Dynamic Model of Growing File-Sharing P2P Network

This work considers the model of developing the P2P file exchange network organized by a torrent tracker. The model is constructed on the basis of ordinary differential equations. The phase variables describing the status of a torrent tracker and the network organized by it (at a first approximation this means the number of the tracker’s users, who actively participate in the information exchange, and the number of active torrents) are defined, and the factors influencing the change in the numbers of users and torrents analyzed. The analysis is used to develop a system of differential equations for describing, at a first approximation, the file exchange network organized by the torrent tracker — the hard dynamic model of the torrent tracker’s evolution. The equilibrium points of the hard model of the tracker’s evolution are analyzed, their possible quantity and type is described. All of the configurations of the general provision, possible in the hard model of the tracker’s evolution are described. The phase portrait of the hard model is represented. The analysis of the hard model is used to derive the system of differential equations that describes the evolution of a file exchange network, considering the dependence of the intensity of the influx of new users on the total size of the torrent tracker’s potential audience, and also the dependence of the torrent dying out rate on the number of users per torrent — the soft dynamic model of the torrent tracker’s evolution. The special points in the soft model are analyzed and their possible number and type described. All of the configurations of the general provision possible in the soft model of evolution are described. The phase portraits of each configuration are represented. The correlation of the parameters necessary for the tracker’s stability is calculated. The influence of various administrative measures on the tracker’s overall stability margin is analyzed. The need for supporting torrents by administrations of highly specialized torrent trackers with a small potential audience is shown.


INTRODUCTION
Nowadays, a huge amount of information is communicated with the help of decentralized networks. In particular, the BitTorrent system [1], broadly used for file sharing, allows sharing information with the help of a complex system (peer network) that does not save the shared files on the server (according to the BitTorrent protocol, sets of several logically bound files shared together are grouped in a continuous flow, which is why we will refer to this set hereafter as a file, too) and includes: -a multitude of users who save files in their local carriers; -a subsystem for each shared file: in its respect, the subsystem includes: -information about the specifications of a shared file (torrent file); -a place for receiving the torrent file (directory entry); -the users, who save the file and provide access to it (seeders), and the users, who download the file (leechers), collectively referred to as peers; -distributed infrastructure that timely updates and shares the information about the seeders and leechers as well as organizes the exchange of information among them; -all of this is covered by the notion of file seeding (this term is also used to refer to file sharing in a peer network); -the global infrastructure called a torrent tracker that is responsible for the exchange of information among the users and the activity of seeds. This notion is often expanded to apply to the community of the tracker's users and to the whole multitude of all supported shared seeds.
The key properties of a torrent tracker and of the file sharing peer network it organizes are the number of active users (not only registered ones but also those who participate in the exchange of information) and the number of shared seeds.
There are major torrent-trackers and their specialized counterparts that share specific kinds of content. In their respect, specialized trackers can be run as aggregators of content of various origins or share proprietary content produced by their administration.
New trackers are regularly opened. Some of them are developed and grow with success; others have a short period of growth with postopening enthusiasm and then lose active users and are closed. So, what is the root of the difference in development among torrent trackers?
1. HARD TORRENT TRACKER EVOLUTION MODEL Now we shall consider the evolution of the P2P file exchange network that runs on the BitTorrent protocol and is organized by some torrent tracker.
The logical choices as phase variables [2,3] to describe the network status are the number of active users and number of active seeds in the tracker. Let us consider the main factors affecting their change.
According to the preferential attachment principle, new users are more eager to sign in at major trackers already with a lot of active users; in addition, new users are attracted by large number of seeds in the tracker ( , the point is for the time derivative). In addition, the users create new seeds ( ). In addition, it always happens that some users get disappointed with the tracker, leave it for good, and take no part in the information exchange any longer. Furthermore, as shown in [4], some seeds lose their seeders and become unfit for downloading (die out). Since the users decide on leaving independently from each other, the seeds die out in a similar manner and the attrition of users and seeds at a first approximation is described by the radioactive decay model recorded as .
Thus we have the first approximation to the torrent tracker evolution model: where and characterize the user and the seed attrition rate, and are the inflow rates. The units of measuring and are . Then (1) is recorded as (2) According to Arnold's classification from [5] this is the rigid model, i.e., applicable within a fairly limited development phase.

Special Points of the Rigid Model
Now we shall find the special points of system (2). It is obvious that at we shall have from the stationarity of second equation (2); i.e., the origin of coordinates will always be a special point. We can say that it will always be the attracting node. At we have from the stationarity of the first equation. Then we have from the stationarity of the second equation. Special point is always a saddle.

Phase Portrait of the Hard Model
According to the general position philosophy [6], the separatrices of saddle must separate from one another the zones of sole attracting node and infinity. This conclusion is confirmed by the phase portrait of the hard system (Fig. 1). In other words, if the torrent tracker suceeds in attracting enough users and seeds on the back of postcreation enthusiasm by the time of reaching the phase described by model (2), its state will be above the separatrix and the number of users and seeds will infinitely rise; if the tracker does not suceed in this, their numbers will go to zero.
However, no infinite growth is possible in practice. Therefore, the rigid model must be refined and turned to soft.

SOFTENING TORRENT TRACKER EVOLUTION MODEL
The growth in the number of the tracker's users is always limited from above by certain number .
In this case, the growth rates are described by the logistic function .
In addition, as shown in [4], the probability of a seed's dying out depends on the its potential audience so that seed loss intensity cannot be constant and rises with a decrease in the number of users per seed.
Taking this into account, we shall record the soft version of system (2) as (3) where is the upper limit for the growing number of users, is the average number of users per seed, at which it is impossible to maintain all seeds active (catastrophic proportion); in this case, characterizes an increase in the dying-out rate. The second equation from model (3) is undefined at (along the axis). Let us complete its definition. At and with tending to zero from quadrants one and three (i.e., at ), the parenthesized value will tend to one. Since all of the physically implementable paths belong to quadrant one, we shall take that at the second equation from (3) turns to and at its right part is equal to zero. Then the system's behavior at all of the physically implementable values of and will be defined and uninterrupted (this continuity is violated with the approach to the axis and the origin of coordinates from the unimplementable second and fourth quadrant).

Special Points of the Soft Model
The number of special points in (3) will never be constant. At we have from the stationarity of the first equation in (3). According to the refined definition from the previous section, the second equation will also be stationary at . Thus we shall consider origin of coordinates the special point of model (3).
and (5) The point corresponds to the hard model's saddle, and corresponds to the tracker's growth limit and is the attracting generalized node.
At the saddle and the node merge into a special degenerated point called saddle node .
If they do exist, all of the nonzero special points of model (3) are located in the first quadrant of the reference frame.

Phase Portrait of the Soft Model
We shall consider the phase portrait of model (3) at various values of (Fig. 2). The saddle node case is not described in the general position philosophy and not shown in Fig. 2. Note that, with the approach to the origin of coordinates from quadrants one and three, origin of coordinates behaves as an attracting node. The paths diverge from the axis (i.e., from the state in which there are no seeds but there is a certain number of users, which is essentially real to implement) at slope .
In quadrants two and four the paths vertically enter the axis; the behavior with the drift of and to zero is undefined.
If the initial state of system (3) lies on the axis (when there are no users but only seeds, which is physically unimplementable), the path lies strictly on the axis and is directed to the origin of coordinates, to embodible state . The behavior of the paths of quadrants two, three, and four is qualitatively independent from the existence and position of special nonzero points. The paths of quadrants two and four end on the axis, whereas the path of quadrant three ends in .

At
, when there are no special nonzero points, all of the paths of quadrant one also end in .
At the separatrices of saddle split quadrant one of the reference frame in the attraction zones of and attracting general node .

Stability of the Soft Model
Unlike for the hard model, the stable existence of torrent-trackers for (3) is possible, but not at all parameter values.
Whatever the number of users and seeds in the tracker, it will get empty and closed over time at (as a rule, the administrators close the tracker without waiting for it to stabilize in zero state).
Though the stabilization in the saddle node is theoretically possible at , a most insignificant change in the state or parameters may result in entering the zone of the origin of coordinates, which means that the tracker will gradually lose of all of its users and be closed.
The stabilization in nonzero state is possible at and a sufficient initial number of users and seeds. It should be noted that if their number is insufficient and the initial state of the torrent-tracker on the way to model (3) is below the saddle's separatrix, stable state will not be reached though it is potentially possible. It is then supposed that the number of users and seeds collected in the initial phase of the tracker's evolution suffices for stabilization, and the stability of state is considered.
When is a little below one, i.e., system (3) has two adjacent special non-zero points, the tracker can stabilize in state ; moreover, this state is reached fairly quickly, but it is not too stable. Like any other system, a torrent tracker is always susceptible to random external influences, which is why it bears the risk of losing some users and leaping from stable state to the zone of the origin of coordinates