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Performance Analysis. Protocol3features lower average execution sequence lengths and much lower standard extension sizes on a star topology network than Protocol 2,

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A neighbour must be assigned to each agent in the scenario, where each agent is represented by a unique node of the network graph.

which indicates a greater adaptability of Protocol 3 over Protocol 2 with respect to a star topology network.

Protocol 4and 5 remain non-terminating on a star topology network. Consider the following infinite execution sequence of Protocol5taken from a scenario with the network topology description shown in Listing 5.8: ab;ac;ba;ca;ba;ca;. . .. That is, in the second round, agentclearns the secret of agentaand agent bfrom agent a. Then, in the fourth round, agent c again considers it possible that it will learn the secret of agent d from agent a(since agent amay have called with agentdin the third round), so agentccalls agent ain what proves to be a redundant fourth round call. In the fifth round, agent b reasons as agentcdid in the fourth round, and likewise makes a redundant call to agent ain the fifth round. And so does agentc again in the sixth round, and the loop goes on infinitely in this sequence. Note that the given execution sequence is also an example of an infinite execution sequence for Protocol4(for example, if agentcconsiders it possible that it will learn the secret of agent dfrom a call with agenta, then it also considers it possible that it will learn some secret from the same call with agenta).

Looking at Table 5.11, Protocol3outperforms Protocol2in terms of scalability. And overall, for both protocols we see the scalability property gets better with increasing scenario size. (Note that Protocol 1 is not successful on a star topology network: the reason is analogous to that given in the preceding section, under theExtension Analysis

for line topology network). Also refer to Appendix B for the empirical results for the scenario with six agents on a star topology network.

Table 5.9: Protocol 2 on Star Topology Network.

Execution Sequence Length Three Agents Four Agents Five Agents

3 16 4 0 5 288 6 768 0 7 12,288 8 61,440 9 196,608 10 294,912

Standard Extension Size 16 1,056 565,248 Average Execution Sequence Length 3 5.72727 9.36957 Successful Sequences 16 1,056 565,248 % Successful Sequences 100.00% 100.00% 100.00%

Table 5.10: Protocol 3 on Star Topology Network.

Execution Sequence Length Three Agents Four Agents Five Agents

3 16 4 0 5 288 6 384 0 7 6,144 8 29,184 9 58,368 10 36,864

Standard Extension Size 16 672 130,560 Average Execution Sequence Length 3 5.57143 8.96471 Successful Sequences 16 672 130,560 % Successful Sequences 100.00% 100.00% 100.00%

For the scalability property of the protocols on a star topology network, we summarise γ for Protocol2 and Protocol3 in Table 5.11.

Table 5.11: Protocol scalability on Star Topology Network. Percentage Increase Protocol2 Protocol3

γ3,4 90.9090% 85.7143%

γ4,5 63.5957% 60.9050%

γ5,6 50.0584% 47.6852%

Extension Analysis. Similar to the line topology network for Protocol 1, there are no successful execution sequences on a star topology network for a scenario with more than two agents. And as on a line topology network this is easy to see: consider a gossip scenario with at least three agents a, b, c, . . . Assume the network topology shown in Listing 5.8 for these agents. So now agent a is in between agent b and agent c in the network, such that, although the epistemic calling condition of Protocol1will be satisfied for agent b to call agent c (or for agent c to call agent b), but since there is no direct link between agent b and c, the only way for both agents to know each other’s secrets is through the subsequence ab;. . .;ac;. . .;ba or ac;. . .;ab;. . .;ca (or their reverse call variants). But such subsequences do not comply with Protocol 1 (since no two agents agents can call each other more than once in an execution sequence of Protocol 1). So in Protocol 1 on a line topology network, one of agentsb and c will never know all the secrets in the scenario. (See also Section 6.4).

For our experiments on the star topology network, both Protocols 4 and 5 feature the same number of successful execution sequences forn= 3,4,5.

Table 5.12: Protocol 4 on Star Topology Network.

Execution Sequence Length Three Agents Four Agents Five Agents

3 16 4 0 5 384 6 768 0 7 18,432 8 110,592 9 405,504 10 1,142,784

Standard Extension Size 16 1,188 1,718,388 Average Execution Sequence Length 3 - - Successful Sequences 16 1,152 1,677,312 % Successful Sequences 100.00% 96.97% 97.61%

Table 5.13: Protocol 5 on Star Topology Network.

Execution Sequence Length Three Agents Four Agents Five Agents

3 16 4 0 5 384 6 768 0 7 18,432 8 110,592 9 405,504 10 1,142,784

Standard Extension Size 16 1,188 1,718,388 Average Execution Sequence Length 3 - - Successful Sequences 16 1,152 1,677,312 % Successful Sequences 100.00% 96.97% 97.61%

On a star topology network, we compare some sample execution sequences. An example of a successful execution sequence in the standard extension of Protocol2 but which is not in the standard extension of Protocol3 isab;ac;ba;ad;ab;ac;ae;ab;ac;ad.

The reason is that after the first two calls in the given sequence, in the third round, agent bknows that agentamust have made a call with some other agent and learnt some new secret (since agent a is the only neighbour that each agent has, agent a is involved in every call in this scenario). So although agent b knows that it would learn some new secret by calling agentain the third round (de dicto), it does not know which new secret it would learn by calling a in the third round (de re). The following is an example of an infinite execution sequence in the extension of Protocol5: ab;ac;ab;ca;ba;ca;ba;. . .. Such sequence shows how an agent at the centre of the star network (in this case agent a) can get into a loop of redundant calls with specific agents (in this case agentb and c) because they always would consider it possible that they will learn some new secret from agenta, whereas agentakeeps alternating between calling both of these agents.

Comparison with Literature. For Protocols 2-5, we do not obtain the shortest successful execution sequence length of 2n −4 for n = 4,5. That is, for n = 4, there is no successful execution sequence of length 4; and for n = 5 there is no suc- cessful execution sequence of length 6. For the protocol in Listing 5.8 (that is, Pro- tocol 2, for five agents, on a star topology network), the shortest successful execu- tion sequence is of length seven (see Table 5.9). An example of such a sequence is ac;ab;ad;ae;ad;ab;ac. An example of a longest successful execution sequence for the same description is ab;ad;ab;ae;ad;ab;ac;ad;ae;ab. See also Table 5.10 for minimum successful execution sequence lengths for Protocol 3. These results for minimum suc- cessful execution sequence lengths all agree with the theoretical results obtained for the minimum successful execution sequence lengths in the non-epistemic traditional gossip literature: Harary and Schwenk [28] showed that if the underlying communication graph is a tree topology network then minimum length of a successful execution sequence is given by2n−3, forn≥2; Bumby [13] and Kleitman [36] confirmed this result by proving a stronger proposition - they showed that for any connected but incomplete (undirected) communication graph without a four-cycle, this minimum is2n−3.