All Peer nodes and the corresponding social links in P2P-iSN form a social graph. A Peer node may be turned on or turned off during the execution of i-Search, and the iSearch re-quest can reach the friends only when the friends are online. In other words, a social link a→ b does not exist if Peer node a or b is turned off (i.e., off-line). Therefore, the
phys-ical network topology of P2P-iSN changes dynamphys-ically when the i-Search mechanism is being executed.
Let Pf be the “path found” probability that a directional social path exists when a Peer node a executes the i-Search mechanism to find a Peer node d. The online status of a Peer node affects the Pf probability significantly. In this chapter, we propose an analytical model to obtain an approximation value for Pf.
To simplify our discussion, we assume that the behaviors of the Peer nodes in P2P-iSN are i.i.d. As discussed in Chapter 3, in this paper, we set ∆ = 0.53 = 0.125 in the i-Search mechanism. In this analytical model, we use the constraint |P| ≤ 3 instead of
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20 CHAPTER 4. ANALYTICAL MODEL
∆ ≤ 0.125, i.e., the i-Search mechanism quits when the path length reaches 3, and no global social path is found.
Assume that a Peer node is turned on (i.e., online) for a time period x (with the density function fx(·) and mean 1/µx), and then it is turned off (i.e., off-line) for a time period y (with the density function fy(·) and mean 1/µy). The Peer node alters between x and y. Suppose that iSearch request arrivals to a Peer node form a Poisson process. Then
ac-cording to the alternating renewal process [12], the probability ponthat an iSearch request arrives when a Peer node is online can be obtained by
pon = E[x]
E[x] + E[y] = µy
µx+ µy (4.1)
Before the derivation, we generate the social graph for P2P-iSN using the W.S. model [14]
with the three parameters α (i.e., the rewire probability), n (i.e., the total number of Peer nodes in P2P-iSN), and m (i.e., the average number of friends of a Peer node). With the setup:
0 < α < 1 and n≫ m ≫ ln n ≫ 1 (4.2)
the W.S. model has the small-world property, including short average length and high clustering. The small-world property can also apply to SNS [11]. The details of the W.S.
model can be found at [14].
Let Ntdenote the expected number of the Peer nodes that receive the iSearch request message during the execution of the i-Search mechanism. Consider the scenario that the Peer node a executes the i-Search mechanism to search a directional social path to d. If d belongs to one of the Ntpeer nodes, then the directional social path from a to d is found.
21 Therefore, we have
Pf = Nt
n . (4.3)
We derive Nt as follows. There are two types of nodes including “far-nodes” and
“near-nodes” defined in the W.S. model. The far-nodes represents the Peer nodes that have social links after rewiring with probability α. The near-nodes represents the Peer nodes that have social links initially.
In the social graph of the P2P-iSN, let Nf and Nnrespectively be the expected num-bers of far-nodes and near-nodes that receive an iSearch request when the i-Search mech-anism is executed. Then we have
Nt = Nf + Nn.
The Nf and Nn are obtained as follows. One round means that the iSearch request is delivered using a directional social link a→ b when both Peer nodes a and b are online.
In the i-Search mechanism, there are at most three rounds to construct a directional social path. In each round, a Peer node that triggers the round can be either a far-node or near-node:
Case 1: The Peer node that triggers the round is a far-node. In this case, there are on average mαpon far-nodes and m(1− α)ponnear-nodes that can receive the iSearch request.
Case 2: The Peer node that triggers the round is a near-node. Because there is high probability that the near-node sends the iSearch request to another near-node that
22 CHAPTER 4. ANALYTICAL MODEL has received this iSearch request previously, we consider that only far-nodes can receive the iSearch request for the approximation. In this case, there are on average mαponfar-nodes that can receive the iSearch request.
We use the following interative procedure to calculate the Nf and Nn.
Procedure 1.
The analytical model is validated by simulation experiments of a discrete event-driven simulation model, which has been widely adopted to simulate the mobile communications network in several studies (e.g., [4]). The simulation model simulates the online/off-line behavior of a Peer node and the behavior of the i-Search mechanism.
In the simulation model, we adopt the discrete event-driven approach in our simulation model, which has been widely applied in many networking studies (e.g., [4]). In our simulation model, we define five types of events listed below:
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• The QUERY ARRIVAL event represents that an online Peer node starts the i-Search mechanism to find another Peer node.
• The QUERY FORWARD event represents that an online Peer node sends a iSearch request to his online friend.
• The QUERY RESPONSE event represents that an online Peer node returns the
results (i.e., a path is found) for the execution of the iSearch algorithm to the Peer node who sends the iSearch request.
• The ONLINE event represents that a Peer node is turned on.
• The OFFLINE event represents that a Peer node is turned off.
We maintain a timestamp tsto indicate the time when an event occurs. The events are in-serted into an event list and deleted/processed from the list in a non-decreasing timestamp order. During execution of the simulation, a simulation clock tc is maintained, which indicates the progress of simulation. The following variables are used in the simulation model:
• Nrindicates the number of rounds that have been executed for an iSearch request.
• a is the ID of the Peer node who triggers the iSearch mechanism.
• d is the ID of the Peer node to be found.
• l indicates whither a social link exists between two Peer nodes.
We use the following counters in our simulation model to calculate the output measure:
24 CHAPTER 4. ANALYTICAL MODEL
• The Cf counter counts the total number of finding a path successfully.
• The Cqcounter counts the total number of the QUERY ARRIVAL events that have been processed.
We repeat the simulation runs until Cq exceeds 100, 000 to ensure the stability of the simulation results. Then we obtain the output measure:
Pf = Cf Cq
Figures 5.1 and 5.2 show the comparison between the analytical and simulation re-sults, whose details of the parameter setups are described in chapter 5. The figures indi-cates that the analysis results approximate the simulation results well.