Moffitt | Risk Study Team (2002) Teen-aged mothers in contemporary Britain[END_REF][START_REF] Maughan | Secular change in psychosocial risks: the case of teenage motherhood[END_REF]. Both teenage mothers [START_REF] Moffitt | Risk Study Team (2002) Teen-aged mothers in contemporary Britain[END_REF][START_REF] Schmidt | Moderate to severe depressive symptoms among adolescent mothers followed four years postpartum[END_REF][START_REF] Henretta | Early motherhood and mental health in midlife: a study of British and American cohorts[END_REF][START_REF] Genna | Risk factors for young adult substance use among women who were teenage mothers[END_REF] and their children [START_REF] Moffitt | Risk Study Team (2002) Teen-aged mothers in contemporary Britain[END_REF][START_REF] Shaw | Teenage children of teenage mothers: psychological, behavioural and health outcomes from an Australian prospective longitudinal study[END_REF][START_REF] Fergusson | Maternal age and educational and psychosocial outcomes in early adulthood[END_REF][START_REF] Ekeus | Psychiatric morbidity is related to parental age: a national cohort study[END_REF] have an increased risk of mental health problems. Children of teenage mothers are more likely to be involved with delinquent behaviour [START_REF] Shaw | Teenage children of teenage mothers: psychological, behavioural and health outcomes from an Australian prospective longitudinal study[END_REF][START_REF] Fergusson | Maternal age and educational and psychosocial outcomes in early adulthood[END_REF] and to have problems in school [START_REF] Moffitt | Risk Study Team (2002) Teen-aged mothers in contemporary Britain[END_REF][START_REF] Shaw | Teenage children of teenage mothers: psychological, behavioural and health outcomes from an Australian prospective longitudinal study[END_REF][START_REF] Fergusson | Maternal age and educational and psychosocial outcomes in early adulthood[END_REF]. They also have a higher risk of teenage pregnancy [START_REF] Woodward | Risk factors and life processes associated with teenage pregnancy. Results of a prospective study from birth to 20 years[END_REF] and motherhood [START_REF] Meade | The intergenerational cycle of teenage motherhood: an ecological approach[END_REF], themselves.

Several studies have been conducted on the factors associated with teenage pregnancy. There is evidence of low socioeconomic status, broken family structure, unsuccessful bonding between parent and child and other family factors [START_REF] Imamura | REPROSTAT 2 group (2007) Factors associated with teenage pregnancy in the European Union countries: a systematic review[END_REF][START_REF] Miller | Family relationships and adolescent pregnancy risk: a research synthesis[END_REF], as well as of psychiatric determinants such as conduct problems and other externalising symptoms [START_REF] Maughan | Secular change in psychosocial risks: the case of teenage motherhood[END_REF][START_REF] Woodward | Risk factors and life processes associated with teenage pregnancy. Results of a prospective study from birth to 20 years[END_REF][START_REF] Fergusson | Educational, psychosocial, and sexual outcomes of girls with conduct problems in early adolescence[END_REF][START_REF] Colman | Outcomes of conduct problems in adolescence: 40 year follow-up of national cohort[END_REF][START_REF] Woodward | Early conduct problems and later risk of teenage pregnancy in girls[END_REF][START_REF] Woodward | Teenage pregnancy: cause for concern[END_REF][START_REF] Miller-Johnson | Motherhood during the teen years: a developmental perspective on risk factors for childbearing[END_REF][START_REF] Bardone | Adult physical health outcomes of adolescent girls with conduct disorder, depression, and anxiety[END_REF][START_REF] Jaffee | Pathways to adversity in young adulthood among early childbearers[END_REF].

In addition, a low level of education and educational expectations, as well as a dislike of school, are known to be associated with teenage pregnancy [START_REF] Imamura | REPROSTAT 2 group (2007) Factors associated with teenage pregnancy in the European Union countries: a systematic review[END_REF]. Underwood et al. suggested that controversial peer status in school is associated with becoming a teenage mother [START_REF] Underwood | Childhood peer sociometric status and aggression as predictors of adolescent childbearing[END_REF]. In another study, however, peer status groups did not differ from each other in this respect [START_REF] Miller-Johnson | Motherhood during the teen years: a developmental perspective on risk factors for childbearing[END_REF].

A school-and peer-related factor, which has not been examined previously in relation to becoming a teenage mother, is bullying. The most widely accepted definition for bullying is the one by Olweus [START_REF] Olweus | Victimization by Peers: Antecedents and Long Term Outcomes[END_REF]: A person is bullied when he or she is exposed, repeatedly and over time, to negative actions on the part of one or more persons. The universality and harmfulness of bullying have been widely recognised. Population-based studies indicate that 20 to 30 % of school children are frequently involved in bullying as bullies and/or victims [START_REF] Forero | Bullying behaviour and psychosocial health among school students in New South Wales, Australia: cross sectional survey[END_REF][START_REF] Dake | The nature and extent of bullying at school[END_REF][START_REF] Kumpulainen | Children involved in bullying at elementary school age: their psychiatric symptoms and deviance in adolescence. An epidemiological sample[END_REF][START_REF] Wolke | Bullying involvement in primary school and common health problems[END_REF][START_REF] Nansel | Bullying behaviors among US youth: prevalence and association with psychosocial adjustment[END_REF]. Bullies and victims of bullying are more likely than other children to suffer from a wide variety of problems, including depression and anxiety [START_REF] Hawker | Twenty years' research on peer victimization and psychosocial maladjustment: a meta-analytic review of cross-sectional studies[END_REF][START_REF] Arseneault | Bullying victimization uniquely contributes to adjustment problems in young children: a nationally representative cohort study[END_REF], low selfregard, [START_REF] Egan | Does low self-regard invite victimization?[END_REF] and violent behaviour [START_REF] Nansel | Relationships between bullying and violence among US youth[END_REF]. Moreover, bullying behaviour during childhood is related to future psychiatric problems [START_REF] Olweus | Victimization by Peers: Antecedents and Long Term Outcomes[END_REF][START_REF] Kumpulainen | Children involved in bullying at elementary school age: their psychiatric symptoms and deviance in adolescence. An epidemiological sample[END_REF][START_REF] Kim | School bullying and youth violence: causes or consequences of psychopathologic behavior?[END_REF]. Previous studies have suggested that bully-victims are the most troubled in terms of outcomes [START_REF] Kumpulainen | Children involved in bullying at elementary school age: their psychiatric symptoms and deviance in adolescence. An epidemiological sample[END_REF][START_REF] Kim | School bullying and youth violence: causes or consequences of psychopathologic behavior?[END_REF][START_REF] Sourander | Childhood bullies and victims and their risk of criminality in late adolescence: the Finnish From a Boy to a Man study[END_REF].

The aim of the present study is to examine whether bullying others or being a victim of bullying during childhood is associated with becoming a mother in adolescence. If an association does indeed exist, then the further aims are to test 1) whether it can be explained by psychiatric symptoms or by risk factors related to family background, 2) whether it is stronger among girls who are bullies, victims of bullying, or those who are both and 3) whether its strength varies according to the informant of bullying (teacher, parent and child).

Methods

The study is a part of the "Finnish 1981 Birth Cohort Study", a multicentre child psychiatric study in Finland. The baseline assessment was conducted in 1989, and the register data for the follow-up were collected from 1994 to 2000.

Informed consent was obtained from the children's parents at baseline. The way in which the combined information from questionnaires and register data was analysed did not allow identification of the subjects. The Ethics Committee of the Intermunicipal Hospital District of Southwest Finland approved the research plan.

Participants

The original study sample was drawn from the total population of Finnish children born in 1981 (n = 60,007). It consisted of 6017 children, which was 10 % of the basic population. Of the selected 6017 children, 5813 (96.6 %) took part in the study in 1989. The number of female participants was 2867. Almqvist et al. [START_REF] Almqvist | Design and subjects of a Finnish epidemiological study on psychiatric disorders in childhood[END_REF] have previously presented the design and subjects of this study and shown that the sample is nationally representative. At follow-up, the girls in the cohort were identified by linking their personal identification number with the data in the Finnish Hospital Discharge Register. Of the 2867 girls studied in 1989, information about both bullying in childhood and births under the age of 20 was obtained for 2507 (87.4 %).

Assessment at age eight Data collection at baseline was organised through teachers. The teachers sent parent questionnaires via the children to the parents, and the parents returned them in a sealed envelope to the teachers. The children filled in their own questionnaires in the classroom.

Bullying

At baseline, three informants were utilised to assess bullying behaviour: the child himself/herself, a parent, and a teacher. The children were asked about bullying by giving them three alternatives from which to choose: 1 = I do not usually bully other children, 2 = I sometimes bully other children, 3 = I bully other children nearly every day. Furthermore, children were asked about being victims of bullying: 1 = Other children do not usually bully me, 2 = Other children sometimes bully me, 3 = Other children bully me nearly every day. Similar questions focusing on bullying and victimization were included in parent and teacher questionnaires, with probe and response items worded as follows: The child bullies other children/is a victim of bullying: 1 = Doesn't apply, 2 = Applies somewhat, 3 = Certainly applies. In the present study, only two categories were used for the frequency of bullying/victimization and therefore options 2 and 3 were pooled together in all of the analyses. The respondents at age eight were hence classified as never bully or ever bully, never victim or ever victim and the combinations of these alternatives. The extent of bullying and victimization was investigated both by combining the information obtained from the child, parent and teacher, and by using the informant-specific variables. When the combined variables were used, bullying or victimization was considered to exist if it was reported by at least one informant.

Confounding variables

The confounding variables used in this study were childhood psychopathology, family structure, and age and educational level of the girl's mothers. The psychopathology score was based on the sum score of the Rutter Parent and Teacher Scales [START_REF] Rutter | A children's behaviour questionnaire for completion by teachers: preliminary findings[END_REF][START_REF]Education, Health and Behaviour[END_REF]. The bullying/victimization items were removed from the sum scale. The Rutter scales comprise conduct, attention and emotional subscales. The psychopathology variable was used as continuous. As regards factors pertaining to family, the intact family structure was defined as living with two biological parents. Any other kind of family arrangement was referred to as a non-intact family structure. The parents' educational level was studied through the mother's completion of upper secondary school only, due to the low number of reports from fathers. Not completing upper secondary school was considered as having low educational level. The girls' mothers who were born in the year 1961 or later were referred to as young mothers and those born in 1960 or earlier were referred to as old mothers.

The outcome

Information concering deliveries in 1994-2000 was based on the Finnish Hospital Discharge Register. The register is maintained by the National Institute for Health and Welfare and is one of its most frequently used registers for research purposes [START_REF] Gissler | Finnish health and social welfare registers in epidemiological research[END_REF]. It covers the diagnoses and the days of admission and discharge in all public and private inpatient care units in Finland. Less than 0.5 % of Finnish babies are born outside hospital [START_REF] Vuori | Newborn children 2006[END_REF], and less than 0.5 % of mothers leave the hospital the same day they give birth [START_REF] Vuori | Parturients, births and newborns 2007[END_REF], which means that more than 99 % of births should be included in the Hospital Discharge Register. The coverage and quality of the register is generally considered to be good [START_REF] Gissler | Finnish health and social welfare registers in epidemiological research[END_REF]. A study on the reliability of the register showed that 99 % of the diagnoses indicating a delivery matched those in the patient records [START_REF] Aro | Reliability of hospital discharge data concerning diagnosis, treatments and accidents[END_REF]. In the present study, the the ICD10 diagnostic codes indicating a delivery represented the information collected. Information pertaining to the mothers' age at birth was only available with an accuracy of one year. Teenage mothers were defined as those who were younger than 20 when they gave birth.

Statistical Analysis

Associations between childhood variables and becoming a teenage mother were quantified by calculating odds ratios (OR) with 95 % confidence intervals (CI) using logistic regression analysis. The odds ratios for continuous variables (related to child psychopathology) were calculated for the change in one standard deviation unit (SD). The reference group was defined as girls who had not given birth under the age of 20. Both single predictor and multipredictor analyses were conducted with combined bullying variables, including information from all informants, and with informant-specific bullying variables. Statistical computations were conducted with the SAS System for Windows, release 9.2/2008.

Results

In the total sample, 51.8 % of girls had never bullied others nor been bullied. The proportion of those who had never bullied others, but had sometimes or frequently been victims of bullying according to at least one informant, was 24.1 %, whereas 8.1 % had sometimes or frequently bullied others, but never been victims themselves. Further, 16.0 % of girls were classified as bully-victims since they had at least sometimes bullied others as well as been bullied. The number of teenage mothers in the total sample was 128 (4.8 %).

The odds ratios for the risk of becoming a teenage mother when the combined bullying variables were used are indicated in Table 1. Both bullies and victims had a higher odds of becoming a teenage mother compared with those who were never victims nor bullies. In the single predictor analysis the odds ratio was highest (2.7) among bullyvictims, but when the three groups were compared with each other instead of being compared with the group "never bully -never victim", no statistically significant differences were found. The associations remained significant when adjusted for the mother's age and educational level and family structure (Model 1 in Table 1), these associations remained significant. However, when also adjusted for childhood psychopathology (Model 2 in Table 1), the associations remained significant for bullies and bully-victims, but not for pure victims.

When the reports of bullying from the parents, teachers and girls themselves were analysed separately, a statistically significant association was found between each informant's report of bullying others and becoming a teenage mother.

Parent and teacher reports but not girl's self-report of victimization were associated with becoming a teenage mother.

When the reports were included in the two multipredictor models, one for bullying and the other for victimization, it was evidnet that the parents', teachers' and children's report of bullying and victimization were all associated with becoming a teenage mother independently of each other (Table 2).

Discussion

It is known from previous research that girls who suffer from psychiatric problems or have a certain type of family background are more likely to become teenage mothers. What this study adds is that bullying in childhood is also associated with becoming a teenage mother. This information contributes to a wider understanding of girls' peer relationships, psychopathology and sexual behaviour.

The first finding of this study indicated that female bullies are at risk of becoming teenage mothers, regardless of the baseline psychopathology or previously known family-related risk factors. Considering that psychiatric problems are known to increase the risk of teenage pregnancy, and to be related to bullying, it is interesting that bullying and psychopathology seem to be independent risk factors. This may suggest that at least the female bullies who become teenage mothers, use more subtle and less physically aggressive ways of bullying and are not necessarily categorised as having conduct problems or other psychiatric symptoms. When adjusted for psychopathology, the association between bullying behaviour and becoming a teenage mother seemed to be stronger for bullies than for bully-victims. This may reflect a more severe psychological disturbance among bully-victims [START_REF] Kumpulainen | Bullying and psychiatric symptoms among elementary school-age children[END_REF].

There are several ways to approach the association between bullying and becoming a teenage mother. Bullying can be considered as a form of interpersonal aggression. Girls' aggression is more often indirect or relational in contrast to overt physical aggression [START_REF] Prinstein | Overt and relational aggression in adolescents: social-psychological adjustment of aggressors and victims[END_REF][START_REF] Crick | Relational aggression, gender, and social-psychological adjustment[END_REF]. Although relational aggression is no less harmful to its victims [START_REF] Prinstein | Overt and relational aggression in adolescents: social-psychological adjustment of aggressors and victims[END_REF][START_REF] Crick | Relational and overt forms of peer victimization: a multi-informant approach[END_REF], it may still be better accepted among peer group members [START_REF] Salmivalli | Aggression and sociometric status among peers: do gender and type of aggression matter?[END_REF]. Girls' use of indirect aggression has been shown to correlate negatively with their rejection among boys in particular [START_REF] Salmivalli | Aggression and sociometric status among peers: do gender and type of aggression matter?[END_REF]. The social position and quality of peer relations may also be reflected in bullies' romantic relationships and their sexual behaviour. Bullies seem to start dating earlier and participate in more advanced forms of dating, but their relationships also tend to be less supportive and more violent relationships [START_REF] Connolly | Dating experiences of bullies in early adolescence[END_REF]. It may be that girls who bully others have learnt that aggression is a way of forming interpersonal relations. This corresponds to research indicating that children's insecure attachment is linked to behaviour problems [START_REF] Greenberg | Correlates of clinic referral for early conduct problems: variable-and person-oriented approaches[END_REF][START_REF] Guttmann-Steinmetz | Attachment and externalizing disorders: a developmental psychopathology perspective[END_REF]. Research has suggested that both bullies [START_REF] Shields | Parental maltreatment and emotion dysregulation as risk factors for bullying and victimization in middle childhood[END_REF] and pregnant teenagers [START_REF] Miller | Family relationships and adolescent pregnancy risk: a research synthesis[END_REF] have experienced family dysfunction more often than their non-bullying or non-pregnant peers. It is also possible that bullies are more willing to have a child than their non-bullying peers. A study by Connolly et al. [START_REF] Connolly | Dating experiences of bullies in early adolescence[END_REF], suggested that bullies are prepared to engage in undesirable activities to keep their partners and consider their relationships very important. The desire for intimacy may also be related to positive attitudes toward parenthood. Unfortunately, differentiating between girls whose baby was intended and wanted, and those for whom motherhood was unplanned or even unwanted was not possible in the present study. Further studies would also be needed to examine if bullies learn different modes of parenting behaviour as compared to their non-bullying peers.

The second finding of this study indicated that victimization at age eight alone is not associated with becoming a teenage mother, after controlling for childhood psychopathology. This may indicate that the association between childhood victimization and later becoming a teenage mother is a function of psychopathology rather than the victimization per se. One possible explanation would be that victims suffer from depression or other psychiatric problems [START_REF] Hawker | Twenty years' research on peer victimization and psychosocial maladjustment: a meta-analytic review of cross-sectional studies[END_REF][START_REF] Klomek | Bullying, depression, and suicidality in adolescents[END_REF], and they may have a tendency to use rapidly initiated sexual activity with multiple partners as a means of coping with feelings of depression and emptiness [START_REF] Becker-Lausen | Integration of teen pregnancy and child abuse research: identifying mediator variables for pregnancy outcome[END_REF]. To be able to feel some control and confidence, the victims, who deal with a situation over which they have no control (peer victimization), put themselves in a controlled situation (pregnancy/being a mother). In other cases, the pregnancy may be a result of sexual victimization, which is a specific extreme form of victimization, frequently neglected in the bullying literature. In these cases, the childhood "no control" situation is repeated in adolescence. This option is in line with previous studies indicating that childhood sexual abuse has been found to place girls at increased risk of subsequent adolescent pregnancy [START_REF] Noll | Childhood sexual abuse and adolescent pregnancy: a meta-analytic update[END_REF].

The third finding of this study involved the fact that reports of bullying and victimization from different informants were independently associated with becoming a teenage mother. Cross-informant agreement about bullying behaviour is typically low [START_REF] Ronning | Cross-informant agreement about bullying and victimization among eight-year-olds: whose information best predicts psychiatric caseness 10-15 years later?[END_REF] and therefore different girls may be considered as bullies or victims by their teachers, parents and themselves. Peer reports of bullying behaviour were not available in the present study, but others suggest that it correlates well with self-report [START_REF] Prinstein | Overt and relational aggression in adolescents: social-psychological adjustment of aggressors and victims[END_REF][START_REF] Crick | Relational and overt forms of peer victimization: a multi-informant approach[END_REF].

There are several strengths in the design of this study. The sample was relatively large and nationally representative, the baseline assessment was founded on multiple informants and conducted as early as at the age of eight, the attrition rate was low and the outcome measure was based on reliable register data. However, there are limitations particularly concerning the data collected in childhood. The information on bullying behaviour was based on only a few questions.

The type of bullying was not specified, and the duration of bullying was not included; nor was there a survey of whether the bullies in the sample bullied girls or boys, or whether the victims in the sample were bullied by girls or boys. In addition, only two categories were used for the frequency of bullying and victimization. However, a post-hoc analysis showed that girls who bullied/were victimized sometimes and girls who bullied/were victimized frequently did not significantly differ in their risk of becoming a teenage mother. The small amount of observations, however, posed a limitation on the analysis for bullies, i.e. there were only three teenage mothers, who had bullied others frequently.

Finally, developmental reasoning was limited because both bullying and psychopathology were assessed only in childhood, and the exact timing of childbirth was not known.

Conclusions

The association between bullying in childhood and becoming a mother in adolescence is a new finding, which may have practical implications in sexual health education and teenage pregnancy prevention, especially in school settings.

School counsellors should be urged to openly discuss involvement in childhood bullying behaviour and target those involved for teenage pregnancy prevention. It is not known whether the prevention of bullying has an impact on the incidence of teenage pregnancies. It has been suggested that improving the school ethos might reduce teenage pregnancy levels, but no strong evidence is yet available [START_REF] Bonell | Improving school ethos may reduce substance misuse and teenage pregnancy[END_REF]. Further studies focusing on possible pathways between bullying and sexual behaviour are needed. Particular attention should be paid to the different types of problems in peer relations, their association with sexual development and the quality of intimate relationships.

Table 1 . Results of single and multipredictor logistic regression models with a binary response (birth < 20 years or not) and combined bullying variables at age eight.

 1 Multivariate model 1 included pooled reports of bullying behaviour, mother's educational level, mother's age and family structure. Multivariate model 2 included pooled reports of bullying behaviour, mother's educational level, mother's age, family structure childhood psychopathology (Rutter total score, parents' and teachers' report combined). The other three groups were compared with the "never bully or victim" group. OR = odds ratio, CI = confidence interval, *p < 0.05, **p < 0.01, ***p < 0.001

		Teenage mothers	Univariate analysis OR (95 % CI)	Multivariate anal
		n	%	Model 1
	Never bully nor victim	1299	3.1	ref

Acknowledgements

Dr. Lehti received a grant from the Finnish Foundation for Pediatric Research (Lastentautien tutkimussäätiö).

Conflict of interest

Dr. Niemelä has consulted Pfizer and been invited as a lecturer by AstraZeneca. These companies have no role in the production of this manuscript. The other authors have no potential conflicts of interest to disclose.