# Structural Gender Imbalances
in Ballet Collaboration Networks

Yessica Herrera-Guzmán Research Center for Social Complexity, Universidad del
Desarrollo, Chile.  Eun Lee Department of Scientific Computing, Pukyong
National University, Republic of Korea.  Heetae Kim Department of Energy
Engineering, Korea Institute of Energy Technology, Republic of Korea.  Data
Science Institute, Faculty of Engineering, Universidad del Desarrollo, Chile
Corresponding Author. Email<EMAIL_ADDRESS>
###### Abstract

Ballet, a mainstream performing art predominantly associated with women,
exhibits significant gender imbalances in leading positions. However, the
collaboration’s structural composition on gender representation in the field
remains unexplored. Our study investigates the gendered labor force
composition and collaboration patterns in ballet creations. Our findings
reveal gender disparities in ballet creations aligned with gendered
collaboration patterns and women occupying more peripheral network positions
respect to men. Productivity disparities show women accessing 20-25% of ballet
creations compared to men. Mathematically derived perception errors show the
underestimation of women artists’ representation within ballet collaboration
networks, potentially impacting women’s careers in the field. Our study
highlights the structural disadvantages that women face in ballet and
emphasizes the need for a more inclusive and equal professional environment to
improve the career development of women in the ballet industry. These insights
contribute to a broader understanding of structural gender imbalances in
artistic domains and can inform cultural organizations about potential
affirmative actions towards a better representation of women leaders in
ballet.

##  1 Introduction

One broadly investigated complex socioeconomic problem, is global economic
inequality [1, 2, 3]. There is growing evidence that economic inequality
affects artists and, more importantly, women artists, to enjoy of economic
growth and access to leading positions in their careers [4, 5]. It is
therefore becoming increasingly important to understand the social dynamics of
gender inequalities in the arts. In specific, ballet is widely recognized and
appreciated around the world, and is assumed as a women-dominated profession
[6, 7]. However, recent reports show considerable gender imbalances where men
specifically dominate leading positions [8].

The lack women’s representation as leaders in ballet has been widely discussed
within the dance community, claiming for more equal professional opportunities
[9, 10]. For example, data from American dance companies reveals the unequal
representation of women (less than 40%) in artistic and executive positions
[11], while the overall participation of women in the workforce is about 70%
[12]. This difference of women’s representation in leadership roles raises the
question of whether or not women face a ‘glass ceiling’ barrier in the ballet
industry [13, 14].

In our complex society, individual characteristics —such as race, religion,
education, or gender— have meaningful effects in social behaviors that shape
structural disparities, which may be a result of homophily, the preference of
individuals to connect with similar others [15]. Network research reveals that
structural properties influence the access to information [16, 17], creativity
[18], productivity [19], and career success [20]. Moreover, homophilic
behaviours embedded to an imbalanced social structure can negatively affect
the ranking of individuals from minority groups by enhancing segregation
effects [21]. In an imbalanced social structure, individuals may inaccurately
estimate the frequency of the minority group, resulting in perception errors
regarding the representation of attributes in a social network [22, 23]. As a
result, the importance of the minority group can be over or underestimated
respect of what can be expected from the real representation in the network
[24]. Since perception errors could reinforce unequal patterns in social
connections, such as collaborations, understanding the role of network
structure regarding gender imbalances can give an insight to an intervention
of equal opportunities in professional positions.

Despite the collaborative nature of ballet creations, previous reports have
primarily focused on quantifying the percentage of women and men artists
involved, while the role of collaboration structures in contributing to gender
imbalances in ballet remains poorly understood. The existing literature
evidences that gendered variations in social network structure contribute to
different professional outcomes for men and women [25], highlighting the
importance of investigating the gender representation in collaborative
structures. Yet, there is a lack of systematic studies exploring the
representation of women and the structural properties of ballet’s professional
network.

In this work, we investigate the social network structure and collaboration
patterns of ballet creations. We hypothesize that, if the network structure is
unbalanced by gender, the imbalanced social structure will align with unequal
collaborative behaviors and the existence of perception errors, which could
explain why females do not undertake or are overlooked from leading positions
in this industry. This research relies on the stable collaborative structure
of ballet, which has remained largely unchanged since its origins, to conduct
a network analysis with scientific validity.

We construct collaboration networks from four renowned ballet companies and
analyze their gender composition. The collaboration structures studied here
mainly comprise a core structure of ballet creators, such as choreographers,
composers, and costumes and light designers. We compare the real-world
collaboration structures with randomized network models. We specifically
explore the structural gendered differences and the labor force composition in
highly central positions. We also measure the formation of perception errors
on the women’s group to examine a possible relationship between gendered
collaboration networks and perceived working environment.

To the best of our knowledge, our study is the first attempt to understand the
structural gender imbalances in major ballet companies. This research will
help understand the underlying social mechanisms driving gender inequalities
in a highly collaborative performing art. We hope that this work will shed
light for more effective interventions to reduce the segregation of women in
creative careers.

##  2 Methods

###  2.1 Network of ballet creators

We construct the collaboration networks of ballet creators from four major
ballet companies —the American Ballet Theatre (ABT) [26], the New York City
Ballet (NYCB) [27], the National Ballet of Canada (NBC) [28], and the Royal
Ballet of the Royal Opera House (ROH) [29]— based on their worldwide prestige
and the availability of their historical repertoire in their website. Company
data are collected using a Robotic Process Automation method for web scraping
[30]. Our data collection and research methods were approved on January 18th,
2023, by the Institutional Research Ethics Committee of Universidad del
Desarrollo, in Chile.

The collected data includes original ballet titles, as stated in each
company’s repository, and refers to ballet works with _artistic elements that
remain constant across time, performances, and productions_ (e.g. creators,
libretto, music, genre). When appropriate, ballet companies list revivals
(recreated works), and/or company premieres (productions that were originally
debuted at a different ballet company, but that are presented for the first
time in the company listing the work).

Collaboration networks are formed from the teams of leading artists working
together to create a ballet work. Teams of ballet creators are formed from
each record of original ballet titles, which includes the credits of leading
artists, such as principal creators (choreographer and composer) and
specialized roles (librettist, costumes and lighting designer), and does not
include the dancers or any other company members. In a few occasions,
companies report the producer, designer (unspecified), and media editor of a
ballet work, and other team structures vary in size by adding multiple
collaborators for the same role (e.g. two or more composers). It is important
to note that ballet is strongly recognized for its stable collaborative
structure, comprising a core structure of leading artists, such as
choreographer, composer, librettist, and costume and light designer. Hence, in
constructing the collaboration network of ballet creators, we consider all
listed artists in each ballet title as equal contributors to the ballet
creation. Therefore, a _ballet collaboration_ is defined as the creative and
collective effort between choreographers, composers, costume designers,
lighting designers, and other artists listed by each company, for the creation
of a ballet title. For further details, please see Section S1.

The processing of the data is as follows. In Fig. 1a there is an illustration
of the data showing a list of ballet titles (as an example, ‘Ballet 1’ and
‘Ballet 2’) with the names of ballet creators (A, B, C, D, and E), and their
roles (e.g. Choreography, Music, and Costumes). Then, all artists who
collaborated in a ballet creation together are part of the same team. To
construct the collaboration network of each company, we first build a
bipartite network between ballet creations and artists, as seen in Fig. 1b,
where left nodes represent ballet titles and right nodes display the artists
that created a ballet title. Next, artists’ collaborations are projected to an
undirected graph, as shown in Fig. 1c, where each node represents one artist,
and a link between two artists denotes their collaboration in the same ballet
creation. An artist who teams up in more than one ballet creation will connect
multiple artists in the same company, becoming a connector in the
collaboration network.

Figure 1: Schematic representation of data processing and network
construction. a The collected data contains ballet titles (the title of a
ballet creation) and artists’ names with their corresponding roles. b The
collected data is transformed to a bipartite network, where artists connect to
ballet titles that they have been participated as creators. c From the
bipartite network, a projected unipartite network representing the
collaboration of artists in a ballet company.

The resulting empirical networks include about 300–560 ballet works, with a
range of 490–850 artists (nodes) and 1900–3100 collaborations (links). In
addition, he time of reported ballet creations ranges from 1930’s to 2020’s,
making the networks comparable in terms of size and longevity. Basic network
properties —such as size of the giant component, average clustering
coefficient [31], average shortest path [32], and small-worldness [33]— can be
seen in Table 1.

Table 1: Data description and basic network characteristics of the four ballet
companies.

| ABT | NYCB | NBC | ROH
---|---|---|---|---
Time window | 1940-2020 | 1935-2020 | 1951-2019 | 1931-2012
No. of Ballets | 491 | 560 | 349 | 449
Artists (nodes) | 779 | 549 | 579 | 847
Collaborations (links) | 2323 | 2202 | 1913 | 3107
Average degree | 5.9641 | 8.0219 | 6.6079 | 7.3365
Density | 0.0076 | 0.0146 | 0.0114 | 0.0086
Clustering coefficient $C_{avg}$ | 0.7556 | 0.7710 | 0.7776 | 0.8033
Average Path Length $L$ | 3.52 | 2.63 | 3.60 | 3.49
Small-worldness, $S$ | 97.73 | 67.57 | 75.07 | 102.99
Giant Component |  |  |  |
Artists (as percentage) | 715 (91.78) | 643 (98.90) | 540 (93.26) | 816 (96.34)
Collaborations (as percentage) | 2251 (96.90) | 2195 (99.68) | 1863 (97.38) | 3071 (98.84)
Density | 0.0088 | 0.0149 | 0.0128 | 0.0092
Diameter | 8 | 5 | 10 | 8

####  2.1.1 Gender inference

Artists’ names were processed for misspelling, middle names, and initials to
distinguish artists’ identity. The names are held constant if reported across
multiple companies. Then, we infer artists’ gender by using gender package for
R [34, 35]. This package contains names from various countries and periods,
and infer names from standardized databases (ssa, ipums, napp, and demo),
making it adequate for this study since the collected data contains names of
artists with diverse nationalities and were born in the 19th and 20th
centuries.

To estimate an artist’s birth year, we assume that each artist was at least 20
years old when they participated in a ballet creation for the first time. Thus
we subtract 20 years from the year of the first ballet production of an artist
in our data as a proxy of the minimum age for a productive life in ballet.
This method considers a range of 10 years ($\pm 5$ years from the estimated
birth date). Then, the gender package estimates a probability that a person
would have certain gender with the name. If the probability is larger than or
equal to $0.7$, the corresponding gender is assigned to each artist. Here, the
assigned ‘gender’ is a binary property (Woman, Man) and does not consider
other gender assignments. Note that the inferred gender does not refer
directly the sex of the artist nor the self-assigned gender chosen by each
artist., but is used as estimate of the social construction of gender. The
names which were not able to assign gender with this method were manually
assigned after a web search of the artist’s identity.

###  2.2 Network analysis and gendered differences in centrality

We measure four network metrics to understand the importance or centrality of
artists in the collaboration networks:

  1. 1.

Degree centrality is computed following [36] to measure the number of total
connections of a node. This metric can capture the individual access to a
richer social capital.

  2. 2.

Harmonic centrality is computed following [37], and is a variant of closeness
centrality created to deal with unconnected graphs to measure the distance one
node has respect to all other nodes in the network. In other words, harmonic
centrality captures the position of nodes to efficiently reach distant parts
of the network.

  3. 3.

Betweenness centrality is computed considering all pair of nodes as described
in [38] to measure the number of shortest paths between two pairs of nodes
that pass through a node in a network. This metric captures what nodes are
best intermediaries or bridges between different parts of the network.

  4. 4.

Eigenvector centrality is computed following [39] and measures the importance
of a node based on the centrality of its neighboring nodes. This centrality
informs about the nodes who are connected to other influential or central
nodes, as these can help gain social prestige in the network.

These metrics are informative on the differential ranking of individuals
embedded in the network [40]. For example, one artist with high centrality
(e.g. degree) should indicate that the artist has multiple connections in the
network, then being well positioned to have more access to information, social
connections, and professional opportunities. In a global sense, these
centrality metrics help identify structural patterns within a network,
providing insights into the underlying relationships between individuals that
ultimately shape the network.

From the centrality metrics, we sort all artists by their centrality in
descending order, and selected a group of top 20 artists, referred as _Top-
Central Artists_ (TCA) in this study. Let consider the ranking of centralities
$C(r)$, where $C$ denotes a corresponding centrality value of an artist at a
given rank $r$, for $r=\\{1,2,\dots,19,20\\}$, so $r=1$ represents the most
central artist having the highest corresponding centrality (e.g. $C(1)=0.8$),
and $r=20$ will have the lowest centrality (e.g. $C(20)=0.2$). We select the
top 20 as this fraction captures the largest observed variation of centrality
values in the empirical networks, and between and within gender groups. Then,
by analyzing the TCA, we capture the artists with best connected individuals
in the network and the differences in network positions across gender
categories.

Next, we implement the TCA ranking to form three independent groups: the first
group is for all artists in a company’s collaboration network, labeled as
$\text{TCA}_{\text{Network}}$, and the other two groups are for a company’s
artists grouped by gender, which results in two separate rankings for
$\text{TCA}_{\text{Women}}$ and $\text{TCA}_{\text{Men}}$. All centralities
are then normalized by the maximum value of the centrality within company
group (Network) and by company gender groups (Women, Men), to have a linear
scaling of $[0,1]$ range. In more detail, the $\text{TCA}_{\text{Network}}$
uses a dense rank function, which generates rank ties for observations with
the same centrality values, so a variation of the total number of artists is
possible if there are artists with equal centrality at each rank. For
$\text{TCA}_{\text{Women}}$ and $\text{TCA}_{\text{Men}}$, the tied centrality
is not considered to keep an equivalent number of women and men artists (i.e.
20 artists per group).

Separately, we quantify the women ratio $R_{\text{Women}}$ in each
$\text{TCA}_{Network}$, computed as
$R_{{\text{Women}}}=\frac{\sum_{i}^{N_{\mathrm{TCA}}}\theta(i)}{N_{\mathrm{TCA}}}$.
Here, $i$ denotes an index for an artist who is in a corresponding TCA, where
$N_{\mathrm{TCA}}$ represents the total number of artists in a
$\text{TCA}_{Network}$, and $\theta(i)=1$ when an artist is woman, or $0$ for
men. Then, $R_{\text{Women}}$ provides the fraction of women artists who
belong to the group of best connected individuals in the collaboration network
of a ballet company. A numerical fraction of women artists at the network
level of $0.5$ is assumed as a gender-balanced collaborations, and we call
this situation as ‘neutral’ composition.

Further, we measure the difference in centrality $\Delta C(r)$ between two
rank-matched artists from each gender group is measured as $\Delta
C(r)=C_{\text{Men},r}-C_{\text{Women},r}$. Here, each woman artist from
$\text{TCA}_{\text{Women}}$ is matched to their corresponding $r$ pair from
$\text{TCA}_{\text{Men}}$. That is to say, if there is a woman artist ranked
$1$ in $\text{TCA}_{\text{Women}}$ with a centrality value of $0.4$, she is at
the most central position in the women’s group, and it can be written as
$C_{\text{Women, 1}}=0.4$. The counterpart of man artist, who is ranked $1$ as
well in $\text{TCA}_{\text{Men}}$ will be $C_{\text{Men, 1}}=0.5$, if he has a
centrality of $0.5$. Then, $\Delta C(1)=0.5-0.4=0.1$. If $\Delta C(r)>0$, it
means that a man artist is located on more central position than the woman
counterpart.

####  2.2.1 Null model analysis

We compute two different null models by simulating 100 synthetic networks
derived from the representation of each company’s empirical collaboration
network. With the help of the null models, we remove the collaborator- or
gender-preferences by shuffling collaborations (links) or artists’ attributes
(gender) in the collaboration network. The overall purpose of the null models
is create a baseline of randomly created networks, which would allow us to
determine the absence or existence of randomness in the observed patterns
respect to the empirical network.

  1. (1)

Edge-shuffled model: In this model, edges are randomly rearranged in the
network while preserving artists’ degrees. This means that the total number of
collaborations per artists are preserved, as well as the total number of
artists (nodes) in the network and artists’ gender. We use the
‘random_reference’ function of NetworkX [41]. From this shuffling, we remove
the gendered correlation from empirical collaboration networks. Therefore, the
resulting synthetic networks show collaboration structures when there is no
gender preference.

  2. (2)

Gender-shuffled model: This model shuffles the gender of artists while holding
all network properties constant. Here, the empirical network structure is used
as a reference, without nodes’ attributes, over which a dictionary containing
the gender of all nodes is used to randomly assign artists’ gender, while
preserving the real fraction of women and men in the network. In this way,
artists’ network position are preserved, but their gender is randomized in
each iteration. Therefore, the resultant networks display an artificial
collaboration pattern without a correlation between an artist’s gender and
position, as well as a gendered collaboration assortativity.

To test a null hypothesis distribution, we compute the Z-score for a
distinction between the centrality values from the empirical networks and
those from the null models. We denote the observed centrality by rank in the
real network as $C(r)_{\text{real}}$, and that of the null model as
$C(r)_{\text{null}}$. Then, the $Z$-score for any TCA group uses the
centrality from the empirical network, $C(r)_{\text{real}}$ , and the averaged
centrality of 100 null models, $\bar{C}(r)_{\text{null}}$, so that
$Z(C)=\frac{C(r)_{\text{real}}-\bar{C}(r)_{\text{null}}}{\sigma(C(r)_{\text{null}})}$.
Z-score of $\Delta C(r)$ in the empirical network is also measured with the
values of the synthetic networks as $Z(\Delta C)=\frac{\Delta
C(r)_{\text{real}}-{\Delta\bar{C}}(r)_{\text{null}}}{\sigma(\Delta
C(r)_{\text{null}}})$.

###  2.3 Perception error on women artists

To understand more about the implications of the gendered differences in the
collaborative environment, we use a mathematical approach to measure the
existence of perception errors based on [24]. Perception errors refer to the
inaccuracy in the estimation of the frequency of an attribute —usually of a
minority group— in a social network, perceived from the frequency of that
attribute within the individual local network [23, 21]. In this research,
perception errors are the difference in the perceived fraction of women
artists from the local network, respect to the fraction in the entire network.
For instance, if in the local network there are mostly women, but in the
entire network there are more men, one individual will have a perception error
above one that over estimates the size of the women’s group, while the
opposite happens for the under-estimation of the women’s group, with a value
below 1. Thus when the perception error is equal to 1, that means that the
perception of the fraction of women in the network is accurate. The perception
error $B$ of an individual artist $i$ is thus computed as
$B_{i}=\frac{W_{i}}{R_{\text{Women}}}$, where $W_{i}$ denotes the local
fraction of women among $i$’s collaborators, and $R_{\text{Women}}$ refers to
the real fraction of women in the network, as noted above.

Based on the individual artist’s perception error, we measure an averaged
perception error by gender group at a network-level, so
$\bar{B}_{Women,Men}=\frac{\sum_{i}B_{i}}{N_{Network}}$, where $N_{Network}$
represents the total number of artists in a ballet company. Consequently, when
$\bar{B}=1$, it means that the overall perception of women on a company is
accurate on average, and when $\bar{B}<1(\bar{B}>1)$, a gender group
underestimates (overestimates) the ratio of women artists on average. In
addition, a gendered homophily is measured following the method in [24] to see
gendered preferences of the collaboration networks.

##  3 Results

Based on previous reports on the lack of representation of women in leading
positions in ballet [11], we explore the general composition of the
collaboration networks of ballet creators and the existence of gendered
collaboration patterns in the professional environment. We also look into the
composition of the most central network positions and the gender gap between
men and women’s centrality in the network; in addition, we measure the
existence of perception errors of the women’s artists group within ballet
companies. We compare network position and perception errors from the
empirical network structures with two null model analyses.

###  3.1 Team structure and collaboration patterns

The most common team size for a ballet creation across companies is three to
four (20–40%), followed by five members (20%), as shown in Fig. S1a. This
evidences that teams of ballet creators are mostly formed by the typical
collaborative structure of leading artists. Fig. 2a shows a sample of the
representation of women in a ballet company (ROH), shows that there are about
50% of teams having 100% men artists, and less than 10% of teams have gender
neutral ratio of 50%. Conversely, the majority of teams are composed with less
than 50% of women artists, regardless of their sizes, and teams having 100%
women artists is almost zero.

Dance communities have specifically reported an overlooking of women in
choreographic leads, and our results suggest that women are less represented
than men in general leading roles. Exploring the team composition by artistic
role, Fig. S2 shows that the proportion of women is considerably low for
Choreographer, Librettist, and Composer. Other positions such as Costumes,
Lighting, and Design have a relatively larger participation of women. These
results suggest variations between women and men regarding artistic roles are
possible. However, because most teams of ballet creators are formed by a core
structure of leading artists, here we focus on the structural representation
of women at both team and network-levels, rather than an individual-artistic
role level.

Further, in Fig. 2b we see that when women collaborate in one team, the
frequency of working with other women in the same team is actually very low
($<30\%$). These results describe that women artists mostly work in men-
dominated environments. In addition, men-alone teams are rather rare
($<10\%$), as they tend to collaborate with at least other three to five men
($>20\%$) and participate in considerably larger teams than women (up to 11
men in one team, at ROH).

Figure 2: Team composition and collaboration patterns by gender. Sample of the
Royal Opera House (ROH). Panel a shows the probability of a gender ratio in
teams: more than 50% of teams are composed only by men, while teams having
only women are nonexistent. Panel b shows the normalized frequency of same-
gender collaborations in a team: women mostly collaborate alone in a men-
dominated teams, while men collaborate more with other 3-5 men and form larger
teams. And panel c shows the number of ballet creations for each artist.
Productivity varies by gender, with less productivity for women artists. Fit
line by gender with 95% confidence intervals.

In terms of productivity, women artists are less involved in ballet creations
than men artists. One ballet creation refers to the participation of an artist
in a team as leading artist for the creation of a ballet work. In NYCB and
ROH, the most productive woman participates in about 20-25% of the creations
of the most productive man artist collaborated (ROH’s maximum collaborations:
Men = 76, Women = 16; NYCB’s maximum collaborations: Men = 211, Women = 54,
see Figures 2c, and S3b). For NBC, the highest productivity is a bit similar
for both genders. Women artists’ highest productivity is just 86% of the most
productive man (NBC’s maximum collaborations: Men=38, Women=33). Only at the
ABT, the most productive woman artist exceeded in 20 collaborations to the
most productive man artist (ABT’s maximum collaborations: Men=35, Women=55).
Despite the exception, most women artists are less productive than their men
counterparts, and the global picture for women is to work in men-dominated
creative environments. Team structures, and collaboration and productivity
patterns, are similar across all companies studied here (for more details and
figures by company, see Section S1).

###  3.2 Centrality differences by gender

So far, we have observed a less frequent participation of women respect to men
in ballet collaborations. These observations raise the question: Does the low
participation of women relate to their network position in the company? To
answer this question, we explore the distribution of artists’ collaborations
in the network. We first compute the fraction of women in the network,
$R_{\text{Women}}$, and the proportion of dyadic interactions (see Table 2),
showing that most companies only have about $20\%$ of women in leading
positions. Figure 3a shows a network sample, where men (in yellow) are not
only a majority but also with higher connectivity respect to women (in
purple). (See all companies’ collaboration networks in S4). Moreover, the man-
man connections are more than 60% across companies (yellow links, Fig. 3b) and
mixed connections are about 30% on average. On the other hand, woman-woman
connections are less than 5% of the total dyadic interactions (purple links,
Fig. 3c). These results inform that, for every 4 men, there is only one woman
in the network, a collaborative structure in which men artists are densely co-
worked with other artists regardless of gender, locating at the center of the
collaboration network, while women artists are sparsely distributed in the
periphery of the network.

Figure 3: Distribution of artists and their collaborations. Sample of ROH’s collaboration network. Panel a shows nodes colored in purple/yellow for women/men. Node size proportional to degree centrality. The dyadic collaborations by gender are shown in: panel b for man-man collaborations; and panel c for woman-woman collaborations. We see that women are visually less central than men, and their collaboration with other women are scarce and peripheral. Table 2: Network composition by gender. $R_{\text{Women}}$ is the fraction of women in a collaboration network. For the collaborations, the number of woman-woman/man-man/mixed dyadic interactions is counted. | ABT | NYCB | NBC | ROH
---|---|---|---|---
Artists (nodes) | 779 | 549 | 579 | 847
$R_{\text{Women}}$ | 0.22 | 0.19 | 0.18 | 0.19
Collaborations (links) | 2589 | 2317 | 1956 | 3385
Woman-woman | 138 (5%) | 64 (3%) | 55 (3%) | 135 (4%)
Man-man | 1603 (62%) | 1576 (68%) | 1336 (68%) | 2353 (70%)
Mixed | 848 (33%) | 667 (29%) | 656 (29%) | 897 (26%)

We then evaluate the proportion of women in the group of top-central artists,
$\text{TCA}_{\text{Network}}$, by their network centrality rank, $C(r)$, and
observe that most companies have a lower representation of women respect to
$R_{\text{Women}}$ in the empirical network. We observe an overall increase of
$R_{\text{Women}}$ in the randomized models for all centralities (see all
companies in Fig. S5). For example, the Edge-shuffled model improves
$R_{\text{Women}}$ for harmonic centrality from $10\%$ to $15\%$, and Gender-
shuffled model raises it up to $19\%$ in the sample of the ROH, a fraction
that matches the $R_{\text{Women}}$ of the total empirical network (Fig. 4).
Note that the Edge-shuffled model keeps $R_{\text{Women}}$ in TCA regarding
degree centrality because the number of collaborations (degree) for an artist
and their inferred gender are held constant in this model. These results
suggest that the low representation of women artists in ballet creations could
be related to gender assortative collaborations, and the current level of
women artists’ centrality is not a deterministic outcome of the small fraction
of women artists in the company. That is to say, even when the fraction of
women remains small in a network, women artists’ representation could be
improved if more equal collaborations for women were encouraged.

The $Z(C)$ reveals a general change in artists’ centrality with the null
models (see ROH’s sample in Fig. 5, all companies in Fig. S6a). For the Edge-
shuffled model, only the harmonic centrality displays a negative Z-score for
both women and men (sample in Fig. S6a). Harmonic centrality denotes an extent
of an artist’s closeness to other artists on average, so small value
represents far distance between artists. The negative $Z$-score suggests that
the distance among artists in empirical collaboration networks falls apart
farther than the expected distance from the null models. In other words, the
distribution of TCA in the empirical networks is more central, suggesting that
TCA can more efficiently reach other artists in the network respect to the
distance observed in the null models. For the Gender-shuffled model, the
negative women artists’ $Z$-scores indicate that their positional importance
can be improved in a synthetic network with collaboration imbalances (sample
in Fig. S6b). Altogether, our results suggest that differences in centrality
among TCA may not be derived by random factors, but there may be underlying
systematic social behaviors limiting women artists’ collaborations and network
position, regardless of their small fraction in the network.

Figure 4: Fraction of women in $\text{TCA}_{\text{Network}}$ from the
empirical network and null models. Sample of ROH. Average $R_{\text{Women}}$
of $\text{TCA}_{\text{Network}}$ in Edge-shuffled and Gender-shuffled models
are shown with standard deviation. The null models show a fairer
representation of women artists than the empirical network.

The difference in degree centrality ($\Delta C$) highlights that a man artist
locates at more central position than the same-ranked women artist in her TCA
group. Figure 5c shows a sample for degree centrality and reveals that the
most central man is considerably more central than the most central woman.
This trend of $\Delta C$ is observed across centralities with slight
variations, confirming that men are considerably better positioned respect to
women across companies (see gender gap in centrality for all companies in Fig.
S7). Interestingly, all empirical $Z$-scores for $\Delta C$ are several
standard deviations away compared to the null models (see all companies in
Fig. S6b). Figure 5c illustrates the variations by null model, and showcases
that a large gender gap is less likely observed when the gender preference
(Edge-shuffled) and gendered productivity correlation (Gender-shuffled) are
destroyed.

Figure 5: Comparison of centralities for artists in
$\text{TCA}_{\text{Women}}$ and $\text{TCA}_{\text{Men}}$ with null models.
These figures are for ROH’s collaboration network. Z-score of centralities
($Z(C)$) compared with a Edge-shuffled model and b Gender-shuffled model. The
red line corresponds to the theoretical mean obtained from the null models
indicating no difference. Panel c shows the gender gap in the degree
centrality (normalized) between $\text{TCA}_{\text{Women}}$ and
$\text{TCA}_{\text{Men}}$, revealing that $\text{TCA}_{\text{Men}}$ have
higher degree centrality than their women counterparts. Panel d shows the
distribution of $Z(\Delta C)$ separated by null model.
---

###  3.3 Perception error of women artists

Given the observed structural imbalances in the ballet collaboration networks,
the low participation of women in professional collaborations could affect the
perceived frequency on women artists in the entire network. Perception errors
are the distorted frequency estimation of an attribute in a social network by
the individual local environment [23, 21]. Here, the perception error is
defined as the fraction of the observed frequency of women in an artist’s
local collaboration network over the real fraction of women in the global
network (see Methods). That is, the perception error denotes a relative
difference of women artists in the local collaboration environment of each
artist and the actual women artists’ frequency in each ballet company. From
the individual-level perception error $B_{i}$, a gender group-level error
$\bar{B}$ compares the average perception error for women and men. If
$\bar{B}>1.0$ ($\bar{B}<1.0$), it means a gender group overestimates
(underestimates) the global frequency of women artist. When $\bar{B}=1.0$, it
denotes an accurate perception on the women frequency (see Methods). We
complement perception error with a measure of homophily.

Figure 6: Average perception error $\bar{B}$ for ballet companies. Average
perception error by gender in each company, compared with those from null
models. a ABT, b NYCB, c NBC, d ROH. Red line indicates $\bar{B}=1$, an
accurate perception of the women’s group size. Line segments in gray guides
the difference in perception error by gender group. Most gender groups
misconceive the real fraction of women in their network, while the difference
is reduced in Edge-shuffled model. The perceived frequency of women is
considerably more accurate in the Gender-shuffled model.

Our results show that the empirical collaboration networks of the ABT, NYCB,
NBC, and ROH, the women (men) artists’ homophily is 0.56 (0.53), 0.47 (0.55),
0.45 (0.57), and 0.56 (0.63), respectively (1 is a perfect homophily, and 0 is
a perfect heterophily situation). The ABT has relatively gender-mixed
environment, resulting in both gender groups having relatively accurate
perception on the global fraction of women artists, as shown in Fig. 6a.
Conversely, the rest of the companies demonstrate a wide difference in the
perception error by gender, shown in Fig. 6b–d. For instance, NYCB’s men group
underestimates women artists about 7%, but their women group underestimates
themselves about 27%, showing a 20% difference in the perception of women
between the two groups. Such difference may be related with men artists’
strong homophily in NYCB collaborations, and women artists’ gender-
heterophilic collaborations (woman-man heterophily $0.53>$, and woman-woman
homophily $0.47$), and indicates a perceived underestimation of women artists
by themselves. In ROH, women artists have a more accurate estimation of women
artists respect to men artists, which aligns with their collaborative
behaviors, where women artists collaborate more with other women artists than
other men artists (woman-man heterophily $0.44<$ woman-woman homophily
$0.56$). Yet, the difference in perception still exists since men artists
collaborate mostly with men artists (man-man homophily $0.63$, man-woman
heterophily $0.37$), and the assortative collaboration widens the difference
in perception between gender groups.

Interestingly, the Edge-shuffled model displays a reduction in perception
error difference between women and men, even though the reduction is limited.
The reduction of the perception error between women and men suggests the
existence of gender-preferred collaborations between gender groups. Moreover,
the Gender-shuffled model not only sensibly reduces the difference in the
average perception error for women and men, but also achieves a nearly
accurate perception on the fraction of women artists. This strongly suggests
that lowering an extent of imbalanced productivity and gendered preferences
altogether boosts the representation of women artists, even considering the
small representation of women artists in the company.

##  4 Discussion

Inequalities have been investigated for different occupations to capture the
gender gap in salary and labor force composition [1]. In this context, we find
that the representation of women artists in ballet creations is about 18–22%,
which is lower than the reported 25% for choreographic leads [11]. These
values are far below a gender-neutral ratio ($0.5$, see Table 2) and are lower
for highly central artists (in this study, TCA groups). In general, we find
that women artists are underrepresented in the overall collaboration network
and all positions as leading artists, not only as choreographers.

In addition to the numerical imbalance, our results suggest that gendered
collaboration structures could potentially aggravate gender imbalances in
ballet creations. Crucial roles of individual’s social network are associated
with the access to information and professional opportunities in creative
collaborations [42, 17, 19, 43, 18, 44]. Thus, our results increase the
understanding of how gendered collaborations could impact artists’
professional experiences and the social perceptions of women as a minority
group. The comparison with null network models gives a hint that the observed
gender imbalances in terms of central network positions could be explained by
systematic inequalities in collaborative behaviors rather than by random
factors.

The social network structures, such as network position and social prestige,
play an important role shaping successful careers [45, 46]. Some studies show
that men and women utilize different social network structures and behavioral
patterns that influence their placement in the job market [25, 47, 48]. Other
studies show that the formation of a personal network and social behaviors
over time are related to reinforced perception errors [49, 50]. Taken
together, these studies suggest the existence of a permanent feedback for the
formation of social relationships, social perception errors, and collaboration
patterns, which in return, can influence individual career decisions. For
women in ballet, a feedback based on a low representation within creative and
men-dominated collaborations could negatively impact their decision to
undertake a career as ballet creators or engaging in multiple collaborative
projects. A future study in this line could provide evidence on why women in
ballet experience ‘glass barrier’ in the field, and whether womens’ network
position facilitates their career success and impact in the long term.

In addition, the collaboration structure can be crucial for teams [51] and
individual performance [52] in terms of creativity and success [18, 53]. A
study demonstrates that diversity can improve creative performance [54], and
emphasizes the participation of women in collaborative environments because
they increase the social sensitivity of the group, making the team
collectively more intelligent and proficient [55]. In view of this, new
policies for more equal collaborations and a more inclusive environment for
women as leading creators should be considered in ballet companies. A more
diverse inclusion would boost creative innovation and impact, which ultimately
benefits the artistic community in general.

At an individual level, perception errors derived from the network structure
in the workplace can affect career decisions. In specific, our results reveal
that most companies experience perception errors on the fraction of women
artists participating at the company. The constant underrepresentation of
women could negavitely impact their visibility as a group, undermining the
motivation of women artists to look for better professional opportunities.
This interplay among working environment, perceived possibility of career
development, and personal decisions could be a pivotal issue to alleviate the
low representation of women in ballet and other industries where women are a
minority.

Our measure of perception errors is a mathematical approach and can be
improved, as there are multiple factors influencing the perception of a local
network structure. That is to say, a local network can be described not only
by its structure, but also to its embedded social mechanisms, like the
strength of relationships formed over time, access to information, formal and
informal norms [16, 56, 57], and individual cognitive processes and
preferences [58, 59]. In addition, ballet is strongly influenced by biological
constraints, such as the physical demands of the art form, including strength,
flexibility, and technical requirements. These constraints, combined with the
distribution of labor in family responsibilities, may be stronger for women
[1] and may contribute to fewer women to overcome social barriers in the
workplace and hinder the professional development of women artists in ballet.

Overall, our results help understand another dimension of gender inequalities
in the ballet industry. Yet we are aware of the limitations of this work. Our
data depends on the archival of the selected ballet companies, which may not
be sufficient to generalize the current results to entire ballet industry.
Moreover, artists may hold different types and duration of contracts within a
company, which can result in variations in observed professional
collaborations. To overcome this, more comprehensive digitized data
collections would be needed. For instance, with the implementation of
computational methods, such as deep learning and network science, it has been
possible to objectively measure the impact of individual performance in
creative domains [60, 61], and similar methods can open the possibility for
future research on the relationship between gender, network centrality, and
actual ballet creators’ impact in the field.

In summary, our research highlights the low representation of women as ballet
creators and sheds light on their peripheral network position and gendered
collaboration preferences within the ballet industry. This investigation can
be extended to explore dynamic network factors shaping gender imbalances to
propose possible and more adequate interventions for the diversity, equity,
and inclusion in cultural organizations. We hope that this work brings
awareness on how social phenomena and inequalities in creative domains can be
systematically studied with network science and data driven methods.

### Data availability

The data used in this study is available under reasonable request.

### Abbreviations

ABT, American Ballet Theatre; NBC, National Ballet of Canada; NYCB, New York
City Ballet; ROH, Royal Ballet of the Royal Opera House; TCA, Top-Central
Artists.

### Availability of data and materials

The data analyzed during the current study are available from the
corresponding author on reasonable request.

## Competing interests

The authors declare that they have no competing interests.

### Funding

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (No.NRF-2022R1C1C1005856), the
National Agency of Investigation and Development, ANID, through the grant
FONDECYT No. 11190096, the KENTECH Research Grant(KRG 2021-01-003), and the
Pukyong National University Research Fund in 2022(202203530001).

### Author’s contributions

All authors contributed to the research design and writing of the paper. YHG
contributed with art-specific knowledge, constructed the data and networks,
developed and performed the models, analyzed the data, and performed data
visualizations; EL was mainly responsible for the measurement of perception
errors and homophily; and HK contributed to data construction and network
analysis. EL and HK supervised the research. All authors discussed the results
and contributed to writing the manuscript. All authors read and approved the
final manuscript.

### Acknowledgements

YHG acknowledges the Centro de Investigación en Complejidad at Universidad del
Desarrollo, Chile, for the financial support to conduct this research.

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Supplementary Information for
Structural Gender Imbalances
in Ballet Collaboration Networks

Yessica Herrera-Guzmán, Eun Lee, Heetae Kim

*Corresponding Author. Email<EMAIL_ADDRESS>
This file includes:

Supplementary Text

Figures S1 to S7

##  S1 Ballet collaborations

We define a _ballet collaboration_ as the creative effort between
choreographers, composers, costume designers, lighting designers, and other
artists listed by each company, for the creation of a ballet work. In
practice, other types of ballet collaborations are possible, such as those
that put the ballet work on stage and are involved in the development of a
production (e.g. production managers, technicians, theatre staff). Also, some
ballet creations require the direct collaboration of the choreographers with
the ballet dancers. Due to limited access to company data, we only consider
the artistic roles as defined in a ballet collaboration.

Figure S1: Team composition. Panel a shows the probability of a team size in
each ballet company. Panel b shows the probability of a gender ratio in a
team.

Figure S2: Representation of women in ballet collaborations by artistic roles.
Fraction of women written inside purple bar. Group size indicated by bar-width
with total count of artists written on top of bar. Figures show women’s
representation by artistic roles for a ABT, b NYCB, c NBC, and d ROH. While
women face less representation in the Composers group, their representation is
consistently low across artistic roles, considering group size.
Figure S3: Collaboration patterns by gender. Purple/green and dots/triangles
represent women/men. Panel a shows the fraction of teams by the number of same
gender artists in their composition. Panel b shows the number of ballet
creations and the count of artists by gender (artists’ productivity).

##  S2 Centrality differences by gender and null models analyses

Figure S4: Distribution of artists in the collaboration networks from ABT,
NYCB, and NBC.

  1. (a)

Degree distribution.

|
---|---

  1. (b)

Network position of man-man collaborations.

|  |
---|---|---

  1. (c)

Network position of woman-woman collaborations.

|  |
---|---|---

Figure S5: Fraction of women of TCA. Mean of fraction of women, $R_{Women}$
with error bars comparing the empirical networks and the $R_{Women}$ obtained
from null models. Figure S6: $Z$-score of $C(r)_{real}$. Boxplot with error
bars. Color code is indicated in each panel. The red line indicates a
$Z$-score equal to zero, corresponding to the mean obtained from each null
model. Panel a shows the distribution of $Z(C)$ for each centrality by gender
group $\text{TCA}_{Women,Men}$. Upper boxes display the Edge-shuffled model,
lower boxes the Gender-shuffled model. Panel b shows the distribution of
$Z(\Delta C)$, the Z-score of the gender gap in centrality by rank, separated
by null model.

Figure S7: Gender gap in centrality of TCA. Reporting normalized values by
company. Artists matched by their rank computed in independent gender groups
(see Methods).