Optimizing Social Network Interventions via Hypergradient-Based Recommender System Design

We thank the reviewer for their feedback and constructive comments. First, we would like to clarify one important point. Our algorithm optimizes only over the network topology, i.e., the network weights $w$, but not the innate opinions $s$. The main optimization problem that we aim to solve, equation (3), has as decision variables the weights $w$ and equilibrium expressed opinions $y$. Since $y$ is uniquely determined by the constraint (3b), we optimize over $w$ solely. We could readily extend our framework to optimize for both $w$ and $s$ simultaneously, as suggested by the reviewer, but this is not something that we investigate in this paper. We will make sure to clearly elaborate on this point in our revised manuscript.

Weakness 1 We thank the reviewer for their insightful comment. Indeed, the main point of Network Administrator Dynamics (NAD) in Chitra & Musco, 2020 is to show how minimizing disagreement can have adverse effects on polarization of opinions. They do have credits for raising this important and critical aspect of this optimization problem, which we will be more careful to acknowledge. In this vein, our goal is to understand the mechanism by which NAD operates and, in turn, demonstrate how our algorithm succeeds by using a different mechanism in the same problem setting. We believe our comparison to be a meaningful one for two reasons:

1. The two algorithms operate in the same fashion: they both propose a model which proceeds in rounds, where the network administrator/leader minimizes the disagreement and then subsequently the opinions converge.

2. our algorithm manages to fix (some of) the issues that Chitra & Musco highlighted with NAD. In fact, in their paper they also propose fixing NAD with NAD*, which we also include in our comparison results. In essence, we provide a potential solution to one of the main issues raised by Chitra & Musco. We will try to clarify these distinctions and elaborate on these points on the revision of our paper by making sure we give Chitra & Musco credits for raising a very important and critical aspect of this optimization problem.

Weakness 2 Regarding the optimization of graph topology and innate opinions, please see our discussion above. Regarding the polarization metric, we have rerun our simulations using the metric proposed by the reviewer and will include them in the corresponding section. We quickly summarize some of them: The reruns of Sections 4.1.2, 4.1.3, and 4.2 with the suggested polarization metric are close to our original results, as can be seen in Figure 1 and 2, and Table 1 on https://imgur.com/a/8GbzUNw. Additionally, BeeRS decreases the polarization in Section 4.1.3 even further with the new polarization metric (-44% compared to -39%).

Weakness 3 We thank the reviewer for their useful suggestion. We have deployed our algorithm on the soc-LiveJournal1 dataset, which is the largest among the ones available in SNAP, and will include the results in the corresponding section. A preview of the results is available on https://imgur.com/a/8GbzUNw. The collected results further confirm the effectiveness of our algorithm.

Weakness 4 We acknowledge the reviewer's suggestion to analyze the rates of convergence of gradient descent. We have performed an empirical analysis of the learning curves of our algorithm, namely objective vs. number of iterations, as also suggested by reviewer pBeX; please refer to that response for more information. Performing a theoretical analysis can be challenging due to the non-convexity of the problem and could potentially require a paper on its own. One such analysis, for example, is the classical result in [1, Ch. 1.2] which proves that vanilla gradient descent converges to an $\epsilon$-stationary point at a rate of $O(\epsilon^{-2})$, for the case of an unconstrained non-convex smooth optimization problem. To the best of our knowledge, convergence rates for projected gradient-descent with momentum on a non-convex objective is an open problem. We consider such theoretical analyses beyond the scope of our paper, and reserve the investigation for future work.

Other Comments or Suggestions 1 Our algorithm can deal with any differentiable objective function, including the one mentioned. We collected new results for the proposed metric as mentioned above, with a preview available at https://imgur.com/a/8GbzUNw.

Other Comments or Suggestions 2 We will highlight this difference in notation.

Other Comments or Suggestions 3 Thank you for mentioning these works. They are indeed very relevant. We will cite them in the revised manuscript and make an accurate comparison.

Questions Please refer to our answers in Weakness 2 and Other Comments or Suggestions for Question 1, and to the first paragraph of our response for Question 2.

[1] Nesterov, Yurii. Lectures on convex optimization. Vol. 137. Berlin: Springer, 2018.

We thank the reviewer for the positive comments and their careful reading.

Methods and Evaluation Criteria The concern raised on "Methods and Evaluation Criteria" is a valid and important one. Indeed, for problems/networks of huge dimensions our algorithm might face computational bottlenecks. Nonetheless, our scheme could be adapted and improved in a number of ways to address such practical problems. We outline some next:

1. We could further exploit the capabilities of GPU computation. In Subsection 4.1, we successfully solve a problem with 3 million decision variables on a GPU that currently is not among the state-of-the-art (NVIDIA GeForce RTX 3060 Ti). By using a more performant GPU, or even a cluster of them, we could definitely handle problems of even larger dimensions. In fact, problems with hundreds of millions of decision variables are routinely solved with GPUs in the context of neural network training. This flexibility is a significant advantage of our algorithm, emanating from the fact that it is a simple, first-order method. (Similar to the ones used in neural network training.)

2. We could reduce the number of weights that we optimize over by employing appropriate heuristics to identify the most important parts of the network, introducing a tradeoff between solution quality and computational requirements. Indeed, our algorithmic approach and our implementation allow the user to specify which weights are immutable and which ones are variable. One meaningful way to choose which weights to optimize (or which areas of the networks to affect) would be to use heuristics to quantify the importance of nodes, and then optimize the weights of edges connected to these nodes. For instance, we could use the degree of a node or, more generally, other centrality measures [1]. These approaches to choosing which weights to update are especially important since we, typically, do not want to modify the weights too much as they affect the users' experience in the network.

3. We could update only a (randomly chosen) subset of the weights at each iteration, thus giving rise to a stochastic/mini-batch version of our algorithm. This is a GPU-friendly way of addressing problems with huge dimensionality.

[1] Bloch, Francis, Matthew O. Jackson, and Pietro Tebaldi. "Centrality measures in networks." Social Choice and Welfare 61.2 (2023): 413-453.

Experimental Designs or Analyses/Question

• We thank the reviewer for their suggestion to include additional datasets in our simulations. Please note that in our original manuscript, beside DBPL, we have also tested our algorithm on the real-world Reddit Dataset. Further, we have also deployed our algorithm on the soc-LiveJournal1 dataset, and will include the results in our revised manuscript. We summarize the results from our additional simulations in Figures 1-3, and Table 1 in https://imgur.com/a/8GbzUNw. We highlight that soc-LiveJournal1 is one of the largest directed social networks on the SNAP platform with 4.8 million users and 69 million edges.

• We thank the reviewer for suggesting an empirical analysis with a uniformed metric. We include an additional simulation where we study the objective $\varphi$ as a function of the iteratation $k$. In particular, we deploy BeeRS on Problem (10) with the polarization metric suggested by Reviewer 5iEL. We use the Reddit, DBLP, and soc-LiveJournal1 datasets, and initialize every simulation i) with initial weights 0, and ii) with 4 additional random initializations. For the total of 5 simulations per dataset we plot the mean cost $\varphi$ with standard deviation as a function of the iteration $k$, as shown in Figure 3 on https://imgur.com/a/8GbzUNw. Please note that there is a large difference in cost between the reddit dataset and the other two datasets, which is explained by the much larger number of users in the latter ones.

  We also rerun the simulations from Sections 4.1.2, 4.1.3, and 4.2 with the polarization metric suggested by Reviewer 5iEL on the additional soc-LiveJournal1 dataset, as can be seen in Figure 1, Table 1, and Figure 2 at https://imgur.com/a/8GbzUNw, respectively.

We thank the reviewer for their positive evaluation and their constructive comments.

Concerning the weaknesses and questions raised by the reviewer:

Question/Weakness 1 Thanks for pointing out this aspect. We foresee using the weights as a penalty (/boosting) factor in a Weighted PageRank (WPR) [1] or EdgeRank [2] algorithm fashion. These algorithms take into account the importance of both the inlinks and the outlinks of the pages and distribute rank scores based on the popularity of the pages: If the network leader (upper level) decreases the weight of the link from user A to user B then this translates into a penalty for all the content shared from A which is a candidate to be visible to B. The details of how this would be algorithmically handled, however, are out of the scope of this work and an interesting direction for future research.

Question 2 We thank the reviewer for their suggestion. Indeed, our algorithmic approach and our implementation allow the user to specify which weights are immutable and which ones are variable. One meaningful way to choose which weights to optimize (or which areas of the networks to affect) would be to use heuristics to quantify the importance of nodes, and then optimize the weights of edges connected to these nodes. For instance, we could use the degree of a node or, more generally, other centrality measures [3]. These approaches to choosing which weights to update are especially important since we, typically, do not want to modify the weights too much as they affect the users' experience in the network.

Another interesting way to optimize for performance is to randomly choose which weights to update at each iteration, giving rise to a stochastic/mini-batch version of our algorithm. This would be particularly helpful when dealing with networks of huge size.

[1] Xing, Wenpu, and Ali Ghorbani. "Weighted pagerank algorithm." Proceedings. Second Annual Conference on Communication Networks and Services Research, IEEE, 2004.

[2] EdgeRank: The Secret Sauce That Makes Facebook's News Feed Tick. techcrunch.com. 2010-04-22. Retrieved 2012-12-08.

[3] Bloch, Francis, Matthew O. Jackson, and Pietro Tebaldi. "Centrality measures in networks." Social Choice and Welfare 61.2 (2023): 413-453.

We thank the reviewer for their very positive evaluation and their kind suggestions. We will make sure to stress the advantage of considering directed networks in the paper and include the references suggested by the reviewer.