# An Efficient Causal Group Communication Protocol for Free Scale Peer-to-Peer Networks
## 3 Results
### Simulations
For communication overheads, the slopes are: 0.990 for internal group, 3.617 for external group, 3.830 for IDR and 7.162 for DS protocol. For storage overheads, the slopes are: 1.325 in internal group, 4.680 in external group and 6.378 for IDR protocol. Scalability Analysis  
To analyze the scalability of the system, we evaluate the control information management of the protocol against the message density, which is defined as the number of messages sent per process per time unit. For this last analysis, we conducted the following experiment. The system model from the previous experiment was modified in the following way: each peer, instead of generating the message every 80 milliseconds was modified to generate messages so that the total number of messages remain fixed for each number of peers in the system. For example, in a system with a generation rate of 1000 messages per second, with 50 peers, each peer generates 20 messages per second; in a system with 100 peers, each peer generates 10 messages per second; and in a system with 500 peers, each peer generates two messages per second. The system is analyzed for the following message generation rates: 2500, 2000, 1500 and 1000 messages per second. The obtained results are compared with the results from the experiment described above. In the experiment above, the number of messages generated in the system is 12.5 per second per peer. By considering channel delays from 0 to 50 milliseconds (see Figure 7), the simulation results show that more messages are generated in the system per second; more control information is required to ensure the causal ordering of messages. In addition, the overhead growth has two different stages: a slow growth for a low number of peers (less than 100), and a linear growth part when the number of peers is high (more that 100-200 in this experiment). Figure 6: Communication (**a**) and storage (**b**) overheads in bytes for a system with delays from 50 to 550 milliseconds. Analyzing the growth of the overhead (linear growth part) in the internal group, the slopes are 0.0612 for 2500 messages/second, 0.0611 for 2000 messages per second, 0.0603 for 1500 messages/second and 0.0583 for 1000 messages/second. In the external group, the slopes are 0.0685 for 2500 messages/second, 0.0728 for 2000 messages/second, 0.0735 for 1500 messages/second and 0.0719 for 1000 messages/second. In the original experiment, the slopes are 0.0979 for the internal group and 1.0177 for the external group. In this scenario, the difference between slopes with the message limit is less than five percent. However, the difference with the original experiment is 1.60 times for the internal group and 13.9 times for the external group. By considering channel delays from 50 to 250 milliseconds (see Figure 8), the results also show that more messages are generated in the system per second; more overhead is required to ensure the causal ordering. The overhead has the same growth stages as for the system with communication channel delays from 0 to 50 milliseconds. Analyzing the growth of the overhead (linear growth part) in the internal group, the slopes are 0.0083 for 2500 messages/second, 0.0041 for 2000 messages per second, 0.0045 for 1500 messages/second and 0.0241 for 1000 messages/second. In the external group, the slopes are 0.4632 for 2500 messages/second, 0.1215 for 2000 messages/second, 0.0147 for 1500 messages/second  
Figure 8: Internal (**a**) and external (**b**) group overheads in bytes for a system with delays from 50 to 250 milliseconds. Figure 7: Internal (**a**) and external (**b**) group overheads in bytes for a system with delays from 0 to 50 milliseconds. and \(-0.0374\) for 1000 messages/second. In the original experiment, the slopes are \(0.4606\) for the internal group and \(3.4804\) for the external group. In this scenario in both groups the slopes have a tendency toward zero, which means that the overhead directly depends on the density of the messages exchanged, and does not depend on the number of peers in the system.