# An Efficient Causal Group Communication Protocol for Free Scale Peer-to-Peer Networks
## 3 Results
### Simulations
For storage overheads, the slopes are: 0.143 in internal group, 2.952 in external group and 5.388 for IDR protocol. In our work in order to reduce the size of the variable \(TT\) in a super peer, we use the fact that a peer in the internal group can only receive a message that has been already received and delivered by a corresponding super peer. If \(m_{1}\) and \(m_{2}\) are messages that have an immediate dependency relationship \(m_{1}\downarrow m_{2}\), and all the peers in the internal group have received and delivered \(m_{2}\), then the information about \(m_{1}\) can be removed from a super peer. This can be seen from two different aspects: message reception and message sending. Message reception: a peer can deliver \(m_{2}\) that \(m_{1}\downarrow m_{2}\) if and only if it has already delivered \(m_{1}\). If a peer received a message \(m_{3}\) that \(m_{1}\downarrow m_{3}\) and has already delivered \(m_{2}\) (which implies delivery of \(m_{1}\)), then it can deliver \(m_{3}\) without any delay. This way the information about dependency on \(m_{1}\) does not affect the \(m_{3}\) delivery in any way. Message sending: if a peer delivered message \(m_{2}\), then no message originated from this peer can carry any dependency on \(m_{1}\) and the information about \(m_{1}\) in super peer is no longer required. Therefore, if all peers have delivered \(m_{2}\) such that \(m_{1}\downarrow m_{2}\), then the delivery of messages depending on \(m_{1}\) will not be affected in the internal group in any way, and a super peer will not receive any message depending on \(m_{1}\) from the internal group. Thus, the information about \(m_{1}\) can be deleted from variable \(TT\) in a super peer. By considering channel delays from 50 to 250 milliseconds (see Figure 5), the results also show that an overhead in an internal group is lower than an overhead in an external group, and the overhead in an external group is lower than an overhead produced by the Immediate Dependency Relation and the Dependency Sequences protocols. For 500 peers, each internal peer requires storing on average 300 bytes of information, and the average overhead in an internal group is around 230 bytes. In the external group, each peer stores on average 2100 bytes and produces an overhead on average of 1700 bytes. The IDR protocol requires storing on average of 3400 bytes on each peer and sending 2400 bytes, while the DS have a communication overhead of 2800 bytes. The results for delays from 50 to 250 milliseconds show that the overhead in an internal group is ten times lower than the overhead of IDR and requires storing eleven times less information. The overhead in an external group is about 1.4 times lower than the overhead of IDR and 1.64 times lower than the one of the DS, and requires storing 1.6 times less information. For communication overheads, the slopes are: 0.461 for internal group, 3.480 for external group, 4.965 for IDR and 5.561 for DS protocol. For storage overheads, the slopes are: 0.603 in internal group, 4.225 in external group and 6.838 for IDR protocol. Figure 5: Communication (**a**) and storage (**b**) overheads in bytes for a system with delays from 50 to 250 milliseconds. In scenario with delays from 50 to 550 milliseconds (see Figure 6) and for 400 peers, each internal peer requires storing on average 550 bytes of information and the average overhead in an internal group is around 400 bytes. In the external group, each peer stores on average 1860 bytes and produces an overhead on average of 1400 bytes. The IDR protocol requires storing on average 2500 bytes on each peer and sending 1500 bytes while the DS requires sending 2850 bytes. These results show that the overhead in an internal group is 3.5 times lower than the overhead of the IDR and requires storing 4.5 times less information. The overhead in an external group is similar to the overhead of the IDR, but requires storing 1.3 times less information. As for the DS protocol, the proposed solution has two times lower communication overhead.