# On Influencing the Influential: Disparity Seeding

Ya-Wen Teng1,2, Hsi-Wen Chen1, De-Nian Yang1,2 and  Yvonne-Anne Pignolet3,
Ting-Wei Li1, Lydia Chen4 1Institute of Information Science, Academia Sinica,
Taiwan 2Research Center for Information Technology Innovation, Academia
Sinica, Taiwan 3DFINITY, Switzerland 4Department of Computer Science, Delft
University of Technology, The Netherlands<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(2021)

###### Abstract.

Online social networks have become a crucial medium to disseminate the latest
political, commercial, and social information. Users with high visibility are
often selected as seeds to spread information and affect their adoption in
target groups. We study how gender differences and similarities can impact the
information spreading process. Using a large-scale Instagram dataset and a
small-scale Facebook dataset, we first conduct a multi-faceted analysis taking
the interaction type, directionality and frequency into account. To this end,
we explore a variety of existing and new single and multihop centrality
measures. Our analysis unveils that males and females interact differently
depending on the interaction types, e.g., likes or comments, and they feature
different support and promotion patterns. We complement prior work showing
that females do not reach top visibility (often referred to as the glass
ceiling effect) jointly factoring in the connectivity and interaction
intensity, both of which were previously mainly discussed independently.

Inspired by these observations, we propose a novel seeding framework, called
_Disparity Seeding_ , which aims at maximizing spread while reaching a target
user group, e.g., a certain percentage of females – promoting the influence of
under-represented groups. Disparity Seeding ranks influential users with two
gender-aware measures, the Target HI-index and the Embedding index. Extensive
simulations comparing Disparity Seeding with target-agnostic algorithms show
that Disparity Seeding meets the target percentage while effectively
maximizing the spread. Disparity Seeding can be generalized to counter
different types of inequality, e.g., race, and proactively promote minorities
in the society.

Glass ceiling, disparity ratio, influence maximization

††journalyear: 2021††copyright: acmlicensed††conference: Proceedings of the
30th ACM International Conference on Information and Knowledge Management;
November 1–5, 2021; Virtual Event, Australia.††booktitle: Proceedings of the
30th ACM Int’l Conf. on Information and Knowledge Management (CIKM ’21),
November 1–5, 2021, Virtual Event, Australia††price: 15.00††isbn:
978-1-4503-8446-9/21/11††doi: 10.1145/3459637.3482375††ccs: Theory of
computation Social networks††ccs: Social and professional topics Gender

##  1\. Introduction

Online social networks are an integral part of people’s lives in today’s
society. People share their experiences and opinions via text, audio and
videos on social media, e.g., Facebook (Bakshy et al., 2012), Twitter
(Nilizadeh et al., 2016), LinkedIn (Tang et al., 2017), and Instagram (Stoica
et al., 2018). Social media grew from a platform to share personal experience
into one of the main channels to disseminate information, e.g., news (Wicks
and Drew, 1991), political campaigns (Allcott and Gentzkow, 2017), and product
reviews (Dang et al., 2010). Influential users who are well connected in their
social networks can critically accelerate the information spread and become
the ideal seeds to affect others (Stoica et al., 2020, 2018).

Thus, it is of great interest to understand the characteristics of influential
users, e.g., demographic traits and educational background. Several studies
(Nilizadeh et al., 2016; Aral and Walker, 2012) point out that there is a
correlation between the (perceived) gender of users and gaining visibility and
influence on social media. Specifically, they have shown the existence of the
_glass ceiling effect_ , which makes it harder for females to become highly
influential (Medel and Pournaghshband, 2017; Imtiaz et al., 2019), observed
from their direct interactions with others, e.g., commenting or liking the
posts. In other words, there are more males than females in the percentiles of
the most popular users (Leavy, 2018; Tang et al., 2017). These studies shed
light on social media usage patterns and show that gender disparity persists
even for the younger generation of users (Aral and Walker, 2012), by
exhibiting the necessary conditions leading to the glass ceiling effect (Avin
et al., 2015; Nilizadeh et al., 2016; Cotter et al., 2001).

An important metric to quantify influence in prior art (Aral and Walker, 2012;
Avin et al., 2018) is a node’s degree, i.e., the number of neighbors a node is
connected to under different types of interactions. Although such a measure
demonstrates well that fewer females reach the highest tier of visibility
compared to males (Avin et al., 2015), the intensity of the ties (Bakshy et
al., 2012) among the users, e.g., the frequency of interaction among connected
users, is often not considered. For instance, Stoica et al. (Stoica et al.,
2018) and Nilizadeh et al. (Nilizadeh et al., 2016) empirically show the
existence of the glass ceiling on Instagram and Twitter, respectively, by
focusing on the network structure or the interaction intensity separately.
This leads us to the question that females still lag behind males in the
highest percentiles when considering their degrees and interaction intensity.

Another essential question revolving around the glass ceiling concerns the
endorsement process: who supports and promotes influential users. Gaining a
deeper understanding of these patterns and structures may unveil solutions to
break the glass ceiling. Parity and diversity seeding (Stoica et al., 2020)
are solutions proposed to maximize the information spread to a target gender
group, e.g., females, by selecting seeds in a biased manner. Thus, females
(forming an under-represented group among the most influential users) are
promoted as seeds to disseminate information with the goal of increasing the
reception of information among females. However, such approaches cannot
explicitly accommodate promoting a specific ratio of the minority group. For
instance, a scholarship program may want to guarantee a certain percentage of
female award candidates and, thus, adequately inform the target audience. A
seeding solution is urgently needed to achieve the target disparity despite
different social network structures and interaction pattern manifestations.

We start by revisiting the glass ceiling effect on two popular platforms,
i.e., _Instagram_ and _Facebook_. In contrast to the prior art, we analyze
each gender’s visibility by both the degree and interaction intensity,
quantifying the number of links and how often they are used, both from a
sender and receiver perspective. We further differentiate between different
interaction types, i.e., likes vs. comments (and tags for Facebook), to
investigate if females face glass ceiling effects that hinder them from
reaching higher visibility in casual social platforms. Our analysis initially
focuses on the one-hop neighborhood in the social network, i.e., user pairs
interacting directly with each other (single-hop analysis). Then, we propose a
new centrality metric, termed _HI-index_ , to quantify the _indirect network
influence_ by considering the multihop impact via friends of friends. Our
detailed analysis demonstrates that there are number of criteria under which
glass ceiling effects are observed, e.g., comment intensity and tag intensity
in the single-hop analysis, and it points out unfavorable metrics with low
visibility ranks in the overall network for females, e.g, the out-degree of
comments.

In the second part of this paper, we tackle the challenge of designing a
seeding algorithm that can achieve a target _disparity ratio_ in groups under-
represented in the highest visibility percentiles while ensuring maximal
information spread. We derive a novel centrality measure, called _Embedding
index_ via state of art graph neural networks. Leveraging our characterization
study, we develop Disparity Seeding, which combines the proposed centrality
measures, Target HI-index and Embedding index, and diffusion simulation in a
novel way. The critical components of Disparity Seeding are the gender-aware
ranking and seed selection methods. Our evaluation results show that Disparity
Seeding can effectively achieve the target gender ratio and maximize the
information spread compared to the baselines. Our contributions are summarized
as follows:

  * •

We conduct a novel gender gap analysis on social media that jointly factors in
the connectivity and intensity from the dual perspectives of the sender and
receiver sides.

  * •

We show where glass ceiling effects exist for different interaction types,
directionality, degree, and link intensity.

  * •

We propose a novel centrality measure, HI-index, and apply it together with
PageRank, to highlight and rank the influence of male/female users on the
entire network.

  * •

We develop a novel seeding framework, Disparity Seeding, that maximizes
information spread and satisfies a target gender ratio that can overcome the
disparity exhibited in the population and compare it to known approaches.

##  2\. Related Work

Prior art has extensively studied gender bias and its implications in the
context of professional network and career paths (McPherson et al., 2001; Avin
et al., 2015). In social network analyses, users’ influence is commonly
quantified by various centrality measures (Avin et al., 2018; Iyengar et al.,
2011; Aral and Walker, 2012), often with an implicit assumption that the links
are undirected and link quality is uniform. Hazard models (Aral and Walker,
2012; Iyengar et al., 2011) are applied to estimate the information spread and
social contagion. On the other hand, several studies stress the importance of
the connection quality. Bakshy et. al (Bakshy et al., 2012) pointed out with a
study on Facebook that users with high interaction intensity form stronger
ties and are more influential than users with low intensity, and thus they are
key in efficiently disseminating information. The increasing popularity of
services to purchase likes and bot activities further raise an alarming
concern on the connection quality and implications on social network analysis
(Schröder, 2019).

Gender Gap Analysis on Social Media. Several studies investigate the glass
ceiling effect on different online social platforms. Typically they find that
males achieve higher visibility and spread information faster (Nilizadeh et
al., 2016; Stoica et al., 2018; Lewis et al., 2008), from the perspective of
either the network connectivity or intensity. Twitter: Nilizadeh et al.
(Nilizadeh et al., 2016) show that the perceived gender affects user
visibility in different metrics, e.g., the number of followers, retweets, and
lists. From the _complementary cumulative distribution function_ (CCDF) of
most of those measures, high percentile males achieve higher visibility than
females. Facebook: In (Lewis et al., 2008), it leverages similarity
calculations to quantify relationships with quadratic assignment procedure
(QAP). More precisely, similarities under gender, race/ethnicity, and
socioeconomic status (SES) are studied. Instagram: Stoica et al. (Stoica et
al., 2018) derived mathematical models to explain how recommendation
algorithms reinforce the glass ceiling gap. Their study does not take the tie
strength into account. By contrast, we analyze influential users by fusing
both tie strength and degree via a novel centrality measures, HI-index, and
observe different glass ceiling effects depending on the interaction types on
Instagram and Facebook.

Influence Maximization. Social influence on online platforms is an active
research topic and one of the main focuses is on maximizing the influence
(Kempe et al., 2003). The key challenge is to pick $k$ seeds which maximize
the number of users receiving information spread by the selected seeds, which
is proved as NP-hard under the Linear Threshold and Independent Cascade
diffusion models (Kempe et al., 2003). Recent studies (Stoica et al., 2020;
Gershtein et al., 2018; Fish et al., 2019; Tsang et al., 2019; Farnad et al.,
2020; Khajehnejad et al., 2020) focus on reducing the gender gap while
maximizing information spread. However, they cannot ensure a user-specified
gender ratio in the influenced users. In this paper, Disparity Seeding takes a
step further to maximize the overall information spread and achieve any
required gender ratio by design.

##  3\. Glass Ceiling on Instagram via Visibility and Endorsement Analysis

We conduct a static characterization of female and male interaction patterns,
answering the questions if there exists a gender gap in terms of users’
visibility and endorsement on Instagram when considering intensity and degree.
The visibility analysis studies appreciation received by posts from different
genders to investigate the evidence of the glass ceiling, i.e., if males are
over-represented in higher percentiles. The endorsement analysis focuses on
how users support posts from others. In contrast to (Avin et al., 2015; Stoica
et al., 2018; Nilizadeh et al., 2016; Aral and Walker, 2012), we take both the
number of interaction partners (_degree_) and the number of interactions
(_intensity_) into account. Moreover, we consider two types of interactions
separately for a more nuanced analysis, i.e., we distinguish likes from
comments since the latter demands higher involvement and effort from the
senders than the former. Our objective here is to revisit the glass ceiling
from multi-faceted perspectives and search for insights to understand and
design potential alleviation measures to reduce gender gaps on social media.

We first introduce the Instagram dataset and then analyze the visibility and
endorsement from users interacting directly with each other, i.e., single-hop
analysis. Afterward, we apply centrality metrics that can factor in both
interaction intensity and degree of the entire network, i.e., multi-hop
analysis. To this end, we extend H-index (Alonso et al., 2009) to propose HI-
index by considering interaction intensity to quantify the overall network
visibility of male/female. Meanwhile, we also investigate the ranking produced
by PageRank.

(a) Received (Likes)

(b) Received (Comments)

(c) Sent (Likes)

(d) Sent (Comments)

Figure 1. Intensity of visibility (received) and endorsement (sent) on
Instagram.

(a) In-degree (Likes)

(b) In-degree (Comments)

(c) Out-degree (Likes)

(d) Out-degree (Comments)

Figure 2. Degrees of visibility (received) and endorsement (sent) on
Instagram.

###  3.1. Instagram Dataset

Table 1. Instagram gender distribution: senders vs. receivers. | Sender | Receiver
---|---|---
| Male | Female | Male | Female
Like | 46.2% (~39k) | 53.8% (~46k) | 41.0% (~1,143k) | 59.0% (~1,648k)
Comment | 45.8% (~38k) | 54.2% (~44k) | 40.7% (~589k) | 59.3% (~857k)
Users | 46.3% (~40k) | 53.7% (~47k) | 41.3% (~1,352k) | 58.7% (~1,921k)

Collection method. Stoica et al. (Stoica et al., 2018) collected data of
users’ reactions with likes and comments on each other’s posts over a $4$ year
period. The data was retrieved with the Instagram API by first gathering a set
of users, starting with the founder of Instagram and recursively retrieving
the list of followers. Subsequently, data from the posters’ perspective and
how such posts receive likes and comments from other users was collected. For
each interaction, the user ID, gender (derived from user names), responding
type, i.e., likes or comments, and timestamps are recorded. Due to space and
computation constraints, a maximum of $5$ interactions per post was sampled.
Hence the interaction intensity is down-sized. The observation period in this
paper spans from Jan 2015 to May 2016, starting right when the number of
active Instagram users surged.

Data characteristics. The dataset comprises around $8$ million unique users
with around $38$ million interactions among each other by considering both
likes and comments before filtering. For both interaction types, we study the
directed network formed by links representing users liking or commenting on
another user’s post.

Prepossessing. Since Instagram emerged to be one of the primary social
platforms, especially for marketing campaigns, there is an increasing number
of bot services (Schröder, 2019) that offer to send “likes” for a payment. To
avoid including such fake interactions in our analysis (Nilizadeh et al.,
2016), we filter out  _inactive_ users with only one interaction across our
17-month observation period. Roughly $57.45\%$ of users are inactive and hence
removed from the following analysis. After the filtering, there are
$1,375,637$ males and $1,499,145$ females. Table 1 summarizes the distribution
of the dataset. There is a significantly higher number of receivers (roughly
30 times more) than senders who respond to posts via likes/comments due to the
collection method, and more female senders/receivers than male ones. In terms
of likes and comments, the number of likes observed is roughly $2.5$ times
higher than the number of comments. The overall dataset is slightly unbalanced
in terms of gender ratio.

Limitations. The data is collected starting from the founder’s posts
recursively (i.e., the receiver of comments/likes); therefore, the dataset
exhibits a bias of this specific group of users, an unavoidable drawback when
crawling information via such an API. Thus, the data may not cover the
diversity of all Instagram users.

###  3.2. Single-Hop Analysis

Table 2. Mann-Whitney U test: the p-value on Instagram. |  | Top 10% | Top 1% | Top 0.1%
---|---|---|---|---
Comment | Rec. intensity | 1.199e-156*** | 0.0*** | 4.285e-116***
Sen. intensity | 1.996e-63*** | 0.003** | 0.102
HI-index | 0.326 | 0.043* | 4.496e-09***
PageRank | 0.0*** | 6.813e-66*** | 9.61e-10***
Like | Rec. intensity | 0.0*** | 3.129e-82*** | 4.484e-33***
Sen. intensity | 3.43e-48*** | 5.96e-05*** | 3.082e-4***
HI-index | 1.649e-212*** | 0.0*** | 3.517e-105***
PageRank | 1.347e-44*** | 1.416e-163*** | 2.493e-26***
*, **, and *** denote $p\leq 0.05$, $p\leq 0.01$, and $p\leq 0.001$, respectively.

####  3.2.1. Interaction Intensity

Using the intensity of comments and likes, we aim to answer which gender is
more _influential_ and which gender is more likely to _endorse_ others. Note
that, typically, nodes’ influence is derived from the degree, often with an
implicit assumption of equal link quality (Dong et al., 2017; Stoica et al.,
2018; Yang et al., 2019; Avin et al., 2018), which is quite imprecise.

Visibility Intensity. Figures 1(a) and 1(b) summarize the total number of
likes and comments received by unique female/male users, respectively. We plot
the complementary cumulative distribution function (CCDF) for females and
males separately. The tail of the CCDF represents the most popular users,
i.e., those who received the most likes/comments (referred to as the top-
ranked users) in their respective gender. By comparing the tails of female and
male CCDFs, we can see if highly visible females receive as many
likes/comments as highly visible males. Following (Stoica et al., 2018), Table
2 shows the statistical confidence for top-ranked user groups (Top $10\%$,
$1\%$, and $0.1\%$).111Due to space constraints, please refer (Teng et al.,
2021) for more details.

We observe that much more likes are received than comments, which is not
surprising as the effort to write a comment is significantly higher than
clicking a like button (Guo et al., 2017). Specifically, the mean number of
likes received per user ($5.16$) is around three times higher than that of
comments received per user ($1.79$). In Figure 1(a), the higher visibility
values for males at the tail indicate that top-ranked males receive more likes
than top-ranked females, though both male and female users receive on average
$5.16$ likes. This observation fits with the glass ceiling effect pointed out
in (Avin et al., 2015; Stoica et al., 2018), i.e., females fall behind males
in top-ranked positions but not in the lower-ranked positions. Specifically,
such a cross point happens around the top $3.2\%$ for likes. However, such a
glass ceiling effect is not observed for comments, shown in Figure 1(b).
Females constantly receive more comments than males at any rank. Consequently,
the average number of comments received by females is $1.95$, around $23\%$
higher than for males ($1.59$). Even though a crossing exists in the highest
percentile, only four males receive more comments than females, which should
probably be considered as outliers.

Endorsement Intensity. Displaying the endorsement activities, Figures 1(c) and
1(d) summarize the number of likes and comments sent by unique female/male
users, respectively. We plot the CCDFs for females and males separately. The
tails of these CCDFs illustrate how highly active female/male users endorse
others.

The intensity of sending likes is almost three times higher than commenting,
matching the CCDF of the receiver side. However, the shapes of the CCDFs are
quite different: the CCDFs of likes/comments sent are higher than the CCDFs
computed for the receiver side. This observation shows a mass of receivers
having low visibility, but senders are more evenly distributed in terms of
their endorsement efforts, i.e., the endorsement intensity varies
considerably. We observe that females are more active than males regarding
both likes and comments, as shown in Figures 1(c) and 1(d), respectively. On
average, females send $117$ likes (around $39\%$ more than males) and $44$
comments (around $69\%$ more than males). Note that the crossing in the
highest percentile in Figure 1(c) is caused by a tiny number ($3$ users) of
males (similar to Figure 1(b)), which should also be regarded as outliers. We
notice that different interaction types in the sender point-of-view are both
led by females.

Takeaway: Females are much more active in giving comments and likes than
males, at almost all ranks. The variability of endorsement intensity is lower
than the visibility intensity.

####  3.2.2. Interaction Degree

Different from the previous subsection, we resort to the interaction degree to
answer the gender difference in achieving high visibility and active
endorsement.

Visibility Degree. Figures 2(a) and 2(b) summarize the numbers of unique users
from whom a particular user receive likes or comments, i.e., in-degree. We
separate males and females and plot the CCDFs of in-degree for likes and
comments. Different from (Nilizadeh et al., 2016), the CCDFs of in-degrees do
not show a strong power-law behavior, i.e., a significant fraction of users
account for a broad spectrum of degree, which can be regarded as a data-
dependence characteristic.

The average degrees of likes and comments are $1.21$ and $0.89$, respectively.
Although the difference between the average like and comment intensity is
around a factor three, the resulting degree difference through these two types
of interaction is lower. This can be explained by the frequency and repetitive
interactions over certain user pairs, indicating a stronger tie. Figure 2(a)
again shows a glass ceiling effect: females attain higher degrees than males
for low and medium percentiles. However, among the $6.4\%$ top-ranked males
($113k$ individuals) have consistently higher degrees than their female
counterparts. As for the degree established through comments (Figure 2(b)),
there is no visible difference between males and females below the top-ranked
$11\%$ ($9k$ users) and a minor gender gap for higher ranked users. Recall
that females receive significantly higher comment intensity than males. Such a
discrepancy can be possibly explained by how such comments are distributed
across different senders. Females appear to receive comments from a smaller
group of users with higher intensity, whereas males receive comments from a
larger group in lower intensity.

Takeaway: For likes and comments, a clear glass ceiling effect can be
observed: top-ranked males receive visibility from a larger set of users than
females.

Endorsement Degree. Figures 2(c) and 2(d) summarize how many unique users are
supported by a particular female or male user, i.e., out-degree, in CCDF
plots.

As the total number of senders is much lower than that of receivers, the
average out-degrees, $45.21$ and $21.27$ for likes and comments, respectively,
are much higher than the average in-degrees per user. With the above
observations, we have valuable insights summarized as follows. In the
intensity analysis, males perceive higher visibility with likes and females
with comments. In the endorsement aspect, females are more active regarding
both interactions. However, recall that in the degree analysis, males dominate
in all scenarios (in/out-degree on like/comment). This implies that females
prefer to interact with a smaller group of users than males do, which means
that males are part of larger groups to which they send or from which they
receive likes and comments.

Takeaway: Top-ranked males support a larger group of users with lower
intensity, whereas females support a small group of users with higher
intensity.

###  3.3. Multi-hop Analysis

The strength of the gender differences varies across the two visibility
measures, i.e., degree and intensity, under the two interaction types, likes
and comments. In this section, we take a complementary perspective to
investigate influence, beyond the one-hop neighborhood. Our objective here is
to combine both the degree and intensity simultaneously for direct and
indirect social ties, i.e., friends of friends. To this end, we propose to use
two centralities to quantify nodes’ influence, a novel centrality _HI-index_ ,
inspired by the H-index ranking of influential authors (Alonso et al., 2009),
and PageRank (Page et al., 1999).

(a) HI-index (Likes)

(b) HI-index (Comments)

Figure 3. HI-index of visibility (received) on Instagram.

####  3.3.1. HI-index

The definition of the HI-index is derived from the H-index (Alonso et al.,
2009), which attributes an author’s productivity and citation impact by the
number of publications that have been cited at least a certain number of
times. In other words, the H-index relies on the degree of publications in the
network formed by citations. We extend it to consider the interaction
intensity as follows.

###### Definition 3.1.

The _HI-index_ of a user $v_{i}$ is defined as the maximum number $H$ such
that $v_{i}$ has at least $H$ neighbors who interact with $v_{i}$ and any
other users in the network at least $H$ times. Let $N(v_{i},n)$ denote the
number of $v_{i}$’s one-hop neighbors who interact with others at least $n$
times. We can formulate the HI-index of $v_{i}$ as

(1) $\small H(v_{i})=\max_{n}\min_{n\in\mathbb{I}^{+}}(N(v_{i},n),n).$

The HI-index goes beyond a single hop analysis by evaluating all interactions
of neighbors, and considers more than the interactions between the author of a
post and its supporters. Thus the two-hop neighborhood of a user determines
its HI-index.

Figure 3 summarizes the CCDFs of the HI-index for males and females for likes
and comments. The tails of the CCDFs represent the highly ranked users who
have large HI-index values. The average HI-index values for likes and comments
are $1.41$ and $1.28$, respectively. We observe that the HI-index values of
males exceed those of females at the same percentiles in Figure 3(a). In other
words, males receive higher visibility than females directly and indirectly in
the social network. On the other hand, in Figure 3(b), the male and female
CCDFs are much closer to each other regarding comments, showing no remarkable
difference. Recall the single-hop analysis in Section 3 that females attain
higher centrality values for comments. Females only attain similar visibility
as males when considering their direct and indirect neighbors. Females’ HI-
index is generally lower than males because of the dual emphasis of
interaction intensity and degree. Remember from the previous analysis that
females tend to establish higher intensity interactions but in lower
quantities than males. As the HI-index considers the interaction in both the
single-hop and two-hop neighborhoods, females reach lower HI-index values than
visibility measures using intensity only.

Takeaway: When combining the number of direct and indirect interactions and
their intensity in the HI-index, males achieve higher or equivalent visibility
compared to females (i.e., the blue line is higher or comparable to the red
line in Figure 3).

####  3.3.2. PageRank

Another widely adopted metric to quantify the influence over an entire network
is the PageRank centrality (Page et al., 1999), devised initially to sort web
pages by their popularity. A page’s popularity is measured by the number of
times it is linked to by other pages, weighed by its popularity in turn. After
that, PageRank recursively computes the steady-state probability of being at a
page when following links at random.222We mainly exploit PageRank for analysis
due to the scalability on massive networks, instead of other centrality
measures, e.g., betweenness and closeness (Page et al., 1999).

(a) PageRank (Likes)

(b) PageRank (Comments)

Figure 4. PageRank of visibility (received) on Instagram.

Both HI-index and PageRank measure the visibility level of nodes taking the
degree and the intensity of interactions into account. However, there are
differences. In HI-index, for a user to achieve a high value, a sufficient
number of highly visible neighbors is necessary. In other words, HI-index
requires to exceed two thresholds, interaction partner quality and quantity.
In PageRank, a user may be highly visible, even if the number of neighbors is
not very high if, in turn, their neighbors are highly visible.

We summarize the PageRank results for likes and comments in Figures 4(a) and
4(b), respectively. In Figure 4(a), one can observe that females attain higher
PageRank visibility than males from the starting up to $5\%$ percentile.
However, medium and top-ranked males outperform females significantly, shown
by their higher CCDF. In contrast to HI-index, top-ranked females attain
higher PageRank visibility than top-ranked males in Figure 4(b). This is
because PageRank considers the visibility of neighbors without any threshold
of degree, which leads to that the females with low degrees but high intensity
can be ranked high in terms of PageRank. Figure 4(b) shows that females’
PageRank visibility for comments is dampened compared to the pure comment
intensity (Figure 1(b)) yet the top-ranked females receive higher visibility
than males due to the higher intensity females receive from their supporters.

Takeaway: When measuring the visibility by the PageRank centrality, females
face glass ceiling effects in the like network, while they reach higher
visibility for comments.

##  4\. Glass Ceiling on Facebook

In this section, we first introduce the second dataset containing user
interactions on Facebook. Then, we will focus more on the intensity of
received interactions and a new interaction type (tag) to complement the
Instagram analysis.

###  4.1. Facebook Dataset

Collection method. Using the Facebook API, we collected data from users who
study at $25$ university departments. The users comprise $1870$ voluntary
senior students of the before-mentioned departments and all interactions
between them are retrieved. For each interaction, the user ID, gender (derived
by questionnaire), interaction type, and timestamps are recorded. In addition,
we also collected user profiles, e.g., academic standing and hometown, by
questionnaires. The period of iterations spans from March 2008 to May 2016,
and $97.26\%$ of interactions are after August 2012.

Data characteristics. The dataset comprises around 20 million interactions by
$1870$ unique users ($765$ males and $1,105$ females). In addition to likes
and comments, Facebook supports a third type of interactions where users can
refer to each other via _tags_ in posts. Different from the Instagram dataset,
we keep all interactions without any filtering. Table 3 summarizes the main
statistics of how males/females interact with each other through likes,
comments and tags. Note that the percentages on the receiver side are very
similar to those observed on Instagram, while we see a higher female
participation among the sender numbers in this dataset.

Table 3. Facebook gender distribution: senders vs. receivers. | Sender | Receiver
---|---|---
| Male | Female | Male | Female
Like | 41.1 % (750) | 58.9% (1079) | 40.9% (743) | 59.1% (1074)
Comment | 40.9%(750) | 59.1% (1087) | 40.7% (735) | 59.3% (1073)
Tags | 40.6% (704) | 59.4% (1031) | 37.3% (540) | 62.7% (910)
Users | 40.8% (763) | 59.2% (1104) | 40.9% (748) | 59.1% (1082)

Limitations. The data is collected over a long time frame but for a rather
small and homogeneous group of students. Thus it is not straight-forward to
generalize from the findings in this dataset to the general population of
Facebook users.

###  4.2. Effect Analysis

Single-hop Analysis. Figures 5(a) and 5(b) present CCDFs for tags. Table 4
shows the statistical confidence for different user groups. Different from
comments and likes, females dominate males in terms of receiving intensity and
in-degree. Females show stronger visibility in any rank than males, almost
twice of males. From the sending side, females are also more active in
endorsing others. As users on Facebook are closer than on Instagram, users
with high visibility are more active in endorsement. However, we observe the
glass ceiling for tag’s outdegree, indicating top-ranked males endorse more
actively than their female counterparts.333Please refer to (Teng et al., 2021)
for the complete results of the Facebook dataset.

Takeaway: In terms of received like, comment and tag intensities, females are
more visible than males in any rank, showing no sign of glass ceiling.

(a) Received (Tags)

(b) In-degree (Tags)

(c) HI-index (Tags)

(d) PageRank (Tags)

Figure 5. Visibility (received) in terms of Tag on Facebook.

Multi-hop Analysis. We focus on the multi-hop analysis for tags due to its
contrasting trend from the single-hop analysis. While females clearly dominate
males in terms of intensity and indegree, strong glass-ceiling effects are
observed in PageRank, shown in Fig. 5(d). Zooming into Fig. 5(c), the mean
value of the HI-index is $4.15$ and $5.33$ for males and females,
respectively, indicating that on average male/female users interact with
slightly more than 4 and 5 other users respectively who also interact with $4$
or $5$ others. However the top $1\%$ of male users have an HI-index of up to
$23.62$ users, while top female users only reach $23.89$.

Takeaway: There is a strong glass ceiling for tags, from multi-hop measures
but not sing-hop on Facebook dataset.

Finally, we also summarize in Table 5 if a glass ceiling effect is exhibited
by the different centrality measures and types of interactions for both the
Instagram and the Facebook datasets.

Table 4. Mann-Whitney U test: the p-value on Facebook. |  | Top 10% | Top 1% | Top 0.1%
---|---|---|---|---
Tag | Rec. intensity | 1.812e-16*** | 7.626e-4*** | 0.0***
In degree | 5.423e-05*** | 0.094 | 0.0***
HI-index | 5.423e-05*** | 0.094 | 0.0***
PageRank | 0.099 | 0.007** | 0.0***
*, **, and *** denote $p\leq 0.05$, $p\leq 0.01$, and $p\leq 0.001$, respectively.
Table 5. Summary of the glass ceiling on Instagram and Facebook.  ✗ indicates
that a glass ceiling effect is observed.

| Instagram | Facebook
---|---|---
Likes |  Comments |  Likes | Comments |  Tags
Single hop |  | Rec. intensity
---
✗ |  |  |  |
2-7[1pt/1pt] |  | Sen. intensity
---
|  |  |  |
2-7[1pt/1pt] |  | In degree
---
✗ |  ✗ |  |  ✗ |
2-7[1pt/1pt] |  | Out degree
---
✗ |  ✗ |  |  |  ✗
Multi hop |  | HI-index
---
|  |  ✗ |  ✗ |
2-7[1pt/1pt] |  | PageRank
---
✗ |  |  |  |  ✗

##  5\. Disparity Seeding

In this section, we address the research question of how to optimize the
information spread via a seeding strategy to the targeted demographic group,
i.e., how to meet a target gender ratio.444We use the target gender ratio and
the target female ratio interchangeably here. The solution can be applied to
commercial and governmental campaigns that aim to optimally reach a certain
percentage of females (True and Mintrom, 2002) or other groups in society.
Concretely, a marketing campaign may aim to select a fixed number of
influencers to try out a product before its formal release such that the
corresponding product information will be maximally spread at a ratio of 30%
females. This is especially challenging when the targeted gender ratio
deviates from the original ratio in the population. Formally, the influence
maximization with the target gender ratio constraint is formulated as follows.

###### Definition 5.1 (Disparity Influence Maximization (DIM)).

Given a social network $G=(V,E)$, a diffusion model, a seed group size $K$, a
target gender ratio $\zeta$, and an error margin $e$, the problem is to select
a seed group $S\subseteq V$ with $\left|S\right|=K$ to maximize the influence
spread under the constraint that the female ratio in the influenced users is
$\zeta$ within an error margin $e$.

######  Theorem 5.1.

It is NP-hard to approximate the Disparity Influence Maximization problem
(DIM) within any factor unless $P=NP$.

###### Proof.

Please refer to (Teng et al., 2021) for the proof. ∎

###  5.1. Disparity Seeding Framework

We propose the Disparity Seeding framework, which selects influential females
and males according to a given seed group size and target gender ratio that
can be far off from the population ratio. Our disparity framework is composed
of two phases, illustrated in Figure 6: ranking users and deciding seeding
ratio. First, we identify influential users by two gender-aware metrics, i.e.,
_Target HI-index_ and _Embedding index_ , to select the most influential node
achieving the target gender ratio. Then, we estimate at what proportion to
allocate the available seeds to males and females, abbreviated as the seeding
ratio, based on their ranks. Our earlier analysis indicates there is a clear
gap between the two. For instance, there is a higher percentage of very
influential males even though there is a lower percentage of males in the
population.555Note that our Disparity Seeding framework is a general method
which can also support other sensitive attributes, e.g., race or age.

Figure 6. The framework illustration of Disparity Seeding.

####  5.1.1. Ranking Users

To rank users, we introduce two gender-aware ranking measures, _Target HI-
index_ and _Embedding index_.

First, Target HI-index, based on HI-index, favors the users having direct
neighbors with a gender ratio close to the given target. In other words, the
users who interact with direct neighbors with a dissimilar female ratio to the
target ratio $\zeta$ are penalized.

###### Definition 5.2.

_Target HI-index_ of a user $v_{i}$ is defined as $v_{i}$’s HI-index but
penalized by the difference between $\zeta$ and the female ratio of $v_{i}$’s
neighbors from which the HI-index is derived. Let $N^{\text{F}}(v_{i},n)$
denote the number of $v_{i}$’s direct female neighbors interacting with others
at least $n$ times. The Target HI-index of $v_{i}$ is formulated as

(2) $\displaystyle\small
TH(v_{i},\zeta)=H(v_{i})\cdot\Big{(}1-|\frac{N^{\text{F}}\big{(}v_{i},H(v_{i})\big{)}}{N\big{(}v_{i},H(v_{i})\big{)}}-\zeta|\Big{)},$

is the penalty for the female ratio of $v_{i}$’s direct neighbors having at
least $H(v_{i})$ interactions with others not satisfying $\zeta$. A larger
difference between $\zeta$ and the female ratio in
$N\big{(}v_{i},H(v_{i})\big{)}$ results in a greater penalty on $H(v_{i})$.

On the other hand, as Graph Neural Networks (GNNs) have been celebrating
successes in various domains by modeling the long-term dependencies between
nodes in a graph, we also introduce the _Embedding index_ with a novel GNN
model, namely _Influence Graph Neural Network (InfGNN)_ model. In contrast to
previous ranking methods, which rely on oversimplified assumptions, remain
highly sensitive to their hyperparameters (Aral and Walker, 2012), and thus
can provide inappropriate estimations compared to actual cascades (Pei et al.,
2018), InfGNN can estimate the influence dynamically and accommodate rich node
attributes to support diverse goal, e.g., target gender ratio.666Note that the
Embedding index is only adopted on Facebook since this dataset contains user
profiles which are required for the initialization of GNN (Kipf and Welling,
2017). In the following, we first present the definition of the _Embedding
index_ and then describe the design of the InfGNN model.

###### Definition 5.3.

_Embedding index_ of a user is learned from the social network by preserving
the proximity between users by InfGNN, which also penalizes the difference
between $\zeta$ and the female ratio.

To effectively estimate the Embedding index, InfGNN consists of two primary
components: 1) stacked GNN layers to derive the node embeddings and 2) an
influence predictor to estimate the influence of the learned embeddings. Given
the $l^{th}$ hidden feature $\mathbf{H}^{l}$ of nodes in the graph, GNNs
update the $(l+1)^{th}$ latent features of node $v_{i}\in V$, denoted by
$\mathbf{h}^{l+1}_{i}\in\mathbf{H}^{l+1}$, by aggregating the features of
$v_{i}$’s neighboring nodes $N(v_{i})$, which can be written as follows.

(3)
$\small\mathbf{h}^{l+1}_{i}=ReLU(Attn^{l}(\mathbf{h}_{i}^{l},\mathbf{h}_{j}^{l},\forall
v_{j}\in N(v_{i}))),$

where $Attn^{l}(\cdot)$ is the attention mechanism (Veličković et al., 2018)
for aggregation, and $ReLU(\cdot)$ is a non-linear activation function (Kipf
and Welling, 2017). After stacking the $L^{th}$ layer, InfGNNs adopts the
final hidden layer $\mathbf{H}^{L}$ as the embedding $\mathbf{h}_{i}$ of each
node $v_{i}$. Then, we adopt skip-gram objective (Mikolov et al., 2013) as our
proximity loss to structural information on graph $G$, i.e.,

(4) $\small L_{prox}=-\sum_{v_{i}\in V}(\sum_{v_{j}\in
N(v_{i})}\sigma(\mathbf{h}^{T}_{i}\mathbf{h}_{j})+\mathbb{E}_{v_{j}\sim
P_{N}(\mathbf{v}_{i})}[\sigma(-\mathbf{h}^{T}_{i}\mathbf{h}j)]),$

where $\sigma$ is the sigmoid function and $P_{N}(\cdot)$ is the distribution
for negative sampling of users. After deriving the node embedding, the next
step is to predict the influence score $s_{i}\in\mathbf{s}$ of each node
$v_{i}$ by

(5) $\small s_{i}=\sigma(\mathbf{c}^{T}\mathbf{h}_{i}),$

where $\mathbf{c}$ is a trainable vector to calculate the importance of each
node. In contrast to traditional GNNs, which usually require a great effort in
collecting labels (e.g., the importance of each node) (Kipf and Welling, 2017;
Veličković et al., 2018), we introduce influence loss $L_{inf}$ to train
InfGNN in a self-supervised manner,

(6) $\small L_{inf}=\sum_{v_{i}\in V}(\left\lVert
s_{i}-s^{\prime}_{i}\right\rVert_{2}-\mathbb{E}_{v_{j}\sim
P_{N}(\mathbf{v}_{i})}(\left\lVert s_{i}-s_{j}\right\rVert_{2})).$

Note that the first part of the Eq. (6) minimizes the error between the self
and estimated scores, and the second part is used to distinguished the
influence between each node by negative sampling. The estimated score
$s^{\prime}_{i}$ is derived from $v_{i}$’s neighborhoods, i.e.,

(7) $\small s^{\prime}_{i}=\sum_{v_{j}\in
N(v_{i})}\frac{\exp(\mathbf{a}^{T}[\mathbf{h}_{i},\mathbf{h}_{j}])s_{j}}{\sum_{v_{k}\in
N(v_{j})}\exp(\mathbf{a}^{T}[\mathbf{h}_{j},\mathbf{h}_{k}])}.$

where $\mathbf{a}$ is a trainable vector to measure the influence between
nodes. Unlike conventional GNN (Veličković et al., 2018) only aggregates the
$1$-hop neighborhood information and thereby suffer from oversmoothing (Kipf
and Welling, 2017), i.e., every node has the same embedding and thereby lead
to a similar influence score, our InfGNN estimate the influence by considering
the $2$-hop information. While previous statistical measurements (Alonso et
al., 2009) require to identify the influence by some fixed hyperparameters to
exploit a specific property of the graph, InfGNN calculates the importance of
each node dynamically and thereby is more general. Our overall objective
becomes,

(8) $\small
L=L_{prox}+\lambda_{1}L_{inf}+\lambda_{2}\left\lVert\mathbf{s}\right\rVert_{1}+\lambda_{3}\left\lVert\mathbf{h}\right\rVert_{2},$

Note that $\left\lVert\mathbf{s}\right\rVert_{1}$ is the $l_{1}$
regularization, which discretizes the output distribution to force the model
to concentrate the influence on a few nodes and
$\left\lVert\mathbf{h}\right\rVert_{2}$ is the $l_{2}$ regularization of node
embeddings. $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ are the
hyperparameters to determine the trade-off between the proximity and the
influence score. We adopt SGD (Bottou, 2012) to minimize the above loss
function and obtain the embedding and influx score of each node. Finally, our
Embedding index (EI) of the node $v_{i}$ is defined as follows.

(9) $\small EI(v_{i})=s_{i}\cdot
D_{in}(v_{i})\cdot\Big{(}1-|\frac{N^{\text{F}}\big{(}v_{i},n)\big{)}}{N\big{(}v_{i},n)\big{)}}-\zeta|\Big{)},$

where $D_{in}(v_{i})$ is the indegree of node $v_{i}$. Intuitively, the
influence score $s_{i}$ represent the spreading intention of a user, while the
indegree $D_{in}(v_{i})$ is the number of connections. Note that
$\Big{(}1-|\frac{N^{\text{F}}\big{(}v_{i},n)\big{)}}{N\big{(}v_{i},n)\big{)}}-\zeta|\Big{)}$
penalizes to $v_{i}$ if it violates the target gender ratio.777The threshold
interacting times $n$ of Embedding index is a hyperparameter which is set to
$3$, empirically.

(a) Target ratio (Likes)

(b) Target ratio (Comments)

Figure 7. The learned scaling function $\omega$.

(a) Instagram (Comments)

(b) Instagram (Likes)

(c) Facebook (Comments)

(d) Facebook (Likes)

(e) Facebook (Tags)

Figure 8. Evaluation on Instagram and Facebook.

####  5.1.2. Deciding Seeding Ratio

Here, we search for a scaling function $\omega:\zeta\rightarrow r$ that can
map the target ratio $\zeta$ into the actual seeding ratio $r\in[0,1]$ based
on the ranking results. We use a simulated diffusion process888To obtain the
dependency properly, the information propagation is simulated according to the
specified diffusion process in the problem. to capture the dependency between
the target and seeding ratio and thus learn $\omega$. Specifically, a number
of seed groups with different gender ratios are selected for information
diffusion. Larger seed groups can better capture the dependency between the
target and seeding ratio, whereas smaller groups shorten the simulation and
learning time. The optimal choice of the size of such seed groups exceeds the
current scope of the paper. The scaling function $\omega$ is determined by the
seeding ratio and the female ratio of users adopting the information in each
simulated result.999Essentially $\omega$ needs to be empirically learned for
each ranking algorithm separately. We explain how to derive $\omega$ from the
simulation data as follows.

Figures 7(a) and 7(b) illustrate the $\omega$ function of Target HI-index for
liking and commenting networks on Instagram, respectively. Each point
represents how to choose a seeding ratio (y-axis) given a target ratio
(x-axis). The point is iteratively determined through simulations. The
$\omega$ function of Target HI-index centrality has a wider range from $0.16$
to $0.86$ of target ratios (in the x-axis), showing higher flexibility in
accommodating different disparity ratios. This observation holds for both
likes and comments.

Finally, we select a total of $K$ seeds based on the learned $\omega$ function
and the target ratio. They are essentially the top $\omega(\zeta)\cdot K$
females and the top $(1-\omega(\zeta))\cdot K$ males in their gender group. We
conduct a final run of simulation using such seeds for both centrality
measures for ranking, namely Target HI-index and Embedding index, and compare
the information spread and resulting in each gender ratio. We return the seeds
selected by the centrality measure to maximize the overall information spread
and achieve the target gender ratio within a certain error margin.

###  5.2. Evaluation

####  5.2.1. Setup

The demo system of Disparity Seeding is online.101010The demo system of
Disparity Seeding: http://bit.ly/DisparitySeeding. We compare two variants of
Disparity Seeding (i.e., Target HI-Index and Embedding index) with the state-
of-the-art approaches: diversity seeding (Stoica et al., 2020) and IM-balanced
(Gershtein et al., 2018).111111The codes and datasets are available in (Teng
et al., 2021). Note that both diversity seeding and IM-balanced are not
designed for the target gender ratio $\zeta$. To be aware of the target gender
ratio, we implement diversity seeding and IM-balanced as follows. Diversity
seeding selects a value between $\zeta$ and the gender ratio of the top $K$
users with the highest in-degree as the seeding ratio such that the influence
is maximized. IM-balanced greedily selects seeds to maximize the influence
while ensuring that the influence on females is at least $\zeta$ of the
optimal influence on females. Following (Stoica et al., 2020), the diffusion
is simulated through the Independent Cascade diffusion model (Kempe et al.,
2003), where the probability for user $v_{i}$ to influence user $v_{j}$ is set
as the number of likes/comments/tags $v_{i}$ gives to $v_{j}$ over the total
number of likes/comments/tags that received by user $v_{i}$ (Nguyen et al.,
2016).

We evaluate the performance with varying target gender ratios. Specifically,
the performance metrics include 1) the absolute error, i.e.,
$\left|s-\zeta\right|$, where $s$ is the spreading gender ratio and 2) the
influence spread. The seed group size $K$ is $5000$ and $100$ for Instagram
and Facebook, respectively; the error margin is $20\%$ for both datasets. For
Disparity Seeding, the size of sampled seeds is $1,000$ and $20$ for Instagram
and Facebook, respectively. Each simulation result is averaged over $10,000$
samples. All experiments are run on an HP DL580 server with an Intel 2.40GHz
CPU and 4GB RAM.

####  5.2.2. Simulation Results on Instagram.

Figures 8(a) and 8(b) manifest the absolute errors (y-axis) under Disparity
Seeding with Target HI-index and diversity seeding with a varying target
gender ratio (x-axis).121212Disparity Seeding with Embedding index is not
adopted on Instagram since the Instagram dataset lacks the user profiles which
are required for the initialization of GNN. IM-balanced is omitted since it
employs the greedy approach and cannot report the results within $12$ hours
due to the large scale of Instagram. Note that an approach with a smaller
absolute error is more capable of fulfilling the requirement of the target
gender ratio. We have two observations. First, for all target gender ratios,
Disparity Seeding has the smallest absolute errors because Disparity Seeding
carefully decides the seeding ratio by capturing the dependency between the
target and the seeding ratios. Besides, target HI-index factors the target
ratio into ranking and penalizes users who do not fulfill the requirement,
adding more flexibility in accommodating extreme target ratios. By contrast,
even though diversity seeding takes the target ratio as its input, it only
looks for a ratio to maximize the influence spread rather than minimizing the
error between the spreading and the target ratios. Second, the absolute errors
are the smallest for both approaches when the target ratio is between $0.5$
and $0.7$ because the female ratio on Instagram is $52.15\%$, making both
methods easier to satisfy the target gender ratio around $50\%$.

####  5.2.3. Simulation Results on Facebook.

Figures 8(c)-8(e) show the absolute errors on Facebook under different target
ratios. In addition to Target HI-index, since Facebook contains user profiles,
we further evaluate Embedding index, which ranks users according to both
network structure and user profiles. The proposed Disparity Seeding with
either of the ranking measures can achieve the smallest absolute errors,
whereas diversity seeding and IM-balanced are unable to achieve. Note that IM-
balanced can only satisfy the target ratio between $0.5$ and $0.8$ since the
female ratio on Facebook is $59.09\%$, showing that the guaranteed ratio of
the optimal influence on females is not related to the target gender ratio.
Comparing Target HI-index and the Embedding index, one can see that Target HI-
index is better than the Embedding index, i.e., Target HI-index is located
within the error margin of $20\%$ for all target gender ratios.

Table 6 compares the influence spread on Instagram and Facebook when the
target ratio $\zeta$ is satisfied within the error margin of
20%.131313Disparity Seeding with Embedding index is “n/a” on Instagram since
this dataset lacks the user profiles for the initialization of GNN. IM-
balanced is not included since it cannot report the results on Instagram
within a reasonable time and its spread on Facebook does not satisfy the
target ratio within the error margin of $20\%$. For Facebook, Disparity
Seeding with Embedding index outperforms the others since it carefully
exploits user profiles on the Facebook dataset to identify influential users.
For Instagram, Disparity Seeding with Target HI-index has the influence spread
comparable to diversity seeding. Nevertheless, Target Hi-index has much
smaller absolute errors than diversity seeding for all $\zeta$.

Table 6. The influence spread on Instagram and Facebook. | Facebook | Instagram
---|---|---
Comment | Like | Tag | Comment | Like
Target ratio $\zeta$ | 0.9 | 0.9 | 0.9 | 0.5 | 0.5
Target HI-index | 222.2 | 202.5 | 241.2 | 8542.6 | 7508.7
Embedding index | 232.8 | 255.3 | 263.8 | n/a | n/a
Diversity seeding | 187.8 | 224.9 | 243.8 | 8516.14 | 7531.73
Agnostic seeding | 175.5 | 182.1 | 229.9 | 8469.28 | 7523.47

##  6\. Conclusion

Leveraging Instagram and Facebook interaction data, we investigated usage
patterns and the manifestation of the glass ceiling in different interaction
types on social media. We discovered correlations between gender and both high
visibility and popular endorsement by jointly considering degrees and
interaction intensity of direct and indirect interactions with traditional and
novel measures. Motivated by the gender disparity exhibited in online social
networks, we proposed a Disparity Seeding framework that aims to maximize the
information spread and reach a gender target ratio, which may differ from the
original ratio in the population. The core step of Disparity Seeding applies
the proposed centrality measures to rank influential users, namely Target HI-
index and Embedding index, and selects a suitable seed set through a
simulation-based learning approach. Our evaluation results show that Disparity
Seeding can achieve the target gender ratio in an agile manner and increase
the information spread better than state-of-the-art parity seeding algorithms.
The general design of Disparity Seeding can be extended to include additional
information. It can also be applied to select seed sets to take under-
privileged minorities into account and enhance their influence.

## Acknowledgement

This work has been partly funded by the Swiss National Science Foundation
NRP75 project 407540_167266, by MOST through grants 110-2221-E-001-014-MY3 and
109-2221-E-001-017-MY2, and by the Institute for Information Industry under
Contract 110-0922. We thank to National Center for High-performance Computing
(NCHC) of National Applied Research Laboratories (NARLabs) in Taiwan for
providing computational and storage resources.

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