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Homophily influences ranking and sampling of minorities in social networks

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Homophily influences ranking and sampling of minorities in social networks

  1. 1. Homophily influences visibility and ranking of minorities in social networks Fariba Karimi GESIS - Leibniz Institute for the Social Sciences, Cologne Computational Social Science Department
  2. 2. Moody, AJS (2001) Baerveldt et al (2004) Political blog sphere, Adamic & Glance (2004)
  3. 3. Moody, AJS (2001) Baerveldt et al (2004) Homophily = attribute assortativity Sexual network at a hgi school Bearman, Moody & Stovel (2004)
  4. 4. Dropout vs. Career Age early career mid career senior Jadidi, Karimi, Lietz, & Wagner. Advances in Complex Systems (2017)
  5. 5. Homophily in Co-authorhship Network Jadidi, Karimi, Lietz, & Wagner. Advances in Complex Systems (2017) 1975 1980 1985 1990 1995 2000 2005 2010 2015 year 0 10 20 30 40 50<z-score(H)> male female homophily
  6. 6. In many networks we observe homophilic / heterophilic interactions and groups with different size. Why does it matter?
  7. 7. Franklin, Anderson J., and Nancy Boyd-Franklin. "Invisibility syndrome: a clinical model of the effects of racism on African- American males." American Journal of Orthopsychiatry (2000). If minorities become less visible, this would create situations in which i) high-ranked minority members become less noticeable globally and therefore less influential in society, ii) minorities feel ignored or overlooked by the wider public, also known as the invisibility syndrome. Visibility Matters!
  8. 8. How does the inherent structure of social networks, (homophily and group size), impact ranking (visibility) of groups (minorities)?
  9. 9. Social Networks with attributes A collaboration network.
  10. 10. Network Growth Model •  2 group of nodes with unequal size •  Arrival nodes connect to existing nodes based on preferential attachment (k) and homophily (h) •  homophily can be asymmetric
  11. 11. BA-Homophily network model h = 0 h = 0.2 h = 0.5 h = 0.8 h = 1 majority minority B C D EA minority size = 0.2 complete homophilycomplete heterophily degreedistributiondegreegrowthnetwork Figure 6 Evolution of the exponents for the degree growth, sym- metrical homophily. The exponents ba (minority) and bb (majority) are defined in eqs. (15) and (17). h = haa = hbb is the homophily parameter and the numbers indicate the fraction of nodes belonging to the minority group (parameter fa). which gives: ⇢ ka µ t fb kb µ t fa (7) Similarly, for haa = hbb = 1 and hab = mophilic network) we get: 8 >>< >>: dKa dt = 2m fa dKb dt = 2m fb and thus for the evolution of the degree of 8 >>>< >>>: dka dt = m fa ka Âi qiki = m fa K dkb dt = m fb kb Âi qiki = m fb K which gives: ⇢ ka µ t1/2 kb µ t1/2 Let’s make the hypothesis that Ka(t) an tions of time, so that Ka(t) = Cmt and K Eq. (2). In the case of two groups, we ca by denoting fa = f and fb = 1 f. Using dKa dt = Cm = m ✓ f ✓ 1+ haaCmt haaCmt +hab(2mt Cmt) ◆ +(1 f) hbaCmt hbb(2mt Cmt)+hbaCmt ◆ which can be rewritten as: (haa hab)(hba hbb)C3 +((2hbb (1 f)hba)(haa hab)+(2hab f(2haa hab))(hba hbb))C2 +(2hbb(2hab f(2haa hab)) 2 fhab(hba hbb) 2(1 f)hbahab)C 4 fhabhbb = 0 are eter nority (7) ⇢ ka µ t1/2 kb µ t1/2 (10) Let’s make the hypothesis that Ka(t) and Kb(t) are linear func- tions of time, so that Ka(t) = Cmt and Kb(t) = (2 C)mt given Eq. (2). In the case of two groups, we can simplify the notations by denoting fa = f and fb = 1 f. Using Eq. (4), we thus have: Cmt b(2mt Cmt) ◆ +(1 f) hbaCmt hbb(2mt Cmt)+hbaCmt ◆ (11) (haa hab)(hba hbb)C3 hab)+(2hab f(2haa hab))(hba hbb))C2 ab)) 2 fhab(hba hbb) 2(1 f)hbahab)C 4 fhabhbb = 0 (12) ges of n the ution Let’s (13) dka dt = m fa haaka Ya +m fb hbaka Yb = ka t ✓ fahaa haaC +hab(2 C) + fbhba hbaC +hbb(2 C) ◆ = ka t ba (15) and thus: ba ka ∝tβa
  12. 12. BA-Homophily network model h = 0 h = 0.2 h = 0.5 h = 0.8 h = 1 majority minority B C D EA minority size = 0.2 complete homophilycomplete heterophily degreedistributiondegreegrowthnetwork h = 0 h = 0.2 h = 0.5 h = 0.8 h = 1 majority minority B C D EA minority size = 0.2 complete homophilycomplete heterophily degreedistributiondegreegrowthnetwork
  13. 13. Exponent of the degree distribution depends on group size and homophily minority fraction --> γ kkp ∝)(--- analytical results . . Simulation
  14. 14. Average degree ranks of minorities vs. Homophily
  15. 15. Minority rank in top d% Our expectation
  16. 16. Information reachability 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 homophily (h) 8 10 12 14 16 18 20 timetoreachthetarget(tij) maj to maj maj to min min to maj min to min A B How long does it take for information to reach a random target from a random source?
  17. 17. Ranking of minorities in Empirical Networks
  18. 18. Measuring Homophily in Empirical Networks •  Assortativity mixing (r), Newman, PRE (2003)- significant level of outgroup mixing compare to configuration •  Minority (m) fraction = 0.2 . •  h_mm = 0.1 ; h_MM = 0.7 ==> r = 0
  19. 19. Asymmetric Homophily •  Number of edges between the group is a function of homophily and group size. •  Given the empirical value of ingroup edges, we can calculate the homophily
  20. 20. Empirical Social Networks - 1 •  Sexual contact network (complete heterophilic) •  N ~17.000 •  Sex-sellers (minority) ; sex-buyers (majority) •  Minority fraction = 0.4 •  Complete heterophily: h(mm) = 0 Rocha, Liljeros, and Holme. PLoS Comput Biol, 2011
  21. 21. Empirical Social Networks (POK) - 2 •  Online dating network (heterophilic) •  N ~20.000 •  men(majority) ; women(minority) •  Minority fraction = 0.4 •  h_mm = 0.19; h_ww = 0.21 Holme, Edling, and Liljeros, 2002
  22. 22. Empirical Social Networks - 3 •  Scientific collaboration(moderate homophilic) •  N ~280.000 •  Men (majority) ; women (minority). Karimi et al. WWW (2016)
  23. 23. Empirical Social Networks - 3 •  Scientific collaboration(moderate homophilic) •  N ~280.000 •  men(majority) ; women(minority) •  Minority fraction = 0.23 •  h_ww = 0.57 ; h_mm = 0.56 Jadidi et al, Advances in Complex Systems, 2017
  24. 24. Empirical Social Networks - 4 •  Scientific citation APS (homophilic) •  N ~1900 •  QSM (majority) ; CSM (minority) •  Minority fraction = 0.38 •  h_cc = 0.8 ; h_qq = 1 www.Aps.org
  25. 25. Empirical Networks – Ranks 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 top d% degree rank 0.0 0.2 0.4 0.6 0.8 1.0 fractionofminoritiesintopd% A) Sexual contacts haa = 0; hbb = 0 data model 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 top d% degree rank 0.0 0.2 0.4 0.6 0.8 1.0 B) POK haa = 0.21; hbb = 0.17 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 top d% degree rank 0.0 0.2 0.4 0.6 0.8 1.0 C) Scientific collaboration haa = 0.57; hbb = 0.56 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 top d% degree rank 0.0 0.2 0.4 0.6 0.8 1.0 D) Scientific citation haa = 0.8; hbb = 1.0
  26. 26. Empirical Networks – Ranks 0.0 0.2 0.4 0.6 0.8 1.0 top d% degree rank 0.0 0.2 0.4 0.6 0.8 1.0 fractionofminoritiesintopd% A) Sexual contacts hab = 1; hba = 1 data model 0.0 0.2 0.4 0.6 0.8 1.0 top d% degree rank 0.0 0.2 0.4 0.6 0.8 1.0 B) Scientific collaboration hab = 0.44; hba = 0.43 0.0 0.2 0.4 0.6 0.8 1.0 top d% degree rank 0.0 0.2 0.4 0.6 0.8 1.0 C) Scientific citation hab = 0.2; hba = 0.0
  27. 27. C. Wagner*, P. Singer*, F. Karimi, J. Pfeffer & M. Strohmaier www 2017 Sampling from Social Networks with Attributes
  28. 28. Sampling from Networks Lee et al PRE (2006) Node sampling Edge sampling Snowball sampling Random walk sampling
  29. 29. Which sampling techniques preserve the true ranking of minorities?
  30. 30. Sampling from Networks with Attributes Original Ranking Homophilic NetworkHeterophilic Network Sample Ranking Original Ranking Sample Ranking
  31. 31. Results 37 Extreme hetero Extreme homo Attributes do not matter Moderate homo Moderate hetero
  32. 32. Thank You References: •  Visibility of minorities in social networks, arXiv: 1702.00150 •  Sampling social networks with attributes, WWW (2017) •  Towards Quantifying Sampling Bias in Network Inference, WWW (2018) Collaborators: Markus Strohmaier Claudia Wagner Mathieu Genois Eun Lee Lisette Noboa Mohsen Jadidi Florian Lemmerich Kristina Lerman Haiko Lietz Philipp Singer

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