Learning Mean Field Games on Sparse Graphs: A Hybrid Graphex Approach

We thank the reviewer for the positive evaluation of our paper and the valuable suggestion to include an empirical comparison with existing LPGMFG and GMFG models. The feedback is answered in the same order that it was given in the review.

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Even though the authors explained in the paper, I didn't like the fact that the proposed GXMFGs have no baseline competitors to compare against. While I agree that one could argue on the contrary that the ability to work with sparse graphs is precisely the unique advantage of GXMGFs, I think that the authors should at least spend some efforts to discuss (if empirical comparison with LPGMFG is indeed unsuitable) how GXMFGs would compare with LPGMFG and GMFG in practice.

A: Thank you for bringing up the important topic of an empirical comparison with existing approaches. As you mention, the ability of GXMFGs to work with sparse realistic graphs can be seen as a major conceptual advantage over existing approaches such as GMFGs and LPGMFGs. We agree that there should be an empirical comparison complementing the discussion on conceptual differences in the first version of the paper. Thus, we have added an empirical comparison of GXMFGs and LPGMFGs on the eight real world networks and three tasks from the first paper version. Due to space constraints, we mention this new comparison in the main text and provide detailed results in the appendix, see Table 3 and the corresponding discussion at the end of the Appendix of the updated paper. The empirical comparison does not include GMFGs because they can be seen as a subclass of LPGMFGs and therefore will not yield better results than the LPGMFG framework.

Since LPGMFGs are not able to depict finite degree agents in the limiting model, all agents in the LPGMFG simulation follow the policy learned for infinite degree agents. The overall result of Table 3 is that our hybrid graphex learning approach clearly outperforms LPGMFGs across all networks and tasks. On some networks and tasks, the improvement is considerable but relatively moderate, for example on Prosper RS (error of 2.80 for LPGMFG vs. 1.58 in GXMFG). On other problems, our approach yields results that are many times better than those of the LPGMFG framework, such as Flickr SIS (16.90 vs. 3.57), Brightkite RS (14.69 vs. 3.37), and Hyves SIR (39.94 vs. 10.06). Thus, the empirical results provide strong evidence that the conceptual advantages of GXMFGs also yield a remarkably better empirical performance compared to previous methods such as LPGMFGs. For more details, please see the updated paper.

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Q: In Figure 3a, it looks like the curves are diverging rather than converging as k increases? Are the curves coloured correctly?

A: The colors in Figure 3a are correct and all curves converge as the graph size $\nu$ increases. For higher $k$ the curves tend to converge slower than for low $k$ which might seem counter-intuitive. The reason for the different convergence speed is that if we sample finite graphs from the power law graphex, with high probability they will have far more nodes with degree $k=2$ than with degree $k=6$. The relatively low number of high degree nodes reflects the power law nature of the network, where many nodes have low degrees and relatively few nodes have high degrees. As a consequence, a larger graph size is required to obtain a sufficiently large, representative subset of nodes with $k=6$ that yields a good mean field estimate. In contrast, relatively small sampled graphs already have numerous vertices with $k=2$ such that the mean field convergence is observed earlier for this subset of nodes.

We thank the reviewer for the positive feedback and the suggestions for improving the accessibility of our work as well as outlining possible extensions of our current approach. The comments are answered in the order they were brought up in the review.

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Q1: Providing an intuitive explanation for assumptions 1(b) and 1(c) would greatly enhance the paper's overall readability and accessibility.

A: Thank you for the valuable suggestion! To increase the accessibility of Assumptions 1 b) and c), we have added a more detailed explanation for the respective assumptions in the updated paper draft. The intuitive interpretation of Assumption 1 b) is that it describes the behavior of $\xi_W$ at infinity. More specifically, for $\sigma \in (0,1)$ Assumption 1 b) states that $\xi_W (\alpha)$ is approximately a power function $\alpha^{- \sigma}$ for large $\alpha$ which aligns with our goal to generate power law graphs. On the other hand, Assumption 1 c) is a technical assumption from the graph theory literature and has no obvious intuitive explanation to the best of our knowledge. Nevertheless, it is especially fulfilled by all separable graphexes which are characterized by the accessible property $W (\alpha, \beta) = \xi_W (\alpha) \xi_W (\beta) / \bar{\xi}_W$. Combining these two findings, one can see that Assumption 1 is especially satisfied by the separable power law graphex used in our paper. We hope that the added intuition and explanation increases both readability and accessibility.

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Q2: While the paper assumes finite state and action spaces, it may be beneficial to explore whether the proposed approach can be extended to scenarios with infinite action spaces

A: In our opinion, it is worthwhile to extend the GXMFG approach to continuous state and action spaces (and also continuous time) to increase the generality of the learning method. Since the extension to a continuous setting will require different and adapted mathematical and algorithmic approaches, it is outside the scope of our paper. We have mentioned the promising research direction of defining a continuous version of GXMFGs in the conclusion of the updated draft. Thanks for the idea!

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Q3: Including the code for the simulations, would enhance reproducibility.

A: We have uploaded the code and will add a link in the final, deanonymized version of the paper.

We thank the reviewer for the positive evaluation of our paper and the helpful comments on how to improve the paper. The feedback and questions are answered in the order they were brought up in the review.

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The model is quite abstract at some places. For the theoretical results, they are mostly about the analysis of the game and I am not sure how relevant they are for this conference (although they are certainly interesting for a certain community). It might have been more interesting to focus more on the learning algorithm.

A: The analysis of the game provides the key insights into complex agent systems that are necessary to eventually provide the equilibrium learning algorithm. Only through a thorough understanding of the core periphery structure and its implications it is possible to state a principled equilibrium learning approach. Therefore, we do believe that the understanding of these complex agent networks and the resulting learning algorithm are relevant for this conference. Nevertheless, we agree that there are various open challenges and hope that our GXMFG learning approach provides a useful framework for future research.

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Q1: I am wondering if some assumptions are missing. For example below Lemma 1, should $f$ be at least measurable (and perhaps more?) with respect to $\alpha$ for the integral to make sense?

A: We thank the reviewer for pointing out the potential issue with the assumptions on $f$. In the theoretical results such as Theorem 1 we state the assumptions on $f$, i.e. measurable and bounded. However, we did not mention these conditions on $f$ when defining the integral you referred to. To avoid the misleading impression that there are no assumptions on $f$, we also added them to the respective definition in the updated draft.

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Q2: Assumption 2 as used for instance in Lemma 1 does not seem to make much sense (unless I missed something): What is $\boldsymbol \pi$? We do not know in advance the equilibrium policy and even if we did, we would still need to define the set of admissible deviations for the Nash equilibrium. Could you please clarify?

A: We completely agree with the reviewer: the policy $\boldsymbol \pi$ should not be part of Assumption 2 and (of course) we do not assume the equilibrium policy to be known in advance; the set of admissible deviation policies from the Nash equilibrium is not restricted. Instead, we have added the Lipschitz condition (up to a finite number of discontinuities) on $\boldsymbol \pi$ to the respective theoretical results, such as Theorems 1-4. Thank you for spotting the mistake in Assumption 2. We have corrected it in the updated paper version.

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Q3: Algorithm 1, line 14: Could you please explain or recall what is $Q^{k, \mu^{\tau_{\max}}}$?

A: In Algorithm 1, $Q^{k, \mu^{\tau_{\max}}}$ is defined similar to $Q_{i,t}^{\pi, \mu}$, except that we substitute the reward function $r$ by $r'_k$ and use the transition kernel $P'_k$ instead of $P$. We have added the definition in the updated paper.

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Some typos: Should the state space be either $\mathcal{X}$ or $X$ (see section 3 for instance)?

A: The state space should be $\mathcal{X}$. We used $X$ in the beginning of Section 3 to denote an arbitrary finite set. Thanks for pointing out the ambiguous notation; we have corrected it in the updated paper version.

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Does $\mathbb{G}^\infty_{\alpha, t}$ depend on $\mu$ or not (see bottom of page 4)? Etc.

A: In our framework, the neighborhood distribution $\mathbb{G}^\infty_{\alpha, t}$ always depends on the mean field $\boldsymbol \mu$. For notational convenience, we sometimes drop the dependence on $\boldsymbol \mu$ in the notations. The lack of any comment on dropping the dependence in the notation led to understandable confusion. We have added an explanation to the updated draft, thanks.