Presented at NetSci satellite 2018

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

3. Moody, AJS (2001)
Baerveldt et al (2004)
Political blog sphere,
Adamic & Glance (2004)

4. Moody, AJS (2001)
Baerveldt et al (2004)
Homophily = attribute assortativity
Sexual network at a hgi school
Bearman, Moody & Stovel (2004)

6. Dropout vs. Career Age
early career mid career senior
Jadidi, Karimi, Lietz, & Wagner. Advances in Complex Systems (2017)

7. 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

8. In many networks we observe
homophilic / heterophilic
interactions and groups with
different size.
Why does it matter?

11. 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!

13. How does the inherent
structure of social networks,
(homophily and group size),
impact ranking (visibility) of
groups (minorities)?

14. Social Networks with attributes
A collaboration network.

15. 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

16. 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

17. 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

18. Exponent of the degree distribution
depends on group size and homophily
minority fraction -->
γ
kkp ∝)(--- analytical results
. . Simulation

19. Average degree ranks of
minorities vs. Homophily

20. Minority rank in top d%
Our expectation

21. 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?

22. Ranking of minorities in
Empirical Networks

23. 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

24. 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

25. 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

26. 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

27. Empirical Social Networks - 3
• Scientific collaboration(moderate
homophilic)
• N ~280.000
• Men (majority) ; women (minority).
Karimi et al. WWW (2016)

28. 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

29. 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

30. 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

31. 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

32. C. Wagner*, P. Singer*, F. Karimi, J.
Pfeffer & M. Strohmaier
www 2017
Sampling from Social Networks
with Attributes

33. Sampling from Networks
Lee et al PRE (2006)
Node sampling Edge sampling
Snowball sampling
Random walk sampling

35. Which sampling techniques
preserve
the true ranking of minorities?

36. Sampling from Networks with
Attributes
Original
Ranking
Homophilic NetworkHeterophilic Network
Sample
Ranking
Original
Ranking
Sample
Ranking

37. Results
37
Extreme hetero
Extreme homo
Attributes do
not matter
Moderate homo
Moderate hetero

38. 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