Optimizing Social Network Interventions via Hypergradient-Based Recommender System Design
Authors: Marino Kühne
Panagiotis D. Grontas
Giulia De Pasquale
Giuseppe Belgioioso
Florian Dorfler
John Lygeros
Venue: ICML 2025 poster (2025)
Keywords: opinion dynamics
recommender systems
hypergradient
social networks
optimization algorithm
Friedkin-Johnsen
polarization reduction
disagreement reduction
Primary area: applications->social_sciences
TL;DR: We propose a scalable algorithm to optimize social network interventions based on any differentiable, eventually non-convex, performance metric with users' opinions adhering to the Friedkin-Johnsen dynamics.

Abstract: Although social networks have expanded the range of ideas and information accessible to users, they are also criticized for amplifying the polarization of user opinions. Given the inherent complexity of these phenomena, existing approaches to counteract these effects typically rely on handcrafted algorithms and heuristics. We propose an elegant solution: we act on the network weights that model user interactions on social networks (e.g., ranking of users’ shared content in feeds), to optimize a performance metric (e.g., minimize polarization), while users’ opinions follow the classical Friedkin-Johnsen model. Our formulation gives rise to a challenging, large-scale optimization problem with non-convex constraints, for which we develop a gradient-based algorithm. Our scheme is simple, scalable, and versatile, as it can readily integrate different, potentially non-convex, objectives. We demonstrate its merit by: (i) rapidly solving complex social network intervention problems with 4.8 million variables based on the Reddit, LiveJournal, and DBLP datasets; (ii) outperforming competing approaches in terms of both computation time and disagreement reduction.

Optimizing Social Network Interventions via Hypergradient-Based Recommender System Design
Marino Kühnee Panagiotis D. Grontas'
Giulia De Pasquale? Giuseppe Belgioioso John Lygeros'
Florian Dörfler!
Abstract
Allcott et al., 2020; Levy, 2021; Guess et al., 2023a;b). The algorithms employed by social media companies to maintain user engagement are thought to contribute to the isolation of the users into echo chambers, where opinions are segre- gated by party lines and ties to opposing views are severed (Pariser, 201 1). Even though the literature suggests that the algorithms of major social media companies may decrease the share of cross-cutting content in users' feeds (Bakshy et al., 2015; González- Bailóó et al., 2023), thereby limit- ing diverse content exposure, the user's choice of engaging with opinion-confirming material tends to be a comparable cause of this isolation (Garrett, 2009; Bonzález-Baildd et al., 2023; Wojeieszak et al., 2022).
Although social networks have expanded the range of ideas and information accessible to users, they are also criticized for amplifying the polar- ization of user opinions. Given the inherent com- plexity of these phenomena, existing approaches to counteract these effects typically rely on hand- crafted algorithms and heuristics. We propose an elegant solution: we act the network weights that model user interactions on social networks (e.g., ranking of users' shared content in feeds), to optimize a performance metric (e.g., minimize polarization), while users' opinions follow the classical Friedkin-Johnn Johnsen model. Our formula- tion gives rise to challenging, large-scale op- timization problem with non-convex constraints, for which we develop a gradient-based based algorithm. Our scheme is simple, scalable, and versatile, as it can readily integrate different, potentially non- convex, objectives. We demonstrate its merit by: (i) rapidly solving complex social network inter- vention problems with 4.8 million variables based the Reddit, LiveJourna,, and DBLP datasets; (ii) outperforming competing approaches in terms of both computation time and disagreement reduc-
The field of opinion dynamics studies how the content pro- vided by social media shapes user opinions and how opin- ions propagate in social network (Noorazar, 2020). Users in the social networks are modeled as nodes in a directed, weighted graph, where the edge orientation represents the direction of influence between users and the edge weights represent the intensity of the social influence. Here we consider the well-established Friedkin-Johnsen (FJ) model (Friedkin & Johnsen, 1990), which has been successfully employed in the study of polarization and disagreement in social networks (Chitra & Musco, 2020; Musco et al., 2018; Chen et al., 2018). In the FJ model, users have both a constant internal opinion (prejudice) and time-varying ex- ternal, declared opinion. The external opinion of each user evolves by repeated averaging of the internal opinion with the external opinions of interacting users, while the preju- dice remains constant. We propose a unified framework to address a broad class of problems involving the influence of user opinions on social platforms, such as polarization miti- gation, through modifications to users' connections when opinions evolve according to the FJ dynamics.
tion.
1. Introduction
Problem description and motivation: Social media plat- forms brought many societal benefits but they have also been associated with the spread of fake news and the polar- ization of opinions (Allcott & Gentzkow, 2017; Bail et al., 2018; Bakshy et al., 2015; González-Bailón et al., 2023;
Literature review: The literature network interventions under the FJ dynamics is vast (Bindel et al., 2015; Wang & Kleinberg, 2023; Gionis et al., 2013; Gaitonde et al., 2020; Ancona et al., 2022; Chen et al., 2018; Musco et al., 2018; Matakos et al., 2017: Zhu et al., 2021; Chitra & Musco, 2020; Cinus al., 2025). Most of these works seek to achieve various objectives by designing interventions on the
'ETH Zürich, Switzerland "Eidhooven University of Tech- nology, The Netherlands 'KTH Royal Institute of Tech- nology, Sweden. Correspondence to: Marino Kühne www.riookucnne@ooooooo.o
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Optimizing Social Network Interventions via Hypergradient-Ban Based Recommender System Design
social network topology or on the internal opinions of users. The most common approach is based adding deleting edges from the network, where the following objectives are considered: minimization of average- and worst-case opinion dynamics metrics (Chen et al., 2018), social cost minimization (Bindel et al., 2015), and polarization and disagreement minimization (Zhu et al., 2021). Differently, (Chitra & Musco, 2020) minimizes disagreement by alter- ing the edge weights, and (Cinus al., 2025) minimizes various objectives by altering the edge weights without prior knowledge of internal opinions. (Wang & Kleinherg, 2023) investigates the effects of link formation on polarization and disagreement. More closely related to our work, (Cinus et al., 2023) minimizes polarization and disagreement using projected gradient scheme. While similar in approach, our algorithm can seamlessly handle more general objectives and constraints the weight modification. A second, less common approach focuses on changing the opinions of spe- cific users (Gionis et al., 2013; Matakos et al., 2017; Musco et al., 2018). To achieve a generally favourable opinion about a product, (Gionis et al., 2013) sets the opinions of a fixed number of users to a specific value. On the other hand, (Matakos al., 2017) sets the opinions of fixed number of users to zero to minimize polarization. Finally, (Musco et al., 2018) modifies opinions to minimize the sum of polarization and disagreement.
computational complexity of such methods (Grazzi et al., 2020). A widely-used technique for obtaining hypergradi- ents involves differentiating through implicit mappings, as explored in (Grontas et al., 2024).
Contributions: We consider the FJ model of opinion dy- namics. We seek seek to drive the equilibrium user opinions to a desired configuration by modifying network weights. Our contribution three-fold:
1. We formulate an optimal intervention task as an op- timization problem with non-convex constraints that describe the equilibria of the FJ dynamics. Our formu- lation can accommodate any (differentiable) objective, e.g.. polarization or disagreement reduction, well as convex constraints modelling intervention limits. Further, we can address both directed and undirected graphs, by appropriately defining the constraints.
2. We propose a first-order algorithm that alternates be- tween modifying the network weights and comput- ing both the equilibrium opinions and their gradient with respect to weights, by efficiently backpropagating through the FJ dynamics. Crucially, both steps are sim- ple and scalable: they consist of a projected gradient update and the solution of two linear systems.
If we view the weights' of the network as hyperparameters of a model, then choosing the optimal ones becomes a hy- perparameter tuning problem. This defines the upper level of bilevel optimization problem. In turn, the equilibrium resulting from the repeated averaging of the FJ dynamics can be interpreted as the Nash equilibrium of a game defined by separate optimization problem (Bindel et al., 2015), forming the lower level. These bilevel optimization prob- lems are difficult to solve- - for example, some formulations in (Bindel et al., 2015) and (Matakos et al., 2017), are shown to be Narhard. A few special problems with tailor-made objectives are shown to be convex, such as (Gaitonde et al., 2020, Section 4) or (Musco et al., 2018). However, this is not the case in general and the problems are solved with problem-specific algorithms or heuristics such as in (Gionis et al., 2013) or (Matakos et al., 2017).
We provide a modular, and well-documented imple- mentation of our algorithm in JAX (Bradbury al., 2018), facilitating the usage of hardware accelerators We then test our algorithm on real-world datasets, with up to 4.8 millions modifiable weights, and obtain a solution in a matter of minutes on a GPU. Additionally, we compare our approach against the nonlinear pro- gramming solver IPOPT, demonstrating up to three orders of magnitude faster runtime while achieving superior solution quality. Finally, compared to an exist- ing heuristic, which is tailored to subclass within our problem formulation, our scheme obtains significantly better solutions.
Notation Let Rzz (R>p) denote the set of non-negative (positive) real numbers. We denote the (i, element of matrix W by Wiz. Let ||:I|r and Il-I12 denote the Frobe- nius and the l norm, respectively. The projection of a vector wo onto a convex set W is denoted by [w[ = argmin w| Let F' R. x =R + R² be a vector- valued differentiable mapping. We denote the partial Jaco- bians of F with respect to the first and second arguments as Jj F(w,y) E Roxm and J2 ((w. y) RR respectively. If = 1, we use F.F, F2F to denote the partial gradients.
For both large-scale convex and non-convex problems. first- order methods have proven highly effective, as demonstrated in works like (Bottou et al., 2018; Bottou, 2010; Kingma & Ba, 2015). These methods are particularly appealing due to their computational efficiency and scalability. Recent advancements have extended these approaches to hypergra- dients (Maclaurin et al., 2015) (the gradient of the objective with respect to hyperparameters) while also analyzing the
'The same principle holds if internal opinions are considered as hyperparameters, but it is not covered in this work.
"Our code is available online at https: http::://gittub.oomm m- kukhneeeeeee BeeRs
2

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Optimizing Social Network Interventions via Hypergradient-Bar Based Recommender System Design
2. Problem Formulation
and we note that the sums are implicitly restricted to the neighbors of i, since Waj 0 if j is not neighbor of i. The update (1) can be understood as repeated weighted averaging between the internal opinion s; and the neighbors' opinions yj. where the weights Wij indicate how susceptible user is to user j's opinion. We can compute the equilibrium of (1) as the solution of the linear system of equations
We consider social networks consisting of interacting users, where a recommendation system (referred to as leader) seeks to influence the users' opinions by controlling the interactions between them. This problem can be framed as a bilevel optimization problem, where the upper level involves the leader's interventions, and the lower level mod- els the users, whose opinions are affected by the leader's decisions. Specifically, the leader aims to shift users' opin- ions toward a desired configuration by modifying network weights. In practice, these weights can represent various metrics-suce as the ranking of users' shared content in feeds, where boosts or penalties influence visibility. This is similar to how algorithms like EdgeRank or Weighted PageRank (Wenpu& & Ghorbani, 2004) adjust ranking scores based on relevance signals. Recommender systems may also weaken connection strength by reducing the frequency or prominence of shared content in users' feeds.
+ ΣjEV Wij
Win
8, (2)
Wa1
1 EjEv Wny
A(w))
with 8 = [.,, E R", y - [y)..... ......... yn]] E IR" and A(w) E R=*=. Note that A(w) depends on the net- work weights w, i... the vectorized form of the adjaceney matrix W. Next, we show that whenever the weights are appropriately chosen, there exists unique vector of equi- librium opinions.
2.1. Lower Level - Opinion Dynamics
Proposition 2.1. For all w E B, the matrix A(w) is invert- ible and the FJ dynamics (1) admits a unique equilibrium that solves (2).
We consider a network G, represented by a directed weighted graph G W) where V = [l,...,,)) n) is the set of nodes, and W E Ruxn is the (weighted) ad- jacency matrix. For notational convenience, we will re- shape W as a vector w E g=² We assume that G con- tains no self-loops and has non-negative weights, formally, w E B = (w E 18= ww = 0, Wij 2 O, Vi,j E V). An edge between nodes i E V exists if and only if way > O, and we say that and j are neighbors. Interpreting G as social network, each of the n nodes corresponds to a user (e.g., individuals, companies, news outlets, etc.), and the edges represent connections between them. For each edge (i,j), the user on the tail i is influenced by the user on the head/ j. The edge is assigned a weight proportional to the influence exerts on i and could, for example, represent the time i spends consuming content shared over this connec- tion, the frequency of interaction, or any other engagement metric. It is natural to assume that a longer exposure re- sults in a greater influence. According to the FJ opinion dynamics (Friedkin & Johnsen, 1990), the opinion of user consists of two parts: the constant internal opinion, or preju- dice, 8 E IR and the time-varying external opinion yı RR. We consider opinions to be in the interval [-1, where -1 1 indicates complete opposition and 1 complete approval. The internal opinion is kept private, while the external opinion y is shared with others. Owing to the interactions between users, the external opinion of user changes over time according to the FJ update rule
The proposition holds since A(w) is strictly row diagonally dominant and therefore invertible (Taussky, 1949, Th. IV),
2.2. Upper Level - Network Intervention
The goal of the leader is to find weights w E RR" that mini- mize cost function y : R=" x RR" R, mapping and the resulting solution of (2) to some upper-level objective. Assumption 2.2. The function ((w, is continuously dif- ferentiable in both arguments.
A possible choice is given by p(w, v) = Ei (v 1 E 3;)2, a classical measure of polarization, previ- ously used by (Chitra & Musco, 2020). In practice, the leader cannot modify the network weights arbitrarily, as such interventions directly impact the users' experience (e.g., consider the case where a connection between two close friends is removed). We model permissible inter- ventions as constraint w E W. obeying the following assumption.
n
Assumption 2.3. The set of constraints W is non-empty, closed, and convex, and W C B.
The set W can be used to add constraints that encode re- quirements such as: (i) some connections cannot be formed or altered, by fixing Way to zero or to its initial value, respec- tively: (ii) ensuring that some connections are maintained by
8; + Σ jev Wiyb)
(1) 1 1+X Ejev Wij
"The matrix A(w) is often represented in the literature as A(w) L(w) + T, where I. is the Laplacian matrix of the graph, and the identity matrix (Cinus et al., 2023).
V/ (1+1)
3

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Optimizing Social Network Interventions via Hypergradient-Ben -Baeed Recommender System Design
imposing lower bounds on some Wig. Further, W ensures that weights are compatible with our assumption w E B.
3. Algorithm Design
To solve (4), we propose a gradient-based algorithm. The main challenge lies in obtaining the hypergradient, i.., the gradient of the objective y(w, y (w)) with respect to w, This can be achieved by using the chain rule as follows:
Note that many of these constraints decrease the number of degrees of freedom in the optimization problem, for exam- ple, by fixing some weights to a given value. In principle, one can maintain all the degrees of freedom and enforce this requirement through the set W. Computationally. however, is more efficient to drop the corresponding decision vari- ables from from the beginning, giving rise to decision vector of dimension lower than n We always drop the weights Wy; from w, as self-loops are absent, reducing the number of decision variables to m = n(n - 1). Throughout the remainder of this paper, m denotes the effective number of decision variables.
Vulp(u, y (w)) =Vip(u, y (w))
(5)
+ Jy V2
Under Assumption 2.2, the partial gradients Viạ and V24 are well-defined. However, ensuring the existence of the Jacobian of the equilibrium mapping, Jy (w), also called the sensitivity, and computing it is a non-trivial task.
To compute (w)) we introduce the auxiliary expression F F(w,, y) A(w)y s. In the following provide an explicit expression for the sensitivity.
This property of our algorithmic approach allows further optimize for performance by deploying it subset of the network. One meaningful way to choose which weights to optimize for (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 node or, more generally, other centrality measures (Bloch et al., 2023).
Proposition 3.1. Under Assumption 2.3, the function (+) is continuously disserentiable and its Jucobian Jy (w) well-defined. Moreover, Jy"(w)
(w) is given by
Jy (w) = J2 yn (w))) F (w)). (6)
Proof. From the definition F(w, y) == A(w)y - 8, it im- mediately follows that J2F y) = A(w), which is contin- uous in both arguments. Furthermore, F1,, y) is continu- ous in both arguments by Lemma A. in Appendix A.1. (3a) Thus, F(w, v) is continuously differentiable. Addition- ally, by Proposition 2.1, A(w) and therefore J2F(w, v) are nonsingular. By invoking the implicit function theorem (3c) (Dontchev & Rockafellar, 2009, Th. 1B.1), the sensitivity exists, is continuous, and given by (6).
Combining the upper-level objective with the equilibrium of the lower level (2) and permissible interventions in W,
we formulate the following optimization problem:
minimize w.
(p(,,)) v)
subject to
A A(w)y= = 8, w E W.
(3b)
The bilinear constraint (3b) renders the feasible set non- convex. We rewrite (3) as
Given Jy" (w), we can evaluate the hypergradient and solve (4) using our preferred first-order method. In this work, we deploy projected gradient descent with momentum (Bolduc et al., 2017) given by
minimize W
p(w,y"(w)
(4a)
subject to w E W,
(4b)
m (+++) = =mm + V a (w(()) -v" (witt)) (x+1) II (w) a (2) (k+1)]
where y" (w) denotes the solution to the system A(w)y = s. We refer to yd y-(+) m R as the equilibrium mapping, which is well-posed under Assumption 2.3. Formulation (4) is favorable since y" (-) is continuously differentiable (see Proposition 3.1), hence motivating us to deploy a first-order solution method.
where a (k) 10, 1] is a possibly cerration-depedden step- size and y 2 0 is a momentum parameter. This formulation contains short-term memory for the descent direction of the previous iteration k in the form of (*), and it can often significantly decrease the runtime for an appropriate choice of y (Goh, 2017). The projection is employed to ensure feasibility of the iterates (k))
Although (4) involves convex constraints, it still challeng- ing to solve, as (-) is an implicit mapping. for which closed-form expression is, in general, not available. More- over, the objective is non-convex.
Note that, in (4a) we optimize solely over the weights w and not the innate opinions s. Our algorithm can readily be extended to jointly optimize for w and s, which we consider as an interesting future work.
3.1. Backpropagation through Equilibrium Opinions
Computing Jy"(w) by solving the matrix equation (6) can be computationally challenging for large-scale problems

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Optimizing Social Network Interventions via Hypergradient-Bar Based Recommender System Design
Algorithm 1 BeeRS
of BeeRS on both a CPU and GPU and the evolution of yp across iterations. In Section 4.2, we compare BeeRS to IPOPT, a state-of-the-art nonlinear programming solver that can directly solve the non-convex program (3), while in Section 4.3 we compare against the algorithm proposed in (Chitra & Musco, 2020). Hardware specifications are given in Appendix A.3, and the choices for the step size a and momentum parameter y are justified in Appendix A.4.
1: Input: Tolerance e, step size (*)), momentum param. 7
2: w(O) < Set initial network weights
3: while True do
4: Compute J2F (w(k))
5: y" (w((k) <- Solve (2)
6: if (f(()) E then Break
Compute J1 (F y" (w)(t))
7:
8:
Compute gradients V 11 (*()) 3 (w(
4.1. Scalability Study
9: (()) + Solve (9)
10:
Twp(w.y"(uu + Compute (8)
4.1.1. PROBLEM SETUP
11:
((++) + Using (7)
We consider a social network with directed connections. Let us consider the problem where a newly founded news agency, labelled as Ç. wants establish itself in the network by sharing content such news articles. The news agency is neutral, which is encoded by the internal opinion sc 0, while all other users (readers) might be biased with internal opinions 8; E [-1,1]. The network operator (the leader) has to decide how to deliver the additional content to the existing users, with the goal of minimizing polarization in the network. Additionally, the operator charges a fee the news agency for distributing its content, which is propor- tional to the amount of content shared over the network. This expense can be viewed as an advertisement fee, reflect- ing the fact that getting space on the platform is costly. The agency's advertising budget is bounded by b E Ryo.
12: end while
13: Output: w*, p("",y"(""" (w*))
(Parise & Ozdaglar, 2021). Concretely, we need to solve
J2 (w. y" (w)) Jy (w)
J1 F1 (w))
Raxn
RaXm
for a given w, and therefore fixed (w:), with matrix sizes as indicated. To reduce the computational bur- den, note that we do not need the sensitivity Jy"( by itself in (5) but rather the vector-Jacobian product Jy" (w) V2sp(u, y" (w)), see e.g., (Grazzi et al., 2020). Substituting (6) in (5), yields:
To model this, we introduce a new connection from every existing user i in the network to the news agency Ç. Since the network is directed and the news agency does not have any outgoing connections, it remains neutral, hence Kc< sc 0. The connections from users to the news agency carry a weight wig 2 0, which is proportional to the share of ('s content in user i's feed. The goal of the leader is to compute weights that minimize polarization, measured as Ei (y] 7 1 v2) under the news agency's 's budget constraint Σx= Wic S
V Vers((n,y (w)) = Vig(w, y² (w)) - ((
(J2F(w, (w) (w)) ( v (w)) -Vig(w,y (w) F1F y (w))
(8)
where R= solves the linear system of equations
J2F(w, v² (w)) V2p(w, y (w)).
(9)
Hence, we can compute the hypergradient by solving (9)
instead of (6), i.e., we solve one linear system instead of m.
We can formalize this problem as
Combining the equilibrium backpropagation (9) with the chain rule (8) yields Algorithm 1, which we refer to as BeeRS (Best Intervention for Recommender Systems). As a termination criterion, we use
minimize yER" .EEB
Σ (v) n2 1=1
1 Σ
03)
(10a)
1=1
subject to
A(w)y = 8,
(10b)
(()) = (k( 1) "y' (tolk- 1))) (A) 10(w0l-11, y (m(A-11)))
.y (M)))) <e
Wi wij (0)
Vi EVAS
(10c)
WC3 = 0, VjeV,
(10d)
for some small tolerance 0. Implementation details are given in Appendix A.2.
ΣE Wiç b,
(10e)
where the variance objective is standard polarization met- ric (Chitra & Musco, 2020), (10b) ensures that opinions are in equilibrium and conform to the FJ dynamics, (10c) fixes the weights that are unrelated to the news agency to their original values w()), (10d) ensures Ç has no outgoing connections, and ( (10e) imposes the budget limit of Ç.
4. Numerical Simulations & Comparisons
In this section, we demonstrate the effectiveness of our algorithm by deploying it on real-world and artificially-
generated datasets. In Section 4. we study the scalability
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Optimizing Social Network Interventions via Hypergradient-Bann Based Recommender System Design
We stress that weights that are fixed by constraints, in (10c) and (10d), shown for clarity of exposition. In our numerical implementation, reduce the dimensionality of the problem. Therefore, (10) ends up with m = n decision variables for the weights.
surprising, given the additional overhead produced by mov- ing computations the GPU and by JAX's just-in-time compilation, among other reasons. However, the GPU im- plementation becomes faster for more than 100,000 users.
remove these weights to
4.1.3. SOLVING ON THE ENTIRE DATASET
4.1.2. NUMERICAL RESULTS
Given the remarkable scalability of BeeRS on the GPU, we deploy on the entire DBLP dataset consisting of n 3, 079, 007 nodes and 25, 166, 994 edges and on the entire LiveJournal dataset consisting of n nodes and 68, 993, 773 edges. We solve (10) ten times with = 10. a = 0.05, 7 = 0.6, and E le-2 on the GPU. BeeRS achieves polarization reduction of 39.5% and 40.0% on the DBLP and LiveJournal dataset, respec- tively. The respective mean runtime until convergence for DBLP and LiveJoumnl is 195.4 and 766.8 s with standard deviation of 9.6 and 5.4 s.
We solve problem (10) on two datasets: the DBLP dataset (Tang al., 2008) and the LiveJournal social network dataset (Backstrom et al., 2006; Leskovee et al., 2008; Leskovec & Krevl, 2014), one of the largest directed cial networks on the SNAP platform (Leskovec & Sosič, 2016). The DBLP dataset describes citation relationships between scientific articles: Nodes represent articles, and edges indicate citations between them, where the weight on the edge from node node defined as wy if article cites article j, and Waj 0 otherwise. By the nature of the dataset, there are self-loops (a paper does not cite itself). We utilize the dataset topology by giving it a social network interpretation. Specifically, we reinterpret the nodes as users and the edges as a "following" relation: user follows user j if and only Wij 0. this con- text, content shared by user j appears in user i's feed. The LiveJournal dataset already contains the topology of cial network. We assign to both datasets internal opinions uniformly at random from the set 1-,
8484 571
4.1.4. ANALYSIS OF THE COST EVOLUTION
We study the evolution of opinion polarization, i.c., the value of the objective function y in problem (10), across algorithm iterations. We simulate problem (10) 5 times for each of the DBLP, LiveJournal, and Reddit datasets (introduced in Section 4.3) with different initializations. We run each simulation for 25 iterations and display the average cost as well = standard deviation k in Figure 2. We set a = 20, 0.6 for the Reddit dataset and a = 0.05, 7 0.6 for DBLP and LiveJournal. For all n datasets we set = T
function of the iteration
We consider (10) for a varying number of users n to study
the computational cost of evaluating the hypergradient. To
do so, we take the first n nodes and the corresponding in-
ternal opinions. We set We execute iteration of BeeRS for every problem size 10 times the CPU and the GPU. We report mean runtime + standard deviation over 10 runs in Figure 1.
n T a 0.05, and > 0.6.
We observe that the cost consistently decreases across all three datasets, with the largest reduction occurring within the first few iterations. Further, note that BeeRS con- verges in all cases in roughly 20 iterations.
CPU vs GPU
Cost vs Iteration
1e6
Execution Time
10a
0.28
Reddit
DBLP
0.9
DRLP Livejournal Cost
LiveJournal
Reddit Cost
0.8
10a
0.26
0.7
10
LiveJournal
10-7
CPU LiveJoumna
0.0
0.2
0.4
0.6
0.8
1.0
# Users
1e6
0
5
10
15
20
25
Tteration
Figure 1. The mean runtime and + standard deviation over 10 runs of 1 iteration of BeeRS on both a GPU and CPU.
Figure 2. The average cost yp + standard deviation as function of the iteration k. The cost is averaged over different initializa- tions of the edge weights wiç.
We observe that for small numbers of users, the CPU im- plementation has a lower per-iteration runtime. This is not
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Optimizing Social Network Interventions via Hypergradient-Ban Basse Recommender System Design
4.2. Comparison with IPOPT
mender systems that naively minimize disagreement, de- fined D(w, v) D ticular, they investigate the change in polarization in the network before and after modifying the weights to minimize disagreement using NAD. In (Chitra & Musco, 2020), the authors raise the concern that attempting to minimize dis- agreement, e.g., through NAD, can have adverse effects on polarization (and even disagreement itself). In our following studies, we provide some insights on this issue, and show how BeeRS circumvents it.
y2)². In par-
We now consider the problem setup from Section 4.1 and solve problem (10) using the same datasets with IPOPT.
4.2.1. 1. NUMERICAL RESULTS
We compare the CPU implementation of BeeRS to the non- linear solver IPOPT (Wächter& & Biegler, 2006), using the cyipopt (cyipopt) wrapper for Python. For IPOPT, we use a tolerance 1e-3 and constraint-violation tolerance of le-4. For BeeRS, we use E Ie-3, 7 = 0.6, and = 0.05. We solve (10) with both methods for varying number of users n, and repeat each simulation 10 times. We set b To
4.3.1. PROBLEM SETUP
The NAD follows an iterative approach, as summarized in Algorithm in Appendix A.6. In a first step, equi- librium opinions y" (w((t) are computed based on the current weights ((k). In second step, disagreement (ww, y (1)( is minimized by tweaking the weights w under the assumption that the opinions remain unaffected by the new weights and under convex constraints. In particular, the modified network should be similar to the original one, and the out-degree of each node should be preserved. The second step is formalized as the convex program
We report the average runtime + standard deviation for dif- ferent problem sizes for both approaches in Figure 3. Note that the number of nodes considered here is significantly smaller than in Figure due to the computational limitations of IPOPT. We observe that BeeRS outperforms IPOPT for all simulated on both datasets, and by larger margin for larger n. The runtime difference ranges between roughly and 3 orders of magnitude for the largest problems.
In terms of solution quality, BeeRS attains the lowest cost in all cases, whereas the IPOPT implementation exhibits rela- tive suboptimality* of more than 5e-3 on the DBLP dataset and of at least 6e-2 on the LiveJournal dataset. Detailed results are reported in Appendix A.5.
minimize ((++)) EB D(w(k+1)) ,03 (w))))
(11a)
subject to Ej (wi) (k+1) ( 0, E (11b) [[((++)) w(') || 8 || (11c)
where 5 > 0 is a parameter and (7 denotes the adjacency matrix of the original network with weights w0 Intuitively, constraint (11c) ensures that users of the social network do not perceive major differences in shared content and (1lb) guarantees that they spend constant time on the network. The iterations continue until (k)) converges.
BeeRS vs IPOPT
10a
Execution Time
10a
IPOPT DBLP
IPOPT LiveJournal
10a
CPU DBLP
(Chitra & Musco, 2020) note that in certain cases NAD fails to reduce disagreement and polarization P = E" 1 ΣØ y2)² increases. To miti- gate the problem, they propose adding a regularization term X||WI with parameter > > O to the objective (Ila), but the method still struggles to reduce disagreement. We aim to improve upon these results.
CPU Livejournal
10-1
500
1000
1500
2000
Users
Figure 3. The mean runtime and 1 standard deviation over 10 runs of BeeRS and TPOPT (until convergence) on CPU.
We observe that BeeRS can be deployed on (11), without the need to assume opinions are fixed during the optimiza- tion iterations. With BeeRS, we can directly estimate the effects of weight change on the equilibrium opinions (and therefore the effect the objective) through the sensitivity Jy (w). We readapt the original problem
4.3. Comparison with the Literature
Next, we compare BecRS to the algorithm in (Chitra & Musco, 2020). There, the authors propose an extension the FJ dynamies, called NAD (Network Administrator Dynamics), to highlight the potential pitfalls of recom-
minimize yER WEB
Σ Σj Wis - n3)
(12a)
subject to
(11b) and (11c), A(w)y =
(12b)
"We define the relative suboptimality as [k-e"] where yp" is [ the best solution achieved by any of the two approaches.
(12c)
7

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Optimizing Social Network Interventions via Hypergradient-Bas Based Recommender System Design
We add constraint (12c) to take into account the dynamic
opinions y- We use 5 = 0.2 for all subsequent results.
0.125
4.3.2. CASE STUDY: TOY EXAMPLE
0.075
To better understand how the two optimization strategies compare with each other, we use a minimal network that al- lows for disagreement. It consists of two nodes with internal opinions 80 = and $1 = 0, connected by an undirected edge with initial weight ( 1. In this network, we aim to solve (12) without constraint (1 (I)b) (as enforeing this con- straint would restrict the allowed weight to w(()) 1). To evaluate the behavior of NAD we first compute the re- sulting equilibrium opinions by solving (2) for appropriate A(w) (line 2). We find yū (w) = 2 and yī (w) = 1. Dis- agreement is given by D(w,2) y/ (w)) 0.11. In a second step, NAD minimizes disagreement by keeping the opinions constant (line 3). Recalling the definition of disagreement, we see that D(w, y) decreases for fixed y if and only if w is decreased. Therefore, NAD decreases w, in order to min- imize disagreement. Then, NAD proceeds by computing the equilibrium opinions under the updated weight w((++)) followed by a reduction of the edge weight w to minimize disagreement for fixed opinions in a second step. It follows that the edge weight is decreased until constraint (11c) is tight, which, for å 0.2, occurs when = 0.8.
0,050
Tnitial Weight
0.025
BeeRS
NAD
0.000
0.0
0.5
1.0
1.5
2.0
Edge Weight
Figure 4. Behavior of BeeRS and NAD (without regularization) in toy example network. We plot disagrecment as a function of weights for internal opinions 8o = and 81 = 0. The feasible region is shown in white.
Table 1. Comparison hetween BeeRS, NAD, and NAD* with X 0.2. The runtime averaged over 10 executions.
BEERS
NAD
NAD*
MEAN RUNTIME, S
141.6
35.9
27.3
MIN. RUNTIME, S
137.7
34.2
25.8
MAX. RUNTIME,
150.4
38.5
28.3
CHANGE IN POL.
-10.2%
+128.7%
-0.7%
CHANGE IN DISAG. D
-1.5%
+11.4%
+0.3%
On the other hand, BeeRS updates the weight in the direc- tion of steepest descent, taking into account the effect on opinions. By deploying it on such a simple problem, we ob- serve that, as opposed to NAD, BeeRS increases the weight to w 1.2. To compare the performance of BeeRS and NAD in terms of disagreement, opinions for all [0, 2] by solving (2), and we illustrate the resulting disagreement in Figure 4. We mark the solution of both algorithms, and we observe that BeeRS outperforms
consists of 556 nodes and 8969 edges. We remove 3 discon- nected users, leading to a total of 553 users. The resulting problem has m = nnn - 1) = 305, 256 upper-level vari- ables. Since the edges are undirected, we set half of the variables through constraint, as shown in the toy example.
compute equilibrium
We deploy BeeRS on CPU with & le-2, 7 0.6. and 1000, (see Appendix A.4.1). We use the default implementation of NAD from (Chitra, 2019), making minor modifications to adapt the code to our specific problem. We run both the default NAD and NAD with regularization À = 0.2, referred to as NAD". We summarize the runtime as well as the change in polarization and disagreement of each algorithm in Table 1. We observe substantial differences in performance among the three algorithms.
NAD. We argue that the reason for this discrepancy comes from the fact that our algorithm is able to anticipate the effect of changing weights on the equilibrium opinions and, hence, move in a direction that minimizes disagreement, By contrast, NAD myopically optimizes weights, neglecting the dynamics of opinions, and thus exacerbates disagreement.
4.3.3. NUMERICAL RESULTS: ADDRESSING A REAL- WORLD PROBLEM
Contrary to (Chitra & Musco, 2020), BeeRS achieves a decrease in disagreement: the objective of NAD and prob- lem (12). Further, the effect on polarization is different, as BeeRS leads to a significant decrease of polarization, whereas NAD without regularization increases polarization. Although NAD* improves on this aspect, it is still outper- formed by BeeRS in both disagreement and polarization reduction. However, the superior performance of BeeRS
Next, we deploy both algorithms on real-world data, We use the undirected Reddit dataset (De et al., 2014), where external opinions y were extracted by sentiment analysis. Analogously to (Chitra& & Musco, 2020), we set the internal opinions E [0, 1] by computing A(w)u y. The dataset
'ee model the undirccted edge weight with the two variables Wol and wto. and the constraint wto1 = w10-
adopt E [0, 1].
"Chitra & Musco, 2020) use E 1-1, 1] in their formulation. However, consistent with their implementation (Chitra, 2019),
"The current implementation of BeeRS does support the Frobenius norm constraint on the GPU.
8

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Optimizing Social Network Interventions via Hypergradient-Barn Based Recommender System Design
Opinion on Head of Edge
1.0
Opinion on Head of Edge
Weight Dift. (ony (iga)
Opinion Huad of Edge
1.0
Woight Diff. (wly ((ga))
0.51
0.51
0.5
0.5
0.50
0.50
0,50
0.49
0.0
Weight Diff.
0.49
0.49
0.0
0.48
0.48
0.5
0.48
0.5
0.47
0.47
-1.0
0.47
-10
0.48
0.50
0.48
0.50
0.48
0.50
Opinion Tail of Edge
Opinion Tail Edge
Opinion Tail of Edge
(a) BeeRS
(b) NAD without regularization
(c) NAD with regularization
Figure 5. Optimal edge weight modification in relation external opinions connected by edge computed by three approaches. The horizontal and vertical axes show the external opinion before intervention the tail and head of an edge, respectively. The color encodes the change of edge weight during the intervention, i.e., Wij "
comes at the price of an increased runtime.
nism followed by NAD does not encapsulate the fact that, when we connect opposing opinions under the FJ dynamics, more moderate opinions are formed through averaging, favorable phenomenon in terms of minimizing disagreement and polarization. This is in line with the weight update rule of NAD (line 3) that ignores the dynamics. Instead, the back- propagation through equilibrium opinions step of BeeRS is able to explicitly incorporate and exploit this effect.
Next, we study the performance mismatch between BeeRS and NAD. To do so, we visualize the intervention mecha- nism, i..., the change of weights Wij wh in Figures 5a, 5b, and 5c for BeeRS, NAD, and NAD", respectively.
All figures show the external opinion of users before the intervention on the horizontal and vertical axis. The x- and y y-coordinates of each point represent the opinion of the user on the tail and head of the edge, respectively. The colour of each point encodes the intervention, i.e., the weight change of the corresponding edge. Since we consider undirected network, the influence over edges is bidirectional and the plots are symmetric. As we can modify the weight all edges (even the ones not present in the initial configuration), there are roughly 300,000 points to be plotted. To enhance the interpretability of the figures, we order all interventions by increasing absolute value and plot them increasingly top of each other, i.e., the largest weight changes are drawn on top and are therefore visible. Note that, large number of edges are not visible as they undergo no change in weight.
5. Conclusion and Future Work
We formulated the task of finding the best network inter- vention given any differentiable objective function under FJ dynamics as an optimization problem with non-convex constraints, and we solved with a gradient- gradient-based algo- rithm. We demonstrated the flexibility and scalability of our method by successfully deploying it on various optimiza- tion problems of different sizes. Our algorithm significantly outperforms state-of oftte-of-the-art solver in terms of runtime, while achieving the same solution quality.
As future work, we would like to make our approach more practical by connecting the weight modifications to action- able interventions in network, such as boost or penalty fac- tors in weighted PageRank fashion (Wenpu Ghorbani, 2004). Further, envision increasing the scalability of our method in two ways: (i) updating only (potentially randomly-chosen) subset of the modifiable weights at each iteration, similar to coordinate descent methods (Wright, 2015), and (ii) solving the linear systems (2) and (9) in a distributed fashion across multiple devices (CPUs or GPUs).
In contrast to Figure 5a and 5b, we cannot identify any ob- vious intervention for N in Figure 5c. The intervention seems lack a specific pattern, although we note that many edges experience large decrease in weight (dark blue dots) and none experience large increase. Hence, subse- quently focus on BeeRS and NAD without regularization.
Notably, BeeRS and NAD seem to behave exactly opposite. BeeRS mainly reinforces connections between above- and below-average opinions and weakens edges connecting opin- ions of the same bias. Conversely, NAD strengthens edges between similar opinions and weakens the ones between opposing views.
One limitation of our algorithm the convexity assumption W, which precludes the use of binary decision variables: is edge present or not? Our framework could incorpo- rate binary decisions by relaxing the binary constraints or employing heuristics. Preliminary numerical experiments indicate this to be a promising future research direction.
This behavior aligns with the observations in Section 4.3.2 and explains the superior performance BeeRS in minimiz- ing disagreement and polarization. The intervention mecha-

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Optimizing Social Network Interventions via Hypergradient-Base Bastd Recommender System Design
Acknowledgements
1804840115. URL https: /www www.w.ppsss.oo....
abs ppas.. 1804840115.
This work was supported by the Swiss National Science Foundation under the NCCR Automation (grant agreement 5INF40 (CNF402225155) and by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
Bakshy, E., Messing, S., and Adamic, I., A. Exposure to ideologically diverse news and opinion on facebook. Science, science. 160. URL https: /www://wwwwww.....w science. org/doi//e abs///b////// / /10. 112111166/6 htt/:cciieeeecce www.eeeeee.alal.....
2015, doi: 10.1126/
Impact Statement
Bindel, D., Kleinberg, J., and Oren, S. How bad is forming your own opinion?
This paper presents work whose goal to advance optimiza- tion methods for recommender systems. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
Games and
Economic Behavior, 92:248-265, 2015. ISSN 0899- 8256. doi:
004. URL httpp::/wwww..ceeeedeeeeeee.eeee
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Optimizing Social Network Interventions via Hypergradient-Ban Based Recommender System Design
A. Appendix
A.I. Computation of J Fu,,w,
Combining the definition of A(w) in (2) with the definition F(w, v) = A(w)y - s, we arrive at
+ Ejev wi
W112
WIn
F(w,y)- =
(13)
1 Σ Waj
A(w)
Now we can compute J1 F(w, y), the partial Jacobian of F(w, y) with respect to w, and show that it continuous both arguments. Recall the definition w = [w12..... Win; W21 .... W2n:..... Wx1..... (n(n-1)] R with n(nn 1)
(there no self-loops, thus drop the entries W/y for computational efficiency).
Lemma A.1. The partial Jacobian of F(w,y) with respect to i.e., J1 (we, v), is continuous in both arguments.
Proof. Considering (13), it is easy to see that J1 (w,, y) is given by
y1
1/2
11
1/3
21
Un 0
0 0
0
J. Fu,,
Raxm (14)
0
0
0
0
Un 1/2
Yn - Yn-
which is continuous.
A.2. Implementation Details
We implement the algorithm in Python 3.12. We provide both GPU- and a CPU-compatible implementation. The GPU version performs most computations with JAX (Bradbury et al., 2018) and has some NumPy (Harris et al., 2020) dependencies. We solve the linear systems (2) and (9) with a jcx.scipy implementation of the GMRES algorithm (Saad & Schultz, 1986). For the projection, we use the jaxopt (Blondel et al., 2022) BoxOSQP solver. JAX functions are just-in-time compiled if applicable. We represent the adjacency matrix of the network and the Jacobians as sparse esr array, which allows efficient handling of large datasets. The upper-level gradients 1,22 are computed with the automatic differentiation (autodiff) functionalities of JAX.
The CPU version is based on NumPy. Unlike the GPU implementation, we allow both sparse and dense representations of the network's 's adjacency matrix and the Jacobians. For dense matrices, we rely on NumPy to solve the linear systems (2) and (9), for sparse matrices, we use a SciPy (Virtanen et al., 2020) implementation of the GMRES algorithm. In both cases, we use CVXpy (Diamond & Boyd, 2016; Agrawal et al., 2018) to perform the projections with the following solvers: Clarabel (Goulart & Chen, 2024) for quadratic programs, and SCS (O' OoDonngh et al., 2016) for second-order cone programs. The upper-level gradients V 1,2 computed with the automatic differentiation (autodiff) functionalities of PyTorch (Paszke et al., 2019).
A.3. Hardware Specifications
Experiments were conducted Windows 11 machine using Windows Subsystem for Linux (WSL) version either on an Intel Core 1177-1170FF 170FF CPU at 2.50 GHz with 8 cores and 16 logical processors and 32 GB of RAM on NVIDIA GeForce RTX 3060 Ti GPU, with GB of GDDR6 video memory, rumning CUDA 12.6 and cuDNN 9.5.1.
A.4. Choosing the Hyperparameters- Ablation Study
As discussed in Section 3, BeeRS employs projected gradient descent with momentum (7). This method uses two hyperparameters, namely step size and momentum parameter We conduct an ablation study to find parameters leading to satisfying results in terms of runtime and minimum cost.
We consider the same setup as in Section 4.1, i..,, problem (10) on the DBLP dataset version 10 (Tang et al., 2008). We consider the first n = 1000 users and set = 100. We initialize variables, i.c., all connections to the news agency, with the value O. We use a tolerance of 1e-3.
14

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Optimizing Social Network Interventions via Hypergradient-Ban Based Recommender System Design
We now solve this problem with BeeRS. We grid the hyperparameters for values of a E (0,500] and 7 [0, 0.95). To enhance visibility, we display only some of the grid points in Figure in particular {0.001, 0.002, ... 0.05, 0.06, ..... 0.1, 0.2, 0.2,..., 1 and E {0.0, 0.3, 0.6, 0.9}. In the left subplot, we show the num- ber of iterations until convergence on the vertical axis as function of on the horizontal axis. According to the legend, we plot distinct line for every choice of y. In the right subplot, we show analogously the minimum achieved cost on the vertical axis. Due to the deterministic nature BeeRS, we run the simulation with every set distinct a and ? exactly once.
100
492.5
Y 0.0
Y
80
0.3
490.0
0.3
0.6
Minimum Cost
0.6
Iterations
60
0.9
487.5
0.9
485.0
40
482.5
20
480.0
10
10-2
10
10-3
10
10
Figure We vary the step size a and display the resulting number iterations (left) and the achieved minimum cost (right) for some momentum parameters 7,
We observe that, for small values of a, we achieve convergence with fewer iterations for larger values of than for smaller values of r. For larger values of a, the number of iteration until convergence first gets closer for different values of y, and some oscillations begin to appear for = 0.9, and later, the number of iterations until convergence starts to increase again.
Regarding the minimum cost achieved, we observe that for small values of a, 7 0.9 clearly outperforms all other choices. When increased, the performance for other values of y improves quickly, and medium value of a, 7 0.6 achieve lower cost than y 0.9. For large values of the minimum achieved cost abruptly rises for some values of and gradually increases for other values of y.
Considering these results, medium value of a, e.g. = 0.05, together with = 0.6 provides a good compromise between the number iterations until convergence and the achieved minimum cost, and effectively avoids numerical instabilitics causing convergence issues for larger values of a.
A.4.1. CHOOSING AND FOR DIFFERENT NUMBER OF USERS
While the ablation study in Section A.4 was only carried out for the first 1000 users of the DBLP dataset, observed that the resulting hyperparameters
0.05 and y 0.6 work well for both other problem dimensions n and for the LiveJournal dataset, as long as we consider problem (10). Differently, due to the change of objective in Section 4.3 and the smaller number of variables m, the previously used step size yielded insufficient progress. We performed a grid search over the step size and selected the best value, a = 1000, while keeping 7 = 0.6 fixed, as determined in the aforementioned ablation study.
A.5. Further Results of Section 4.2
In addition to the results provided in Section 4.2, we provide Table 2 the relative suboptimality for the compared approaches. We define the relative suboptimality as
l
1p"
where yp the best solution achieved by any of the two approaches.
15

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Optimizing Social Network Interventions via Hypergradient-Baxe Recommender System Design
Table Comparison between BeeRS and POPTT in terms of relative suboptimality.
NUMBER USERS
100
200
300
400
500
1000
2000
DBLP
REL. SUBOPT. IPOPT
3.04e-2
1.53e-2
1.35e-2
1.02e-2
1.06e-2
2.30e-2
5.59e-3
REL. SUBOPT. BEERS
LIVEJOURNAL
REL. SUBOPT. OF IPOPT
6.96e-2
7.89e-2 0
8.31e-2 0
9.09e-2 0
9.31e-2
8.74e-2
1.15e-1
REL. SUBOPT. OF BEERS
0
A.6. Network Administrator Dynamics Alg.
In the following, we repeat the algorithm of (Chitra Musco, 2020).
Algorithm 2 NAD (Network Administrator Dynamics)
while not converged do
2: Vk For given network weights we compute the equilibrium opinions with (2)
3: Wi For fixed opinions y. solve (11)
4: A +- k: +
5:
end while
16


Reviews:
[{"review_id": "WautP7XaYX", "reviewer": "Reviewer_nKxb", "summary": "This paper proposes a gradient descent based method for modifying network weights towards an optimal (general) downstream performance metric, under the framework of Friedkin-Johnsons opinion dynamics. Experiments show significant improvement on computation time on large-scale real-world dataset.\n\n## update after rebuttal\nI have read all of the rebuttals, and would like to keep my current rating.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their very positive evaluation and their kind suggestions. \nWe will make sure to stress the advantage of considering directed networks in the paper and include the references suggested by the reviewer.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Not all, but those in Sections 3 and 4.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "This work has substantial contribution on top of the existing literature.", "essential_references_not_discussed": "I feel that the references being cited in the paper are relatively old. There are a few more recent papers working on similar topics  - though none of them are as generalized as this work, but it would be nice to mention them since  it would help further show the great contribution in this paper.", "other_strengths_and_weaknesses": "**Strengths**\n\nThis is a very nice paper on Friedkin-Johnson opinion dynamics, definitely among the best I have seen in a while. I especially like two things in this paper, the first one not even highlighted by authors themselves:\n\n- It gets rid of the the symmetry assumption made the vast majority of papers working on FJ model. Previous works did so mainly because they needed the symmetry condition to derive the closed-form equilibrium state (i.e. (I+L)^-1 s), which is the basis for many further derivations. This work, however, starts by assuming the network to be directed, and cleverly avoids having to explicitly work with the closed-form solution by denoting it as y*(w). Prop. 2.1 is also good observation.\n\n- It observes and successfully addresses a very important problem, which is that existing studies based on closed-form sensitivity analysis are very computationally expensive, since they usually involves inverting a gigantic matrix. This works elegantly solves this problem with gradient descent. It is also nice to see Assumption 2.2 which generalizes some of the specific performance metrics that people have been studying for years into a more general form.\n\n\n**Weakness**\n- The references being mentioned in \"literature review\" are mostly before 2020. This might give the readers an impression that the study of FJ model is less active in most recent years, which is not true. For example, [1, 2] involve a similar (albeit narrower) topic of sensitivity analysis on network weights. I encourage the authors to also include these more recent papers, which should further help highlight the broader contribution of this paper.\n\n[1] On the Relationship Between Relevance and Conflict in Online Social Link Recommendations, Wang et al., NeurIPS 2023.\n\n[2] Minimizing Polarization and Disagreement in the Friedkin–Johnsen Model with Unknown Innate Opinions, ArXiv, 2025.", "other_comments_or_suggestions": "n/a", "questions_for_authors": "n/a", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "z1uDvAFgos", "reviewer": "Reviewer_soGD", "summary": "This paper proposes a novel method, Best Intervention for Recommender Systems (BeeRS), which provides a method of using the hypergradient to optimize connection weights in a social network for certain objectives (for example, reducing polarization). The paper models the social network via the Friedkin-Johnson (FJ) model, where each actor has a constant internal opinion and evolving external opinion that is updated and influenced by neighbors in the network. The problem is formulated as bilevel optimization problem, where the lower level models the opinion dynamics among users and the upper level models the recommendation algorithm (termed the “leader”) that adjusts network weights to achieve an objective. Numerical results show an improvement over IPOPT on runtime for the optimization problem. The algorithm is also compared to the Network Administrator Dynamics (NAD) algorithm on real world data from Reddit, and it shows improvement on both the change in polarization and change in disagreement (at the price of an increased runtime).", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their positive evaluation and their constructive comments.\n\nConcerning the weaknesses and questions raised by the reviewer:\n\n**Question/Weakness 1** Thanks for pointing out this aspect. We foresee using the weights as a penalty (/boosting) \nfactor in a Weighted PageRank (WPR) [1] or EdgeRank [2] algorithm fashion.\nThese algorithms take into account the importance of both the inlinks and the outlinks of the pages and distribute rank scores \nbased on the popularity of the pages:\nIf the network leader (upper level) decreases the weight of the link from user A to user B then this translates into a penalty \nfor all the content shared from A which is a candidate to be visible to B. \nThe details of how this would be algorithmically handled, however, are out of the scope of this work and an interesting direction for future research.\n\n**Question 2** We thank the reviewer for their suggestion. Indeed, our algorithmic approach and our implementation allow the user to specify which\nweights are immutable and which ones are variable.\nOne meaningful way to choose which weights to optimize (or which areas of the networks to affect) would be to use heuristics to quantify\nthe importance of nodes, and then optimize the weights of edges connected to these nodes. \nFor instance, we could use the degree of a node or, more generally, other centrality measures [3].\nThese approaches to choosing which weights to update are especially important since we, typically, do not want\nto modify the weights too much as they affect the users' experience in the network.\n\nAnother interesting way to optimize for performance is to randomly choose which weights to update at each iteration, giving rise\nto a stochastic/mini-batch version of our algorithm.\nThis would be particularly helpful when dealing with networks of huge size.\n\n[1] Xing, Wenpu, and Ali Ghorbani. \"Weighted pagerank algorithm.\" Proceedings. Second Annual Conference on Communication Networks and Services Research, IEEE, 2004.\n\n[2] EdgeRank: The Secret Sauce That Makes Facebook's News Feed Tick. techcrunch.com. 2010-04-22. Retrieved 2012-12-08.\n\n[3] Bloch, Francis, Matthew O. Jackson, and Pietro Tebaldi. \"Centrality measures in networks.\" Social Choice and Welfare 61.2 (2023): 413-453.", "claims_and_evidence": "The claims in the submission are supported by clear evidence. In particular, the experiments are comprehensive and show the improvement of BeeRS over the existing NAD algorithm.", "methods_and_evaluation_criteria": "The methods used generally make sense for the problem at hand. I do note some concerns about the practicality of the simulation being conducted (see Other Strengths and Weaknesses below), but overall the methods work well for the problem setup and assumptions.", "theoretical_claims": "I did not verify the proofs of any theoretical claims.", "experimental_designs_or_analyses": "The experimental designs do seem sound, and I especially appreciate the use of large, real-world datasets for the work.", "relation_to_broader_scientific_literature": "This work is a useful extension of existing work on social network interventions, specifically motivated by polarization and disagreement. The advantage of this work comes from the flexibility of the method, given that any differentiable objective can be plugged into the BeeRS algorithm.", "essential_references_not_discussed": "I am not aware of any essential references that were not discussed.", "other_strengths_and_weaknesses": "The primary strengths of this paper are in the problem formulation and algorithm construction. The paper provides a flexible framework for interventions on social networks. It is also written clearly and has thorough experiments to validate the claims that the authors make regarding the algorithm’s performance.\n\nI think the primary weakness of the paper is a lack of connection back to real-world recommendation systems. While the paper does a good job of using real-world networks from DBLP and Reddit, it is difficult to see how the idealized “leader” in this setup translates to a practical multi-level recommender system that exists out in the wild on a social network. For example, in this work, the leader accomplishes goals by adjusting network weights between users, thereby changing how users influence one another. The paper states that such weights could represent the volume of content from one user that another sees or similar metrics. In practice, social media recommender systems work by ranking content to be seen by a user. How does a ranker system like that “adjust” the weights between users? Does it discount or boost the ranks of certain posts? Does it have to maintain an additional influence model that feeds into the ranking algorithm? This gap is not fatal to the paper, but to me it is a missing piece that makes the difference between a good paper and a great one.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. How would weight adjustments work in a practical real world system? Consider for example a timeline ranking system on a network like Facebook or X. \n\n2. Can the algorithm be further optimized for performance by considering updates in portions of the graph rather than updating the weights of the entire network? Perhaps there is some way to tell which areas of the network will be most influential for the objective overall (e.g. particularly influential users).", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "inkmkwnWhM", "reviewer": "Reviewer_pBeX", "summary": "This paper investigates opinion polarization in social networks using a large-scale optimization approach to modify network interactions based on the Friedkin-Johnsen model. The authors propose a gradient-based algorithm designed to address this problem in a scalable and computationally efficient manner. Some empirical analysis results are presented to demonstrate the effectiveness.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the positive comments and their careful reading.\n\n**Methods and Evaluation Criteria** The concern raised on \"Methods and Evaluation Criteria\" is a valid and important one.\nIndeed, for problems/networks of huge dimensions our algorithm might face computational bottlenecks.\nNonetheless, our scheme could be adapted and improved in a number of ways to address such practical problems.\nWe outline some next:\n\n1. We could further exploit the capabilities of GPU computation. In Subsection 4.1, we successfully solve a problem\n    with 3 million decision variables on a GPU that currently is not among the state-of-the-art (NVIDIA GeForce RTX 3060 Ti).\n    By using a more performant GPU, or even a cluster of them, we could definitely handle problems of even larger dimensions.\n    In fact, problems with hundreds of millions of decision variables are routinely solved with GPUs in the context of neural network training.\n    This flexibility is a significant advantage of our algorithm, emanating from the fact that it is a simple, first-order method.\n    (Similar to the ones used in neural network training.)\n\n2. We could reduce the number of weights that we optimize over by employing appropriate heuristics\n    to identify the most important parts of the network,\n    introducing a tradeoff between solution quality and computational requirements.\n    Indeed, our algorithmic approach and our implementation allow the user to specify which\n    weights are immutable and which ones are variable.\n   One meaningful way to choose which weights to optimize (or which areas of the networks to affect) would be to use heuristics to quantify\n  the importance of nodes, and then optimize the weights of edges connected to these nodes. \n  For instance, we could use the degree of a node or, more generally, other centrality measures [1].\nThese approaches to choosing which weights to update are especially important since we, typically, do not want\nto modify the weights too much as they affect the users' experience in the network.\n\n3. We could update only a (randomly chosen) subset of the weights at each iteration, thus giving rise \n    to a stochastic/mini-batch version of our algorithm.\n    This is a GPU-friendly way of addressing problems with huge dimensionality.\n\n[1] Bloch, Francis, Matthew O. Jackson, and Pietro Tebaldi. \"Centrality measures in networks.\" Social Choice and Welfare 61.2 (2023): 413-453.\n\n**Experimental Designs or Analyses/Question** \n\n- We thank the reviewer for their suggestion to include additional datasets in our simulations.\nPlease note that in our original manuscript, beside DBPL, we have also tested our algorithm on the real-world Reddit Dataset.\nFurther,  we have also deployed our algorithm on the soc-LiveJournal1 dataset, \nand will include the results in our revised manuscript.\nWe summarize the results from our additional simulations in Figures 1-3, and Table 1 in https://imgur.com/a/8GbzUNw.\nWe 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.\n\n- We thank the reviewer for suggesting an empirical analysis with a uniformed metric.\nWe include an additional simulation where we study the objective $\\varphi$ as a function of the iteratation $k$.\nIn 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.\n\n    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.", "claims_and_evidence": "In general, this paper features a clear motivation of problem definitions and an intuitive proposed method, is generally well written. Assumptions are properly outlined and theoretical derivations are properly provided to motivate the proposed solution. In particular, I like the style that authors offer detailed derivations in Sections 2 and 3, which can offer a clear view on the conditions as well as prerequisites for their contents to be valid.", "methods_and_evaluation_criteria": "The proposed BeeRS algorithm is intuitive and well-motivated, backed by straightforward derivations. On the other hand, as mentioned by the authors in their experiments, \"the superior performance of BeeRS comes at the price of an increased runtime\", indicating that while the proposed hypergradient method can be time consuming. While this problem can be somewhat alleviated by GPU parallel computing, it could still be a problem given a large number of users or nodes.", "theoretical_claims": "The main derivations are in the main body of the paper and I found them to be quite intuitive and clear.", "experimental_designs_or_analyses": "Some empirical results are provided to demonstrate the characteristics of the proposed BeeRS method, including running time, GPU-aided computation, and the distribution of edge weights after optimization. However, it seems that the empirical analysis lacks a uniformed metric in terms of the performance, such as the direct illustration of the cost ($\\phi$) and the performance tradeoff. In particular, only one real-world dataset DBLP is involved in the experiments, which can be insufficient in terms of validating the empirical effectiveness.", "relation_to_broader_scientific_literature": "The paper extends existing literature on opinion dynamics and network interventions by formulating social network influence optimization as a scalable hypergradient-based optimization problem under the classical Friedkin-Johnsen model. The proposed hypergradient method can be of independent interest to other audience apart from the Social Sciences community.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "Please see my comments above in terms of the experiments, in \"Experimental Designs Or Analyses\". In particular, experiments on additional real-world datasets are encouraged to comprehensively demonstrate the effectiveness of the proposed method.", "other_comments_or_suggestions": "None", "questions_for_authors": "Please refer to my comments in terms of scalability, efficiency, and experiment comprehensiveness. In particular, please see my comments above in terms of the experiments, in \"Experimental Designs Or Analyses\". In particular, experiments on additional real-world datasets are encouraged to comprehensively demonstrate the effectiveness of the proposed method.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xxIFG2BX72", "reviewer": "Reviewer_5iEL", "summary": "The paper studies interventions in the Friedkin-Johnsen opinion formation model. This model is defined by a graph in which each node has a fixed innate opinion and a time-dependent expressed opinion; the model is often used to study polarization and disagreement in networks from a mathematical angle. Many recent works studied interventions in this opinion formation model by considering optimization problems which *either* change the graph topology *or* the innate opinions of the nodes.\n\nThe submission's main contribution is to show that one can *simultaneously* optimize for the network topology *and* the opinions. This is done via gradient descent by considering the hypergradient; the main technical advantage here is that this allows to deal with the expressed opinions $y$ which are given by an equation system $Ay=s$ where $s$ is part of the input and $A$ depends on the graph topology.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their feedback and constructive comments. \nFirst, we would like to clarify one important point.\nOur algorithm optimizes only over the network topology, i.e., the network weights $w$, but not the _innate_ opinions $s$.\nThe main optimization problem that we aim to solve, equation (3), has as decision variables\nthe weights $w$ and equilibrium _expressed_ opinions $y$.\nSince $y$ is uniquely determined by the constraint (3b), we optimize over $w$ solely.\nWe could readily extend our framework to optimize for both $w$ and $s$ simultaneously, as suggested by the reviewer, but\nthis is not something that we investigate in this paper.\nWe will make sure to clearly elaborate on this point in our revised manuscript.\n\n**Weakness 1** We thank the reviewer for their insightful comment. \nIndeed, 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.\nIn this vein, our goal is to  understand the mechanism by which NAD operates and, in turn, \ndemonstrate how our algorithm succeeds by using a different mechanism in the same problem setting. \nWe believe our comparison to be a meaningful one for two reasons:\n\n1. 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.\n\n2. our algorithm manages to fix (some of) \nthe issues that Chitra \\& Musco highlighted with NAD. \nIn fact, in their paper they also propose fixing NAD with NAD*, which we also include in our comparison results. \nIn essence, we provide a potential solution to one of the main issues raised by Chitra \\& Musco. \nWe 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.\n\n**Weakness 2** Regarding the optimization of graph topology and innate opinions, please see our discussion above.\nRegarding the polarization metric, we have rerun our simulations using the metric proposed by the reviewer\nand will include them in the corresponding section.\nWe 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. \nAdditionally, BeeRS decreases the polarization in Section 4.1.3 even further with the new polarization metric (-44\\% compared to -39\\%).\n\n**Weakness 3** We thank the reviewer for their useful suggestion.\nWe 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.\nThe collected results further confirm the effectiveness of our algorithm.\n\n**Weakness 4** We acknowledge the reviewer's suggestion to analyze the rates of convergence of gradient descent.\nWe have performed an empirical analysis of the learning curves of our algorithm, namely objective vs. number of iterations,\nas also suggested by reviewer pBeX; please refer to that response for more information.\nPerforming a theoretical analysis can be challenging due to the non-convexity of the problem and could potentially\nrequire a paper on its own.\nOne such analysis, for example, is the classical result in [1, Ch. 1.2] which proves that _vanilla_ gradient descent \nconverges to an $\\epsilon$-stationary point at a rate of $O(\\epsilon^{-2})$, for the case of an unconstrained non-convex smooth optimization\nproblem.\nTo the best of our knowledge, convergence rates for projected gradient-descent with momentum on a non-convex objective is an open\nproblem.\nWe consider such theoretical analyses beyond the scope of our paper, \nand reserve the investigation for future work.\n\n**Other Comments or Suggestions 1** Our algorithm can deal with any differentiable objective function, including the one mentioned.\nWe collected new results for the proposed metric as mentioned above, with a preview available at https://imgur.com/a/8GbzUNw.\n\n**Other Comments or Suggestions 2** We will highlight this difference in notation.\n\n**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.\n\n**Questions** Please refer to our answers in **Weakness 2** and **Other Comments or Suggestions** for Question 1,\nand to the first paragraph of our response for Question 2.\n\n[1] Nesterov, Yurii. Lectures on convex optimization. Vol. 137. Berlin: Springer, 2018.", "claims_and_evidence": "I think it is mostly fine, but the comparison with the work of Chitra and Musco is inappropriate. See my more detailed comments below.", "methods_and_evaluation_criteria": "The definition that was used for polarization in the experiments is non-standard by now. See my comments below.", "theoretical_claims": "The theoretical derivations appear to be correct, but I did not check in detail.", "experimental_designs_or_analyses": "See below.", "relation_to_broader_scientific_literature": "I think the main contribution of optimizing opinions and graph topology simultaneously is definitely interesting and something I have not seen before. I appreciate these results. However, in terms of motivation, I think it could be improved (see below).", "essential_references_not_discussed": "Some more papers could be cited, see my comments below.", "other_strengths_and_weaknesses": "Strenghts:\n* The theoretical results are interesting and a good contribution to the literature.\n* The paper is easy to follow.\n\nWeaknesses:\n* The comparison with the Chitra and Musco paper in the experiments is somewhat misleading. The point of the Chitra and Musco paper was not to minimize the overall disagreement; instead, they proposed a model which proceeds in rounds, where the network administrator minimizes the disagreement and then subsequently the opinions converge. Their goal was to study how this particular process might lead to increased polarization. Thus, stating that the algorithm from the paper achieves better objective function values because it can simultaneously minimize over the graph topology and the opinions simultaneously is just not a meaningful comparison.\n* Typically, papers in this domain give a model of interventions and then provide optimization algorithms. While the paper does give a quite general optimization approach, it would be better if there were concrete examples in which settings one would expect to optimize over the graph topology and the innate opinions simultaneously. I am also not convinced by the example in Section 4.1.1 because it uses a polarization-metric that is non-standard in the meantime.\n* The paper is only evaluated on two datasets, only one of which actually corresponds to a social network. I was surprised that the DBLP dataset (which is a citation network) was used, given that platforms like SNAP make a lot of social network datasets available.\n* There is no analysis on how quickly the gradient descent algorithm converges to an optimal solution.\n\n**Update after rebuttal:** The authors have addressed several of my concerns in their rebuttal and thus I have increased my score. I still think that the optimization problems they study could be better motivated, though.", "other_comments_or_suggestions": "* Recent papers in this line of work define the polarization as the variance of the opinions, i.e., $\\sum_u (y_u - \\bar{y})^2$ where $\\bar{y}$ is the average opinion over all nodes. This is not the same as defined in the paper. Sometimes people do consider polarization as $\\sum_u y_u^2$ but only if the average opinion is $0$, in which case both notions coincide; here, the assumption of mean-centered opinions is crucial but it is not made in the paper. I would find it more interesting if the paper could show results for this notion of polarization.\n* The notation used in the paper is quite non-standard. Typically, the expressed opinions are referred to as $z$ and not as $y$. Also, what is $A$ in the paper is typically referred to as $I+L$ (identity matrix + Laplacian).\n* There are quite a few references that could be added. For instance, recent works from Aris Gionis and Sijing Tu gave convergence analysis for gradient descent-based methods in the FJ model and also studied a similar setting with news agencies.", "questions_for_authors": "* Does your approach also work for the more common notion of polarization that I mentioned above?\n* In what kind of model would it make sense that one can simultaneously impact the graph topology and the innate opinions?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper proposes a general and flexible optimization framework in the context of Friedkin-Johnsen opinion dynamics model. The framework allows the use of gradient-based algorithm to solve the optimization tasks, with applications in polarization and other network related tasks. After the rebuttal period, all reviewers are convinced on the contribution of the paper and support the acceptance of the paper. Therefore I recommend the paper be accepted at ICML'2025. 

The authors need to seriously address all reviewer comments in their camera-ready version. In particular, I share the concern of one reviewer on the feasibility of intervention in the model: the intervention is by changing the edge weights, but how could it be realized in an actual social platform or recommendation system? More discussions and perhaps examples are needed to support this intervention model.
Decision: Accept (poster)