User-Creator Feature Polarization in Recommender Systems with Dual Influence
Authors: Tao Lin
Kun Jin
Andrew Estornell
Xiaoying Zhang
Yiling Chen
Yang Liu
Venue: NeurIPS 2024 poster (2024)
Keywords: recommender systems
performativity
preference dynamics
diversity
polarization
Primary area: algorithmic_game_theory
TL;DR: We show that recommender systems with dual influences on users and creators are guaranteed to polarize, and discuss how to prevent it.

Abstract: Recommender systems serve the dual purpose of presenting relevant content to users and helping content creators reach their target audience. The dual nature of these systems naturally influences both users and creators: users' preferences are affected by the items they are recommended, while creators may be incentivized to alter their content to attract more users. We define a model, called user-creator feature dynamics, to capture the dual influence of recommender systems. We prove that a recommender system with dual influence is guaranteed to polarize, causing diversity loss in the system. We then investigate, both theoretically and empirically, approaches for mitigating polarization and promoting diversity in recommender systems. Unexpectedly, we find that common diversity-promoting approaches do not work in the presence of dual influence, while relevancy-optimizing methods like top-$k$ truncation can prevent polarization and improve diversity of the system.

User-Creator Feature Polarization in Recommender Systems with Dual Influence
Tao Lin"
Kun Jin Google kun funjjin@oogge.coo
Harvard University
tlin@g.hher hww.ard.
Andrew Estornell ByteDance andrew woddre..ettoneelle. settoreell@byteeaneeee
Xiaoying Zhang ByteDance
zhangxiaoying www.eyyyyyeyeee..ooom
Yiling Chen
Yang Liu University of California, Santa Cruz
Harvard University yilingdseas. harvard. edu
yangl iuducac.edu
Abstract
Recommender systems serve the dual purpose of presenting relevant content users and helping content creators reach their target audience. The dual nature these systems naturally influences both users and creators: users' preferences affected by the items they are recommended, while creators may be incentivized alter their content attract feature dynamics, to capture the dual influence of recommender systems. We prove that recommender system with dual influence is guaranteed to polarize, causing diversity loss in the system. We then investigate, both theoretically and empirically, approaches for mitigating polarization and promoting diversity in recommender systems. Unexpectedly, we find that common diversity-promooing approaches do not work in the presence dual influence, while slevancy-optimizing methods like top-k truncation can prevent polarization and improve diversity of the system.
We define model, called user-creator
Introduction
From restaurant selection, video watching, to apartment renting, recommender systems play a pivotal role across plethora of real-world domains. These systems match users with content they like, and help creators (those producing the content) identify their target audiences. Nevertheless, behind such success, concerns have emerged regarding possible harmful outcomes of recommender systems, in particular, filter bubbles [32 5 and polarization [36] outcomes with insufficient recommendation diversity and creation diversity. Recommendation diversity, meaning the diversity of the contents recommended to user, key to users' engagement and retention on the platform. Meanwhile, creation diversity, meaning the variety of content created on the platform, is determinant of the platform's long-term health. extreme cases, insufficient creation diversity can lead to consensus or polarization, where the latter cause conflict and hatred, diminish people's 's mutual understanding, and cause societal crises. Therefore, from both business and social responsibility perspectives, championing and improving diversity in recommender systems is equally important as optimizing recommendation relevancy.
'The first two authors contributed equally to this work. And this work was done when Kun was at TikTok.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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There is increasing emphasis in academia and industry on investigating and improving the diversity of recommender systems, combating filter bubbles and polarization. Popular
approaches include applying post-processing procedures such as re-ranking [
47 and setting
diversity-aware objectives in addition to relevance maximization 38 22
39 12 These methods
aim to increase the recommendation diversity for users. Assuming that the contents on the platform are static, these methods have been shown to bring diversity gain to the system.
However, an important aspect that is overlooked in the aforementioned approaches is that: users and contents on a recommendation platform are not static entities they can be influenced by the recommendation made by the system. In content creation platforms like YouTube, TikTok, and Twitter. recommendations naturally affect both content users and content creators. It is well known that the exposure to recommended items can shift user's 's preference (24 26 14] On the other hand, the creators have the incentive to change their creation styles constantly to attract their audience better (and to make more profits from the platform) [1] 20 23). While the effects of recommendation on either users or creators have been investigated separately, to our knowledge no previous work considers both effects. The dual influence of recommendation causes complicated dynamics where users and creators interact and their preferences evolve together. Such evolution might exacerbate filter bubble and polarization effects. Whether the aforementioned inversiyy-boosting approaches still work in a dynamic environment with dual influence is questionable.
users and creators
The first contribution of our work is to define a novel, natural dynamics model that captures the dual influence of a recommender system on users and creators, which we call user-creator feature dynamics (Section 2). We leverage the users' and items' /creators' embedding vectors to represent their preferences and creation styles, and use cosine similarity to characterize the relevance of creations and users' interests (which is common in the recommender system literature and practice). This model allows us to formally reason about the impact of various design choices on the long-term diversity of a recommender system with dual influence.
Our contributions
Our second contribution is demonstrate that, under realistic conditions, the user-creator feature dynamics of any recommender system with dual influence must unavoidably converge to polarization (Section 3. i.e., the preferences of users and the contents of creators will become tightly clustered into two opposite groups, significantly reducing the diversity of the system. We demonstrate that this phenomenon still occurs even after applying diversity-boosting interventions to the system.
Then, (in Section 4 we investigate some real-world designs of recommendation algorithms in order to look for techniques that mitigate polarization. Interestingly, we find that some common efficiency- improving methods, such as top-k truncation, can both prevent the system from polarization and improve the creation diversity. We also provide empirical results (Section 5) on both synthetic and real-world (MovieLens) data. As predicted by our theory, we find that systems with dual influence more easily converge polarization under Eiversity-boosting designs, while fffciency--rientee and relevance-optimizing designs can in fact improve the long-term diversity of the system. This could explain why polarization does not always happen in reality. Section 6 conclude..
1.1 Related Work
Diversity in recommendations Diversity, filter bubbles, and polarization in recommendations have been important research topics in recent years, and they are closely related concepts with different focuses. On the one hand, filter bubbles are frequently defined as decreasing recommendation diversity over time (5), which describes both the process and the outcome of insufficiently diverse recommendations. On the other hand, polarization describes the negative outcome of insufficient mutual understanding between people [36.. In content platforms, example of polarization is people creating content with strong agreement or disagreement with other content under the same topic, e.g., political opinions. To combat these negative outcomes, previous works propose diversity-boostinn approaches including re-ranking [1 [47 and diversity-aware objective optimization 38 22 39
12 45). Despite having positive effects in situations where user preferences and creation styles
fixed, these approaches overlooked the dynamic nature of recommender systems and our work shows that certain approaches will make long-term outcomes worse under the dual influence.
Opinion dynamics study the effect of people exchanging opinions with others on social networks [7 29 4. Our model of recommender system with dual influence users
Opinion dynamics

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and creators resembles a bipartite social network, and our conclusion that the system converges to polarization conceptually similar to people reaching consensus on social networks [ 31 46]. However, the technique we use to prove our conclusion (absorbing Markov chain) significantly differs from the main technique (stability of ODE) in the mentioned works.
Performative effects of recommender systems The phenomenon that predictive systems like recommender systems can impact the individuals interacting with those systems (e.g.. users and creators) is related to the literature of performative prediction [4 [8]. These impacts can be direct, such as individuals ostensibly modifying their features in order to obtain more desirable outcomes
27
Prior works on the performative effects of recommender systems (e.g., 7 24 14
41 15 42 2 23 only consider one-sided impact, either on users or on creators. Differing from
35 20 3
them, our work studies two-sided impacts, i.e., on both users and creators. We provide table to compare our work with previous works in Appendix A
Model: User-Creator Feature Dynamics
We define dynamics model for user preferences and content/creator features in recommender system. Let U = =[u]] = [ui...., unm E [dx be a population of m users and V = [.]][= = [uf.-.., "0 E R be population of creators at time 4, where each vector u/, of E $d-1 represent the preference/feature vector of each user and creator respectively, assumed to be on the unit sphere şd-1 with 22-norm. Then (U", Vt) denotes the state of the dynamics at time t. The dynamics evolve as follows at each time step 2
1) Recommendation: Each user j E [m] is recommended a creator, where creator E [n]] is chosen with probability
ply - P',("U V*).
(1)
While we allow wide array of different functions pi,() (-), common example of such functions is the so-called softmax function:
exp(B(u)) x/))
pij = softmax(u', V = Σ exp(3(u), "())
(2)
A larger 3 means that the recommendation is more sensitive to the relevance of a creator to a user, measured by (uf, of).
2) User update: After recommendation, each user E [m] updates their feature vector u),, based on which creator, say it. was recommended them:
P(u + I. vo)))))
Here, 7/ [0, 1] is parameter controlling the rate of update, f u,) a function that quantifies the impact of creator i's content on user j (discussed in detail later), and P(x) = T is the projection back onto the unit sphere. Our user update model generalizes [14] which considers u 1+1 P(u! The (v u))et)) by replacing the inner product with general function f.
3) Creator update: Creators also update their feature vectors based on which users are recommended their content. For each creator i E [n=), let J = (i i i} be the set of users being recommended creator i, then of is updated by:
1+1 P
of +
Σ
slul of)ut).
(4)
J
JEJS
where Nc E [0. 1] is a parameter controlling the rate of update, and g(u,, vs) is function that quantifies the impact user j on creator i.
Impact functions f and g Our results apply to any impact functions f and that satisfy the following natural assumptions. First, f(v;, u;) and the inner product (v,, ux) have the same sign:
> [e; my) >
f(u, u,) [f; uz) < This means that a user likes the content (of, ut) > 0), then the
(r uy) =0.
3

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user vector 2I will be updated towards the direction of the creator vector vj. If the user dislikes the content (off uf) < 0), then the user vector 3 will move away from vf. Such "biased assimilation" user behavior is well documented the literature 14. Further, we assume upper and lower bounds on [f]:
[f(; u2)) 1, (f u3)l 2 L, > O whenever (v; un) # 0.
The lower bound (f(,,, u;)l 2 F means that the exposure to an item that a user likes or dis- likes always has some non-negligible impact on the user's preference. For example, f(v; uz) Sigu((u,, uz))a b(0; أz} satisfies both assumptions when L. = a > 0 and 6 2 0.
0 (My 0y) >0
For g, likewise assume that its sign is the same as (u), guu, vi) is
<0 if [My. ez) < Intuitively, [= (f vz)
this captures the incentive of creator who aims to maximize the average ratings from users who are recommended their items. On video platforms for example, if the creators are rewarded based on the average rating of their videos, they will try to reinforce their creation styles based on the users who give positive feedback ((uj, u4) 0) so that their creations are more likely to be recommended to those users. Meanwhile, the creators will also change their creation styles based on negative feedback (ut,, (2)) < 0), but in the opposite direction of the negative-feedhar users' interests, so that their creations are less likely to be recommended to those users. Taking both scenarios into account, the creator moves towards the weighted average of user preferences Σjent g(u), vf)u
which is captured by our update rule (4). A particular example of g the sign function g(u,, 0t) = (ign((uj, m:)) {-1,0, 0, 1}. We will only consider the sign function g in order to simplify the theoretical presentation. We believe that all our results can be generalized to other g functions satisfying similar conditions as fi the details are left as future work.
3 Unavoidable Polarization
Having defined the user-creator feature dynamics in recommender system with dual influence, we now theoretically study how such dynamics evolve. Our main result is: if every creator can be recommended to every user with some non-zero probability, then the dynamics must eventually polarize.
Definition 3.1 (consensus and bi-polarization). Let R 0. The dynamics (U*, Vt) said to reach:
R-consensus if there exists vector E Rd such that every feature vector is R-close to c: Vul, Iuu - cllp S and Vol, || - cll2 S R.
R-bi-polarization if there exists a vector e Rdd such that every feature vector is R-close to +c Ror
or -c: Vu', |/u - c|l2 S R luu; cli2 S R, and Vol, ||l| - cll2 S R or ||l! + clla S R.
The dynamics is said to reach (R, c)-consensus (or (R, cbbbbi-oolaaiiiiion if the dynamics reaches R-consensus (or bbbapoolarzzation) with the vector c.
Consensus any state where all users and creators have similar feature vectors (with maximum difference R), implying that they have similar interests or preferences. Bi-polarization is any state where all users and creators are clustered into two groups with exactly opposite features (e.g.. Republicans Democrats). Mathematically, consensus is special case of bi bolaaaaiization.
Proposition 3.2. Bi-polarization states are absorbing: once the dynamics reaches (R, c)-bi- polarization with some R e [0,1] and E Sd-1, it will satisfy (R, bolpoolaiization forever. The same holds for consensus.
A natural property of recommender system is that every creator can be recommended to every user with some non-zero probability: ply 2 po > with some constant po. This is satisfied by the softmax function, which a rough model of real-world recommendation algorithms
[ 26] pl, =
En (M(.
exp(0(uf oft)
exp(-8
Ta eXpp(2
Po
Moreover, many large-scale real-world recomendation
systems (e.g.. Yahoo! to improve recommendation diversity or explore users' interests (25 40, which will cause all
28 and Kuaishou [16] intentionally insert small random traffic attempting
recommendation probabilities to be non-zero. We show in Theorem 3.3 that, however, a recommender
system satisfying ply 2 Po > 0 must converge to polarization, under some additional conditions on
the users' and creators' update rates:

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Theorem 3.3. Suppose g v.) = sign( v,)), the update rates ne S Tulss and < 2' and the recommendation probability ptg 2 Po > 0, j. Then. from almost all initial states, the dynamics (U".V) will eveniually reach R-consensus R-bi-polarization for any R > 0.
In other words, if the users' and creators' updates are not too fast and all recommendation probabilities non-zero, then all users and creators will eventually converge to most clusters (regardless of the feature dimension d). Since creators in one cluster produce similar contents, users in such a polarized system
never receive diverse recommendations. This means that the na've attempt of imposing Piş 2 Po > 0 cannot improve the diversity of recommender system with dual influence. The conditions on the update rates Tas Te are only assumed simplify the proof of Theorem 3.3 Our experiments (in S will show that polarization still occurs even without those conditions.
Theorem does not characterize the rate of convergence of the user-creator feature dynamics to polarization, which we leave as an open question.
The proof of Theorem 3.3 is an absorbing Markov chain argument. It uses the following lemma:
Lemma 3.4. Suppose Tec S Dakt and Thu < 1.2 For any R > for almost every state (U*, V²) in 2
the state space, there exists a path (U', V + (U++), vv+1) + (U++, VI+T) of finite length that leads to R-bi-poll b-bi-polarization state (U++, V(TT).
The proof of this lemma (in Appendix F) is involved. It uses induction on the number of creators The base case of n 1 proved by a potential function argument. For n 2 we first construct path that leads the subsystem of n - creators and all users to R- B- bi-polarization, Then, depending on where the remaining creator is, we construct a sequence of recommendations that leads the remaining creator to one of the two clusters formed by the 1 creators and all users. Such recommendations will move some users out of the formed clusters, which requires extra care in the proof.
For any state (U', V*) in the state space, by Lemma 3.4 there exists a path (U++, V(+T) of length T that leads to R-bi-polarrizatonn Because every creator can be recommended to a user with probability least Po. each transition (U", V²) (UL'1, vt++1) happens with probability least pp" So. the path of length T has probability least phT > 0, and the probability that the dynamics does not reach R- rrbii-polarization after KT steps is at most (1 >* which -> as oc. Therefore, with probability the dynamics will reach R-bi-polarization eventually.
Proof of Theorem 3.3 (U', V*)
4 Discussions on Real-Worle Designs
Next, we discuss how 4 types of real-world recommender system designs affect the user ceererreator feature dynamics: top-k truncation, threshold truncation, diversity-boosting and uniform traffic.
(1) Top-k Truncation A prevalent practice in modern two-stage recommendation algorithms large-scale platforms, such as YouTube [13), is to first filter out items that are unlikely to be relevant to user, then make recommendations from the remaining items. In particular, truncation policy: for every user j. find the l most relevant creators, namely, the & creators whose inner products with the user (ui, uz)) are largest (equivalently, the is creators whose probabilities Ply of being recommended to user j are highest), then recommend one of those l creators to user j with probability proportional to ply- The other creators will not be recommended. This practice significantly reduces the computation cost and improves the relevancy of recommendations. Interestingly, we show that such a practice also has the potential to improve the long-term diversity of recommender system with dual influence.
consider the top-k
Definition 4.1 (clusters). We say state (U*, v') forms g clusters if there exist CL...., Cg. E Rd such that every feature vector in the Eg ball of some e; with radius
and a small number (denoted by B(ce, R) {æ: | call2 R)), and B(cs, 2R) n B(cg, 2R) = 0 for # C.
is clear that consensus has a single cluster, and bi polariizzation has two. Proposition 4.2. With top-k truncation, there exist states (U', Vt) that form \n/ clusters and are absorbing (i.e., once the system forms [n/k] clusters, forms [n/ k] clusters forever).
5

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This result is in contrast with Theorem 3.3 which shows that a recommender system where every creator can be recommended every user (plj 0) is doomed to polarize. With top-k truncation where some pi O, polarization can be avoided. Experiments in Section 5.3 support our prediction that top-k truncation can reduce polarization and improve diversity.
(2) Threshold Truncation Besides top-k truncation, threshold truncation is another way to filter out irrelevant creators: set threshold [-1, 1] such that any user-creator pair with inner product (u:; vÿ) < T is not recommended. A natural choice = 0, meaning that users will not receive recommendations predicted to be "disliked" by them. Increasing 7 is similar to increasing the B in the softmax function, which improves recommendation relevance.
Proposition 4.3. In d-dimensional feature space, if user-creator pairs with (ui, Uz) < O are not recommended,, then there exist stable states with d + clusters.
Although truncation at r = allows stable states with d + 1 clusters to exist, the dynamics does not necessarily converge such states; still end up with stable states with fewer clusters. In fact, experiments (in Appendix [.2) show that truncation at 7 = 0 is not good for diversity and causes severe polarization, while truncation large threshold like 0.707 is better at reducing polarization.
(3) Diversity Boosting Diversity boosting aims to explore users' interests and improve users' experience by diversifying recommendation. For example, when making recommendations, the model optimizes the objective:
(5) where fayet, haw rewards the recommendation relevance and diversity respectively and list; records the recent list of recommended items user i. have can take a simple form of Ey 1 (My, uz)) and p > controls the strength of diversity-boosting. Despite being successful when users' preferences and items are fixed, this design alone cannot prevent bi- bi-polarization in our dual-influence dynamics, since the conditions in Theorem are are still satisfied and the users' and creators' update rules remain the same. Experiments in Section 5.2 support our claim.
(u; m,)) + phaio ())
(4) Uniform Traffic Adding a small fraction of uniform traffic to the personalized recommendations is another method proposed in previous works to improve recommendation diversity or to explore user preferences 16 g g 30 This method gives a non-zero lower bound on the probability of
every creator being recommended to every user. So, as corollary of our Theorem 3.3 it causes recommender system with dual influence polarize. Such an observation is striking as demonstrates that optimizing for recommendation diversity in static setting can ultimately lead to huge loss of the system diversity in the long run.
5 Experiments
We present experimental results on the behavior of user-creator feature dynamics on synthetic data and real-world (MovieLens 20M) data and the effect of top-k truncation and threshold truncation on the dynamics.
5.1 Synthetic Data Experiments
Setup The dynamics is initialized by randomly generating user and creator features on the unit sphere in Rd. We pick 10, number of creators n = 50, number of users m = 100. We use the softmax recommendation probability function 2 We simulate the dynamics for T = 1000 steps, repeated 100 times each with a new initialization. We choose the sign impact function g u, v.) sign((u, v,)) for creator updates. For user updates, we choose inner product f(,, u,)) (v, uz)- The inner product function studied in previous works on users' preference dynamics (but not creators') 14. Note that the inner product does not satisfy the condition If u,)) L, needed in Theorem 3.3 However, we still observe convergence to polarization in nearly all experiments. Thus, even when this condition does not hold, users and creators still tend towards polarization in practice.
Three key parameters in our model are (sensitivity of the softmax function), ne (creator update rate), and T/ (user update rate). We set them to 3 = 1, Me Th 0.1, and change one parameter at

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100
t 200
t= 100
t= 200
t=
(Initial state)
B =
= 3:
B 1:
* creator
user
Figure Snapshots of the dynamics simulated with the same initialization but different recommendation sensitivity B. A larger / resulted in more clusters time step 200.
time to see its effect on the dynamics. We also test what happens when some dimensions of the user features are fixed features that are not updated.
Measures To quantify the behavior of the dynamics, given user and creator feature vectors (U, V) we compute the following measures, which cover diversity, relevancy, and polarization the system:
Creator Diversity (CD); diversity of the creator features, measured by their average pairwise
distance 47 3
CD(V) m)n- E Ejfss ||e Byyl.
Recommendation Diversity (RD): diversity the contents recommended to a user, measured by the weighted variance of the contents: RD(U,, V; = # > Piyllo; ,, 112, where
exp(B(u,, サ Ei=1 Ui and Pij E (xp()(u,,
Recommendation Relevance (RR): relevance of the contents recommended user, measured by the weighted average of inner products: RR(U,
bu Σy=E Pij (uj v.).
Tendency to Polarization (TP): This novel measure we propose to quantify how close the system to consensus or bi bi-oolari,,tion, measured by the average absolute inner products between the creators: TP(V) = पス Ei= i (v,, v.)). TP(V) being closer to means that the system is more polarized. because the term (U,, vx)) is iff the two vectors 8, U are equal or opposite to each other.
It is worth noting that high creator diversity is necessary for simultaneously achieving high recom- mendation relevance and high recommendation diversity. For example, they cannot be simultaneously achieved in polarized state.
Sensitivity Parameter A larger means that a user will be recommended more relevant con- tent/creator with a higher probability. 3 = O, on the other hand, means that the user receives uniform recommendations across all creators. Our main observation from the experiments a larger leads to higher creator diversity and alleviated polarization in the long
Figure shows snapshots of the dynamics different time steps under different 3 values. Here, choose dimension = 3 instead of 10 so the feature vectors can be visualized on a 3d sphere. We see that the system tends form more clusters at time 220 as 3 increases.
Figure 2 shows the changes the 4 measures CD, RD, RR, TP over time under different 3 values. We see that more diverse recommendation policy smaller 3) leads to lower creator diversity and a higher level of polarization the long run. In particular, while Creator Diversity reaches a similar level under different / in the end, it drops at slower rate with larger / (see 3 = 5, 6). Moreover, from the plot of Tendency to Polarization, we see that a larger alleviates polarization, which means improvement in the diversity of the whole system.
explanation for our observation is the following: When 8 smaller, each user receives more uniform recommendations across all creators. So, for different creators, the sets of users recommended
7

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0
8=2
4=4
B 8=8
B 10
Creater Dieessty
Accomrentatenn Diverstyy
Recommeercatiinn Relevance
Jendenccy Polanzaticn
14
1.
1.2
0.6
04
065
202
3002
Time
Time
Timoe
Figure 2: Changes of measures over time under different sensitivity parameter 5, on synthetic data. means uniform (non-personalized) recommendation. tion: only recommend the single most relevant creator to user. Larger / reduces the tendency to polarization.
means hard-max recommenda-
those creators have larger intersections. Since the creator updates are based on the sets of recommended users, different creators will be moving towards more similar directions. This leads to faster polarization. One can also predict this observation from Theorem 3.3] when 3 is large, minimum recommendation probability po of the softmax function tends to 0, so it might take long time for the system to converge to polarization, while with a small the system polarizes quickly.
Update Rates 1)e and thu A larger Tee means that creator features are updated faster, and intuitively should lead to faster polarization. This is validated in experiments: Figure shows that larger Ve indeed causes more extreme polarization and lower diversity (both CD and RD). A larger 7hu means that user features are updated faster. It has similar effect of exacerbating polarization T/e does, as shown in Figure
0.3
7: - 0.1
4g 0.03
0.01
4. - 0.003
2: n 0.001
fg = 0.0
Diversity
Recormmendetion Diversity
Tendency
50
07
05
855
08
64
Tinge
Figure Changes of measures over time under different creator update rate Tc. on synthetic data
A
02
n2
Creater Diversity
Recommeencatio Diversily
Recormmenciation Relevence
Tendency Polaticsiion
0.7
0.2
650
67
06
0.3
02
Figure 4: Changes of measures over time under different user update rate The on synthetic data
Number of Fixed Dimensions We also consider the scenario where some dimensions of the user feature vectors are fixed features and thus not updated from round to round (e.g.. age, gender), which is a realistic scenario. Detailed results are in Appendix B.I The main observation is: as the number of fixed dimensions increases, the diversity of the system improves and the degree of polarization is reduced. This is similar to the effect of decreasing user update rate nue The observation that

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fixed dimensions of user features help to improve diversity might be reason why the recommender systems practice are not as polarized as our theoretical prediction.
5.2 Real- World Data Experiments
In this part, we conduct experiments on the MovieLens 20M dataset 19. We use a real-world two- tower recommendation model with 16-dimensional tower tops as the user and creator embeddings (Figure The model is initialized by fitting a two-tower model [21] on the existing MovieLens rating data and using the tower tops the initial user and creator embeddings. Then we follow Algorithm to simulate the dynamics.
Figure | shows the effect of the recommendation sensitivity parameter 8 the system. Similar to the synthetic data experiments, a smaller 3 (more diverse recommendation for the users in the short term) results in faster polarization. We note that the joint results on CD and TP are more informative than each one alone: despite 3 has a higher creator diversity than 3 = 2 at T 500, the system reaches polarization more quickly under B 0. The higher creator diversity under B 0 is because the two clusters in the bi- bi-polari state are more balanced so the average pairwise distance between the creators is higher under / than under 2.
Figure 6 shows the effect of using diversity-awan objective (Eq. 5 for diversity boosting. We see that myopically promoting the short-term recommendation diversity (using larger results higher creation diversity but also higher tendency to polarization. possible explanation for this phenomenon is, similar to the case with 3, the system polarizes into two balanced clusters which actually have a large average pairwise distance. In this case, Tendency to Polarization a better measure for diversity loss than Creator Diversity (average pairwise distance).
B 0
B 1
f 2
Creator Diversity
Recommendation Diversity
Recommenndttion Relevance
Tendency to Polarization
1.3
0.0
0.4
0.4
0.3
300
200
200
400
300
100
200
300
400
300
300
200
300
400
500
300
200
300
300
Figure 5: Experiment on MovieLens 20M dataset under different recommendation sensitivity /
0.0
p= 0.5
p=
Diversity
Diversity
Recommendattio
Tendency
12-
0.5
13
0.4
1.0
0.5
Time
Time
Time
Time
Figure 6: Experiment on MovieLens 20M dataset with diversity-aware objective under different p
5.3 Top-k Truncation and Threshold Truncation
We experimented with top-k truncation on the synthetic data (Table T and the MovieLens dataset (Appendix C). Our main observation is: a small i improves the diversity of the recommender system and reduces polarization. This is consistent with our theoretical prediction (Proposition 4.2). However, there is a tradeoff between the diversity of recommendations to users (RD) and the diversity of creations in the system (CD and TP). A top-k truncation policy with small & is "not diverse" for users because it exposes a user only to small set of contents. However, such a policy can lead to a more diverse outcome in the whole system. This tradeoff is worth further studying.

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Table 1: Diversity improvement by top-k truncation on synthetic data
g k
Creator Diversity
Recommendation Diversity Recommendation Relevance Tendency to Polarization
50
1.00..0s
0.42
0.76....0
1.00
2.5
0.03
0.97
0.91
20
2.91
0.00
1.00
0.6820.12
10
1.17#.06
0,00
1.00
0.50
5
1.31
1003
1.00+ 10-3
0.35
1
1.40 =10
0.00
1.00
0. 0.27 1:
50
0.95+t114
0.02
0.99
0..10
0.80+.24 25
0.00
1.00
0.77
20
0.
0.00+10-3
1.00+10-3
0.74
3
10
1.18
0.00
1.00
0.49
5 #.02
0.00
1.00
0.34
1
1.40
0.00
1.00
0.27
We also experimented with threshold truncation synthetic data (Appendix B.2) and MovieLens data (Appendix C). The effect of a large truncation threshold 7 is similar to the effect of a small &s in top-k truncation.
6 Conclusion
Our work defines dynamics model to capture the dual influence of recommender systems on user preferences and content creation. Although our model theoretical abstraction, believe that it captures the essence of a real-world recommender system, and our effort is an important initial endeavor to study diversity in recommender systems with dual influence. (See Appendix H for some additional discussions on real-world recommender systems.) We specifically point out different concepts of diversity in recommender systems (creation diversity, recommendation diversity, and tendency to polarization) and provide theoretical and empirical evidences to show that, due to dual in- fluence, myopically optimizing recommendation diversity might hurt the long-term creation diversity and result in polarization of the system. We also explore popular design choices in recommender systems and show an interesting and somewhat counter-intuitive result that designs purely targeting efficiency improvement (e.g., top-k truncation) from our work are valuable to building healthy and sustainable recommender systems, and our results inspire more sophisticated solutions for improving the long-term diversity of recommender systems to be developed.
alleviate polarization. We believe that the insights
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14

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A
Additional Discussions on Related Works
Table 2: Comparison between our work and some previous works on performative effects of recom-
mender systems
Adaptive
Adaptive
Dynamics or Equilibrium?
Works
Creator Reward
Content Adjustment Model
Users?
Creators?
Conditioned on previous time step; implicit cost of content adjustment
Ours
Yes
Yes
User engagement
Dynamics
Conditioned on previous time step;
13
No
Yes
Exposure
Dynamics
explicit cost of content adjustment
42
No
Yes
User engagement
Dynamics
Freell choose without cost
35
No
Yes
User engagement
Dynamics
Freely choose without cost
23
No
Yes
Exposure
Equilibrium
Freely choose with cost
20
No
Yes
Exposure
Equilibrium
Freely choose without cost
No
Yes
Exposure
Equilibrium
Freely choose without cost
No
Yes
User engagement
Equilibrium
Freely choose without cost
Designed by welfare-
No
Yes
Dynamics
Freely choose without cost
maximizing platform
14
Yes
No
N/A
Dynamics
N/A
41
Yes
No
N/
Dynamics
N/A
Adaptive and
No! N/A
Dynamics
N/A
adversarial
1: These works study the design of recommendation algorithms for the platform with fixed set of content, without explicitly modeling the content creators.
B Additional Experiments on Synthetic Data
B.1 Number of Fixed Dimensions
In this part, we consider the case where some dimensions the user feature vectors are fixed features and thus not updated from round to round (e.g., ages, genders), Formally, we fix the first dimensions u [k + d = (u² +
S d dimensions. The remaining
1}.... ...... uf([]]
are updated according to the following rule: u + +1 d] = || +1 d]l PPu [k + 1
d f.. ut, [of[ : dl) The multiplication by [| 1 d]] ensures unit norm |
|
The effect of the number of fixed dimensions on the dynamics is shown in Figure 7 We see that the diversity of the system improves as the number of fixed dimensions increases, and the degree of polarization is reduced. This is similar to the effect of decreasing user update rate Th in Figure 4
The observation that fixed user features encourage diversity might reason why the recommender systems in practice are not polarized as our theoretical prediction.
laad =
Txxed =
hsed 2
Toed
feted
ftred
fixed =
fixed
foed
Creator Diversity
Recommendation Diversity
Recommendetton Relevance
Tendency Polerization
1.4
10
0.7
1.2
12:
11
1.6
1.0
200
400
$00
300
3090
200
400
400
300
1000
204
000
000
3000
200
400
600
000
2000
Figure 7: Changes of measures over time under different numbers of fixed dimensions, on synthetic data
15

---

B.2 Threshold Truncation
Table shows that the effect of different thresholds in threshold truncation on the long-term diversity of the system. We see that truncating at O. which corresponds to 90" angle between uj and vs is not good for diversity, resulting the lowest creator diversity measure (CD) and highest tendency to polarization (TP). Truncating at large threshold like 0.707 is good for diversity, instead.
Figure shows how the diversity measures change time, under different truncation thresholds.
Table Diversity improvement by threshold truncation on synthetic data
8
threshold
CD
RD
RR
TP
cos 60%)
1.00 0.03
0.00 + 10
1.00 10-3
0.99 10
c0s(72")
0.309
0.96 + 0.06
0.01 0.02
1.00
0.92 + 0.10
c0s(90%) =
0.03 0.16
0.00 100:10 10
1.00 10
0.99
0
cOs(72%) 0.309
0.72 0.30
0.00 10--
1.00 10
0.11
cos(60%) 0.5
1.16 0.11
0.00
1.00 1000=10-
0.47
cos(459) = 0.707
1.37 + 0.02
0.00 10-3
1.00
0.33
cos(30%) 0.866
1.30 +.00
0.00 1
1.00 10
0.55
cos( 60") ) 0.5
0.98 0.04
0.02
1.00
0.96
cos( 722) ) = 0.309
0.92
0.02
0.99 0.02
0.10
cos(90%) 0
0.13
0.00
1.00 1.00 10-3
0.97
cos cos(72") 0.309
0.85 1.16
0.00
1.00 10-3 10-3
0.76
cos(60" (60") = 0.5
1.00
0.00
1.00
0.08
cos(45%) 0.707
1.38 0.001
0.00 10-3
1.00 10
0.30
cos(30%) ) = 0.866
1.33 0.02
0.00 10-3
1.00 10-3
0.04
cos 60%) ) = -0.5
0.188
0.01
1.01
0.100
cos(72") = -0.399
0.25
0.00 10-
1.00 10-3
0.78
cos(90%) = 0
0.64 0.33
0.00
1.00 10- 10-3
0.81
3 cos cos(72) (72) = 0.309
1.01 0.14
0.00
1.00
0.64
cos(60% 0.5
1.26
0.00
10-0
0.38
cos cos(455 (45% = 0.707
+.39
0.00
1.00 +00
0.28+0.00
cos = 0.866
1.37 0.01
0.00
1.00 10-3
0.34
16

---

Creator Diversity
Recommendation Diversity
-0.500
12
-0.309
0.000
0.309
0.500
0.8
== 0.866
0.6
0.2
Time (0)
Time (0)
Recommennaation Relevance
Tendency Polarization
1.0
1.0
0.5
02
0.3
200
400
630
800
1000
200
400
500
800
1000
Time (t)
Time (t)
Figure 8: Changes of measures over time under different truncation threshold 7, with B 1, on synthetic data
17

---

C Additional Experiments on Real-World Dataset
C.1 Details of the Recommendation Algorithm
Predictions on User-item Pairs
Predicted Item/Creator Embedding
Predicted User Embedding
tem/Creator Tower
User Tower DNN
DNN
Item/Creator
Figure 9: Two tower model for the MovieLens experiment, where the two towers both have size 16 16 with linear layers and ReLu activations.
Algorithm 1 Real-world Recommendation with Dual Influence
Input: O, actual embedding (1))), y(),, true labels y = s(u(") (0) (0))), initial parameter g(0) (which includes the predicted embedding (10),, (00)
repeat
Let temporary parameter (0) (())
Compute loss L(0()),)) ( for
to m 1 do
w(++1) <- ((+) - V (on[(wl))))
end for @(++1)
(m)
Deliver recommendations based on ((++1), ((+11
Update (t-1)), v(+11) and ((+1)
t +
until ||000 0 1) 2 < 5
18

---

C.2 Top-k and Threshold Truncations
We also try top-k and threshold truncations (Section 4 on the MovieLens 20M dataset. Here, we have 2000 creators and 2000 users, with feature dimension d = 16. The results for top-k: truncation are in Table and Figure T0 Similar to the experiments with synthetie data, we see that smaller improves Creator Diversity (CD) and Recommendation Relevance (RR), reduces Tendency to Polarization (TP), yet worsens Recommendation Diversity (RD).
Table 4: Diversity improvement by top-k truncation on MovicLens 20M dataset g
k
CD
RD
RR
TP
2000
1.00 - 10-3 3
1,00 10 10-4
0.00 + 10
1.00 10
1000
0.30 + 0.04
0.03 + 0.01
0.88
1.00 + 10-3
500
1.10+0.00 0.06
0.00 10-3
1.00
0.43 + 0.03
0
100
1.36 - 10-3
10
1.00
0.28 0.01
10
1.40 10-3
0.00
1.00
0.20 10-3
1.40 + 10
0.00 1:00::00 10-3
10-3
1.00
0.20 + 10-3
2000
1.00 + 10-3
0.42
0.90
1.00
1000
+ 0.16
0.03 1.03:000.
0.97
500
1.14 0.04
0.00
1.00
0.41 0
100
1.35 = 0.01
0.00=10--
1000
0.27 10-3
10
1.40 = 10-3
0.00 10
1000
0.20 = 10
1.40 + 10-3
0.00 + 10-3
1.00 10
0.20 10-3
2000
0.92 + 0.07
0.02 0.01
0.99
0.91 0.05
1000
0.65 + 0.18
0.00 10-3
1.00
0.69 0.14
500
1.07 + 0.07
0.00 1000=10-
1.00
0.48 +.11
3
100
1.36 - 0.01
0.00=100
1.00
0.27
10
1.40 + 10-3
0.00 - 10-3
1000
0.20 + 10-3
1.40 + 10-3
0.00 = 10
1.00 10-3
0.20 + 10-3
k=
k= 2000
Creator Diversity
Recommendation Diversity
1.4
1.0
12-
0.8
1.0
0.6
0.0
0.4
0.6
0.2
0.4
0.0
100
200
300
400
500
100
230
300
400
500
Time
Time
Recommendation Relevance
Tendency to Polarization
1.0
0.9
0.8
0.0
0.4
0.5
6.2
0.2
0
100
200
300
400
500
100
200
300
400
500
Time
Time
Figure 10: Changes of measures over time under different k with B = on MovieLens 20M dataset
19

---

Results for threshold truncation are in Table 5 and Figure I1 Similar to synthetic data, we see that large (but not large) threshold like 0.707 good for improving CD and TP.
Table 5: Threshold truncation with different thresholds on MovieLens 20M dataset
B
threshold 7
CD
RD
RR
TP
cos 60%) 0.5
1.00 10
0.00 = 10
1.00 10
1.00 10
(7))) 0.309
1.00 + 10-3
0.00 = 10
1.00 = 10-3
1.00 = 10
(9s()00) = 0
0.01 + 0.01
0.00 + 10-3
1.00 10-3
1
0
cos(72") 0.309
0.83 0.08
0.00 10
1000
0.72
cos(60%) 0.5
1.20 = 0.05
1.00:1100
1.00 10
0.46 0.46
) = 0.707
1.39 + 10
0.00 = 10
1.00 = 10
10
c3s(30%) = 0.866
+
0.00 + 10-3
1.00 - 10
0.01
cos(60%) -0.5
1.00 10
0.00 : 10
1.00 10
1.00
cos(72") = 0.309
0.96
0.00 = 10
1.00 10
0.95
cos(909) = 0
0.02
0.00
1.00 10
0.90
cos(72%) 0.309
0
1.00:10-0
1.00 10
0.10
cos = 0.5
1.18 0.06
0.00::00 10-3
1.00 - 0.01
0.07
cos(459) = 0.707
1
0.00 10
1.00 10
0
cos(30%) 0.866
1.35 0.01
0.00 10
1,00 = 10
0.40
cos(60%) -0.5
0.77 0.27
10
1.00 = 10
0.86 0.09
cos(72") = 0.309
0.80 - 0.24
0.00 10
1.00 10
0.79
cos(90%) = 0
0.00
1.00
0.98
3
cos(72* 0.309
0.99 = 0.11
0.00
1.00
0.13
cos(60% (60%) = 0.5
1.26 + 0.05
= 0.00
1.00 10
0.06
cos( 45%) 00.707
1.40 + 10-3
0.00 + 10-3
1.00 + 10-3
0.20+100 10-3
cos(30%) (30%) 0.866
1.36 + 10-3
0.00 + 10
1.00 + 10-3
0.39 0.01
0.309
Creator Diversity
Recommendation Diversity
1.4
1.2
0.8
1.0
0.6
0.6
0.2
0.2
0.0
100
200
300
400
500
100
200
300
400
500
Time t
Time
Recommendation Relevance
Tendency to Polarization
1.0
1.0
0.9
0.8
0.8
0.6
0.4
0.5
0.4
0.2
0.3
0.2
100
300
400
500
400
Figure 11: Changes of measures over time under truncation with different threshold 7, with 3 on MovieLens 20M dataset
20

---

D Useful Lemmas
This section provides some lemmas that will be used in the proofs. They are mainly about some properties of the dynamics update rule.
Claim D.I. For vectors æ; y E TRd with unit norm ||ar||2 = ||y/|2 = 1, we have:
* o - -212 = 2(1 (z. v)).
(a, y) -1 x y
Lemma D.2 (Convex Cone Property). Let 2k. E Rd be vectors with norm || 1. .....,, k for some y E Rd. Let x P(E a,zi) for some a1....,. 2 O (namely, a is the normalization of some vector in the convex cone formed by
2k
Suppose (z,. y) > 0 for every i 1,...,k
21....; 2x). Then, we have
(a, 2 min(zi, v) >
and || 2l1 S
max || i=1
v112 > 0.
[=1
Proof.
k 0
(x,, 4) =
3)
as (zi, y)
Σ 0i 112
E azzill i=1
Σ i== ai
a,
min (en(zz, (zi, 8) = min y)
E a;zil1 112 1=1
1=1
D 0;2i
Σi= ai
Al min 2. y)
min(z.. v)
This proves the first inequality. To prove the second inequality, we use Claim D.I inequality:
and the first
| vll2 -22(11 - /2(1-(x,8)) s / min(zinn y)) max 22 (2,, v)) max || y 8112-
Lemma D.3. Let x², u. x' E It be vectors with norm || = 1. 11all2 2 1121 S 1. Suppose (xe y) 2 0, (z', y) 2 After the update 21+1 P(æ + n122), we have
(x x;, y) 2 1 n||22112
17
(1- r) - 1122 ,y)
As a corollary, ify = z' and ||01||2 = then
1 1+ (x x², x²) 2
11 17(11 (2", 20)
Proof. By definition,
xx + nz²
(x x,, y)
x², 3))
10 + 1/22112
|00 + 12 1|2 - 1) (x0, ,y) +
2| + nz (2", .))
17
(because ||= + 122112 5 +11221112) 2
1
- 1) (2x" 3) +
1 + +nllallla (222 4)
1
+ 17 |
"
((2 (aa..)) - ||="|12 ..)))
1 nl|
Lemma D.4. Let xx', ,²' E RRd be vectors with norm || 1, ||21||2 S 1. Suppose (at, 2 and 1j > O. Then the update +++1 P(x' + nz²) satisfies
21

---

(otth - x², 2²) 2 2 | x'122
1 20112 S 1/|12112
Proof. Let wc+1 = xe + nz', so x.' =P(# and z' 1 ( x²).
Then have
(x x', x²) - (x ++ x', xt x²).
Because (x', 2 O. the vector ā = x² + nz has length and hence is outside (or on the surface) of the d-dimensional unit ball. Since x²' = P( is the projection of z++1 onto the unit
ball, and zf is another vector inside the unit ball, by the "Pythagorean property" (Proposition 2.2 in [6), we must have (x² x+1 g
xx+1)== I 0. This implies
(o x², x²) 2 (ztth1 x², a x²) + (x² - w xet1 - x xl-1))
(+ - x', x - x²) _ +| x 11.
which proves the first claim. To prove the second claim, we use Cauchy-Schward inequality:
||x| x' 112 I (x x", x²) I || x x 1|21111112
This implies || æ2112 S 1112"112.
Lemma D.S. Consider a creator ot and a user u/y- Suppose the user is always recommended creator (so the user is updated by m1tt P(u' + maf u))o)),, and creator is updated by
b P(w + maa4)) with ||a2[ll S 1 and (o) 2 0 at each time step. Assume:
The inner product (u), v.)) > 0 initially. (Note that (ulf. un.)) needs not hold.)
There exists some constant L > 0 such that f(us, uz) Lg 0 whenever (u,, u.) > 0.
* < Mu and 0 S Ta < 1
Then, we have (u'y, wi) > 0 in all time steps.
Proof. We prove by induction. Suppose (u), of) > 0 already holds. We prove that (u,t2 oft1) > 0
will also hold. Take the difference between (u) +1 , and (uf, of):
(ugt ++ of + - (u'). vt) = (u oott - -vi) + (u u², v/)-
For (u ut. vt), using Lemma D.3 with xf =u,, u²/=² = =o; and 1j = no. [ uf), , we get
(u,t - u/. vi) 2 I+m ftn (1 (uf, ol)) 2 mukte (1 - (u)j u()).
For (u +1 wi vi). by Cauchy-Schward inequality and Lemma D.4
(u) oft1 o!) 2 -1lu,t1 |o 2fll2 2 1 -en.nellaa nellagl12 2 nc.
If (u), 0/) [1 + mrLL)) then we have
(u of 1/22 - (u', o/) > yuLss 1/2 - 11 20 0
by the assumption of no < nnLy
If (ut, of) S (1 + nn LL), ), then have
(n) +1 of ++ (n// o/) 2 0 1/c
(1 11 2 (uj. w/} - nc 2 2
1
L. - >
2
under the assumption of ne < Naks and nu 1.
22

---

The above two cases together ensure (uf"), ,
> O.
Lemma D.6. Consider a system of one user and one creator that satisfies (u,: 02) and (ulz. v) > (v.. > 0 for some y E R with llvil I 1 initially. The creator is always recommended the user (so the updates are u P(u) P +nff( uf)of) and u PPet + neul)). Suppose Me Juhe and 0 1hu z/2 Then, we have:
(u/, v) (wf: v) > 0 for all t 2 1.
Suppose (ul, y) (w.. v) = D O. For any R < D, after T have (oy y) (of. y) 2 nu t%
8 In Ωे steps, we 3muley
Te (D R).
Proof. We prove the first item by induction. Suppose (u), v) (vf, y) > holds. Consider t + 1. First, by Lemma D.2 (oft) .y > O holds. Then, we prove (ut) y) > (0) y). Let =f(uf u²).
(ugt . (w/tt .v) = <
| nh S 11112" ..)
uj + fof
of + neul
| nouull2 11)
1
ne
1
nuf
(0%. y)
1 + -T Sufll2
|| + n 11
|0 + nou2l12
|| + The S 1
Te
(u%, v)
nuf
(00 y)
luu + +nfuull2
-10g + u3 ' 1
||o + nouill
llee + MuJ 10/l122
+ nof
1 1 n7c
(ut 4)
22 + fuflll
all| + n/ 11
1 + nu.
+
(v). v).
/ 1 + 2naf (ul) u/) + (naf)²
/1 2 w/) (ne))
Let a = (u), o/) $ 1. We note that the function
1+
1+ 2n + 112
(2 - (2-2a)m
2(1 - a)
((T)
v1 + 2nja + n/2
V
+ 2nja + 1/2
+
1 2nja + 112
1 + I 2a 1
7f
is increasing in E [0, 1]. Under the assumption of no _ nuke f(....) and hence
< nul 2
nuf. we have (1(1))) S
2
(u)t) y) - (ott1 () > (h(nuff) - h((o))))))))))))()))) (ut, 3) 2 0.
We then prove the second item. Using Lemma D.3 for of =P(of + n,u)) get
(ot - v5,4) v) 2 1 + me
Te
(uu), y)
(v v))
Using Lemma D.3 for u (+1 = P(u') + nnl( ub)) and using the fact (vl v) (u), y) < 0 proved in item I,
ma Muf(ot ut)
(u ++1 uS. v) 2
(1o v) - (u/, v1)
2
(10% y) - (ul) v1)
1+ Th (ot ut)
1
Rearranging the above two inequalities:
1 1 ne
(oo y) - (1) 8) 2 {ui y) - (00 y);
Te
1 + The
((utt ./) (u/ v) (of, v) (u, v)-
nu
23

---

Summing the above two inequalities over t = =0,1, O. T - 1:
Tle (oy y) (o%: +
1 nu (uu) 3) - (u) 2 0.
(6)
?c
7/u
According to Lemma F.I after at most T = 3nuly
In 1pz
2 steps, we have l/u] o 12 1 R. This
implies (u,, 8) (o) y = {u] r/- 3) S |uj 1 S and hence
(1o² u) -)) (o%.1) m) - (uu² v) - (w)) 3/ (uu) y) - -(of. v) - (u)) y) - -1² m1) 2 D R.
Multiplying 7 by hh and adding to ( Tu 1 + Te
Thu
(of 3) (0f. 2))
1 + Thu (D R).
To
7/
This implies
1+7
(o) y) - 10 2
Yhu
(D-R)
No + Tu)
(D-R)= 2 Tee (D- RR).
I+
1+1
1u (1 ne) no([1 7aa)
The 7c
1/2
given S Tu.
The following lemma shows that, when we reflect some of the feature vectors in system (U*, V1) ([u)) Jefm]- [o)) reflected vectors and the original system.
helm)) ). there is a correspondence between the behaviors of the system with the
Lemma D.7 (Reflection). Let (U", V*) = (eem) (uf)ret)) be a system of m users and n vi E [n]. Vi E [m] be some binary
creators with impact functions J.9. Let oz, bz,by E constants. Define:
ú = b, +u..
= = = +
and impact functions
f(0,, ū.)) a uz),
O(ūj, 0.) aub wi)-
Then:
There is "ccrrrepondence" " between the evolution of the system (U*, Vt) with impact functions f.g and the evolution of the system (U+, V*) impact functions f.g. Formally, suppose every user is recommended the same creator in the two systems, then the updated vectors in the two systems still satisfy the relations: i++1 = bb
(o) he(m)) with
2+1
a,m,
If the system (Ü*, vV) is in R-bi--olbraiti --biipollarizatinn then the original system (U', Vt) is also in R-bi- R--bi-polarization.
Proof. Consider the first item. Suppose user is recommended creator j at time step t in the two systems. Then by definition, the updated user vectors in the two systems satisfy
u P(ū + fof ()) P(b,ut + mua;b u))a,f) = P(b,ut n,b, uf)w() = 6,P( + +m nuf( uf)o!) = byutt1 Suppose creator is recommended to the set of users J at time step in the two systems. Then, Σ
5ه+1 = P(o) + "/
Caui )})))))
jEJ
P(aoo + 1
Σabbo(uf u()6,u;)
JEJ
P(a,of + 1
Σ Laglu uf)ut))
JEJ
aPPlo + D)
Σ ouul.o()u of)ut) azttt
JEJ
24

---

This means that the evolution of the system (Ü V1 has correspondence to the evolution of the original system (U*, V*).
Consider the second item. Suppose (Ü*, Vt is in Jt-bi-ollarizatinn so oj f is R-close ie and ü = is R-close to ie with some vector E S This implies that of is R-close to t and uz is R-close ic. So, the system (U*, Vt) satisfies Rt-bi-pole hi-bipollaizztion.
E Proof of Proposition 3.2
state with R E [0, 1] and e E S where all uj We show that, after step of update, u/ and t still within distance R to +o or -c, so (U++1 V*+1) still satisfies (R, cib-l-polarizattonn
Proof. Let (U", V*) be an (R,
and of within distance R
Consider wi. Without loss of generality, suppose uf is close to +c, so llef cll S R. Suppose user j is recommended creator at step I. Let o = of if (vf, uf) 2 0 and of uf if ut) < O.
Then, the user update is
u =P(ui; + tnf((e) (of,, P P(ui, nu of 2.),, u))lo
Since ūi is close to +c or -c, (0ū ut) > O. and u/ is close to +c, it must be that of is close to +c, so |f - c/l2 R. Then, since ui / is the normalization of vector in the convex cone formed by uz and i6, by Lemma D.2 we have
| clla S max { [lus - c/12, || cll2) S R.
Consider of. Suppose ||l c|12 S R. Let J the set users that are recommended creator at step t. For each J. let úi u if (u', of) update is
and utj -u3 if (u)j vf) O. Then, the creator
-+ = s
Te
E(u u ui)ut) p(ot
The Σ la(uy, of)laat) [J
We note that every ū. satisfies |Íū cl2 S R (by the same reasoning as above). Then, since o/ is the normalization of vector in the convex cone formed by of and ( Jee, by Lemma D.2 we have I
|| c||2 z min {loe
cl22
min ύ cll2 s R. jEs
Proof of Lemma 3.4
Lemma 3.4 is proved by induction the number n of creators. We first show that any system with
creator and multiple users must converge to R- polliooaarization in finite steps for any R > 0. Using the result for creator, we then construct a finite length path that leads to 77-bi-oolarization for any system with n 2 creators.
F.1 Base Case: Convergence Results for 1 Creator
We prove some convergence results for the special case of only one creator. This will serve as the basis for the proof for creators. Recall that have the following dynamics update rule:
User: uit P(u! + nuf u()x!)
where of is the creator recommended to user
f(v;, uz) satisfies:
if (v,, uz) < (v., Uz) if (u,, Uy) = 0.
f(v,, u;) is
(8)
Creator: uf+1=10 P(0) + [}] EJEJ olui, u))uz)) where J is the set of users being recom-
mended creator i.
25

---

Lemma F.1. Consider system of 1 creator vf and [J] users {ujhee, where the creator is recommended all users every time step. Assume:
Initially, Vj E J, (ul, 2 > 0.
There exists some constant L. >> 0 such that f(0,, 2 Lg > 0 whenever (u,, u2) > 0.
g(uj, vi) when (uz, v;) O.
Te S Tikks and S 7a < 3/2
Then, for any O, after at most 3n. Ly In 3l particular, each user vector will satisfy llus 2 I R.
R steps. Σ5er 1lu) v/13 R² will hold forever. In
Proof. We first note that, by Lemma D.5 all user vectors satisfy (uf, o/) of) 0 in all time steps
t > O. Hence, the creator update is always oft1 P(o + 1i Eees s(u), vf)us) = =P
MepT Ejenuh u/))
Let a, = 1/(1 3r. )t. Define the following potential function:
Φ'
ae
luy
w2l12 ae (u,j
(9)
jEJ
We will show that Φf monotonically decreasing. Take the difference between gi+1 and $C:
rc+1_ - O² ax+1
Σ
(u ot) (u) ++1 ,,, + (a++1 - az)
Σ (1 (u/, w())
jEJ
JEJ
Σ
(u u - -ust!) + E(uj. of - -oft1) > + Σ (u/tt) - u/j 2t - -0, 0/+11)
a++1
jEJ
jEJ
jEJ
+ (at+1 a4) Σ (1 - (ur). uf))
jEJ
Using Lemma D.3 with x' = u/y. x² v/j and 17 m. (v], ut), ) we get
1 + nof Tuf( ug) (1 (u) ofi) S Mull
na (o]( u1)))
(vf. uz - u 1 S
(1 -2/,, o!).
2
Using Lemma D.4 with x. = u/j x" = =j and = muf(fff ut), we get
(vf, u - 1 S Muf(uf u²) lu - u2'12 < | 1+1 u3.12.
Using Lemma D.4 with x. = -// x' 111 Ejes ugy and 17 Te. get
[J
Σ u of o/41) -J1{ Σ u/ of 21t1) s
1 0//12.
ne
jeJ
jeJ
26

---

Using the above three inequalities, we can upper bound d Φ²:
3
(uf, uj - u +
Σ
(0/
u - ujj 2
-c++1
1
JEJ
jEJ
Σ (ujj of - -ott)
Σ
[uj
1+1
/ 2/. 01 0,t1)) + (a(+1 ar) Σ (1
(u), nf))
JEJ
jEJ
jEJ
$ at+1
3 Σ
Yules (1 - (z)j o0²))
Σ
|
u/12
4
2
4
jEJ
jEJ
|J| |
- -f112 +
Σ
f|2 1100 0fl12 + (at+1 -a;)
(1) of))
1)c
jEJ
jEJ
37
Σ
(1- - (ujj w1))
ac+1
8
jEJ
1
1
[/
++1
u/i12
11 / 1/12
|1u] + uill2 -|0o² oflle)
41/u
17c
jEJ
22v
|1
11
o
Amune
+ (0141 az)
Σ
[1
(u)j oft))
JEJ
39abss
S at+1
(uj. o1))) -
Σ
1 1+ 2z 12)122 11 ++ 0//12
Tuue
JEJ
JEJ
20
+ (0++1 a+))
(u) of))
jEJ
37a L
(u) v1i) + 0) + (ai+1 - aa)
Σ
(1 2f)
ac+1
jEJ
JEJ
(
31
Tuf
Σ (1 (u)j ofi))
aa+1
at)
8
jEJ
where the last step is because (1 3r,
)a++1 az.
8
We have shown that o' is monotonically decreasing, Thus,
T
T
Ejer (1
/ T
1 |J| S R
2
0T
aT
aT
8
[J] S
e
2
JEJ
whenever T
In
2 72
L.
Corollary F.2 (of Lemma
F.1 Consider system of creator of and [J] users (uf where the
creator is recommended all users at every time step. Assume:
every Initially, (u,, v # e.rorv j E J.
There exists some constant L, > 0 such that (f u;)l 2 L, whenever (vi, un) # g(u,, v.)) sign( (u, v.)).
Me S nuL; and 0 < nu < }
Then, for any R. > 0, after at most Ln In
2|J1 R steps, the system will reach R-bi-polarization.
27

---

Proof. Let J+ = [j E J: (u]] (u), b9) > 0f be the set of users with positive inner products with creator i initially; let J = e j Je Then, the system consisting of [a) and vj satisfies the initial condition (ū.. eg!) in Lemma F.1 So, by Lemma F.1 it reaches R-consensus after at most 3n,2,
= J (u.; e.) < 0}. Let ū³ -u} for E J and ū = u² for
8 In Re steps. Then by
the reflection lemma (Lemma D.7, the original system, consisting of (ujbes and vî, must reach R- bi-ppoaarization
E22 Inductive Step: Proof of Lemma 3.4
Lemma F.3. Consider a system of 2 1 creators {ui...., 0 and [J] users {u) buj.e.. Assume:
Initially, (uf, vQ) > 0 for every i, 1, and (oo ul)) > 0 for every i.
Assumptions of Lemma R.
Then, for any R E (0, 1). there exists a path of finite length that leads the initial state (U², Vo) to R-consensus.
Proof. Fix any R E (0, 1). Choose Ri such that ( +24RR R. Clearly, Ri R. We construct a path that leads the state (U", VO) to R-consensus as follows.
Step (I): Consider the subsystem of the first n -1 creators and all users J. By induction, there exists a path of length Ti Ln-11 < +00 that leads the subsystem to (R1, c²i cconsenuuss with some c"i 5d-1, So, after these Ti steps, all creators i E {l...., n 1} and all users j E J satisfy || Ti - || S Rí and ||u. - - < R1. Creator n does not update during these Ti steps, so v/' = wn and it still has positive inner products with the first n- creators and all users by the convex cone property (Lemma D.2). Let's then consider the distance between creators n and the consensus center cm: ||l c |. If || Il < R, then the system has satisfied (R, c conssnnsss so our construction finished. Ötherwise, | cti R. We continue the construction as follows:
Step (2): Pick any user jo E J, recommend creator n to user jo for To 3n.L, In steps, while Ii
recommending creator all other users. From the (R1, cTh ) in step (1) we know
|1/] - cTill R1,
(u]) (uf,c"i) ) = = 11 j0
1 2
R²
1 1- R 2
2
1-
cti 12 (0 ,ct).
Thus, we can apply Lemma
D.6 with y
c.i to derive that, after these T2 steps,
(0 To .1
(u Ti Tiy 2
y
(u
T: jo
c
f c TT) ) - Rı
qu +%
Ve
R²
(0
,.t)) )- Rı
4%
2
(u Ti +T Fity > (0 Ti
Ti
+
de
1
n² Ti c.) - R1).
(0
(10)
t/e
2
For the inner product between creator n and user jo. by Lemma F.1
+1's
u
Ti+T'a
< R1, so
(0) Ti 77
Ti+T2) = 1 - 11lo% T
Ti+1a jo
2
R 2
(11)
Consider the inner products between creator n and the first n creators and the users in J (jo)- Because the first n 1 creators and the users in J {io) form (R), edh c-onsensuus time step T', by
Observation 3.2 they still form (R1, "e.c"""")oonssste at time step T2 T2, so 1/ +i+ cTi < R and || Ti +2
cill < Ry. This implies, for + n.
(0 T v Ti TTyy
(u +T Ti - -|0 Ti TitTz cTill
(u
Ti +T; ,,TT) Rı
10
(w Ti ci + De
1 - R2
(122
T' c. C..)) R1
-R1, (12)
nut
28

---

and for j E J {iot.
(v , ug
T2
2 (b c²1)- ||uu ||
Ti+Ts - c²:l
(02 23's +1- ,.7)) - Ri
To
(0 Ti c
+
Th
(1- R2. 2
(u cT - Rí Rl.
(13)
Tu +
Step (3): Consider the subsystem of the first n creators and all users J. By induction, there exists
path of length T3- = Ln,, < +00 that leads the subsystem to (R1, c cTi+Ta+Ta [ cconsensus with
some c <T+TT++ +72+Ts Ts E gd-1, So, we have || 1o/ - (Ti+T; T5 S R, for every E f n- 1} 22+ S Rı for every E J, and vn +
w oiit Consider the inner product between creator n and any of the first n creators i E {I,..., - 13. By the convex cone property (Lemma D.2).
and || [u3 [1 +s
(v
+3 T3+ (DTI+T U; +
0
Lemma D.2 2 min (0, tits Th (2
,
m +it
Uj TT+T
jEJ
by
12
13
min {(u c1
1
n²
(o c2i) - - R1, 1- B2
2
(u ci +
4o ne the
[
R
(v
TT)) Rs R (14)
2
where the last equality is because, under the assumption of c
|
R
[u 22)
(1 F..
1
R.
T:
Rτ
Rı
Nu +
2
Mu
(u
1
R
R2
t/e
2
1/u
1 111122" Ti
cT²
The
(1
11,
Rı
Tu 1o
the
2
Tu
11 1 (
2)4RR
Tle
R
R
R
Th +1e
tu
2
S max
1 Th 2)4 1
Rı
Rı
2 me
Rº Ri R1 < 1
n²
= 1
2
From 14p and 10
2 +2+++ < R.,
(0 Ts TT++T+Taa 2 (u v. T. Ts+To+Ta - Ti - -17++7+733
TitT's
Ti c.ty + nu to
(1
R²
(n T: rTi) R7 ) -2
Using 1 to minus the above inequality, we obtain
1 (v T2+ tfit <
Tu tme (1 (1- 1100 ,00%,c221) .221) +
ne
[1 + R1 + 2R
motme
7
Let Fr = 1 - -(0 c²), then
<
Yu
F
+
R2 + +R1) 2R
(15)
nu 1o
nu ne
29

---

Repeat steps (2) and for K times. Then, using 15 for K times,
F +Tttk(TT (((T(T+)Ta))))
(KI++(x1))7277 +
The
2 + R1
+ 2R
1/ to
the
the
Tu (K)+((=))(7++++
+
No 71
ne
R
2
+ Rí +2R1
+
Nu th
T
( RR)) + 2R
Dlu
K
F
(1
+...
) )
R5 +Ri) + 2R)
+7 T/e
1/
K
+
Tre Tue
Ri 2
+ Rí) + 2R)
+le
Yu
Nu te
K
+
R
+ R1 +
tue 2R1
2
+ R1
+
Nu + 2
I
( qu + 2)2R1,
The
2 In
by choosing K times, we must have
1 tu In
This means that, after repeating steps (2) and (3) for K
You Lic
R
TM
Ti T+T((T) c c
22(1
(un
Ti (T+K(T
cTL+K((T++T5))
(2
2( the nec
+ 2)2R1 R.
The above inequality, together with the fact that other creators * n and all users in J already satisfy (R, < R. cfl+ --cottsesssss after step (3), implies that the whole system has reached (R, +.IK(((T))))) 3 conssessu.
The length of the path constructed above is at most:
T1 + K(T2 + Ts) s In-1,1t, + the In 2 8 In 2|J1 RG + Ln-1111a)
Ln, < + 00,
ne Rf 3ndLy
which is finite.
Lemma E.4. Consider a subsystem of n creators {uf,.... v and [J] users (ujbes. Assume:
Initially, the first n - 1 creators and all users are in Ro-consensus: !oll - cll S Ro. |u] S Ro, with Ro 5mn+ta)
(0h ) for some jo J.
و(س,, v;) = sign((u,, vz))-
Assumption of Lemma F.
Then, for any R E (0, 1), there exists a path of finite length that leads the initial state (UO, Vo) to R-consensus.
Proof. First, we recommend creator user jo for T
Ir In steps, while recommending
other creators to other users arbitrarily. Applying Lemma L. y u%. get
(u ) - (o ) 2
Tie t1-
(w4) j 4% (v 2 Ro =
n tno
(1
(w)
u - Ro
(16)
On the other hand, because the first n - creators and all users in J \ (io) form an (Ro, c)-coneensus at time step 0, according to Observation 3.2 they still form an (Ro, -)coosensuus at time step T, so
|!o - cll S Ro for every E (l,...,7 n This implies, for every E (l,...,, n -1]. 1}.
(o v. (u 3 2 - -110," 2 2 - -1o² cl ||c u% > - -2Ro.
(17)
30

---

Adding (6) and 17
and moving (i 2//o) ) to the right side, we get
(0 oft)
(o +
" nuttle
(0 u 30 Ro) -Roo
tu nu (nn 300 +
Ru) 2Ro
Tu +1
0 +
na the -Ro)
Tee
2Ro > 0,
under the condition of Ro
me
Moreover, for every j E \\ {io). because ||u] 09
5(% 5 tul
|uy ell
||e of 2Ro.
{0) u
2 (0) v
1/ oi 2
nc +ie
(1
Ro)
4Ro > 0.
For jo. by Lemma F.1
I u 30
T. s Ro, so
(w 2)
Jol = 1 11 1o 112 2
R 2
> 0.
For the inner product between any creator E (1..... n-1 1} and the users:
(oy " 2 (w) "²") - | 4 2
te (1 Ra) - 2Ro Rs = nu the
(1 Ro) - -3Ro > 0; n6
tie
Vj E J (j0). (of ug)) = 1- bo ut 12 2 1 (1l cll ||c. uz'l)
1 (2R)) > 0.
All of the "> O" inequalities above show that the system of {ut ]ven) and (u. JeJ satisfies the condition of Lemma F33 So, there exists a path of finite length T2 < too that leads the system
to R-consensus by Lemma F3 The total length of path T' T2 3m R
In 2 T2 < t00 is
finite.
Lemma 3.4. Suppose The S Take and The 1/4 For any R > 0. for almost every state (U+, Vt) in the stute space, there exists a path (U*,V)) (U++1 V++1) + (U++,, V+T) of finite length that leads to Rt-i-pollariztton state (U++,, V(T)..
Proof. We prove this lemma by induction on the number of creators n. The case for n = 1 directly follows from Corollary F2 which shows that, for any system of 1 creator and |J] users with
no (uf, u²) 0, there exists a path of length at most L 3n LF R-bi R--i-pon po-poaarrzztion.
8 In 2|J +o0 that leads to
R
Consider 2 2. Consider the subsystem consisting of the first n - creators {0í, w and all users. Let Ro 6m.. By induction, there exists path of finite length Ti Ri < +o0 that leads the subsystem to Ro -olbbi-oolarizatio,, with some vector co _d-1, every ui is Ro-close to +co -c,, for i o n, and every wi²t is Ra-close + or Cp. Define:
of
if o is Ro-close to +c if u is Ro-close to -c
if To is Ro-close to +C T: if u is Ro-close to - VjEJ
Vi Vi nn,
i
=
u
By definition, we have
|| coll S Ro, Vi Vi n,
a coll Ro: VjEJ.
This means that {o7)) Jigh and (ū hee form an (Rg, con-consssuus. Consider creator n. Let
5% v -
if (uii) û TTy 0 for some jo E J 1p 0 for all j E J.
if (o u
(The case where (o ) 0 for some j is ignored because the initial states that can lead to such states have measure 0.) By definition, we have
(o uifi)) 0 for some ju E J.
Note that, at time step Ti, the system consisting of {τ, ]ee(x) and (ū² hee satisfies the condition of Lemma F.4, so there exists a path of length T3 Σ < too that leads the system to R-consensus. Then by the reflection lemma (Lemma D.7). the original system [u/hemm) {u² beJ must reach
R- b-bi-polarization. The total length of path that leads to this R-bi-polarization is L = Ti + T2 RR + L < +o0.
31

---

G Missing Proofs from Section
G.1 Proof of Proposition 4.2
Let R > 0 be any small number. Let Ci...., C[n/k] E R be [n/k] vectors that satisfy B(cr, 2R) n B(er, 2/t) 0 for E + f", where Bic, Rt) is the ball centered at с with radius R: {or E R ||e c||2 < R}. Consider user and creator features (U", V*) that satisfy: every ball B(cr, R) -f= 1,...; [n/k]) contains & creator vectors, and every user vector uz is in one of the balls B(ce, R). By definition, (U", V! form |n/k] clusters. We show that, after step of update, the new state (Uf++1 V(+1) must still form |n/k] clusters. Consider any user j. Suppose u/ E B(c,, R), then the distance from uz to any creator of E B(c,, R) is at most 2R:
[u]j v/l S 2R.
The distance from u/ to any creator vs not in (cc,, Rt) is greater than 2/::
|uy ot
2R
because bb, is in some other ball B(cr, R)) that satisfies B(er, 2R) B(ce, 2 0. This implies that the inner products between user j and the creators in ball Bic,, R) are greater than that with the creators in other ball:
1 2 Ilu of 2 (2R) > 2 |lu] 2/ (u)) of), Voy E B(ce. R).
VeL E B(ce, R), (m)). 05) =1 (
Since (ce, Rt) contains k creators, these & creators are the k-most relevant ones to user j. user will only be recommended these creators. Then, by applying Observation 3.2 to each of the [n/k]] balls separately, we see that each ball is a Rtconsensus and hence absorbing. So, the new state (U++), V*+1) still forms |n/k] clusters with these [n/k] balls.
G.2 Proof Proposition 4.3
The d-dimensional simplex centered at the original has d+ vectors with negative inner products with each other. They form d + 1 clusters. Since user-creator pairs with negative inner product (u., U.y are not recommended, recommendations only happen within each cluster. By Observation 3.2 each cluster is absorbing, so the whole system is stable, keep forming d + clusters forever.
H Additional Discussion on
Recommender Systems
Here we further discuss real-world recommender systems' properties and designs that are currently not covered in our main paper. We plan to generalize our model in the future to further capture these features and discuss insightful findings, but having them in the current paper may be a distraction to our main findings.
H.1 User and Creator Retention and Activeness
In our current model, the users and creators will stay in the system from the start to the end. However, real-world recommender systems, users and creators may leave the platform either permanently or for certain period. Meanwhile, new users and creators will join the platform. Such join and leave dynamics are also influenced by the recommendations' relevance and diversity, which further complicate the problem. Morcover, users and creators have different activeness levels the platform, c.g., some users may watch a lot more videos than others, and some creators may post a lot more creations, these effects will also be strongly correlated with the dual influence of the recommender system.
H.2 Creation Quality
Creation quality is a major factor influencing users' feedback in addition to the creation style, e.g., well-made cuisine videos could also be fun and liked by gamers and pet lovers, which we need more than collaborative filtering type of modeling like our current model to capture such features. A
32

---

potential solution to boost both long-term system diversity and single-shot recommendation diversity to design mechanisms that can incentivize creators create higher-quality videos instead of changing their creation styles.
H.3
Cold Start
Cold Start is widely used in real-world recommender systems for newly published items. Due to the lack of user-item interactions on new items, the systems randomly recommend these new items users and collect data for collaborative filtering. In our current model, if we consider the creators creating new items in each time step under their current time creation style, then cold start guarantees the conditions in Theorem 3.3 But we consider the system have good enough content understanding ability and can accurately predict the new creations' embeddings, the cold start is not necessary and our model and results in the top-k truncation and threshold truncation parts are valid. We also highlight a subtle difference between cold start and random traffic, if cold start is used on creators instead of items, then after the creator is exposed to users certain number of times, the system will not guarantee to provide a non-zero probability of recommending this creator, and thus the conditions Theorem 3.3 may not hold.
33

---

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Reviews:
[{"review_id": "y6WCTXsGmt", "reviewer": "Reviewer_EsTG", "summary": "This paper studies how recommendations become polarized over the long run when user and creator features dynamically change over time. The authors theoretically prove that, under the assumption that every creator can be recommended to every user with some non-zero probability, recommender systems will eventually converge to polarization. They also simulate some real-world models, including top-k recommendation, truncation, diversity boosting, and lower-bounding probabilities in a long-term setting. The key observation is that top-k recommendation (i.e., only recommending top-k items to users) can reduce polarization to some extent, while existing diversity-boosting methods will worsen polarization when user/creator features dynamically change over time in the system.", "questions": "1. Please address the points I raised in Weaknesses.\n2. (Minor) Are both consensus and bi-polarization conceptually polarization?\n3. (Minor) How are the initial user/creator embeddings initialized in the Movielens experiment?", "limitations": "The authors adequately addressed the limitations.", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The authors provide both theoretical and empirical evidence showing that relevance-focused recommendations (as opposed to diversity-focused recommendations), which harm diversity in a static setting, are actually effective in improving diversity in the long term. This observation is somewhat counter-intuitive to previous beliefs, making it very interesting.\n2. The authors conducted simulations with both synthetic data and real-world data (i.e., Movielens) using four diversity and relevance-related measures. Additionally, the analysis with sensitivity parameters in softmax is insightful and supports the authors' main claim.\n3. Studying diversity in a dynamic setting is novel.", "weaknesses": "1. Despite the novelty and interestingness, I have concerns about the key assumptions of the theoretical and empirical analyses. The assumption that all items can be recommended to users is not realistic. In practice, almost all recommender systems rely on top-k recommendations for either effectiveness or resource constraints like screen size. For example, on platforms like Netflix or Amazon, customers can only see a certain number of items on the webpage (i.e., p=0 for items that users can't see). Even if they can scroll down and the system continually recommends new items, they cannot physically see all items in the system. Thus, I believe the top-k setting is the most realistic and natural for real-world scenarios, and this seems like a hole in the authors' analyses. In this sense, the measures for empirical analysis should also only consider top-k items, not all items.\n2. For the real-world designs, it would be more extensive if users included trustworthiness-aware recommender systems that consider dynamic/continual settings. For example, [1] consider performance difference between two different user groups when the user/item features are continually updated over time in the systems.\n3. For the analysis with Movielens, considering the interaction timestamp in the simulation would more accurately reflect real-world scenarios, for example, for determining the true labels.\n\n[1] Yoo et al., Ensuring User-side Fairness in Dynamic Recommender Systems, WWW'24", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1: Please address the points I raised in Weaknesses.\n\n> W1: The assumption that all items can be recommended to users is not realistic. ... Customers can only see a certain number of items on the webpage (i.e., p=0 for items that users can't see).\n\n\nFirst, we note that customers not seeing some items due to screen size limit does not mean that those items are recommended with probability p=0. The set of items seen by a customer is randomly drawn from a distribution over items which can have positive probability on all items, even if only $k$ items are shown.\n\n\nSecond, we argue that non-zero probability of recommendation is realistic, even in large-scale real-world recommendation systems used by Yahoo [[Marlin et al, 2009]](https://dl.acm.org/doi/10.1145/1639714.1639717), [[Li et al, 2010]](https://dl.acm.org/doi/10.1145/1772690.1772758), and Kuaishou [[Gao et al, 2022]](https://arxiv.org/pdf/2208.08696.pdf). As noted in our Section 4.4, practical recommendation systems insert small random traffic (uniformly random recommendations, or missing-at-random MAR [[Yang et al, 2018]](https://dl.acm.org/doi/10.1145/3240323.3240355) data) to improve recommendation diversity [[Section 2.2 of Moller et al, 2018]](https://www.tandfonline.com/doi/full/10.1080/1369118X.2018.1444076) or to explore users' interests for unseen contents [[Gao et al, 2022]](https://arxiv.org/pdf/2208.08696.pdf) or for debias purposes [[Liu et al, 2023]](https://dl.acm.org/doi/10.1145/3582002). These interventions will cause all recommendation probabilities to be non-zero (although they may be very small). \n\n> ... In this sense, the measures for empirical analysis should also only consider top-k items, not all items.\n\nNote that our measures RD and RR include the recommendation probability $p_{ij}$ (where $p_{ij} = 0$ for non-top-k creators), so RD and RR do consider top-k items. Our other two measures, CD and TP, aim to measure the diversity of the entire creator pool, independent of the recommendation scheme, so they do not consider the recommendation probability for users. \n\n\n> W2: For the real-world designs, it would be more extensive if users included trustworthiness-aware recommender systems that consider dynamic/continual settings. For example, [1] consider performance difference between two different user groups when the user/item features are continually updated over time in the systems.\n\nWe thank the reviewer for the suggestion and we agree that fairness-aware recommendation systems are important in practice. However, we want to make the best use of the limited space in the main article to introduce the notion of dual influence and outline its relationship with polarization in recommendation systems. Whether a system polarizes its users or creators is by itself an interesting trustworthiness question, separated from fairness considerations. Moreover, [1] does not study strategic content creators in recommendation systems and has fundamental differences with our work. Applying this work in case of adaptive users and creators will require a careful re-designing of the method proposed in [1]. We see the consideration of fairness-aware systems as a deeply important area, and hope that our work on dual influence will inspire future research to consider both fairness-aware recommendation and dual influence (as we fully agree with you that most realistic settings would consider both aspects). We are happy to add further discussions on fairness-aware recommendation systems and how our work relate to them in the appendix in the final version.\n\n> W3: For the analysis with Movielens, considering the interaction timestamp in the simulation would more accurately reflect real-world scenarios, for example, for determining the true labels.\n\nWe agree with the reviewer that this will be an interesting simulation. However, we need to model the dual dynamics (that include both user updates and creator updates) in the system, such dynamics are not currently captured in any publicly available dataset that we are aware of. Since we need to model dual dynamics, solely relying on existing MovieLens data (or any other dataset) is not enough, we need to create pseudo-real-world data which simulates these dual dynamics.\n\n\n> Q2: (Minor) Are both consensus and bi-polarization conceptually polarization?\n\nYes. \n\n> Q3: (Minor) How are the initial user/creator embeddings initialized in the Movielens experiment?\n\nThe initialization is done by fitting a two-tower model [Huang et al.] on the existing MovieLens ratings data and we use the tower tops as the initial user and creator embeddings.\n\n----\n[Huang et al.] Learning Deep Structured Semantic Models for Web Search using Clickthrough Data.  CIKM, 2013.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "klRQoAUejv", "reviewer": "Reviewer_Gmcw", "summary": "The paper explores the dynamics between users and content creators in recommender systems, highlighting the dual influence where users’ preferences are shaped by recommendations and creators modify their content to align with what is more likely to be recommended. The study defines a model called user-creator feature dynamics to capture these interactions, demonstrating that such systems are prone to polarization, resulting in a loss of diversity. The paper then examines various approaches to mitigate polarization and improve diversity, finding that relevancy-optimizing methods, such as top-k recommendations, can prevent polarization more effectively than traditional diversity-promoting approaches.", "questions": "1. In theorem 3.3, how does the convergence rate depending on the temperature parameter $\\beta$? I ask this because when $\\beta\\rightarrow +\\infty$, the softmax recommendation strategy is equivalent to the top-1 recommendation strategy. In this case, Proposition 4.2 predicts that the top-1 recommendation lead to $n$ clusters rather than bi-polarization, which seems to contradict theorem 3.3.\n2. I do not fully get why the specific forms of function $f$ and $g$ do not affect the analysis. Is it because your main results only depend on the range of $f$ and $g$?\n3. In the experiments, the range of $\\beta$ is quite conservative. I'm curious about the result under a larger range of $\\beta$.", "limitations": "yes", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper provides an interesting perspective by addressing the mutual influence between users and creators in recommender systems. The theoretical results and experimental validation using both synthetic and real-world data look credible. The writing is overall easy to follow.", "weaknesses": "1. There are two lines of works focusing on modeling content creator dynamics and user preference evolving dynamics that are neglected by the authors. I listed several representative works and it would be great to include a comprehensive literature review regarding these works in the related work section.\n\n2. One of your main observation (larger $\\beta$ leads to higher creator diversity and alleviated polarization) is actually pointed out in [1] under a similar model, where content creators compete for a fixed user population (see section 3.2 in [1]). And another main observation in section 5.3 that smaller $k$ improves diversity does not echo the result in [2], which shows that larger $k$ improves the total creator utilities. It would be better to include some detailed discussions regarding these two works. \n\n3. The user/creator preference updating dynamics need more justifications and empirical evidence.\n\n4. The dynamical model makes some sense to me, but it would be more interesting to understand whether the observations still hold in the presence of noise. If the noisy version is hard to analyze theoretically, additional simulation results could also be valuable.\n\n\n[1]. Modeling Content Creator Incentives on Algorithm-Curated Platforms\n[2]. How Bad is Top-K Recommendation under Competing Content Creators?\n[3]. Online recommendations for agents with discounted adaptive preferences\n[4]. Recommender systems as dynamical systems: Interactions with viewers and creators\n[5]. Learning from a learning user for optimal recommendations\n[6]. Supply-side equilibria in recommender systems", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Weakness 1: related works.\n\nThank you for listing those related works! We will discuss them in the revision. We also provide comparisons between some of those works and our work in a table in our global response. \n\n\n> Weakness 2: It would be better to include some detailed discussions regarding these two works [1] [2].\n>\n> [1] Modeling Content Creator Incentives on Algorithm-Curated Platforms\n> \n> [2] How Bad is Top-K Recommendation under Competing Content Creators?\n\n[1] indeed also observes that a larger $\\beta$ leads to higher creator diversity. We view our work as complementary to [1]. Our findings corroborate those of [1] despite two key differences between our setting and that of [1]: in [1], creators maximize exposure while we consider creators who maximize user engagement; second, [1]'s user population is fixed while ours is adaptive. This helps us understand that there may be some fundamental mitigation strategies (such as selecting $\\beta$), which have similar effects regardless of changes to the underlying problem formulation.\n\n[2] shows that a larger $k$ improves the social welfare, defined to be the total utility/relevance of the users (which is also the total utility of creators in their model), assuming a fixed user population. In contrast, we focus on the diversity/polarization of creators, and we have adaptive users. So the $k$ plays a different role in our setting: a larger $k$ leads to worse creator diversity. \n\n> Weakness 3: the user/creator preference updating dynamics need more justifications and empirical evidence.\n\nOur user preference update model is a generalization of [[Dean & Morgenstern, EC 22]](https://arxiv.org/pdf/2205.13026), which models user update as \n$u^{t+1}_j = \\mathcal{P}(u^t_j + \\eta \\langle v_i^t, u_j^t\\rangle v_i^t)$.\nOur work replaces the inner produce with a general function $f(v_i^t, u_j^t)$ (our Equation 3), with some constraints outlined on lines 113 - 121 of our paper. The motivation for this update rule, as outlined by [[Dean & Morgenstern, EC 22]](https://arxiv.org/pdf/2205.13026), is the \"biased assimilation\" phenomenon, which is in turn inspired by the opinion polarization literature.  (We mentioned this in Lines 107, 115, and 241.)\n\nOur creator update model, as we wrote in Lines 122 - 129, is motivated by rating-maximizing creators. This means that the $g(u_j, v_i)$ function has the same sign as $\\langle u_j, v_i\\rangle$. Intuitively, for all the users $J_i^t$ being recommended creator $i$, if the user likes the item, $\\langle u_j^t, v_i^t\\rangle > 0 \\implies g(u_j^t, v_i^t) > 0$, then the creator is incentivized to move towards that user to receive more positive rating from that user. Otherwise, $\\langle u_j^t, v_i^t\\rangle < 0 \\implies g(u_j^t, v_i^t) < 0$, the creator is incentivized to move away from the user in order to be recommended less often to that user (to avoid being negatively-rated by the user). Taking both scenarios into account, the creator moves toward the weighted average $\\sum_{j\\in J_i^t} g(u_j^t, v_i^t) u_j^t$, which gives our update rule (4). \n\nIf any additional details would help improve the justification of the user/creator update rules, please let us know. This is an important aspect of our work and we would like to polish it as much as possible.\n\n\n> Weakness 4: it would be more interesting to understand whether the observations still hold in the presence of noise.\n\nIn the newly uploaded PDF, we provide simulation results where the user and creator updates include normally distributed unbiased noises inside the projection operator: \n$u_j^{t+1} = P (u_j^t + \\eta_u f( v_{i_j^t}^t, u_j^t) v_{i_j^t}^t + \\eta_u \\epsilon_j^t)$ where $\\epsilon_j^t \\sim Normal(0, \\sigma^2I)$\nand\n$v_i^{t+1} = P ( v_i^t + \\frac{\\eta_c}{|J_i^t|} \\sum_{j \\in J_i^t} g(u_j^t, v_i^t) u_j^t + \\eta_c \\epsilon_i^t)$ where $\\epsilon_i^t \\sim Normal(0, \\sigma^2 I)$. \n\nWe observe that a small noise still leads to near polarization, while a large noise (large $\\sigma$) reduces the tendency to polarization. And the observation that top-k recommendation reduces polarization still holds. (see Figures R2 - R5 in the uploaded PDF.)\n\n> Q1: how does the convergence rate depend on the temperature parameter $\\beta$? I ask this because when $\\beta\\rightarrow +\\infty$, the softmax recommendation strategy is equivalent to the top-1 recommendation strategy, which by Proposition 4.2 leads to $n$ clusters rather than bi-polarization, which seems to contradict theorem 3.3.\n\nFinite $\\beta$ and infinite $\\beta$ are qualitatively different. Finite $\\beta$ leads to bi-polarization, while $\\beta = +\\infty$ is equivalent to top-1 recommendation and does not lead to bi-polarization.  Indeed, the rate of convergence to bi-polarization with a large but finite $\\beta$ might be slow (see the \"Tendency to Polarization\" plot in Figure R1 in the PDF we uploaded during rebuttal).  Our Theorem 3.3 is more of an asymptotic result.  Analyzing the convergence rate would be an interesting yet challenging direction for future work.\n\n> Q2: I do not fully get why the specific forms of function do not affect the analysis. \n\nFor $f$, our results and analysis only require the assumptions in Lines 114 - 121 that $f(v_i, u_j)$ has the same sign as $\\langle v_i, u_j\\rangle$ and is two-sided bounded $L_f \\le |f(v_i, u_j)| \\le 1$.\n\nFor $g$, our current analysis assumes the specific form of $g(u_j, v_i) = sign(\\langle u_j, v_i\\rangle)$. We believe our analysis (in particular, Lemma E.1) can be generalized to other $g$ functions satisfying similar assumptions as $f$ (but as of now this generalization remains an open problem). \n\n\n> Q3: results under a large range of $\\beta$. \n\nIn Figure R1 of the uploaded PDF, we provide experiment results for $\\beta \\in [0, 10]$ and $+\\infty$. We note that $\\beta = 10$, although not very large, has similar effects as $\\beta=+\\infty$ (top-1 recommendation) in 1000 time steps, because the softmax probability of non-max creators when $\\beta =10$ is very close to 0.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "lOaIhR0ul3", "reviewer": "Reviewer_JScD", "summary": "This paper models dynamics of both users and creators in a recommender system. The user features shift in the direction of the content recommended to them. The creator dynamics are strategically motivated i.e. they try to align content to attract their audience, to increase profit. \n\nThe authors then provide sufficient conditions for this model of dual dynamics to converge to polarization under a natural assumption that each creator has some non zero probability of being recommended to every user.\n\nThe paper then discusses four real world recommendation designs, and whether they cause polarization or multiple clusters etc. They also provide results on synthetic and Movielens data complementing theory results and show that certain recommender designs do lead to polarization vs diverse clusters.", "questions": "-  I understand the motivation for the form of user update in equation (3) . In each recommendation step an item is recommended and the user preference shifts in that direction, this is like in [Dean and Morgenstern]. Can you motivate the update in Eq (4), is this myopically optimal for the creator to do, and how does it generalize [Eilat & Rosenfeld]?\n- Minor Typo? For Figure 6,  larger $\\rho$ seems to lead to higher creator diversity (green curve).", "limitations": "The authors discuss limitations of their results in the paper.", "rating": 7, "confidence": 5, "soundness": 4, "presentation": 2, "contribution": 4, "strengths": "- This paper is the first to consider dynamics of both users and creators in a recommender systems and provides sufficient analytic conditions for polarization\n- They apply this theory to 4 natural designs: (1) Top-k,  (2) Truncation, (3)Diversity boosting and (4) Lower bounding probability. They show that rules (3, 4) lead to polarization and rule (1) leads to diverse clusters. This section is particularly insightful.\n- The experimental evaluation with synthetic and Movielens data is also insightful and complements the theory. The softmax probability leads to diminishing creator and recommendation diversity over time. They also study top-k probability and show how lower k is better for higher creator diversity, recommendation relevance.", "weaknesses": "A criticism I had while reading the paper are gaps in literature for the discussion on dynamics in recommender systems. In addition to [Eilat & Rosenfeld] referenced in the introduction, [1,2,3,4,5,6] consider creator dynamics in recommender systems. These works assume static user features and provide results on content at equilibrium and user welfare. In the context of these works, it would be beneficial to highlight how your work is the first to consider both creator and user dynamics.\n\n[1] A Game-Theoretic Approach to Recommendation Systems with Strategic Content Providers (Ben-Porat and Tennenholtz)\n\n[2] Supply-side equilibria in recommender systems (Jagadeesan et al)\n\n[3] How Bad is Top-k Recommendation under Competing Content Creators? (Yao et al)\n\n[4] Modeling content creator incentives on algorithm-curated platforms (Hron et al)\n\n[5] Producers Equilibria and Dynamics in Engagement-Driven Recommender Systems (Acharya et al)\n\n[6] User Welfare Optimization in Recommender Systems with Competing Content Creators (Yao et al)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1: Can you motivate the update in Eq (4)? Is this myopically optimal for the creator to do, and how does it generalize [Eilat & Rosenfeld]?\n\n[[Eilat & Rosenfeld]](https://arxiv.org/pdf/2302.04336) assumes that creators aim to maximize exposure (defined as the sum of inner products between user embeddings and the creator embedding) minus the content adjustment cost (see their Section 2.1 and equation (6)). They show that such creators will move towards the average of the user embeddings by some step size (their equation (11)). In our notations, their update is $v_i^{t+1} = \\mathcal P(v_i^t + \\frac{\\eta_c}{|J_i^t|} \\sum_{j\\in J_i^t} u_j^t)$, which is the special case of our equation (4) where $g$ is the constant positive function $g(u_j, v_i) = 1$.\n\nNevertheless, as we wrote in Lines 122 - 129, we are motivated by a different type of creators: rating-maximizing creators. This means that the $g(u_j, v_i)$ function should have the same sign as $\\langle u_j, v_i\\rangle$. Intuitively, for all the users $J_i^t$ being recommended creator $i$, if the user likes the item, $\\langle u_j^t, v_i^t\\rangle > 0 \\implies g(u_j^t, v_i^t) > 0$, then the creator is incentivized to move towards that user to receive more positive rating from that user. Otherwise, $\\langle u_j^t, v_i^t\\rangle < 0 \\implies g(u_j^t, v_i^t) < 0$, the creator is incentivized to move away from the user in order to be recommended less often to that user (to avoid being negatively-rated by the user). Taking both scenarios into account, the creator moves toward the weighted average $\\sum_{j\\in J_i^t} g(u_j^t, v_i^t) u_j^t$, which gives equation (4). Under this particular assumption on $g$, our equation (4) does not capture [[Eilat & Rosenfeld]](https://arxiv.org/pdf/2302.04336). \n\n\n\n> Q2: Minor Typo? For Figure 6, larger $\\rho$ seems to lead to higher creator diversity (green curve).\n\nYes, this is a minor typo. The figure shows that a higher $\\rho$ leads to higher creator diversity but also higher tendency to polarization at the same time.  A possible explanation for this phenomenon is (similar to our reasoning in Lines 307 to 309): the system polarizes into two balanced clusters which actually have a large average pairwise distance.  So in this case, Tendency to Polarization may be a better measure for diversity loss than the Creator Diversity measure (average pairwise distance).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:
We thank the reviewers for the helpful comments, especially the provided related works. Here, we provide a table to compare our work with those works (and some works that were already cited in our paper). We will add this table to an additional related work section in our appendix. We want to highlight that, while previous works have studied dynamic creators and dynamic users separately, "our work is the first to consider both creator and user dynamics", as pointed out by reviewer JScD.

| Works            | Adaptive Users | Adaptive Creators | Creator Reward  | Characterizing Dynamics or Equilibrium Behavior | Creation Change Model |
| ---------------- | ---------|-----  | -------------- | --------------------- | ------------------- |
| Our Work         | Yes | Yes | User Engagement | Dynamics | Conditioned on previous time step; implicit cost of content adjustment| 
| Eilat & Rosenfeld [1] | No | Yes | Exposure        | Dynamics | Conditioned on previous time step; explicit cost of content adjustment |
| Yao et al [2]      | No  | Yes | User Engagement | Dynamics | Freely choose without cost |
| Jagadeesan et al [3] | No  | Yes | Exposure        | Equilibrium  | Freely choose with cost    |
| Hron and Krauth et al [4] | No | Yes |  Exposure    | Equilibrium  | Freely choose without cost |
| Ben-Porat and Tennenholtz [5] | No | Yes |  Exposure    | Equilibrium  | Freely choose without cost |
| Acharya et al [6] | No | Yes |  User Engagement    | Equilibrium  | Freely choose without cost |
| Yao et al [7] | No | Yes |  Reward designed by a social-welfare-maximizing platform    | Dynamics  | Freely choose without cost |
| Dean and Morgenstern [8] | Yes | No* |  N/A    | Dynamics  | N/A |
| Yao et al [9] | Yes | No* |  N/A    | Dynamics  | N/A |
| Agarwal and Brown [10] | Adaptive and adversarial | No* |  N/A    | Dynamics  | N/A |

*: These works study the design of recommendation algorithms for the platform with a fixed set of items, without explicitly modeling the content creators. 

---

[1] Eilat and Nir Rosenfeld. Performative Recommendation: Diversifying Content via Strategic Incentives. ICML 2023.

[2] Yao et al. How Bad is Top-K Recommendation under Competing Conten Creators? ICML 2023.

[3] Jagadeesan et al. Supply-Side Equilibria in Recommender Systens. NeurIPS 2023.

[4] Hron and Krauth et al. Modeling Content Creator Incentives on Algorithm-Curated Platforms. ICLR 2023.

[5] Ben-Porat and Tennenholtz. A Game-Theoretic Approach to Recommendation Systems with Strategic Content Providers. NeurIPS 2018. 

[6] Acharya et al. Producers Equilibria and Dynamics in Engagement-Driven Recommender Systems. ArXiv 2024. 

[7] Yao et al. User Welfare Optimization with Competing Ccontent Creators. ArXiv 2024.

[8] Dean and Morgenstern. Preference Dynamics Under Personalized Recommendations. EC, 2022.

[9] Yao et al. Learning from a Learning User for Optimal Recommendations. ICML 2022.

[10] Agarwal and Brown. Online recommendations for agents with discounted adaptive preferences. ALT 2024.

Meta-review: The paper studies the dual influence of recommender systems on users and creators, and provides theoretical proof on the convergence towards polarization under the model and certain conditions, and presented findings showing relevance focusing systems such as top-K recommendations rather than diversity promoting systems are more effective at preventing polarization.  Reviewers find the work solid and the result interesting. Some minor comments are missing references and clarification on assumptions were addressed during rebuttal.
Decision: Accept (poster)