Biased Assimilation, Homophily, and the Dynamics of Polarization

Transcription

1 Biased Assimilation, Homophily, and the Dynamics of Polarization Pranav Dandekar joint work with A. Goel D. Lee

2 Motivating Questions Are we as a society getting polarized? If so, why? Do recommender systems polarize?

3 Debate on Polarization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olarization? Yes

4 Homophily & Polarization Literature argues that Homophily (greater interaction with like-minded individuals) Social Corroboration Extreme Opinions Polarization

5 Debate on Polarization Polarization? No

6 Debate on Polarization Polarization? No

7 Our Contributions Define polarization. Show: DeGroot s opinion formation process is not polarizing, even in arbitrarily homophilous networks. Model biased assimilation in an opinion formation process. Show: Biased opinion formation results in polarization. Make connection between biased assimilation and the polarizing effects of recommender systems. Punchline: Homophily alone, without biased assimilation, is not sufficient to polarize society.

8 Polarization Definition Polarization = Increasing Divergence in Opinions

9 Polarization Definition Polarization = Increasing Divergence in Opinions

10 Model A connected, undirected, weighted graph G = (V, E, w). Node i has opinion x i (t) [0, 1] at time t = 0, 1, 2,.... Network Disagreement Index (NDI): η(g, x) := w ij (x i x j ) 2 (i,j) E An opinion formation process is polarizing if NDI increases.

11 DeGroot s Process Definition (DeGroot s Process) x i (t + 1) = w iix i (t) + s i (t) w ii + d i (1) where s i (t) := j N(i) w ijx j (t), and d i := j N(i) w ij. Variants used to explain lack of consensus (e.g., stubborn agents, intrinsic opinions).

12 DeGroot is not Polarizing Theorem At each timestep, DeGroot s process can only decrease NDI even if the network is arbitrarily homophilous.

13 Biased Assimilation aka confirmation bias (aka cherry-picking evidence) Lord et al. : People who hold strong opinions on complex social issues are likely to examine relevant empirical evidence in a biased manner. They are apt to accept confirming evidence at face value while subjecting disconfirming evidence to critical evaluation, and as a result to draw undue support for their initial positions from mixed or random empirical findings. A well-known phenomenon in social psychology.

14 Connection to Urn Models x i (t): fraction of RED balls in i s urn. 1 x i (t): fraction of BLUE balls in i s urn. x i (t + 1) 1 x i (t + 1) x j(t) 1 x j (t)

15 Connection to Urn Models DeGroot x i (t): fraction of RED balls in i s urn. 1 x i (t): fraction of BLUE balls in i s urn. x i (t + 1) 1 x i (t + 1) x j(t) 1 x j (t)

16 Connection to Urn Models DeGroot x i (t): fraction of RED balls in i s urn. 1 x i (t): fraction of BLUE balls in i s urn. x i (t + 1) 1 x i (t + 1) x j(t) 1 x j (t)

17 Connection to Urn Models DeGroot x i (t): fraction of RED balls in i s urn. 1 x i (t): fraction of BLUE balls in i s urn. x i (t + 1) 1 x i (t + 1) x j(t) 1 x j (t)

18 Connection to Urn Models Biased Assimilation x i (t): fraction of RED balls in i s urn. 1 x i (t): fraction of BLUE balls in i s urn. x i (t + 1) 1 x i (t + 1) x i (t)x j (t) (1 x i (t))(1 x j (t))

19 Connection to Urn Models Biased Assimilation x i (t): fraction of RED balls in i s urn. 1 x i (t): fraction of BLUE balls in i s urn. x i (t + 1) 1 x i (t + 1) x i (t)x j (t) (1 x i (t))(1 x j (t))

20 Connection to Urn Models Biased Assimilation x i (t): fraction of RED balls in i s urn. 1 x i (t): fraction of BLUE balls in i s urn. x i (t + 1) 1 x i (t + 1) x i (t)x j (t) (1 x i (t))(1 x j (t))

21 Connection to Urn Models Biased Assimilation x i (t): fraction of RED balls in i s urn. 1 x i (t): fraction of BLUE balls in i s urn. x i (t + 1) 1 x i (t + 1) x i (t)x j (t) (1 x i (t))(1 x j (t))

22 Connection to Urn Models Biased Assimilation x i (t): fraction of RED balls in i s urn. 1 x i (t): fraction of BLUE balls in i s urn. x i (t + 1) 1 x i (t + 1) x i (t)x j (t) (1 x i (t))(1 x j (t))

23 Connection to Urn Models Biased Assimilation x i (t): fraction of RED balls in i s urn. 1 x i (t): fraction of BLUE balls in i s urn. x i (t + 1) 1 x i (t + 1) x i (t)x j (t) (1 x i (t))(1 x j (t))

24 Connection to Urn Models Biased Assimilation x i (t): fraction of RED balls in i s urn. 1 x i (t): fraction of BLUE balls in i s urn. x i (t + 1) 1 x i (t + 1) x i (t)x j (t) (1 x i (t))(1 x j (t))

25 Biased Opinion Formation x i (t + 1) 1 x i (t + 1) (x i (t)) bi x j (t) (1 x i (t)) b i (1 xj (t))

26 Biased Opinion Formation x i (t + 1) 1 x i (t + 1) (x i (t)) bi x j (t) (1 x i (t)) b i (1 xj (t)) Definition (DeGroot s Process) x i (t + 1) = w iix i (t) + s i (t) w ii + d i where s i (t) := j N(i) w ijx j (t), and d i := j N(i) w ij.

27 Biased Opinion Formation x i (t + 1) 1 x i (t + 1) (x i (t)) bi x j (t) (1 x i (t)) b i (1 xj (t)) Definition (Biased Opinion Formation Process) x i (t + 1) = w ii x i (t) + (x i (t)) b i s i (t) w ii + (x i (t)) b i si (t) + (1 x i (t)) b i (di s i (t)) (2) where, s i (t) := j N(i) w ijx j (t), and d i := j N(i) w ij.

28 Two-island Network ps ps pd V1 h = ps/pd V2

29 Biased Assimilation and Polarization Theorem Assume for all i V 1, x i (0) = x 0 where 1 2 < x 0 < 1. Assume for all i V 2, x i (0) = 1 x 0. 1 (Polarization) If b 1, i V 1, lim t x i (t) = 1, and i V 2, lim t x i (t) = 0. 2 (Persistent Disagreement) if 1 > b 2 h+1, then there exists a unique ˆx ( 1 2, 1) such that i V 1, lim t x i (t) = ˆx, and i V 2, lim t x i (t) = 1 ˆx. 3 (Consensus) if b < 2 h+1, then for all i, lim t x i (t) = 1 2.

30 Recommender Systems and Polarization Do recommender systems polarize? Recommender systems Echo-chamber effect Extreme Opinions Polarization

31 Recommender Systems and Polarization n m... n

32 Recommender Systems and Polarization n m... n

33 Recommender Systems and Polarization n m... n

34 Recommender Systems and Polarization n m... n

35 Recommender Systems and Polarization n m... n

36 Recommender Systems and Polarization Definition Consider a book recommended to an individual i V 1. We say that i is unbiased if i accepts the recommendation with the same probability independent of whether the book is RED or BLUE. We say that i is biased if 1 i accepts the recommendation of a RED book with probability x i, and rejects it with probability 1 x i, and 2 i accepts the recommendation of a BLUE book with probability 1 x i, and rejects it with probability x i.

37 Recommender Systems and Polarization Definition Consider a recommender algorithm and an individual i V 1 that accepts the algorithm s recommendation. The algorithm is polarizing with respect to i if 1 when x i > 1 2, the probability that the recommended book was RED is greater than x i, and 2 when x i < 1 2, the probability that the recommended book was RED is less than x i. Recommender algorithms should be depolarizing at least when users are unbiased

38 Recommender Systems and Polarization Algorithm 1: Find the most relevant item R for a user, using a random walk based collaborative filtering algorithm.

39 Recommender Systems and Polarization Algorithm 2: Find a random item X that the user bought, and then use a random walk based collaborative filtering algorithm to find the most similar item R to X. Informally: The first algorithm causes polarization even with unbiased users, whereas the second causes depolarization.

40 Thank you Questions?