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Let us consider the Copy Centric Downloading case in a bit more detail.
Suppose the attackers inject N polluted copies of the file in the network. Fur-
thermore, suppose that there are M (assumed to be a constant throughout)
benign peers in the network and each peer's time spent to inspect a down-
loaded file (to see if it is a good copy) is exponentially distributed with rate
m. Now, we can use x and y to denote the number of benign peers with good
and corrupted copies, respectively. Using the tuple (x, y) as a system state,
Figure 7.1 depicts the state transitions of the system. Specifically, starting at
state (x, y), the system can change to one of the following states:
•(x + 1, y): a peer with no copy gets a good copy;
•(x, y + 1): a peer with no copy downloads a corrupted copy;
•(x + 1, y−1): a peer with a corrupted copy gets a good copy; and
•(x, y): a peer with a corrupted copy downloads a polluted copy again.
x+1, y
x+1, y-1
x, y
x, y+1
FIGURE 7.1: System state transitions in the Copy Centric Downloading
model [Kumar et al., 2006].
Unfortunately, this Markov process is intractable to solve because M is
typically a very large number (e.g., 100,000 or more). Thus, Kumar et al.
[Kumar et al., 2006] resorted to tackling the state transition modeling from
an individual peer's perspective. Specifically, at time t, the probability that a
peer chooses a corrupted file to download is given by:
y(t) + N
x(t) + y(t) + N
p(t) =
(7.4)
A useful insight is that the number of good copies, i.e., x(t), increases when a
peer with no copy gets a good copy. This event occurs at the following rate:
[M−x(t)−y(t)]m(1−p(t))
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