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In Tables 5.1-5.5 we present the protocol characteristics obtained from the EGP tool for Protocols 1 through 5, respectively. The tables show the number of lengthx sequences for 3,4,5 agents, where 3 ≤ x ≤ 10. We also show the standard extension size and average execution sequence length for the protocols.

Performance Analysis. Our empirical results for scenarios with 3, 4 and 5 agents show that Protocols 1, 2 and 3 are all successful after n(n−1)/2 rounds. From the numerical results for average execution sequence length and standard extension size, respectively, we observe that Protocol 1 is more time and space efficient than Protocol

2 and Protocol3. We also note that Protocol3 proves significantly more space efficient and slightly more time efficient than Protocol2.

As shown in the proof of Proposition 3.29, Protocols 4 and 5 are non-terminating because there is the possibility of infinitely looping execution sequences in these protocols. But we see that for the scenario comprising of three agents, Protocol 4 and Protocol 5

terminate and are 100% successful. For the scenario comprising of four agents we have up to 94.48% successful sequences, although the resulting standard extension size for each of both protocols in the four-agent scenario is about five times the standard extension size of the corresponding scenario size for Protocol3, about four times the standard extension size of the corresponding scenario size for Protocol2and about sixty-three times that of the corresponding scenario size for Protocol1. Note that in accordance with the theory of Chapter 3, every successful sequence in Protocol4is also in Protocol5, and vice versa (the number of successful sequences for the three- and four-agent scenarios of Protocols

4 and5 are the same).

To measure the scalability of the protocols on a complete topology network we con- sider the percentage increase in the average execution sequence length of the protocols as a result of increasing the size of the gossip scenario by one (that is, adding one more agent to the scenario, and correspondingly, one more secret). Let the average execution sequence length of a protocol for a scenario consisting ofn agents be αn, then the per- centage increase γ of the average execution sequence length as a result of introducing one more agent is given by:

γn,n+1 =

αn+1−αn αn

Table 5.2: Protocol2 on Complete Topology Network.

Execution Sequence Length Three Agents Four Agents Five Agents

3 96 4 384 5 15,744 6 64,896 195,840 7 7,958,400 8 61,155,840 9 220,404,480 10 472,988,160

Standard Extension Size 96 81,024 762,702,720 Average Execution Sequence Length 3 5.79621 9.51833 Successful Sequences 96 81,024 762,702,720 % Successful Sequences 100.00% 100.00% 100.00%

Table 5.3: Protocol3 on Complete Topology Network.

Execution Sequence Length Three Agents Four Agents Five Agents

3 96 4 384 5 13,824 6 53,952 149,760 7 5,798,400 8 37,975,680 9 172,362,240 10 325,891,200

Standard Extension Size 96 68,160 542,177,280 Average Execution Sequence Length 3 5.78592 9.50882 Successful Sequences 96 68,160 542,177,280 % Successful Sequences 100.00% 100.00% 100.00%

Table 5.4: Protocol4 on Complete Topology Network.

Execution Sequence Length Three Agents Four Agents Five Agents

3 96 - 4 384 - 5 31,488 - 6 297,984 - 7 - 8 - 9 - 10 -

Standard Extension Size 96 349,134 - Average Execution Sequence Length 3 - - Successful Sequences 96 329,856 - % Successful Sequences 100.00% 94.48% -

Table 5.5: Protocol5 on Complete Topology Network.

Execution Sequence Length Three Agents Four Agents Five Agents

3 96 - 4 384 - 5 31,488 - 6 297,984 - 7 - 8 - 9 - 10 -

Standard Extension Size 96 349,134 - Average Execution Sequence Length 3 - - Successful Sequences 96 329,856 - % Successful Sequences 100.00% 94.48% -

We summarise γ for Protocols 1,2and 3in Table 5.6.

Table 5.6: Protocol scalability on Complete Topology Network. Percentage Increase Protocol1 Protocol2 Protocol3

γ3,4 80.4597% 93.2070% 92.8640%

γ4,5 60.5834% 64.2164% 64.3441%

Looking at Table 5.6, we observe that Protocol 1 is the most scalable of the three protocols. However we observe that Protocol 2 shows a greater growth than Protocol3

when increasing the size of the scenario from three to four agents, but the reverse is the case when increasing the size of the scenario from four to five agents. So it is not yet clear which is more scalable between Protocol 2 and3. We will also carry out a similar analysis for line, star and binary tree topology networks (we will skip this analysis for the circle topology network because our empirical results show that on a circle topology network most of the protocols are unsuccessful in scenarios with more than three agents). Notice also that we skipped the scalability analysis for Protocols 4 and 5 because they are both non-terminating.

Extension Analysis. Let us consider some extension analysis for a scenario that con- sists of four agents a, b, c, d. Example execution sequences of Protocol 1 are ab;ac;de; ad; bd;ce, ab;ac;de;ad;ce;bc and ab;ac;de;ad;ce;bd. The execution sequence ab;ac;bd; ad;ab;bc is an execution sequence of Protocol 2 but not an execution sequence of Pro- tocol 3. Not all execution sequences of Protocol 4 and Protocol 5 are successful. Some examples of unsuccessful execution sequences of Protocol 5 are: ab;cd;ab;bc;ab;cd;ab, ad;bc;ad;bc;ad;bc;ad and bd;ac;bd;ac;bd;ac;bd. Examples of successful execution se- quences of Protocol 5 are: cd;ab;bd;ac,ab;ac;bd;cd;ab andbc;cd;bc;ab;bd;bc.

Not every execution sequence of Protocol 2 is an execution sequence of Protocol 1, and not every execution sequence of Protocol 3 is an execution sequence of Protocol 1. For example, the following execution sequences are execution sequences of Protocol 2 but they are not execution sequences of Protocol 1: ab;ac;ad;ac;ab, bd;ad;bd;ac;ab;cd and cd;bd;bc;ab;ac;bd. Not every successful execution sequence of Protocol 5 is an execution sequence of Protocol 3, for example: ac;ab;ad;ac;bc, bd;cd;ad;bd;bc and ab;ac;bd;ad;ac;bd. Also, not every successful execution sequence of Protocol 4 is an execution sequence of Protocol 2, examples are: ad;bd;ac;ad;bc, bc;ab;ad;cd;bc and cd;bd;ac;cd;ac;bc. Examples of execution sequences of Protocol 2 that are not execu- tion sequences of Protocol 3 are: ab;ac;bd;ad;ab;bc, bd;cd;bc;ab;ad;acandcd;ad;ab;cd; ac;bd.

Comparison with Literature. From Tables 5.1-5.5 we see immediately that the min- imum successful execution sequence lengths agree with the theoretical result obtained for the minimum successful execution sequence length in the non-epistemic traditional gossip literature for the complete topology network [13, 27, 61, 63]. Furthermore, recall that for a random protocol (that is, wherein a pairwise call is staged at random in each round),

Protocol 1 Protocol 2, Protocol 3 0 5 10 15 20 25 30 3 4 5 A ve ra ge E xe cut ion S eque nc e L engt h Number of Agents Random Protocol Protocol 1 Protocol 2

Figure 5.1: Comparing Protocols 1, 2 and 3 to a random protocol.

Boyd and Steele [11] showed that the average number A(n) of pairwise communicative interactions needed for successful gossiping among nagents is:

A(n) = 3

2n ln n+O(n(ln n)

0.5_{) (5.6)}

The graph shown in Figure 5.1 compares the average execution sequence lengths obtained for Protocols1−3with those obtained from Equation 5.6 for a random protocol. Clearly the epistemic gossip protocols (Protocols1,2 and 3) outperform the random protocol.