Source-linked AI summary
Biased Assimilation, Homophily and the Dynamics of Polarization
Pranav Dandekar, Ashish Goel, David Lee
TL;DR
The paper asks why society may become polarized and whether homophily and personalized recommender systems contribute. It generalizes DeGroot’s opinion model to include biased assimilation and finds that homophily alone does not polarize, whereas biased assimilation can produce polarization, persistent disagreement, or consensus.
Problem
The paper addresses conflicting evidence about societal polarization and whether homophily and personalized information systems contribute to it.
Method
The paper generalizes DeGroot’s repeated-averaging model to represent biased assimilation in weighted social networks and analyzes related recommender algorithms.
Results
DeGroot-like averaging is never polarizing, while biased opinion formation yields polarization for b ≥1 and otherwise persistent disagreement or consensus depending on homophily.
Takeaways & Limitations
Homophily alone is insufficient to polarize society; biased assimilation is central to the model’s polarization results and to the analyzed recommender effects.
Takeaways & Limitations
The network disagreement index does not fully capture issue alignment, such as opposing ideologically coherent factions with diverse internal opinions.
Abstract
from arXiv · showhide
Are we as a society getting more polarized, and if so, why? We try to answer this question through a model of opinion formation. Empirical studies have shown that homophily results in polarization. However, we show that DeGroot's well-known model of opinion formation based on repeated averaging can never be polarizing, even if individuals are arbitrarily homophilous. We generalize DeGroot's model to account for a phenomenon well-known in social psychology as biased assimilation: when presented with mixed or inconclusive evidence on a complex issue, individuals draw undue support for their initial position thereby arriving at a more extreme opinion. We show that in a simple model of homophilous networks, our biased opinion formation process results in either polarization, persistent disagreement or consensus depending on how biased individuals are. In other words, homophily alone, without biased assimilation, is not sufficient to polarize society. Quite interestingly, biased assimilation also provides insight into the following related question: do internet based recommender algorithms that show us personalized content contribute to polarization? We make a connection between biased assimilation and the polarizing effects of some random-walk based recommender algorithms that are similar in spirit to some commonly used recommender algorithms.
1 Introduction
The paper studies whether polarization is increasing and how interaction patterns and biased opinion formation may produce it. It argues that homophily alone is insufficient: repeated averaging is non-polarizing, while biased assimilation can yield polarization, disagreement, or consensus.
- Polarization is debated empirically because studies reach different conclusions depending on context and measurement.
- Homophily—greater interaction with like-minded individuals—has been argued to produce polarization and echo-chamber effects.
- The model generalizes DeGroot by weighting confirming and opposing evidence differently through individuals’ bias parameters.
- DeGroot-like repeated averaging cannot polarize opinions, even when individuals are arbitrarily homophilous.
- With homophilous networks, biased opinion formation produces polarization when b ≥1, but persistent disagreement or consensus when b < 1, depending on homophily.
- The paper connects biased assimilation to recommender algorithms, finding SimplePPR always polarizing while SimpleSALSA and SimpleICF polarize only when individuals are biased.
2 Model
The model measures polarization as increased opinion divergence and generalizes DeGroot averaging to represent biased assimilation, where individuals overweight evidence supporting their existing views. For a fixed environment, bias above one drives opinions toward extremes, while lower bias stabilizes the threshold.
- Measuring Polarization: Polarization is defined as an opinion-formation process increasing network disagreement, measured by the weighted squared differences between connected individuals.The NDI captures issue radicalization but does not fully capture issue alignment between opposing factions.
- DeGroot’s Model: DeGroot’s process updates each individual by repeatedly averaging neighbors’ and their own previous opinions.The model represents influence through weighted social-network edges and individual self-weights.
- Biased Assimilation: Biased assimilation makes individuals accept confirming evidence more readily and scrutinize disconfirming evidence, producing more extreme opinions from mixed evidence.The proposed update weights a neighbor’s support for an individual’s current position by (x_i(t))^b_i.
- Biased Assimilation: When b_i = 0, the biased process reduces to DeGroot’s unbiased averaging process.More generally, the update can use a non-decreasing bias function to weight support according to the individual’s current opinion.
- Single-Agent Dynamics: When b > 1, the polarization threshold is unstable: opinions above it move toward 1, while opinions below it move toward 0.At the threshold itself, the opinion remains unchanged; when b < 1, the threshold is stable.
3 DeGroot’s Repeated Averaging Process is not Polarizing
The paper shows that repeated averaging cannot increase network disagreement, even on arbitrarily homophilous weighted networks. Because the NDI may miss disagreement between opposing factions, the authors also introduce a global measure and analyze a flocking process that decreases it.
- DeGroot’s Process: DeGroot’s repeated averaging weakly decreases the network disagreement index for every connected weighted undirected graph when all bias parameters are zero.Thus, repeated averaging is depolarizing even when the underlying network is arbitrarily homophilous.
- DeGroot’s Process: The result also applies when self-weights become infinite, corresponding to repeated averaging with zealots.In that limit, an individual’s opinion remains unchanged.
- Alternative Polarization Measure: The NDI may remain small for two densely connected opposing factions with sparse cross-links, motivating the global disagreement index as an alternative measure.The GDI is independent of edge weights, unlike the NDI.
- Flocking Process: The flocking process moves selected individuals toward their group’s average opinion while leaving others unchanged.The authors state that each such update can only lower the global disagreement index.
4 Polarization due to Biased Assimilation
In a two-island network, biased assimilation produces polarization, persistent disagreement, or consensus depending on bias strength, while homophily sets the network structure.
- Network model: The two-island network uses stronger within-type connectivity than cross-type connectivity, with homophily hG defined as ps/pd.Higher hG means nodes are more likely to connect to others of their own type.
- Equilibrium outcomes: When b ≥ 1, opinions polarize: type V1 converges to 1 and type V2 converges to 0.The result holds for the stated initial conditions and symmetric bias parameter.
- Equilibrium outcomes: When 1 > b ≥ 2/(hG + 1), the two types converge to distinct complementary opinions, producing persistent disagreement.There is a unique equilibrium x̂ ∈ (1/2, 1), with V1 converging to x̂ and V2 to 1 − x̂.
- Equilibrium outcomes: When b < 2/(hG + 1), all individuals converge to the same opinion, yielding consensus.The equilibrium disagreement index is zero in this regime.
- Implications: The bias parameter determines whether the process polarizes, sustains disagreement, or depolarizes; b = 1 causes polarization for arbitrarily small homophily.Moderate bias may avoid consensus without necessarily increasing divergence, whereas low bias produces consensus.
5 Recommender Systems and Polarization
The paper models personalized recommender systems as opinion-formation processes on user-item graphs and analyzes when random-walk algorithms amplify users’ existing preferences.
- Model: A recommender algorithm maps a bipartite user-item graph and an individual to one recommended item.Edges represent ownership, and items include books, webpages, news articles, and products.
- Model: Buying an item requires both recommendation and user acceptance, with item colors used to measure opinion movement.Each user’s opinion xi is the fraction of owned RED books, while 1 − xi is the BLUE fraction.
- User behavior: Biased users accept RED recommendations with probability xi and BLUE recommendations with probability 1 − xi, whereas unbiased users accept both colors equally often.The biased-assimilation rule makes acceptance depend on the user’s current opinion.
- Results: In the limit as n →∞ and T →∞, SimplePPR is polarizing even when the user is unbiased.The proofs use the Strong Law of Large Numbers to replace relevant random quantities by their expectations.
- Results: SimpleSALSA and, as T →∞, SimpleICF are polarizing with respect to a user if and only if that user is biased.Thus these algorithms do not polarize unbiased users under the model.
6 Discussion of Various Measures of Opinion Divergence
The paper checks its polarization claims under several divergence measures, finding that DeGroot-like averaging is not polarizing while biased assimilation can be.
- Convex divergence: For any symmetric convex divergence measure, each flocking update is depolarizing because the new opinion vector is majorized by the previous one.This generalizes the conclusion beyond a single disagreement index.
- Alternative measures: Under second-order stochastic dominance, individual DeGroot and flocking updates are generally neither polarizing nor depolarizing, but both converge to consensus under broad conditions.Consequently, both processes are depolarizing in equilibrium under those conditions.
- Alternative measures: The three recommender-algorithm results also hold when polarization is defined using second-order stochastic dominance.The alternative definition therefore preserves the reported recommender-system conclusions.
- Stepwise divergence: Under the strongest definition, which tracks movement away from the average at every step, biased assimilation polarizes a two-island network when b ≥ 1.DeGroot and flocking are neither polarizing nor depolarizing under this stepwise definition.
7 Conclusion
The paper concludes that homophily alone cannot polarize society in DeGroot-like averaging, whereas biased assimilation can generate polarization, disagreement, or consensus and helps explain recommender effects.
- Main conclusions: DeGroot-like repeated averaging cannot polarize, even when individuals are arbitrarily homophilous.The conclusion identifies biased assimilation as necessary for polarization in the paper’s network model.
- Main conclusions: In a two-island network, biased opinion formation yields polarization when b ≥ 1, persistent disagreement when 1 > b ≥ 2/(h + 1), and consensus when b < 2/(h + 1).The outcome depends on how strongly individuals are biased.
- Recommender systems: SimpleSALSA and SimpleICF polarize only biased individuals, whereas SimplePPR polarizes even unbiased individuals under the user-item graph model.These results connect biased assimilation with the effects of three recommender algorithms.
- Future work: The authors identify human-subject experiments on homophily and biased-assimilation strength as a direction for further investigation.They also describe the recommender analysis as a first step toward systems that counteract polarization and facilitate consensus.
A Proof of Theorem 1
The update rule makes the threshold equilibrium stable when b < 1 but unstable when b > 1. Above the threshold, opinions move upward; below it, they move downward, and sufficiently biased opinions converge to an extreme.
- Update rule: The update rule expresses the next opinion as a weighted combination of current support terms raised to bias parameter b.The ratio form rewrites the update using powers b−1, exposing how bias changes relative support.
- Threshold dynamics: When b > 1, the threshold ˆx is an unstable equilibrium.The threshold itself remains fixed, but any initial opinion away from it moves farther in the corresponding direction.
- Extreme outcomes: When b > 1, limt→∞x(t) = 1 or limt→∞x(t) = 0.Monotonicity and boundedness rule out convergence to an interior value.
- Stable regime: When b < 1, ˆx is a stable equilibrium and limt→∞x(t) = ˆx.Opinions above the threshold decrease and opinions below it increase toward the threshold.
B Proofs of Section 3
These proofs establish contraction properties for graph-based opinion dynamics using Laplacian quadratic forms and doubly stochastic averaging matrices. The key inequalities show that the relevant disagreement measure does not increase under the update.
- Laplacian analysis: The weighted graph Laplacian and diagonal normalization reduce the required bound to y⊤D^1/2LGD^1/2y ≤ 2||y||^2.Lemma B.1 supplies this bound for arbitrary vectors on the weighted undirected graph.
- Disagreement contraction: To show η(G, x(t + 1)) − η(G, x(t)) ≤ 0, it suffices to apply the normalized-Laplacian bound.The disagreement measure is represented as y⊤LGy and compared across successive updates.
- Matrix update: The opinion update under the flocking process is represented in matrix form using A(t).The matrix averages opinions within the active set while leaving other nodes unchanged.
- Disagreement contraction: A(t) is doubly stochastic, so γ(A(t)x(t)) ≤ γ(x(t)).The proof uses majorization and Schur-convexity of γ.
C Proof of Theorem 4
The proof analyzes symmetric two-community dynamics through invariant opinion relationships and a one-dimensional equilibrium function. Depending on b relative to the graph threshold, the system moves toward extremes, a stable interior equilibrium, or consensus-like behavior.
- Symmetry reduction: Nodes within each community retain identical opinions, while corresponding nodes across communities satisfy xi(t) = 1 − xj(t).These symmetries reduce the analysis to one representative opinion in V1.
- Equilibrium structure: For 1 > b ≥ 2/(hG + 1), there is a unique equilibrium ˆx in (1/2, 1).The equilibrium is defined through equation (C.3), with uniqueness following from strict monotonicity of f.
- Outcome regimes: The persistent-disagreement case is analyzed as central, with polarization and consensus described as limiting cases as b approaches 1 and 2/(hG + 1).The proof explicitly treats the stable interior equilibrium regime and identifies the other outcomes as limits.
- Persistent disagreement: For 1 > b ≥ 2/(hG + 1), limt→∞xi(t) = ˆx.Bounded monotonicity and the equilibrium characterization establish convergence to the interior value.
D Proofs of Section 5
The recommender-system proofs compare random-walk recommendation probabilities across RED and BLUE books. SimplePPR is polarizing regardless of whether the individual is biased, whereas SimpleSALSA and SimpleICF are polarizing exactly when the individual is biased.
- SimpleSALSA: SimpleSALSA is polarizing with respect to i if and only if i is biased.The result is established in the limit as n →∞ and T →∞.
- SimpleSALSA: For SimpleSALSA, the proof first establishes pr > 1/2 and pr ≤ xi before analyzing biased acceptance.These bounds are used to connect recommendation probabilities with polarization under biased assimilation.
- Biased acceptance: For a biased individual, accepting a RED recommendation raises the conditional probability that the recommendation was RED above its unconditional probability.The comparison follows from the individual accepting RED with probability xi and BLUE with probability 1 − xi.
- SimpleICF: SimpleICF is polarizing with respect to i if and only if i is biased.The same limiting characterization holds for the two-step random-walk recommender.
- SimplePPR: SimplePPR is polarizing regardless of whether the recommended book is accepted by a biased or unbiased individual.The accepted recommendation has RED probability pr in either case, so acceptance does not alter the recommendation’s partisan skew.