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Performance Analysis. On a binary tree topology network, there are no successful execution sequences of Protocol 1 for a scenario with more than two agents. For the scenario with four agents on the binary tree topology network, Protocol2and Protocol3

have the same standard extension sizes and average execution sequence lengths. Notice that the average execution sequence length of Protocol2for the scenario with four agents is the same for both the star topology network and the binary tree topology network (see Table 5.9 and Table 5.14).

Table 5.14: Protocol 2 on Binary Tree Topology Network.

Execution Sequence Length Three Agents Four Agents Five Agents

3 16 4 0 5 192 6 512 0 7 4,992 8 30,080 9 75,904 10 100,864

Standard Extension Size 16 704 211,840 Average Execution Sequence Length 3 5.72727 9.28701 Successful Sequences 16 704 211,840 % Successful Sequences 100.00% 100.00% 100.00%

From Table 5.14 and Table 5.15 we see that for a scenario with five agents, Protocol

3 exhibits a lower average execution sequence length than Protocol 2 on a binary tree topology network. From our experiments, Protocol 2 features lower average execution sequence lengths and lower standard extension sizes on the binary tree topology network than it does on the star topology network. On the contrary, Protocol 3 features lower average execution sequence lengths on the star topology network than it does on the bi- nary tree topology network. For a scenario with five agents, Protocol 3features a larger standard extension size on the star topology network than it does on the binary tree topology network, but the reverse is the case for a scenario with four agents. Therefore from our experiments, Protocol 2 is more adaptable on the binary tree topology net- work than on the star topology network with respect to both time efficiency and space

efficiency, and Protocol 3 is more adaptable on the star topology network than on the binary tree topology network with respect to time efficiency.

Table 5.15: Protocol 3 on Binary Tree Topology Network.

Execution Sequence Length Three Agents Four Agents Five Agents

3 16 4 0 5 192 6 512 0 7 4,992 8 20,480 9 51,456 10 48,896

Standard Extension Size 16 704 125,824 Average Execution Sequence Length 3 5.72727 9.14649 Successful Sequences 16 704 125,824 % Successful Sequences 100.00% 100.00% 100.00%

For the scalability property of the protocols on a binary tree topology network, we summarise γ for Protocol 2 and Protocol3 in Table 5.16.

Table 5.16: Protocol scalability on Binary Tree Topology Network. Percentage Increase Protocol2 Protocol3

γ3,4 90.9090% 90.9090%

γ4,5 62.1542% 59.7007%

γ5,6 48.8971% 50.4550%

Looking at Table 5.16, Protocol3outperforms Protocol2in terms of scalability. And overall, for both protocols we see the scalability property gets better with increasing sce- nario size. (Again, note that Protocol1is not successful on a star topology network). So, similar to the scalability results on complete, line and star topology networks, Protocol3

proves to be more scalable on a binary tree topology network than Protocol2. Also refer to Appendix B for the empirical results for the scenario with six agents on the binary tree topology network.

Table 5.17: Protocol 4 on Binary Tree Topology Network.

Execution Sequence Length Three Agents Four Agents Five Agents

3 16 4 0 5 192 6 768 0 7 7,424 8 67,072 9 278,016 10 807,424

Standard Extension Size 16 974 1,194,434 Average Execution Sequence Length 3 - - Successful Sequences 16 960 1,159,936 % Successful Sequences 100.00% 98.56% 97.11%

Protocol 4 and 5 remain non-terminating on a binary tree topology network, with both protocols featuring the same number of successful and unsuccessful execution se- quences, as well as the same standard extension size. An example of a non-terminating sequence in Protocol 4 on the binary tree topology network shown in Listing 5.9 is bd;be;bd;eb;db;eb;db;. . ., with the subsequenceeb;dbrepeating infinitely.

Extension Analysis. As on the star topology network, we do not obtain the short- est successful execution sequence length for n = 4,5, for Protocols 2-5 on the bi- nary tree topology network. The shortest successful execution sequence for a scenario with four agents is of length five (rather than 2n−4 = 4), and the shortest suc- cessful execution sequence for a five-agent scenario is of length seven. An example of a shortest successful execution sequence for the description given in Listing 5.9 is ac;ab;bd;be;ab;ac;bd. An example of a longest successful execution sequence (length of n(n−2)/2) is: ab;db;ca;ba;bd;ac;be;ab;db;ca. For the description given in Listing 5.9 the following execution sequence is in Protocol2but not in Protocol3: be;ca;db;eb;ab;ac;bd;be. After the second and third rounds, in third round, agenteknows that given the network topology, agent b must have made a call (either with agent d or with agent a) in the second round or in the third round - if agent b did not call with agent a in the second round it was because agent b was calling agent d, or because agent a was calling with agent c in the second round in which case agent b will call agent a or agent d in the third round. Moreover in the fourth round agent e does not know a particular secret that it would learn by calling agent b. Finally, the average execution sequence length of Protocol 2 is better on the binary tree topology network than on the star topology network.

Comparison with Literature. Similar to how the results for the star topology net- work, results shown in Tables 5.14 and 5.15 for minimum successful execution sequence lengths also agree with the theoretical results obtained for the minimum successful exe- cution sequence lengths in the non-epistemic traditional gossip literature.