Mitigation of Routing Congestion on Data Networks: A Quantum Game Theory Approach
Round 1
Reviewer 1 Report
Rewrite the abstract that what is the purpose of doing this work.
Highlight the contributions of the proposed work in introduction.
How this work is different from existing one?
What is the novelty?
Don't write conclusion in points.
Recent references of 2021 & 2022 can be added of relevant journal.
Author Response
Response to Reviewer 1 Comments:
First of all, we want to thank reviewer 1 for the constructive review. We believe that our revised manuscript has incorporated all the comments/requirements improving the quality of our article. As such, we sincerely hope that the revised manuscript can now be accepted for publication. Thanks.
1. Rewrite the abstract that what is the purpose of doing this work.
The abstract was rewritten focusing more on the objective of our article both in the present and in the future. Added or changed sentences:
(line 4) “In order to contribute to solve the congestion problem on communication networks, a novel framework based on a quantum game model is proposed, where network packets compete selfishly for their fastest route.”
(line 12) “Due to its generality, this game approach can be applied both in classical complex networks and in future quantum networks in order to maximize the performance of the quantum internet.”
2. Highlight the contributions of the proposed work in introduction.
We add several paragraphs in the introduction in order to all this suggestions:
(line 59) “In this work, we start by modeling a network that allows both classical and quantum packets and then propose a routing protocol designed using the Game Theory formalism. After that, different variables with information about the dynamics of the network were measured and we point out that the protocol that makes use of the quantum game theory rules [13] significantly outperforms its classical equivalent.”
3. How this work is different from existing one?
(line 67) “There have already been efforts to try to mitigate congestion in networks using quantum technologies (mainly coming from the automotive industry [14–16]) but they focus on a centralized optimization approach using quantum annealing.”
4. What is the novelty?
(line 69) “. In this work, we propose a decentralized self-organization approach using gate-based quantum computers.”
5. Don't write conclusion in points.
(line 292) The outlines were rewritten without the bulleted points.
6. Recent references of 2021 & 2022 can be added of relevant journal.
We add some references, these are the newest references:
- Szopa, M. Efficiency of classical and quantum games equilibria. Entropy 2021, 23, 506. (line 48)
- Bassoli, R.; Boche, H.; Deppe, C.; Ferrara, R.; Fitzek, F.H.; Janssen, G.; Saeedinaeeni, S. Quantum Communication Networks; Vol. 23, Springer, 2021. (line 58)
- Ikeda, K.; Aoki, S. Infinitely repeated quantum games and strategic efficiency. Quantum Information Processing 2021, 20, 1–24. (line 258)
Reviewer 2 Report
The data presented in the manuscript give valuable contribution to the further progress in communication networks and mitigation of the routing congestion problem.
I cannot agree with the terms “quantum game theory”, “quantum strategy”.
In fact, authors consider “classical” computation of an optimal strategy (to reduce sum of the traveling time and routing time) with setting a fixed probability p (of choice of the other, non-shortest, channel), on the one hand, and finding of the best strategy among the all-possible strategies (when the probability p is not fixed), on the other hand. The latter can be performed with the most efficiency by means of quantum computation. I recommend authors to give this meaning of the used terms “quantum game theory”, “quantum strategy”.
Author Response
Response to Reviewer 2 Comments
First of all, we want to thank reviewer 2 for the constructive review. We believe that our revised manuscript has incorporated all the comments/requirements improving the quality of our article. As such, we sincerely hope that the revised manuscript can now be accepted for publication. Thanks.
1. The data presented in the manuscript give valuable contribution to the further progress in communication networks and mitigation of the routing congestion problem.
Thank you very much.
2. I cannot agree with the terms “quantum game theory”, “quantum strategy”. In fact, authors consider “classical” computation of an optimal strategy (to reduce sum of the traveling time and routing time) with setting a fixed probability p (of choice of the other, non-shortest, channel), on the one hand, and finding of the best strategy among the all-possible strategies (when the probability p is not fixed), on the other hand. The latter can be performed with the most efficiency by means of quantum computation. I recommend authors to give this meaning of the used terms “quantum game theory”, “quantum strategy”.
We re-read our article and found that we may have abused the term "quantum game theory" a bit. Therefore, we rewrote or deleted the parts where this happened. On the other hand, the term "quantum strategy" is based on the concept proposed by J Eisert, M Wilkens and M Lewenstein in their 1999 paper "Quantum games and quantum strategies". However, in order to provide clarity we re-wrote the sections "4.1.2. Pure Quantum Strategies" and, mainly, "4.1.3. Mixed Quantum Strategies" where we clarify that in both the classical and quantum case simulations, all players apply the same strategies, p is always fixed for all players.
Reviewer 3 Report
In the manuscript entitled "Mitigation of Routing Congestion on Data Networks: a Quantum Game Theory Approach", the authors Silva et al. study a routing problem using Game theory approach. They compare the performance of a classical strategy and a quantum strategy. As quantified by the total transmission time, the quantum strategy is advantageous over the classical one. In addition, the trade off between routing time and traveling time in the classical game theory does not apply to the quantum case. Noise in the real-world quantum computers is also taken into account.
I find the topic of applying quantum game theory in real work application important. The result that trade off is absent in quantum case is interesting. Unfortunately, some fundamental part of this work is not explained well, which will cause difficulties for readers outside this specific field.
I suggest the authors revise the manuscript, by clearly defining each term, preferably with examples. Here are some points that I find confusing:
1. What is routing time?
This is an important quantity centered in this work. However, it is not properly defined. The authors stated "The routing time measures how long it takes for a packet to find a path for going from origin to destination." This description gives a intuitive understanding, which is good and necessary. But a more mathematical and rigorous definition is also needed.
2. What are the possible actions?
The authors try to explain this by saying "The two possible strategies are: choosing the preferred shortest channel and the search for another one, and will be called options 1 and 0 respectively." I find this sentence very confusing. What do you mean by the preferred shortest channel? What happens next if the player choose to search for another one? How is this related to the example the authors give -- "A value of p closer to one will create players more patiently looking for another clearer route". In particular, I do not see the connection between choosing to search for another route and the route being clear.
I suggest the authors to explain these two questions with a concrete example: given a simple graph and two players, explain what happens if they choose 00, 01, 10, and 11. Then explain which part is considered as routing time and which part is considered as traveling time.
Beside the major comment, I also suggest the authors to give the reason for choosing this particular classical strategy. Given that you want to show the advantage of quantum strategy over the classical one, it is important that you choose a reasonably good classical strategy for comparison. The classical strategy used in this work is very simple, so readers may doubt whether the quantum advantage is merely due to the fact that you are using an inefficient classical strategy. I understand it is impractical to find the best classical strategy, but brief reasoning or references showing the performance of this classical strategy is needed.
Typo: The caption of Figure 1 says N=10. Should be n1=10.
In summary, the topic of this work is interesting, but some basic setting of this routing game is not clear to me. I cannot recommend publication until the authors address the points above.
Author Response
Response to Reviewer 3 Comments
First of all, we want to thank reviewer 3 for the constructive review. We believe that our revised manuscript has incorporated all the comments/requirements improving the quality of our article. As such, we sincerely hope that the revised manuscript can now be accepted for publication. Thanks.
1. In the manuscript entitled "Mitigation of Routing Congestion on Data Networks: a Quantum Game Theory Approach", the authors Silva et al. study a routing problem using Game theory approach. They compare the performance of a classical strategy and a quantum strategy. As quantified by the total transmission time, the quantum strategy is advantageous over the classical one. In addition, the trade off between routing time and traveling time in the classical game theory does not apply to the quantum case. Noise in the real-world quantum computers is also taken into account. I find the topic of applying quantum game theory in real work application important. The result that trade off is absent in quantum case is interesting. Unfortunately, some fundamental part of this work is not explained well, which will cause difficulties for readers outside this specific field. I suggest the authors revise the manuscript, by clearly defining each term, preferably with examples. Here are some points that I find confusing:
Thank you very much for your comments.
2. What is routing time? This is an important quantity centered in this work. However, it is not properly defined. The authors stated "The routing time measures how long it takes for a packet to find a path for going from origin to destination." This description gives a intuitive understanding, which is good and necessary. But a more mathematical and rigorous definition is also needed
We dig a little deeper into our definitions of routing and traveling time and add the mathematical equation we use to measure both quantities. Added or changed sentences:
(line 84) “More precisely, in our model, the routing time is a quantity proportional to the number of games a packet must play before finding its final path. The greater the number of possible paths that a packet considers, the longer the routing time.”
(line 89) “This means, the sum of the weights of all the edges that a final path has.”
(line 90) “Therefore, TotalTimei = KG Gi + KE ∑Eie=1 w(e, e + 1), where Gi is the number of games played by the player i, Ei is the number of edges of the final path of player i, w(e, e + 1) is the weight of the edge connecting the nodes e and e+1 and KG and KE two model-dependent design constants.”
3. What are the possible actions? The authors try to explain this by saying "The two possible strategies are: choosing the preferred shortest channel and the search for another one, and will be called options 1 and 0 respectively." I find this sentence very confusing. What do you mean by the preferred shortest channel? What happens next if the player choose to search for another one? How is this related to the example the authors give -- "A value of p closer to one will create players more patiently looking for another clearer route". In particular, I do not see the connection between choosing to search for another route and the route being clear. I suggest the authors to explain these two questions with a concrete example: given a simple graph and two players, explain what happens if they choose 00, 01, 10, and 11. Then explain which part is considered as routing time and which part is considered as traveling time.
Our explanation regarding the two possible strategies was rewritten giving more detail to each. And we also added a table with the situation proposed by the reviewer, this was a very good suggestion and we think this adds a lot of understanding to the article. Added or changed sentences:
(line 132) “If more than one packet is interested in a channel, because it is part of their currently shortest path, each packet has two possible strategies: choose this preferred channel (risking that other packets will also select it and then congest the channel) or search for another (possibly longer but idler).”
(line 142) “In table 1, an example of a situation where two packets are interested in the same channel, is presented.”
4. Beside the major comment, I also suggest the authors to give the reason for choosing this particular classical strategy. Given that you want to show the advantage of quantum strategy over the classical one, it is important that you choose a reasonably good classical strategy for comparison. The classical strategy used in this work is very simple, so readers may doubt whether the quantum advantage is merely due to the fact that you are using an inefficient classical strategy. I understand it is impractical to find the best classical strategy, but brief reasoning or references showing the performance of this classical strategy is needed.
We clarify that the strategies we analyzed are exactly the same for both (classical and quantum), the reason why this particular set of strategies was selected was only for the sake of simplicity. We wanted both to have the same set of strategies so we could effectively quantize the classical algorithms and create a clear comparison. Added or changed sentences:
(line 6) “Simulations show that final network routing and traveling times achieved with the quantum version outperform those obtained with a classical game model with the same options for packet transmission for both.”
(line 145) “The first step is to assign a quantum state to each of the possible strategies. The quantum protocol is exactly the same as the classical one, the only difference being that strategies 0 and 1 were previously represented on a bit and now they are represented on a qubit. Strategy 0 (leave the preferred route) is mapped to the quantum state | 0 > and strategy 1 (take the preferred route) to the quantum state | 1 >.”
5. Typo: The caption of Figure 1 says N=10. Should be n1=10.
Done.
Round 2
Reviewer 3 Report
The authors have improved the presentation and clarified many of the confusing terms. However, I still find the following point missing.
- What's the value of K_E and K_G in line 90?
- Consider the example in Table 1. If both take option 1, then one has to wait for another one to pass. This waiting time is not included in the definition of total time given in line 90. Based on the discussion of the trade off I guess the authors count the waiting time as part of traveling time, but the authors should state that explicitly without letting the readers guess.
- Option 0 is still not clearly defined. A longer and idler path is still not specific. Here are examples that are specific: the shortest idle path; the second shortest path.
A general suggestion to the authors: the paper should contain enough information so that any readers can reproduce your result. The authors should consider this standard when describing the model.
Author Response
Response to Reviewer 1 Comments
First of all, we want to thank reviewer 1 again for the constructive review. We believe that his insights were of great help in making our article more comprehensive. Therefore, our revised manuscript has incorporated all the comments/requirements improving the quality of our article. As such, we sincerely hope that the revised manuscript can now be accepted for publication.
1. The authors have improved the presentation and clarified many of the confusing terms. However, I still find the following point missing.
Thank you very much.
2. What's the value of K_E and K_G in line 90?
We added the meaning of those two constants in the text.
[line 90]: Therefore, $TotalTime_{i} = K_{G} G_{i} + K_{E}\sum_{e=1}^{E_{i}} w(e,e+1)$, where $\mathbf{K_{G}}$ is the time it takes to play one game, $\mathbf{G_{i}}$ is the number of games played by the player $i$, $\mathbf{K_{E}}$ is the maximum distance between two nodes in the network, $\mathbf{E_{i}}$ is the number of edges of the final path of player $i$ and $\mathbf{w(e,e+1)}$ is the weight of the edge connecting the nodes $e$ and $e+1$.
3. Consider the example in Table 1. If both take option 1, then one has to wait for another one to pass. This waiting time is not included in the definition of total time given in line 90. Based on the discussion of the trade off I guess the authors count the waiting time as part of traveling time, but the authors should state that explicitly without letting the readers guess.
Yes, you are right. We modify the weight of the channels depending on how congested they are. We made it explicit in the text now.
[line 99]: This decline will be reflected in our model by increasing the corresponding weight, $w(e,e+1)$, of each channel proportionally to the number of packets going through it.
4. Option 0 is still not clearly defined. A longer and idler path is still not specific. Here are examples that are specific: the shortest idle path; the second shortest path.
Option 0 means that the player leaves their current path and goes to the following shortest path hoping that it is idler. We try to clarify this by rewriting this paragraph:
[line 135]: If more than one packet is interested in a channel, because it is part of their currently shortest path, each packet has two possible strategies: choose this preferred channel (risking that other packets will also select it and then congest the channel) \textit{or} search for its following shortest path (longer but possibly idler). These two strategies will be called option 1 and 0 respectively. For example, in the first game of a player, he has two options: \textit{1)} take the shortest path risking that the congestion increases his traveling time significantly. \textit{0)} try with his second shortest path, where there might be no congestion. In table \ref{table:packets}, an example of a situation where two packets are interested in the same channel, is presented.
5. A general suggestion to the authors: the paper should contain enough information so that any readers can reproduce your result. The authors should consider this standard when describing the model.
Thank you very much for your kind words, after our revisions we believe the manuscript now contains enough detail about our model for readers to reproduce the results.
