# Understanding the Signature of Controversial Wikipedia Articles through
Motifs in Editor Revision Networks

James R. Ashford<EMAIL_ADDRESS>School of Computer Science and
Informatics, Cardiff University,
UK  ,  Liam D. Turner<EMAIL_ADDRESS>School of Computer Science and
Informatics, Cardiff University,
UK  ,  Roger M. Whitaker<EMAIL_ADDRESS>School of Computer Science
and Informatics, Cardiff University,
UK  ,  Alun Preece<EMAIL_ADDRESS>Crime and Security Research
Institute, Cardiff University,
UK  ,  Diane Felmlee<EMAIL_ADDRESS>Dept. of Sociology & Criminology,
Pennsylvania State University,
USA  and  Don Towsley<EMAIL_ADDRESS>College of Information and
Computer Sciences, University of Massachusetts Amherst, USA

(2019)

###### Abstract.

Wikipedia serves as a good example of how editors collaborate to form and
maintain an article. The relationship between editors, derived from their
sequence of editing activity, results in a directed network structure called
the revision network, that potentially holds valuable insights into editing
activity. In this paper we create revision networks to assess differences
between controversial and non-controversial articles, as labelled by
Wikipedia. Originating from complex networks, we apply motif analysis, which
determines the under or over-representation of induced sub-structures, in this
case triads of editors. We analyse 21,631 Wikipedia articles in this way, and
use principal component analysis to consider the relationship between their
motif subgraph ratio profiles. Results show that a small number of induced
triads play an important role in characterising relationships between editors,
with controversial articles having a tendency to cluster. This provides useful
insight into editing behaviour and interaction capturing counter-narratives,
without recourse to semantic analysis. It also provides a potentially useful
feature for future prediction of controversial Wikipedia articles.

Network Motifs, Wikipedia, Controversy, Complex Networks

††journalyear: 2019††copyright: iw3c2w3††conference: Companion Proceedings of
the 2019 World Wide Web Conference; May 13–17, 2019; San Francisco, CA,
USA††booktitle: Companion Proceedings of the 2019 World Wide Web Conference
(WWW ’19 Companion), May 13–17, 2019, San Francisco, CA, USA††doi:
10.1145/3308560.3316754††isbn: 978-1-4503-6675-5/19/05

##  1\. Introduction

Wikipedia has become a tremendous platform for crowdsourcing knowledge,
representing a cornerstone of the World Wide Web (Doan et al., 2011). It
allows the ”wisdom of the crowd” (Surowiecki and James, 2005) to potentially
emerge, providing intelligence on a vast range of topics (Brabham, 2008).
However, complex dynamics support the emergence of content, since the
formation of Wikipedia articles involves both human cooperation and human
conflict, based on the extent of convergent and divergent views. Narrative and
counter-narrative frequently jostle for presence in a articles, representing a
source of friction that is seen through editor interaction (Sepehri-Rad and
Barbosa, 2015a) and in the semantics of article content (Rad and Barbosa,
2012a). Wikipedia conveniently provides a list of controversial content that
are labelled by the Wikipedia community
themselves111https://en.wikipedia.org/wiki/Wikipedia:List_of_controversial_issues.

In an age of misinformation (Acemoglu et al., 2010; Mocanu et al., 2015;
Vicario et al., 2016), understanding characteristics of controversial articles
has increased in importance. Because of the controversial nature of some
topics, the narrative in a Wikipedia article may contain misleading
information that stops a neutral consensus emerging. Prior work in this area
has established insights such as the predictability of controversy from editor
behaviour (Rad and Barbosa, 2012a), such as deletions, reversions, and
statistics from the collaboration network, prediction of article quality
taking insights from multiple models (Wu et al., 2012), and interactions
between users, bots, admin and pages (Jurgens and Lu, 2012). There has also
been a number of different types of network developed to assess Wikipedia
articles, including collaboration networks (Brandes et al., 2009) that capture
the positive or negative relationship between editors, edit networks that
capture ”undoing” of edits by a third party (Kittur et al., 2007) and
affiliation networks (Kane and Ransbotham, 2016).

Our focus in this paper is to further understand the relationship between
small groups of editors, as induced by their editing sequences, by using a
revision network. This does not require information on the nature of the
editing undertaken - it simply captures the ordering in which editing occurs
and is therefore a simple metric to infer. Editors are represented by nodes
and a directed edge from node $A$ to $B$ indicates that ” _Editor A edits the
article after Editor B_ ” (see Figure 1). From this, we seek to determine the
extent to which controversial articles exhibit a distinctive signature
relative to those that are deemed non-controversial.

There has already been some consideration of revision networks in the
literature (Iba et al., 2010; Keegan et al., 2012), where more recently the
emphasis (Wu and Cunningham, 2015) has been to combine them with other network
representations. However, given the fundamental nature of revision networks,
it is interesting to question the extent to which they hold sufficient
information to characterise controversial Wikipedia articles. Currently this
is not well-understood, and motivates our work.

Editor AEditor BEditor CEditor D Figure 1. A network generated using the
editor order $A,B,D,A,C,A$, from newest edit to oldest. Each editor is
characterised by a letter $A$ to $D$, and each occurrence in the list marks a
single revision of the article by the corresponding author. An edge is formed
between the current editor and its adjacent neighbour in the sequence, forming
the directed edges $(A,B),(B,D),(D,A),(A,C),(C,A)$.  Figure 2. 13 possible
combinations of connected triads in directed networks.

###  1.1. Hypotheses

We hypothesise that interaction differences between small sub-groups of
Wikipedia editors is sufficient to distinguish between controversial articles
and non-controversial articles. To address this we consider the extent to
which revision networks of Wikipedia articles have different local induced
substructures based on their having controversial classifications. Our
approach is based on techniques from complex networks (Milo et al., 2002,
2004), that have been successful in classifying diverse and complex biological
networks based on their latent induced subgraphs.

To achieve this at scale, and in contrast to previous literature (Sepehri-Rad
and Barbosa, 2015a; Rad and Barbosa, 2012a; Jurgens and Lu, 2012), we assess a
relatively large sample of Wikipedia articles, involving over 21,000 Wikipedia
articles, by determining their subgraph ratio profiles. Each such profile
represents the under and over representation of induced triads in the revision
network of a Wikipedia article using 13 dimensions of connected triads, while
also normalising for differences in network size.

Understanding the relationship between Wikipedia articles is important, as it
allows us to determine the extent to which they may cluster. The sample we are
considering represents 21,631 articles each represented in 13 dimensions.
Therefore we perform dimensionality reduction, and project the subgraph ratio
profiles into lower dimensional spaces. This allows us to examine the
relationship both within and between controversial and non-controversial
articles.

The results draw attention to distinctive clustering patterns concerning
controversial Wikipedia articles. We further assess the results by examining
the correlation with a range of variables, allowing to understand the role of
substructures. The findings reaffirm that the sequence of editing provides an
important mechanism to understand Wikipedia articles, independent of an
article’s topic and without recourse to semantic analysis.

##  2\. Related Work

Understanding the content of crowd-sourced platforms such as Wikipedia and the
behaviour of their contributors is of wide research interest (Surowiecki and
James, 2005). Wikipedia represents a dynamic network of articles with a
structure resembling that of the World Wide Web (Zlatić et al., 2006), whereby
dominant articles act as connectivity hubs. Dynamics also exist within the
formation and maintenance of individual Wikipedia articles, through open and
collaborative editing.

Interactions between editors range from positive to negative, where debates
and arguments lead to different patterns of revision (e.g., (Rad and Barbosa,
2012a; Laniado et al., 2011)), capturing behaviours such as vandalism
(Potthast et al., 2008) and the propagation of disinformation (Kumar et al.,
2016). Characterising articles and contributors through revisions provides a
means for Wikipedia to manage and review its content. This is potentially
labour intensive and has led to interest in creating and exploiting methods to
detect issues (e.g.,(Adler et al., 2011; Sepehri-Rad and Barbosa, 2015b)).

Controversial articles have become an increasing point of focus, and
characterised as such by Wikipedia. Automated methods for classifying articles
have received much attention (e.g., (Rad and Barbosa, 2012b; Wu et al., 2012;
Wu and Cunningham, 2015). The associated revision log for Wikipedia articles
has been shown to provide a basis to examine potential controversy through
examining the collaborative behaviour of individual editors within an article
(Sepehri-Rad and Barbosa, 2015b) or across multiple articles (Wu et al.,
2011). An article’s revision log identifies the structure underlying temporal
interactions (Wu and Cunningham, 2015), and provides insight into how articles
and contributors’ habits may evolve over time (Jurgens and Lu, 2012). Features
from the aggregation of this, such as number of edits, revision, and previous
version restorations have been shown to correlate (e.g., (Sepehri-Rad and
Barbosa, 2015b)).

Treating the revision log as a network between editors (Sepehri-Rad and
Barbosa, 2015a) has been shown to provide additional useful features using
graph theory and social network analysis techniques (e.g., (Rad and Barbosa,
2012b)). This has ranged from global features such as the degree distribution
(e.g., (Sepehri-Rad and Barbosa, 2015b)), through to analysis of local sub-
structures concerning the articles with which editors interact (e.g., (Wu et
al., 2011)).

However, there has been little investigation of controversial articles based
on the under or over representation of local-substructures. Called network
motif analysis, this approach originated from biology (Milo et al., 2004;
Braines et al., 2018), and has been used to good effect in characterising
other complex networks, including technology (e.g., (O’Callaghan et al.,
2012)). In terms of Wikipedia, motif analysis has been used to determine how
articles point to each other (Zlatić et al., 2006) and in assessing
interactions between editors and different Wikipedia articles (Wu et al.,
2011). Our contribution is two-fold: firstly to use motif analysis to
understand the fundamental revision networks, and secondly to consider the
role of motifs in differentiating between controversial and non-controversial
articles.

##  3\. Methods

###  3.1. Dataset

We collected revision history logs and article meta-data of a sample of
Wikipedia articles ($N=21,631$) through Wikipedia’s web
API222https://en.wikipedia.org/w/api.php. The revision logs contain time-
series events and meta-data attributing the revision to a particular user at a
given time. Within this set of articles, a subset ($N=2,661$) are considered
to be ’controversial’ as they are listed in Wikipedia’s ”List of controversial
issues”1. The remaining articles ($N=18,970$) are random articles that are not
contained in the controversial issue list to serve as a basis for comparison.
These were taken from an original sample of 23,000 articles (20,000 non-
controversial and 3,000 controversial), from which articles were removed if
they did not contain sufficient information for motif analysis.

(a) A revision network taken from the non-controversial The Web Conference
Wikipedia article.

(b) A revision network taken from the controversial Brexit Wikipedia article.
The large volume of editors produces a highly dense cluster of interactions.

Figure 3. Revision network of two articles - one non controversial and one
controversial.

###  3.2. Network Construction

For each article, we construct a revision network (in the same manner as
(Keegan et al., 2012; Wu and Cunningham, 2015)) where nodes represent unique
editors and directed edges show that an editor added a revision after another
editor. We traverse the revision log list to build a network that spans the
article’s lifetime, adding nodes and edges as they appear in each event.
Specifically let the revision network of a Wikipedia article be defined by
$G=(V,E)$, where each editor is represented by a node $v\in V$. An edge
$(v_{i},v_{j})\in V$ indicates that editor $v_{i}$ edits the article after
editor $v_{j}$. This excludes self-loops and editor $v_{i}$ editing after
editor $v_{j}$ multiple times does not result in a multi-edge.

A simple example can be found in Figure 1, which describes how the network is
constructed. Two extreme examples from the dataset shown in Figures 3a and 3b.
This form of representation is potentially useful as large articles do not
typically follow a linear or incremental structure. For example, it is highly
likely that users will restore work back to an earlier revision should a
revision become vandalised or irrelevant. Editors are likely to refer back to
previous editor’s work. These behaviours result in complex sequential patterns
that are captured through revision networks.

###  3.3. Network Motif Analysis

Network motif analysis focuses on determining the under or over representation
of induced subgraphs (Milo et al., 2002; Braines et al., 2018), as compared to
an alternative sample of graphs (i.e., a null-model that acts a relevant
baseline for comparison). Dyads, triads and tetrads are often considered as
induced subgraphs such over and under-represented induced subgraphs are called
network motifs or anti-motifs. In this analysis, we examine each article’s
revision network using triads, representing how all possible triples of
editors may sequentially interact. Triads are sufficiently large enough to
capture both direct and indirect reciprocity between editors, while not being
of a scale that is impeded by combinatorial complexity - there are 13 possible
connected triads, shown in Figure 2. The coded names listed in Figure 2 are
provided as part of the convention used in the triad census algorithm
(Batagelj and Mrvar, 2001).

For each article, we calculate the subgraph ratio profile (SRP) as defined by
Milo et al (Milo et al., 2004). This accounts for variations in network size.
This is achieved by determining the relative abundance of each type of triad
compared to random graphs generated by the null-model. For each type of triad
$i$, we firstly calculate $\Delta_{i}$:

(1)
$\Delta_{i}=\frac{(Nreal_{i}-<Nrand_{i}>)}{(Nreal_{i}+<Nrand_{i}>)+\epsilon}$

where $Nreal_{i}$ is the number of such triads observed in the graph under
observation, and $Nrand_{i}$ is the average frequency of such triads seen
across the sample of networks composed by the null model. In this case the
null model uses 100 random graphs with the same number of nodes and edges as
the graph under observation. The value of $\epsilon$ is set to $4$ (Milo et
al., 2004; Tu et al., 2018) to prevent the result from being misleadingly
large when a subgraph rarely appears. This process is repeated for each triad
$i$ and normalised across triads to form the subgraph ratio profile (SRP) for
a given network. The $i^{th}$ SRP, denoted $SRP_{i}$, denotes the extent of
under or over representation of the triad $i$, and is defined by:

(2) $SRP_{i}=\Delta_{i}/(\sum\Delta_{i}^{2})^{1/2}.$

###  3.4. Article comparison and dimensionality reduction

The SRP composed for each article provides a 13-dimensional vector whose
components indicate the extent of triad representation relative to networks in
the null model. To assess these collectively, we use principal component
analysis (PCA) that allows the SRPs to be considered in a lower dimensional
space. We apply three and two dimensional PCA. Finally, we make comparison
with a number of external variables (number of editors/nodes, age of article,
and revision rate) to understand potential correlations with motifs.

##  4\. Results

###  4.1. Subgraph Ratio Profiles

We first examine the SRPs that arise from both controversial and non-
controversial articles in isolation (Figures 4 and 5 respectively). To find
significant triads, we use a cutoff value of +0.3 and -0.3. We determine that
controversial articles are strongly represented by triads `111D`, `111U` and
`201`, which attain average SRP scores of 0.382, 0.375, 0.372, with relatively
low dispersion (SDs of 0.136, 0.149 and 0.124 respectively). Interestingly,
together these represent a chain of three nodes, where one edge is
reciprocated, with the other edge covering all possible directional types
(i.e., reciprocated, directed in, directed out).

In contrast, the results for the non-controversial articles provide a
different profile. Here `021D` and `021U` are significantly under represented
(average SRP scores of -0.511, -0.485), albeit with higher standard deviations
present (SDs of 0.192, 0.2). Interestingly these anti-motifs (`021D` and
`021U`) relate to a lack of subgraphs where directed edges either emanate from
or are received by a single node in the triad. Such configurations relate to
the role of a mediating editor that may be presented with or respond to the
editing of others. In other words, such mediators have a reduced role in non-
controversial articles.

Figure 4. Subgraph ratio profiles of all controversial articles. Average
displayed in red. Figure 5. Subgraph ratio profiles of all non-controversial
articles. Average displayed in red. Figure 6. Subgraph ratio profiles of all
articles, with controversial articles displayed in red and non-controversial
in black.

The comparison between these subgraph ratio profiles is shown in Figure 6.
These profiles are quite distinct. We also calculate the Pearson correlation
coefficient for each distinct pair of articles in three groups - controversial
articles, non-controversial articles and all articles. Controversial articles
provide the greatest correlation to each other (M=$0.41193$, SD=$0.41262$).
Non-controversial articles have a lower mean correlation (M=$0.37498$,
SD=$0.36749$) which is similar to the result when considering all articles
together (M=$0.37569$, SD=$0.36844$).

###  4.2. Principal Component Analysis

We analyse the 21,631 subgraph ratio profiles to determine the relationship in
terms of relative clustering. We apply PCA in order to reduce the 13
dimensions of the SRPs down to a more manageable form. We initially project
the SRPs into 3-dimensional space for clarity, as seen in Figure 7. This
presents a distinctive region where controversial articles are dominant. This
provides evidence for a distinction between the controversial and non-
controversial articles, consistent with the variation in motifs identified in
the previous section.

Figure 7. PCA scatter plot repression of the 13-point feature vector in 3D
clustering space.

The PCA coefficients (Table 1) that define the three dimensions reveals that
principal component one ($x$ axis in Figure 7) primarily depends on triads
`111D` and `111U`. Principal component two ($y$ axis in Figure 7) primarily
depends on triad `021C`. The third principal component primarily depends on
triads `021C` and `030C`. However, we also note that when represented in the
three dimensional space, the revision networks have limited dispersion in the
third dimension (i.e., vertical dimension as plotted).

Table 1. PCA coefficients displaying the strongest triads | 021D | 021U | 021C | 111D | 111U | 030T | 030C | 201 | 120D | 120U | 120C | 210 | 300
---|---|---|---|---|---|---|---|---|---|---|---|---|---
PC-1 | 0.332 | 0.321 | -0.126 | 0.509 | 0.522 | 0.145 | 0.205 | 0.318 | 0.078 | 0.079 | 0.204 | 0.117 | 0.065
PC-2 | 0.427 | 0.438 | 0.643 | -0.278 | -0.242 | 0.207 | 0.104 | -0.052 | 0.059 | 0.06 | 0.047 | 0.074 | 0.06
PC-3 | 0.074 | 0.127 | -0.609 | -0.398 | -0.332 | 0.136 | 0.441 | 0.115 | 0.131 | 0.130 | 0.19 | 0.167 | 0.112

Calculating the percentage of explained variance by principal component
confirms that the first principal component produces 53.7% of the shared
variance, the second produces 22.3% and the third produces the least with
6.7%. This confirms that the third principal component provides a limited
contribution to representation of the total variance across the significant
ratio profiles. This supports representation through two principal components,
as plotted in Figure 8, with the relative composition of each principal
component being near identical to PC-1 and PC-2 in Table 1. As anticipated,
this is similar dependency on the first and second primary components in the
three dimensional representation.

Representation in two dimensions further clarifies the distinction between
controversial and non-controversial revision networks. In particular, from
Figure 8 we note that both classes of article exhibit a similar maximum and
minimum range against principal component two, which is primarily defined by
the linear path between three nodes (`021C`). However, it is the variation in
the first principal component, dominated by `111D` and `111U`, which represent
linear paths with reciprocation on one edge, that differentiate the non-
controversial from controversial. High values in principal component one
correlate with controversial articles - in other words, controversial articles
exhibit more reciprocation on top of possible linear paths.

Figure 8. PCA scatter plot repression of the 13-point feature vectors in 2D
clustering space.

###  4.3. Additional Metrics

We examine the relationship between revision networks and primary external
variables (number of editors/nodes, age of article, and revision rate) using
motifs. Specifically, using the dimensions of two-dimensional PCA analysis, we
examine the correlation with the external variables, and how this differs
between controversial and non-controversial articles. The results are shown in
Figures 9, 10 and 11.

The greatest differences between controversial and non-controversial articles
occur with respect to article age (Figure 10). Here, controversial articles
with high age cluster against high values of principal component one, and to
some extent this occurs for principal component two. This contrasts against
the clustering seen for non-controversial articles.

Figure 10 also shows that controversial articles have a tendency to be older.
The high values in principal component one and two which align with dense
clustering of controversial articles show that while such articles accumulate
the linear revision path between authors (`021C` which dominates principal
component two), controversial articles also accumulate instances of linear
paths where one edge is reciprocated (i.e., `111D` and `111U` which dominate
principal component one).

Figure 9. Scatter plot of each principal component combined with node count
for both non-controversial and controversial articles. Figure 10. Scatter plot
of each principal component combined with article age for both non-
controversial and controversial articles. Figure 11. Scatter plot of each
principal component combined with edit rate (mean number of edits per month)
for both non-controversial and controversial articles.

##  5\. Discussion

Motif analysis of revision networks gives insight into how the temporal
editing relationship between small groups of Wikipedia authors create
signatures that allow controversial articles to be distinguished. In contrast
to previous work, we have investigated this using a relatively large sample of
Wikipedia articles, where distinct patterns emerge. This provides strong
support for our hypothesis, and reaffirms the importance of the revision
network as a simple but fundamental element in editing Wikipedia.

Through motif analysis, we identify that reciprocation on linear paths among
triads of editors in the revision network is over represented in controversial
Wikipedia articles. These motifs are defined by the triads `111D`, `111U` and
`201`. In contrast, the revision networks from non-controversial articles
exhibit two anti-motifs, involving the under representation of triads
involving two directed edges either arriving at or emanating from a mediating
node (triads `021D` and `021U`). These motifs and the underlying subgraph
ratio profiles represent an unusual and distinctive profile that we believe
represent distinctive ”super-families” beyond those seen in other
technologically related networks, such as the world wide web (Zlatić et al.,
2006).

Performing dimensionality reduction upon the subgraph ratio profiles from each
revision network allows us to further understand the relationships between
Wikipedia articles. Our analysis shows that the structure of the data is
amenable to reduction to two dimensions, where the principal components are
dominated by triads `111D` and `111U` in the first component, and mainly
`021C` in the second component, but with lesser contributions from triads
`021D` and `021U`.

The results from two-dimensional principal analysis are illuminating - the
dominant triads in both components, as listed above, correspond to linear
paths, i.e., open triads which represent sequences of editing without indirect
reciprocity. The extent and format of reciprocated (i.e., bidirectional) edges
on these open triads is sufficient to define the two principal components. The
dominant triads in the first principal component each involve reciprocation on
one edge, where as interestingly, in the second principal component, the
dominant triads are open triads with no reciprocated edges. From this we
deduce that short paths, rather than short loops of editing that represent
indirect reciprocity, are important features in characterising Wikipedia
revision networks.

We also observe through two-dimensional principal component analysis that it
is the first principal component that strongly distinguishes between the
revision networks of controversial and non-controversial articles. The
dominant triads defining this capture the extent of direct reciprocation being
present. Finally, through consideration of the principal components against
additional external variables, we find in particular that article age plays a
role in distinguishing the controversial articles. High values in both
principal components aligns with strongest clustering of controversial
articles, which is not the case for non-controversial articles.

##  6\. Conclusion

The analysis has given insights into the structure underlying revision
networks from Wikipedia articles, and has shown that a relatively small number
of features, in terms of substructures in revision networks, characterise
controversial Wikipedia articles. The results have identified key clusters of
editorial interactions to this effect, in support of the hypothesis. These are
distinctive and indicate that the revision networks for controversial and non-
controversial Wikipedia articles have differentiated subgraph ratio profiles.
Our study gives understanding as to how prediction or classification of
articles can be enhanced using the latent structures relating to editor
behaviour. This also reaffirms the importance of the revision network as a
simple but useful representation for assessment of Wikipedia articles.

## Acknowledgements

This research was sponsored by the U.S. Army Research Laboratory and the U.K.
Ministry of Defence under Agreement Number W911NF-16-3-0001. The views and
conclusions contained in this document are those of the authors and should not
be interpreted as representing the official policies, either expressed or
implied, of the U.S. Army Research Laboratory, the U.S. Government, the U.K.
Ministry of Defence or the U.K. Government. The U.S. and U.K. Governments are
authorized to reproduce and distribute reprints for Government purposes
notwithstanding any copyright notation hereon.

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# Complexity of universal access structures

László Csirmaz This research was partially supported by the “Lendület Program”
of the Hungarian Academy of Sciences and by the grant NKTH OM-00289/2008

(Central European University, Budapest
Rényi Institute, Budapest)

###### Abstract

An important parameter in a secret sharing scheme is the number of minimal
qualified sets. Given this number, the universal access structure is the
richest possible structure, namely the one in which there are one or more
participants in every possible Boolean combination of the minimal qualified
sets. Every access structure is a substructure of the universal structure for
the same number of minimal qualified subsets, thus universal access structures
have the highest complexity given the number of minimal qualified sets. We
show that the complexity of the universal structure with $n$ minimal qualified
sets is between $n/\log_{2}n$ and $n/2.7182\dots$ asymptotically.

Keywords: secret sharing; complexity; entropy method; harmonic series.

MSC numbers: 94A62, 90C25, 05B35.

##  1 Introduction

In a secret sharing scheme the access structure defines which subsets of the
participants should recover the secret – these are the qualified subsets.
Unqualified subsets are also called independent. The collection of qualified
subsets is determined uniquely by the minimal qualified sets. Suppose in an
access structure we have $n$ minimal qualified sets: $A_{1}$, $\dots$,
$A_{n}$. For $\varepsilon_{i}=0$ or $1$ let $A_{i}^{\varepsilon_{i}}$ be
$A_{i}$ when $\varepsilon_{i}=1$, and the complement of $A_{i}$ when
$\varepsilon_{i}=0$. The access structure is universal if none of the $2^{n}$
possible intersections $\bigcap A_{i}^{\varepsilon_{i}}$ is empty. In this
note $\mathcal{U}_{n}$ denotes any universal access structure with $n$
qualified subsets.

Universal structures with at most three minimal qualified sets were
investigated in [4], and for four qualified sets in [5]. For $n=2$ and $n=3$
the exact complexity is known: it is $1$ and $3/2$, respectively. For $n=4$ it
is proved in [5] that the complexity of $\mathcal{U}_{4}$ is between $7/4$ and
$11/6$, the exact value is not known. This paper provides the first bounds for
the complexity of $\mathcal{U}_{n}$ for larger values of $n$. The upper bound
follows from a secret sharing scheme defined recursively for each $n$. The
lower bound is a well-tailored application of the independent sequence method
described, e.g., in [1].

##  2 Reduction

In this section we show that all universal structures with $n$ minimal
qualified subsets have the same complexity, and pinpoint a particular
structure which will be investigated later in details. Result of this section
were also proved in [5].

First we note that the number of participants in each class $\bigcap
A_{i}^{\varepsilon_{i}}$ is irrelevant: the complexity of the scheme does not
depend on the number of participants in the classes, it depends only on which
classes are empty and which are not. Consequently, for each $n$, we can define
a unique “general” universal access structure with $n$ minimal qualified sets.
As a first step, we can evidently discard all participants who are not in any
of the qualified sets as they can play no role in recovering the secret. Less
trivially we can also discard those participants which are in all of the
qualified sets. Details follow.

Let $\mathcal{U}^{*}_{n}$ be the normalized structure with $n$ minimal
qualified sets $A_{1}$, $\dots$, $A_{n}$, where the intersection $\bigcap
A_{i}^{\varepsilon_{i}}$ has exactly one element when not all the
$\varepsilon_{i}$ are equal, and is empty otherwise. In particular, the
structure $\mathcal{U}^{*}_{n}$ is connected, no participant is in all, or
none, of the minimal qualified subsets, and has exactly $2^{n}-2$
participants.

###### Claim 1

The complexity of the structures $\mathcal{U}_{n}$ and $\mathcal{U}^{*}_{n}$
are equal: $\sigma(\mathcal{U}_{n})=\sigma(\mathcal{U}^{*}_{n})$

Proof Let $\mathcal{S}$ be any scheme realizing $\mathcal{U}_{n}$. Pick one
participant from each class $\bigcap A_{i}^{\varepsilon_{i}}$ except when all
the $\varepsilon_{i}$’s are equal, $2^{n}-2$ participants in total. Make the
shares of all other participants public. This defines a scheme
$\mathcal{S}^{*}$ realizing the access structure $\mathcal{U}^{*}_{n}$, and it
is clear that the complexity of $\mathcal{S}^{*}$ is less than or equal to the
complexity of $\mathcal{S}$. This establishes
$\sigma(\mathcal{U}^{*}_{n})\leq\sigma(\mathcal{U}_{n})$.

To see the other direction, let $\mathcal{S}^{*}$ be a scheme realizing
$\mathcal{U}^{*}_{n}$. For the sake of simplicity we assume that the secret
for $\mathcal{S}^{*}$ is a single random bit. Define the scheme $\mathcal{S}$
for $\mathcal{U}_{n}$ as follows. Write the random bit $s$ as the mod 2 sum of
two random independent bits $r_{1}$ and $s_{1}$:

$s=r_{1}\oplus s_{1}.$

The secret for $\mathcal{S}$ will be $s$. Use any perfect ideal $k$ out of $k$
threshold scheme to distribute $r_{1}$ among participants in the intersection
$\bigcap A_{i}$. That is, all participants from this set together can
reconstruct $r_{1}$, but no proper subset of them (or any any other subset not
containing all participants from $\bigcap A_{i}$) has any information on
$r_{1}$. Next, distribute $s_{1}$ according to the scheme $\mathcal{S}^{*}$.
Participants in the intersection $\bigcap A_{i}^{\varepsilon_{i}}$ (not all
$\varepsilon_{i}$ are equal) will be able to recover the share what
$\mathcal{S}^{*}$ assigns to them using again an independent realization of a
perfect ideal $k$ out of $k$ threshold scheme (different $k$, of course). It
is clear that qualified subsets of $\mathcal{U}_{n}$ can recover the secret
$s$, while unqualified subsets have no information on it. The complexity of
$\mathcal{S}$ is $\max\\{1,\sigma(\mathcal{S}^{*})\\}$ (here $1$ comes from
distributing $r_{1}$) as threshold schemes have complexity $1$. This
establishes $\sigma(\mathcal{U}_{n})\leq\sigma(\mathcal{U}^{*}_{n})$ as
required. $\square$

###### Claim 2

$\sigma(\mathcal{U}_{2})=1$.

Proof In $\mathcal{U}^{*}_{2}$ there are $2^{2}-2=2$ participants, and both of
them form a one-element qualified set. Simply give the secret to both of them.
$\square$

##  3 Upper bound

In this section we define a perfect scheme realizing $\mathcal{U}^{*}_{n}$ by
recursion on $n$. Using this scheme we establish an upper bound for the
complexity of the universal scheme. We call participants $p$ and $q$ of
$\mathcal{U}^{*}_{n}$ equivalent if there is a permutation $\pi$ on the full
set of participants which sends $p$ to $q$ and maps all qualified subsets into
qualified subsets, i.e., $\pi$ is an automorphism of the structure
$\mathcal{U}^{*}_{n}$. It is easy to see that $p$ and $q$ are equivalent if
and only if both of them are members of exactly the same number of minimal
qualified sets. Thus there are $n-1$ equivalence classes, one for each
$i=1,\dots,n-1$.

Suppose $\mathcal{S}_{n}$ realizes $\mathcal{U}^{*}_{n}$. Using the standard
symmetrization technique we may assume that $\mathcal{S}_{n}$ is fully
symmetrical, that is, equivalent participants receive shares of the same size.
We denote by $f_{n}(i)$ this common size of the shares of those who are in
exactly $i$ qualified subsets divided by the size of the secret. The
complexity of the scheme $\mathcal{S}_{n}$ is then

$\sigma(\mathcal{S}_{n})=\max\\{f_{n}(i)\,:\,1\leq i<n\\}.$

Let us also define $f_{n}(0)=0$ and $f_{n}(n)=1$.

###### Lemma 3

Suppose $\mathcal{S}_{n}$ realizes $\mathcal{U}^{*}_{n}$ with $f_{n}(i)$ as
defined above. Then there is a perfect scheme $\mathcal{S}_{n+1}$ realizing
$\mathcal{U}^{*}_{n+1}$ such that

$n\cdot f_{n+1}(i)=(n+1-i)\cdot f_{n}(i)+i\cdot f_{n}(i-1)~{}~{}\mbox{ for all
}~{}~{}0<i\leq n.$ (1)

Proof We will use Stinson’s decomposition technique from [7]. Given
$U^{*}_{n+1}$ define $n+1$ access structures $\Gamma_{1},\dots,\Gamma_{n+1}$
such that the set of participants is the same, while in $\Gamma_{j}$ the
$j$-th minimal qualified subset is left out. By the assumption and by Claim 1
above there are perfect schemes realizing $\Gamma_{j}$ such that participants
in $i$ out of the $n$ qualified subsets of $\Gamma_{j}$ receive
$f_{n}(i)$-size shares for all $0\leq i\leq n$.

Now use all of these $n+1$ schemes simultaneously. Each (minimal) qualified
subset of $U^{*}_{n+1}$ recovers exactly $n$ out of the $n+1$ distributed
secrets, while an unqualified subset has no information on any of those
secrets. Using Stinson’s trick from [7] we can define a secret of size $n$ and
$n+1$ shadows of size $1$ each so that the secret can be determined by any $n$
of the shadows. Thus the composite scheme will distribute a secret of size
$n$. A participant who is in exactly $i$ of the minimal qualified subsets will
receive shares of size $f_{n}(i-1)$ from $\Gamma_{j}$ when $j$ is one of the
minimal qualified subsets he is in ($i$ in total), and shares of size
$f_{n}(i)$ from $\Gamma_{j}$ when $j$ is one of the minimal qualified subsets
he is not in ($n+1-i$ in total). This establishes the claim of the lemma.
$\square$

###### Lemma 4

Suppose $f_{n}(0)=0$, $f_{n}(n)=1$, $f_{2}(1)=1$ and for $n\geq 2$ the
function $f_{n+1}$ satisfies the recursion in (1). Then

$f_{n}(i)=(n-i)\big{(}h(n)-h(n-i)\big{)}~{}~{}\mbox{ for all }~{}~{}0\leq
i<n,$ (2)

where $h(1)=0$ and $h(n)=\sum_{0<j<n}1/j$ otherwise.

Proof When $i=0$ then, by definition, $f_{n}(0)=0$, and (2) also yields $0$.
(2) also gives

$f_{2}(1)=h(2)-h(1)=1-0=1.$

Suppose (2) holds for $n\geq 2$ and $0<i<n$. Equation (1) can be rewritten as

$\frac{f_{n+1}(i)}{n+1-i}=\frac{n-i}{n}\cdot\frac{f_{n}(i)}{n-i}+\frac{i}{n}\cdot\frac{f_{n}(i-1)}{n+1-i}$
(3)

Now $h(n+1)=h(n)+1/n$, and $h(n+1-i)=h(n-i)+1/(n-i)$. Consequently

$\begin{array}[]{r@{\;}c@{\;}l}\displaystyle h(n+1)&=&\displaystyle
h(n)+\frac{1}{n}=\frac{i}{n}\cdot h(n)+\frac{n-i}{n}\cdot
h(n)+\frac{1}{n},\\\\[8.0pt] \displaystyle
h(n+1-i)&=&\displaystyle\frac{i}{n}\cdot
h(n+1-i)+\frac{n-i}{n}\cdot\big{(}h(n-i)+\frac{1}{n-i}\big{)},\end{array}$

that is

$h(n+1)-h(n+1-i)=\frac{n-i}{n}\big{(}h(n)-h(n-i)\big{)}+\frac{i}{n}\big{(}h(n)-h(n+1-i)\big{)}.$

This shows that the function $h(n)-h(n-i)$ satisfies the recursion (3), that
is $f_{n}(i)=(n-i)\big{(}h(n)-h(n-i)\big{)}$ for all $n$ and $i$ by induction.
$\square$

###### Theorem 5

The complexity of $\mathcal{U}_{n}$ is asymptotically at most $n/e$ where
$e=2.7182\dots$ is the Euler number.

Proof By Lemma 3 there is a scheme $\mathcal{S}_{n}$ realizing
$\mathcal{U}_{n}$ with complexity

$\sigma(\mathcal{S}_{n})=\max\\{f_{n}(i)\,:\,\ 1\leq i<n\\}$

where, by Lemma 4 $f_{n}(i)=(n-i)\big{(}h(n)-h(n-i)\big{)}$. Now

$f_{n}(i+1)-f_{n}(i)=h(n-i)+1-h(n).$ (4)

As $h(n-i)$ is strictly decreasing, $f_{n}$ increases while $h(n-i)\geq
h(n)-1$, and decreases after this value of $i$, and takes its maximum when (4)
changes sign. Using the approximation $h(x)=\gamma+\log(x-1/2)+O(x^{-2})$ from
[6] where $\gamma$ is the Euler-Mascheroni constant, this happens when

$n-i=\frac{n}{e}+\frac{e-1}{2e}+O(1/n),$

and then the maximal value of $f_{n}$ taken at this $i$ is

$f_{n}(i)=(n-i)\big{(}1+O(1/n)\big{)}=\frac{n}{e}+\frac{e-1}{2e}+O(1/n).$

From here the theorem follows. $\square$

A more careful analysis shows that

$\max_{i}f_{n}(i)<\frac{n}{e}+\frac{e-1}{2e}+\frac{1}{3n}<\frac{n}{e}+0.5$

for all $n\geq 2$ which gives a non-asymptotic estimate on the complexity of
$\mathcal{S}_{n}$.

##  4 Lower bound

As mentioned in the Introduction, for the lower bound we use a variation of
the independent sequence method from [1]. For a general description how the
method works, see, e.g., [1, 2]. Briefly, for any subset $A$ of the
participant one considers the relative entropy $f(A)$, which is simply the
entropy of all the shares given to members of $A$ divided by the entropy of
the secret. The function $f$ satisfies a certain set of linear inequalities
derived from the Shannon inequalities for the entropy function. For example,
$f(\emptyset)=0$, for arbitrary subsets $A$ and $B$ of the participants

$f(AB)\leq f(A)+f(B),$

or, $f(B)\leq f(A)$ when $B\subseteq A$. This latter inequality is called
monotonicity. The strict monotonicity property says that whenever $B$ is
independent and $B\subseteq A$ is qualified then not only $f(A)$ is larger
than $f(B)$, but the difference is at least one: $f(B)+1\leq f(A)$. As a
particular case,

$f(A)\leq f(a_{1})+f(a_{2})+\cdots+f(a_{k})$ (5)

when $A=\\{a_{1},\dots,\allowbreak a_{k}\\}$. Here and the in the sequel, as
usual, we write $f(a)$ instead of $f(\\{a\\})$, and write $AB$ and $Aa$
instead of $A\cup B$ and $A\cup\\{a\\}$, respectively.

The entropy method works as follows. To prove that the complexity of an access
structure is at least $\kappa$ it is enough to show that for all non-negative
functions $f$ satisfying all of the above inequalities there is a participant
$p$ with $f(p)\geq\kappa$.

The independent sequence method relies on the following lemma. For the sake of
self-containment we supply a proof.

###### Lemma 6

Suppose $A$ is qualified, $B\subseteq A$, $p$ is a participant not in $A$, and
for some $B\subseteq C\subseteq A$, $C$ is independent, while
$Cp=C\cup\\{p\\}$ is qualified. Then $f(A)-f(B)\geq\allowbreak f(Ap)-f(Bp)+1$.

Proof As the function $f$ is the relative entropy, it is submodular, i.e.
$f(X)+f(Y)\geq f(X\cup Y)+f(X\cap Y)$ for any two subsets $X$ and $Y$. In
fact, this is the same inequality which says that the mutual conditional
information is non-negative, see [3]. If $s$ denotes the secret, then
$f(Xs)=f(X)$ if $X$ is qualified, as $X$ can determine the secret, furthermore
$f(Xs)=f(X)+1$ if $X$ is unqualified, expressing the fact that the information
members of $X$ have is independent of the secret.

By the assumptions of the lemma, $A$ and $Cp$ are qualified, while $C$ is not,
thus by the submodularity

$f(A)+f(Cp)=f(As)+f(Cps)\geq f(Aps)+f(Cs)=f(Ap)+(f(C)+1),$ (6)

as $C\subseteq A$. Using submodularity for the sets $C$ and $Bp$ yields

$f(C)+b(Bp)\geq f(Cp)+f(B),$ (7)

here we used that $B\subseteq C$. The sum of (6) and (7) gives claim of the
lemma immediately. $\square$

The next lemma summarizes the method itself.

######  Lemma 7 (Independent sequence method)

Suppose $A_{0}$ is a qualified set, $B_{0}$ is the empty set, and $b_{1}$,
$\dots$, $b_{n}$ are participants not in $A_{0}$. Let
$B_{i}=\\{b_{1},\allowbreak\dots,b_{i}\\}$, and $A_{i}=A_{0}\cup B_{i}$.
Suppose that for all $0\leq i<n$ there is a subset $C_{i}\subset A_{0}$ such
that $B_{i}C_{i}$ is independent, while $B_{i+1}C_{i}=b_{i+1}B_{i}C_{i}$ is
qualified. Then $f(A_{0})\geq n$.

In the usual terminology the sequence $A_{0}$, $A_{1}$, $\dots$ is made
independent by the sequence $B_{0}$, $B_{1}$, $\dots$ from where the method
got its name.

Proof By the monotonicity of the function $f$, $f(A_{n})-f(B_{n})\geq 0$ as
$A_{n}\supseteq B_{n}$. By Lemma 6,

$f(A_{i})-f(B_{i})\geq f(A_{i+1})-f(B_{i+1})+1$

which is shown by the set $C=B_{i}C_{i}$ and the participant $p=b_{i+1}$.
Putting all of these inequalities together we get

$f(A_{0})-f(B_{0})\geq n.$

As $B_{0}$ is the emptyset, $f(B_{0})=0$ and the claim of the lemma follows.
$\square$

As it was remarked in the Introduction, among all access structures with $n$
minimal qualified subsets, $\mathcal{U}_{n}$ has the highest complexity.
Consequently to show that $\mathcal{U}_{n}$ has complexity at least $\kappa$
it is enough to find any access structure with $n$ minimal qualified subsets
with complexity ${}\geq\kappa$. This is exactly what we will do.

###### Theorem 8

For each $n\geq 2$ there is an access structure with $n$ minimal qualified
subsets and complexity at least $n/(1+\log_{2}n)$.

Proof Let $k\geq 1$ to be chosen later and $X$ be a set with exactly $k$
elements. Enumerate the proper subsets of $X$ in decreasing order as $C_{0}$,
$C_{1}$, $\dots$, $C_{\ell-1}=\emptyset$ where $\ell=2^{k}-1$. “Decreasing”
means that sets with larger index does not have more elements; in particular
$C_{i}\not\subset C_{j}$ whenever $0\leq i<j<\ell$.

Let $b_{1}$, $\dots$, $b_{\ell}$ be new participants not in $X$, and define
$B_{0}=\emptyset$, and for $1\leq i\leq\ell$ let $B_{i}$ be the set
$\\{b_{1},\allowbreak b_{2},\dots,b_{i}\\}$. There will be $n\leq\ell$ minimal
qualified subsets: $B_{1}C_{0}$, $B_{2}C_{1}$, $\dots$, $B_{n}C_{n-1}$. These
sets can indeed form a collection of minimal qualified subsets as none of them
is a subset of any other. Also we remark that the sets $B_{i}C_{i}$ for $0\leq
i<n$ are independent as none of $B_{j+1}C_{j}$ is a subset of $B_{i}C_{i}$.
Indeed, if $j\geq i$ then $B_{j+1}$ is not a subset of $B_{i}$, and if $j<i$
then $C_{j}$ is not a subset of $C_{i}$. Thus we can apply Lemma 7 with
$A_{0}=X$ giving

$f(X)\geq n.$

As $X$ has $k$ elements, say $a_{1}$, $\dots$, $a_{k}$, applying property (5)
we get

$f(a_{1})+\cdots+f(a_{k})\geq f(X)\geq n,$

meaning that some participant in this access structure has an $f$-value at
least $n/k$. Thus the complexity of the structure is also at least $n/k$. Our
aim is to choose $k$ as small as possible. We have only a single restriction,
namely that $n$ should not exceed the value $\ell=2^{k}-1$. Thus there is
always an appropriate $k$ which is less than or equal to $(1+\log_{2}n)$,
proving the theorem. $\square$

As a consequence of Theorem 8 the universal structure $\mathcal{U}_{n}$ also
has complexity at least $n/(1+\log_{2}n)$, proving that the complexity is at
least $n/\log_{2}n$ asymptotically.

## Acknowledgment

The author would like to thank one of the referees for pointing out an
omission in Lemma 6, and for the suggestions which helped to improve the
paper.

## References

  * [1] C. Blundo, A. De Santis, R. De Simone, U. Vaccaro:  Tight Bounds on the Information Rate of Secret Sharing Schemes,  Designs, Codes and Cryptography, vol 11(2) (1997), pp. 107–110
  * [2] L. Csirmaz:  An impossibility result on graph secret sharing,  Designs, Codes and Cryptography, vol 53 (2009), pp 195–209
  * [3] I. Csiszár and J. Körner:  Information Theory. Coding Theorems for Discrete Memoryless Systems,  Academic Press, New York, 1981.
  * [4] J. Martí-Farré, C. Padró.  Secret sharing schemes with three or four minimal qualified subsets.  Designs, Codes and Cryptography 34 (2005) 17-34.
  * [5] J. Martí-Farré, C. Padró, L. Vázquez.  Optimal Complexity of Secret Sharing Schemes with Four Minimal Qualified Subsets.  Designs, Codes and Cryptography, vol 61 (2011), pp 167–186
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PASA 2024

# Bridging the Ultraviolet and Optical Regions: Transformation Equations
between GALEX and UBV Photometric Systems

S. Bilir1<EMAIL_ADDRESS>N. Alan1 S. Tunçel Güçtekin1 M. Çelebi2 T.
Yontan2 O. Plevne2 S. Ak1 T. Ak1 S. Karaali1
1Istanbul University, Faculty of Science, Department of Astronomy and Space
Sciences, 34119, Beyazıt, Istanbul, Turkey  2Istanbul University, Institute of
Graduate Studies in Science, Programme of Astronomy and Space Sciences, 34116,
Beyazıt, Istanbul, Turkey

###### Abstract

We derive transformation equations between GALEX and UBV colours by using the
reliable data of 556 stars. We present two sets of equations; as a function of
(only) luminosity class, and as a function of both luminosity class and
metallicity. The metallicities are provided from the literature, while the
luminosity classes are determined by using the PARSEC mass tracks in this
study. Small colour residuals and high squared correlation coefficients
promise accurate derived colours. The application of the transformation
equations to 70 stars with reliable data shows that the metallicity plays an
important role in estimation of more accurate colours.

###### doi:

10.1017/pas.2024.xxx

###### keywords:

techniques: photometric - catalogue - surveys

##  1 Introduction

Reliable spectroscopic, photometric and astrometric data are important for
understanding the structure, formation and evolution of our Galaxy. Today, sky
surveys are systematically carried out in a wide range of the electromagnetic
spectrum from X-rays to the radio. In some sky surveys such as ROSAT (Snowden,
1995), GALEX (Martin et al., 2005), and SDSS (York et al., 2000) which are
carried out between the X-ray and optical regions of the electromagnetic
spectrum, the interstellar absorption prevents to obtain accurate magnitude
and colours. However, this problem has been overcame by means of the sky
surveys which are defined over the infrared region of the star spectrum, i.e.
2MASS (Skrutskie et al., 2006), UKIDSS (Hewett et al., 1996), VVV (Minniti et
al., 2010), VISTA (Cross et al., 2012), WISE (Wright et al., 2010), and AKARI
(Murakami et al., 2007). Thus, considerable information have been obtained,
especially on the bulge, bar structure and Galactic stellar warp in our Galaxy
(Dwek et al., 1995; Lopez-Corredoira et al., 2005, 2019a, 2019b; Benjamin et
al., 2005).

Today, photometric sky survey observations are systematically performed to
cover the ultraviolet (UV) and infrared regions of the electromagnetic
spectrum. In spectroscopic sky surveys, spectroscopic observations are made
for a limited number of objects with different luminosities which are
classified according to their positions in colour spaces obtained from the
photometric observations. The main-sequence stars provide information about
the solar neighbourhood and the evolved stars provide information about the
old thin disc, thick disc and halo populations of our Galaxy beyond the solar
vicinity. The number of stars observed on current spectroscopic sky survey
programs do not exceed one million in total. As this number is less than
needed for a detailed study of the Galactic structure, precise measurements of
photometric sky surveys, which contain billions of bright and faint objects,
are still important in testing models about the structure and evolution of the
Galaxy.

Figure 1: Normalized transmission curves of the GALEX $FUV$, $NUV$ and
Johnson-Morgan $U$, $B$, $V$ filters.

Photometric sky surveys performed in different regions of the electromagnetic
spectrum are designed to include shallow or deep magnitudes, according to the
purpose of the researchers. However, transformations between different
photometric systems can be used as a tool to combine shallow and deep
magnitudes. These transformation equations can also be produced in terms of
luminosity and metallicity. In the literature, transformation equations are
given for main-sequence stars (Smith et al., 2002; Karaali et al., 2005; Bilir
et al., 2005, 2008a, 2011; Rodgers et al., 2006; Jordi et al., 2006; Covey et
al., 2007; Chonis & Gaskell, 2008) and evolved stars (Straizys & Lazauskaite,
2009; Yaz et al., 2010; Bilir et al., 2012, 2013; Ak et al., 2014).
Transformation equations between the photometric systems have been obtained
for approximately 20 years and used effectively to investigate the structure
and evolution of our Galaxy. UBV is one of the important photometric system in
the optical region (Johnson & Morgan, 1953) which is used to determine the
photometric metallicities of the stars and the interstellar absorption. The
$U-B$ colour index plays an important role in these determinations, while its
combination with the colour index $B-V$ can be used in determination of the
reddening of stars. In the UBV photometric system, the colour excess $E(B-V)$
of a star can be determined by the $Q$ method or by shifting its observed
$B-V$ colour index along the reddening vector in the $U-B\times B-V$ two-
colour diagram up to the intrinsic colour index $(B-V)_{0}$ (Johnson & Morgan,
1953). While the colour excess $E(U-B)$ can be calculated by the equation of
the reddening line, i.e. $E(U-B)=0.72\times E(B-V)+0.05\times E(B-V)^{2}$.
Roman (1955) discovered that stars with weak metal lines have larger
ultraviolet (UV) excesses than the ones with strong metal lines. Schwarzschild
et al. (1955), Sandage & Eggen (1959), and Wallerstein (1962) confirmed the
work of Roman (1955). Thus, metal-rich and metal-poor stars could be
classified not only spectroscopically, but also photometrically, i.e. by their
UV excesses. Sandage (1969) noticed in a $(U-B)_{0}\times(B-V)_{0}$ two-colour
diagram of a set of stars in solar neighbourhood that stars with the same
metallicity have a maximum UV excess at the colour index $(B-V)_{0}=0.6$ mag,
and introduced a procedure to reduce the UV excesses of the stars to the one
of colour index $(B-V)_{0}=0.6$ mag. The relation between the UV excess and
metallicities of stars have been used by the researchers for their metallicity
estimation via photometry i.e. (Carney, 1979; Karaali et al., 2003a, b, c;
Karataş & Schuster, 2006; Karaali et al., 2011; Tunçel Güçtekin et al., 2016;
Çelebi et al., 2019). Similar calibrations have been developed for the SDSS
photometric system and applied to faint stars (Karaali et al., 2005; Bilir et
al., 2005; Tunçel Güçtekin et al., 2017). These calibrations were used to
calculate metal abundances (Ak et al., 2007; Ivezic et al., 2008; Tunçel
Güçtekin et al., 2019) and estimation of the Galactic model parameters for
different populations (Karaali et al., 2004; Ak et al., 2007; Bilir et al.,
2008b; Juric et al., 2008; Yaz & Karaali, 2010).

The UBV photometric system provides reliable data for the bright stars which
occupy the solar neighbourhood. However, for the faint stars the same case
thus not hold. Additionally, the low transmission of the Earth’s atmosphere
limits the number of stars with reliable $U$ magnitudes. This problem could be
solved by measurements performed outside of the atmosphere such as the
satellite Galaxy Evolution Discovery (GALEX) (Martin et al., 2005).

The GALEX satellite was launched in 2003 and continued its active mission
until 2012. It is the first satellite to observe the entire sky with the two
detectors, i.e. far ultraviolet ($FUV$, $\lambda_{eff}$ = 1528Å; 1344 - 1786Å)
and near ultraviolet ($NUV$, $\lambda_{eff}$ = 2310Å; 1771 - 2831Å). The
passbands of GALEX and UBV photometric systems are shown in Fig. 1.
Measurements in the far and near UV bands of approximately 583 million objects
obtained from the reduction of 100865 images from satellite observations are
given in DR 6+7 versions of GALEX database (Bianchi et al., 2017). In our
study, transformation equations between the colour indices of GALEX and UBV
photometric systems are derived in terms of the luminosity class. These
equations provide us empirical $U-V$ colour index and UV excess for stars
which can be used in photometric metal abundance estimation. We organized the
paper as follows. Section 2 is devoted to the selection of the calibration
stars in our study, derivation of the calibrations is given in Section 3, and
finally the results and discussion are presented in Section 4.

Figure 2: $\log g\times T_{eff}$ diagram of the stars in the sample. The
diagram is colour-coded for the metallicity of 556 stars.

##  2 Data

In this study, we prioritized the stars that have precise spectroscopic,
astrometric and photometric data in the literature. In this context, we used
the spectroscopic data from 14 studies (Boesgaard et al., 2011; Nissen &
Schuster, 2011; Ishigaki et al., 2012; Mishenina et al., 2013; Molenda-
Zakowicz et al., 2013; Bensby et al., 2014; da Silva et al., 2015; Sitnova et
al., 2015; Jofré et al., 2015; Brewer et al., 2016; Kim et al., 2016;
Maldonado & Villaver, 2016; Luck, 2017; Delgado Mena et al., 2017). 6149 stars
with atmospheric model parameters ($T_{eff}$, $\log g$ and [Fe/H]) could be
provided from these studies (Table 1). The photometric data are supplied from
GALEX DR7 (Bianchi et al., 2017) and UBV (Oja, 1984; Mermilliod, 1987, 1997;
Ducati, 2002; Koen et al., 2010; Carrasco et al., 2010), while the
trigonometric parallaxes are taken from Gaia DR2 (Gaia Collaboration et al.,
2018). The photometric and astrometric data of 5593 stars were not available
for the original set of stars (6149 stars). Hence, our sample reduced to 556
(Table 2).

The $\log g\times T_{eff}$ diagram of the sample stars is shown in Fig. 2 with
the colour coded for metallicity [Fe/H]. PARSEC mass tracks for different
metal abundances are used to determine the luminosity classes of the sample
stars (Bressan et al., 2012; Tang et al., 2014; Chen et al., 2015). The
evolutionary tracks generated for different heavy element abundances ($Z$ =
0.040, 0.030, 0.020, 0.017, 0.014, 0.010, 0.008, 0.006, 0.004 and 0.002) are
converted to [Fe/H] metallicities using the formulae given by Jo
Bovy111https://github.com/jobovy/isodist/blob/master/isodist/
Isochrone.py (see also, Bostancı et al., 2018; Eker et al., 2018; Yontan et
al., 2019; Banks et al., 2020). Then, zero age main sequence (ZAMS) and
terminal age main sequence (TAMS) evolutionary tracks corresponding to the
metal abundance ranges for mean $Z$ values were established (Fig. 3). The
luminosity classes of the sample stars were determined by the metallicity
intervals as indicated in Fig. 3 and marked on the $\log g\times T_{eff}$
diagrams. Stars between the ZAMS and TAMS curves are classified as main-
sequence stars, while the ones above the TAMS curve are adopted as evolved
stars, i.e. those with $\log g\geq 3.5$ as sub-giants and the ones with $\log
g<3.5$ as giants. Thus, the number of stars with different luminosity classes
turned out to be as follows: 245 main-sequence, 187 sub-giants, and 124
giants.

We used the atmospheric model parameters to classify the luminosity class of
each star. The giant stars tend to be well separated from the sub-giants,
while the sub-giants are very close to the main-sequence stars. Hence, we
investigated the uncertainty of the atmospheric model parameters to reveal any
contamination of the sub-giants into the main sequence region and vice-versa,
as explained in the following. As the uncertainty of the atmospheric model
parameters were not considered in some studies which cover our sample stars
(556 stars), we used the uncertainty of the atmospheric model parameters in
Bensby et al. (2014) which contains approximately 22% of the stars in the
sample, for our purpose. The median errors of $T_{eff}$, $\log g$ and [Fe/H]
in Bensby et al. (2014) are 56 K, 0.08 cm s-2 and 0.05 dex, respectively. The
luminosity classes of the stars in the sample were determined by comparing the
atmospheric model parameters of the stars with the ZAMS and TAMS curves
designated from the PARSEC mass tracks. The median errors of the atmospheric
model parameters were added to the original parameters of the stars their
luminosity classes were re-asigned. Stars, whose luminosity classes were
changed, were considered as contamination. It is found that, the contamination
of the main-sequence stars by sub-giant stars is 2.9%, the contamination of
sub-giant stars by main-sequence is 3.7%, and the contamination of giant stars
by sub-giant stars is 3.2%. Hence, one can say that our transformation
equations can be considered for the luminosity class of the sample stars in
question.

Table 1: Spectroscopic data used in this study. $N$ denotes the number of
stars, $R$ spectral resolution, $S/N$ signal-to-noise ratio. Observatory,
telescope and the spectrograph used in the observations are also noted.

ID | Authors | $N$ | $R$ | $S/N$ | Observatory / Telescope / Spectrograph
---|---|---|---|---|---
1 | Boesgaard et al. (2011) | 117 |  $\sim$42000 | 106 | Keck / Keck I / HIRES
2 | Nissen & Schuster (2011) | 100 | 55000 | 250-500 | ESO / VLT / UVES, ORM / NOT / FIES
3 | Ishigaki et al. (2012) | 97 | 100000 | 140-390 | NAOJ / Subaru / HDS
4 | Mishenina et al. (2013) | 276 | 42000 | $>100$ | Haute-Provence / 1.93m / ELODIE
5 | Molenda-Zakowicz et al. (2013) | 221 | 25000-46000 | 80-6500 | ORM / NOT / FIES, OACt / 91cm / FRESCO, ORM / Mercator / HERMES
|  |  |  |  | OPM / TBL / NARVAL, MKO / CFHT / ESPaDOnS
6 | Bensby et al. (2014) | 714 | 40000-110000 | 150-300 | ESO / 1.5m and 2.2m / FEROS, ORM / NOT / SOFIN and FIES,
|  |  |  |  | ESO / VLT / UVES, ESO / 3.6m / HARPS, Magellan Clay / MIKE
7 | da Silva et al. (2015) | 309 |  $\sim$42000 | $>150$ | Haute Provence / 1.93m / ELODIE
8 | Sitnova et al. (2015) | 51 | $>60000$ | 70-100 | Lick / Shane 3m / Hamilton, CFH / CFHT / ESPaDOnS
9 | Jofré et al. (2015) | 223 | $30000-120000$ | $>150$ | ESO / 3.6m / HARPS, ESO / 2.2m / FEROS, OHP / 1.93m / ELODIE,
|  |  |  |  | OHP / 1.93m / SOPHIE, CASLEO, 2.15m / EBASIM
10 | Brewer et al. (2016) | 1615 |  $\sim$70000 | $>200$ | Keck / Keck I / HIRES
11 | Kim et al. (2016) | 170 | 10000 | $>100$ | KPNO / Mayall 4m / Echelle spectrograph
12 | Maldonado & Villaver (2016) | 154 |  $\sim$42000-115000 | 107 | La Palma / Mercator / HERMES, ORM / NOT / FIES,
|  |  |  |  | Calar Alto / 2.2m / FOCES, ORM / Nazionale Galileo / SARG
13 | Luck (2017) | 1041 | 30000-42000 | $>75$ | McDonald / 2.1m / SCES, McDonald / HET / High-Resolution
14 | Delgado Mena et al. (2017) | 1059 |  $\sim$115000 | $>200$ | HARPS GTO programs

CASLEO: Complejo Astronomico El Leoncito, EBASIM: Echellede Banco Simmons,
CFH: Canada-France-Hawaii, CFHT: Canada-France-Hawaii Telescope, ESO: European
Southern Observatory, ESPaDOnS: an Echelle SpectroPolarimetric Device for the
Observation of Stars at CFHT, FEROS: The Fiber-fed Extended Range Optical
Spectrograph, FIES: The high-resolution Fibre-fed Echelle Spectrograph, FOCES:
a fibre optics Cassegrain echelle spectrograph, FRESCO: Fiber-optic Reosc
Echelle Spectrograph of Catania Observatory, GTO: Guaranteed Time
Observations, HARPS: High Accuracy Radial velocity Planet Searcher, HERMES:
High-Efficiency and high-Resolution Mercator Echelle Spectrograph, HET: Hobby-
Eberly Telescope, HDS: High Dispersion Spectrograph, HIRES: High Resolution
Echelle Spectrometer, KPNO: Kitt Peak National Observatory, MIKE: Magellan
Inamori Kyocera Echelle, MKO: Mauna Kea Observatory, NAOJ: National
Astronomical Observatory of Japan, NOT: Nordic Optical Telescope, OACt:
Catania Astrophysical Observatory, OPM: Observatorie Pic du Midi, ORM:
Observatorio del Roque de losMuchachos, SCES: Sandiford Cassegrain Echelle
Spectrograph, SOFIN: The Soviet-Finnish optical high-resolution spectrograph,
SOPHIE: Spectrographe pour l’Observation des Phenomenes des Interieurs
stellaires et des Exoplanetes, TBL: Telescope Bernard Lyot, UVES: Ultraviolet
and Visual Echelle Spectrograph, VLT: Very Large Telescope,

Table 2: The basic parameters of 556 sample stars; ID, star, equatorial coordinates in J2000 ($\alpha$, $\delta$), photometric data ($FUV$, $NUV$, $V$, $U-B$, $B-V$), reduced colour excess ($E_{d}(B-V)$), atmospheric model parameters ($T_{eff}$, $\log g$ and [Fe/H]) and their references, and trigonometric parallaxes ($\pi$) with the errors taken from Gaia DR2. ID | Star | $\alpha$ | $\delta$ | $FUV$ | $NUV$ | $V$ | $U-B$ | $B-V$ | $E_{d}(B-V)$ | $T_{eff}$ | $\log g$ | [Fe/H] | Reference | $\pi$ | $\sigma_{\pi}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | (hh:mm:ss) | (dd:mm:ss) | (mag) | (mag) | (mag) | (mag) | (mag) | (mag) | (K) |  (cm s-2) | (dex) |  | (mas) | (mas)
1 | Hip 80 | 00 00 58.28 |  $-$11 49 25.50 | 20.026 | 13.123 | 8.400 | -0.080 | 0.550 | 0.012 | 5856 | 4.10 | -0.59 | (6) | 13.9286 | 0.0691
2 | HD 225197 | 00 04 19.79 |  $-$16 31 44.50 | 21.159 | 14.675 | 5.780 | 1.054 | 1.080 | 0.012 | 4778 | 2.66 | 0.11 | (7) | 9.8054 | 0.0946
3 | HD 249 | 00 07 22.56 | +26 27 02.20 | 22.101 | 15.353 | 7.381 | 0.750 | 0.995 | 0.015 | 4775 | 2.95 | -0.04 | (10) | 7.6866 | 0.0371
4 | HD 870 | 00 12 50.25 |  $-$57 54 45.40 | 19.946 | 13.660 | 7.226 | 0.344 | 0.774 | 0.001 | 5381 | 4.42 | -0.10 | (14) | 48.4741 | 0.0263
5 | Hip 1128 | 00 14 04.48 |  $-$11 18 41.70 | 21.027 | 13.565 | 8.360 | 0.015 | 0.655 | 0.009 | 5522 | 4.37 | -0.64 | (6) | 23.3418 | 0.0534
… | … | … | … | … | … | … | … | … | … | … | … | … | … | … | …
… | … | … | … | … | … | … | … | … | … | … | … | … | … | … | …
… | … | … | … | … | … | … | … | … | … | … | … | … | … | … | …
552 | Hip 117526 | 23 50 05.74 | +02 52 37.82 | 21.482 | 14.599 | 8.339 | 0.362 | 0.744 | 0.010 | 5540 | 4.35 | 0.20 | (6) | 21.3273 | 0.0588
553 | HD 223524 | 23 50 14.73 |  $-$09 58 26.90 | 21.009 | 14.832 | 5.941 | 1.150 | 1.130 | 0.015 | 4656 | 2.58 | 0.10 | (13) | 10.4176 | 0.0925
554 | Hip 117946 | 23 55 26.60 | +22 11 35.80 | 20.229 | 16.266 | 8.770 | 0.810 | 1.020 | 0.006 | 4790 | 4.52 | 0.04 | (10) | 39.1780 | 0.0580
555 | Hip 118115 | 23 57 33.52 |  $-$09 38 51.10 | 20.277 | 13.434 | 7.863 | 0.146 | 0.641 | 0.011 | 5833 | 4.39 | 0.02 | (6) | 19.4670 | 0.0642
556 | Hip 118278 | 23 59 28.43 |  $-$20 02 05.00 | 20.803 | 14.054 | 7.470 | 0.290 | 0.740 | 0.004 | 5533 | 4.41 | -0.07 | (11) | 38.1312 | 0.0519
Figure 3: $\log g\times T_{eff}$ diagram of the stars with different metallicity intervals. Blue circle: main-sequence, red circle: sub-giants and cyan circle: giant stars. Green solid and dashed curves represent the ZAMS and TAMS, respectively. Figure 4: Distribution of the spectroscopic metal abundances for all sample, main-sequence, sub-giant, and giant stars. Figure 5: Histograms of the original $E_{\infty}(B-V)$ (a) and reduced $E_{d}(B-V)$ (b) colour excesses of 556 stars. Table 3: Distribution of 556 sample stars according to the luminosity classes and the metallicity intervals. | ${\rm[Fe/H]}\leq-1$ | $-1<{\rm[Fe/H]}\leq-0.5$ | $-0.5<{\rm[Fe/H]}\leq+0.5$ | Total
---|---|---|---|---
| (dex) | (dex) | (dex) |
Main-sequence | 7 | 26 | 212 | 245
Sub-giant | 38 | 35 | 114 | 187
Giant | – | 5 | 119 | 124

The sample stars were also separated into different population types according
to their metallicities, i.e. thin disc ($-0.5<{\rm[Fe/H]}<+0.5$ dex), thick
disc ($-1<{\rm[Fe/H]}<-0.5$ dex), and halo (${\rm[Fe/H]}<-1$ dex), and they
were listed in Table 3. The metallicity distribution for all stars and for
different luminosity classes are given in Fig. 4.

We estimated the interstellar absorption in the UBV system for the sample
stars by using the dust map of Schlafly & Finkbeiner (2011) and reduced it to
the distance of the star in question by the following equation of Bahcall &
Soneira (1980):

$A_{d}(b)=A_{\infty}(b)\Biggl{[}1-\exp\Biggl{(}\frac{-\mid
d~{}\sin(b)\mid}{H}\Biggr{)}\Biggr{]}.$ (1)

Here, $d$ and $b$ are the distance and Galactic latitude of a star,
respectively, and $H$ indicates the scaleheight of the Galactic dust ($H=125$
pc; Marshall et al., 2006). Distances of the stars are calculated by using the
trigonometric parallaxes in Gaia DR2 via the equation
$d{\rm(pc)}=1000/\pi$(mas). The median distances of the main-sequence, sub-
giant and giant stars are 33, 49 and 83 pc, respectively. Thus, the colour
excesses $E_{d}(B-V)$ and $E_{d}(U-B)$ could be calculated by replacing the
total absorption $A_{d}(b)$ in the following equations of Cardelli et al.
(1989) and Garcia et al. (1988):

$\displaystyle E_{d}(B-V)=A_{d}(b)/3.1$ (2) $\displaystyle
E_{d}(U-B)=0.72\times E_{d}(B-V)+0.05\times E_{d}(B-V)^{2}$

Then, we estimated the intrinsic colours and magnitudes by using the following
equations. Extinction coefficients in the equations are taken from Cardelli et
al. (1989) and Yuan et al. (2013):

$\displaystyle(B-V)_{0}=(B-V)-E_{d}(B-V)$ (3)
$\displaystyle(U-B)_{0}=(U-B)-E_{d}(U-B)$ $\displaystyle V_{0}=V-3.1\times
E_{d}(B-V)$ $\displaystyle(FUV)_{0}=FUV-4.37\times E_{d}(B-V)$
$\displaystyle(NUV)_{0}=NUV-7.06\times E_{d}(B-V)$

Distribution of the colour excesses $E_{\infty}(B-V)$ and $E_{d}(B-V)$ are
plotted in Fig. 5. Small colour excesses indicate that our star sample consist
of solar neighbourhood stars. The two-colour diagrams,
$(U-V)_{0}\times(B-V)_{0}$ and $(U-V)_{0}\times(FUV-NUV)_{0}$, of the sample
stars are plotted in Fig. 6 and Fig. 7 with colour coded for the luminosity
class and metallicity, respectively.

Figure 6: Distribution of the sample stars in the $(U-V)_{0}\times(B-V)_{0}$
(a) and $(U-V)_{0}\times(FUV-NUV)_{0}$ (b) two-colour diagrams, colour coded
for the luminosity class as indicated. Figure 7: Distribution of the sample
stars in the $(U-V)_{0}\times(B-V)_{0}$ (a) and $(U-V)_{0}\times(FUV-NUV)_{0}$
(b) two-colour diagrams, colour coded for the metallicity as indicated.

##  3 Transformation Equations

We derived transformation equations between the $(FUV-NUV)_{0}$ and
$(U-V)_{0}$ colour indices as a function of luminosity class as well as
metallicity.

###  3.1 Transformation Equations According to Luminosity Classes of Stars

We adopted the following equation to transform the $(FUV-NUV)_{0}$ colour
index to the $(U-V)_{0}$ as a function of the luminosity class,

$(U-V)_{0}=a(FUV-NUV)_{0}+b(B-V)_{0}+c$ (4)

The numerical values of the coefficients $a$, $b$ and $c$ estimated for 245
main-sequence stars, 187 sub-giants and 124 giants by using multiple
regression method are given in Table 4. $T$ and $p$ values corresponding to
the sensitivity of the coefficients are given in the third and fourth lines
for each coefficient, while the squared correlation coefficients and the
standard deviations are given in the last two columns of the table. One can
see that the correlation coefficients are rather high, while the standard
deviations are small. The residuals for the colour index $(U-V)_{0}$, the
differences between the original colour indices and the estimated ones are
rather small and no systematic differences can be seen in their distribution
(Fig. 8). However, there is an exception for the giants, i.e. the uncertainty
for the coefficient a is larger than itself, additionally the corresponding
$p$ value is greater than the usual one, $p=0.05$.

Table 4: Coefficients derived from Eq. (4) and the corresponding squared correlation coefficient ($R^{2}$) and standard deviation ($\sigma$), for the sample stars of different luminosity classes. The metallicities are not considered in these calculations. $N$ indicates the number of stars. The remaining symbols are explained in the text. Luminosity Class | $N$ | $a$ | $b$ | $c$ | $R^{2}$ | $\sigma$
---|---|---|---|---|---|---
|  | -0.042159(0.003659) | 2.65427(0.02125) | -0.63791(0.02827) |  |
Main-sequence | 245 |  $T$ = -11.52 |  $T$ = 124.90 |  $T$ = -22.57 | 0.985 | 0.055
|  |  $p$ = 0.000 |  $p$ = 0.000 |  $p$ = 0.000 |  |
|  | -0.020667(0.004340) | 2.79537(0.02161) | -0.88118(0.03417) |  |
Sub-giant | 187 |  $T$ = -4.76 |  $T$ = 129.35 |  $T$ = -25.79 | 0.990 | 0.055
|  |  $p$ = 0.000 |  $p$ = 0.000 |  $p$ = 0.000 |  |
|  | 0.002630(0.007415) | 3.18474(0.04374) | -1.37664(0.07691) |  |
Giant | 124 |  $T$ = 0.35 |  $T$ = 72.82 |  $T$ = -17.90 | 0.982 | 0.050
|  |  $p$ = 0.723 |  $p$ = 0.000 |  $p$ = 0.000 |  |
Figure 8: Colour residuals in terms of $(FUV-NUV)_{0}$ (left column) and $(B-V)_{0}$ (right column) for three luminosity classes as indicated in six panels. Metallicity is not considered in calculation of the residuals. Dashed lines denote $\pm 1\sigma$ prediction levels. Table 5: Coefficients derived from Eq. (4) and the corresponding statistical results for sample stars of different luminosity classes and metallicities. $N$ indicates the number of stars. The remaining symbols are explained in the text. Luminosity Class | [Fe/H] | $N$ | $a$ | $b$ | $c$ | $R^{2}$ | $\sigma$
---|---|---|---|---|---|---|---
| (dex) |  |  |  |  |  |
| (0, +0.5] | 110 | -0.021402(0.004368) | 2.62304(0.02228) | -0.70015(0.03360) | 0.993 | 0.043
|  |  |  $T$= -4.90 |  $T$= 117.72 |  $T$= -20.84 |  |
|  |  |  $p$= 0.000 |  $p$= 0.000 |  $p$= 0.000 |  |
Main-sequence | (-0.5, 0] | 102 | -0.034443(0.004107) | 2.58820(0.02807) | -0.66283(0.03011) | 0.989 | 0.038
|  |  |  $T$ = -8.39 |  $T$ = 92.22 |  $T$ = -22.01 |  |
|  |  |  $p$ = 0.000 |  $p$ = 0.000 |  $p$ = 0.000 |  |
| (-1, -0.5] | 26 | -0.03266(0.01207) | 2.4366(0.1004) | -0.63172(0.05707) | 0.979 | 0.031
|  |  |  $T$ = -2.71 |  $T$ = 24.27 |  $T$ = -11.07 |  |
|  |  |  $p$ = 0.013 |  $p$ = 0.000 |  $p$ = 0.000 |  |
| (0, +0.5] | 50 | -0.002031(0.000630) | 2.76866(0.03730) | -0.92365(0.05313) | 0.992 | 0.040
|  |  |  $T$ = -2.48 |  $T$ = 74.22 |  $T$ = -17.38 |  |
|  |  |  $p$ = 0.014 |  $p$ = 0.000 |  $p$ = 0.000 |  |
| (-0.5, 0] | 64 | -0.012175(0.005475) | 2.77782(0.02874) | -0.93117(0.04992) | 0.996 | 0.036
|  |  |  $T$ = -2.22 |  $T$ = 96.65 |  $T$ = -18.65 |  |
|  |  |  $p$ = 0.030 |  $p$ = 0.000 |  $p$ = 0.000 |  |
Sub-giant | (-1, -0.5] | 35 | -0.014890(0.008362) | 2.04629(0.08635) | -0.52484(0.04988) | 0.959 | 0.029
|  |  |  $T$ = -2.15 |  $T$ = 23.70 |  $T$ = -10.52 |  |
|  |  |  $p$ = 0.039 |  $p$ = 0.000 |  $p$ = 0.000 |  |
| (-3, -1] | 38 | -0.004800(0.000810) | 1.9139(0.1289) | -0.59024(0.04176) | 0.926 | 0.030
|  |  |  $T$ = -1.59 |  $T$ = 14.85 |  $T$ = -14.13 |  |
|  |  |  $p$ = 0.038 |  $p$ = 0.000 |  $p$ = 0.000 |  |
| (0, +0.5] | 57 | 0.003814(0.007765) | 3.23331(0.04865) | -1.39634(0.07903) | 0.989 | 0.034
|  |  |  $T$ = 0.49 |  $T$ = 66.47 |  $T$ = -17.67 |  |
Giant |  |  |  $p$ = 0.625 |  $p$ = 0.000 |  $p$ = 0.000 |  |
| (-0.5, 0] | 62 | -0.004697(0.006729) | 3.07446(0.04036) | -1.24802(0.07181) | 0.993 | 0.034
|  |  |  $T$ = -0.70 |  $T$ = 76.17 |  $T$ = -17.38 |  |
|  |  |  $p$ = 0.488 |  $p$ = 0.000 |  $p$ = 0.000 |  |
Figure 9: Colour residuals for the sample stars in terms of $(FUV-NUV)_{0}$
and $(B-V)_{0}$ colours for different luminosity classes and metallicities, as
indicated in the panels. Dashed lines denote $\pm 1\sigma$ prediction levels.

###  3.2 Transformation Equations According to the Luminosity Classes and
Metallicities of Stars

The sample stars are separated into different metallicity intervals and
transformation equations are derived for three luminosity classes of stars in
each metallicity interval, as explained in the following. Main-sequence and
sub-giant stars occupy the metallicity intervals $0<{\rm[Fe/H]}\leq+0.5$ dex,
$-0.5<{\rm[Fe/H]}\leq 0$ dex, $-1<{\rm[Fe/H]}\leq-0.5$ dex, additionally sub-
giants occupy the interval $-3<{\rm[Fe/H]}\leq-1$ dex, while giants cover the
metallicity intervals $0<{\rm[Fe/H]}\leq 0.5$ dex, and $-0.5<{\rm[Fe/H]}\leq
0$ dex. We used the Eq. (4) and estimated the numerical values of the
coefficients $a$, $b$, and $c$ for each metallicity interval and luminosity
class by the procedure explained in Section 3.1. Result are tabulated in Table
5. The squared correlation coefficients in this table are higher than those in
Table 4. Also, the standard deviations in Table 4 reduced by 30%. Comparison
of the observed $(U-V)_{0}$ colour indices and the estimated ones via Eq. (4)
are given in Fig. 9. As it can be seen easily, there is no any systematic
deviations in the distribution of residuals for $(U-V)_{0}$ colour index. They
are smaller than the those plotted in Fig. 8, as well. However, we should note
that the coefficient $a$ estimated for the metallicity intervals of giants
does not promise accurate $(U-V)_{0}$ colour index estimation.

##  4 Summary and Discussion

In this study, we used 556 stars with accurate spectroscopic, photometric and
astrometric data and derived transformation equations between GALEX and UBV
colours. Thus, the $U$ magnitudes of the stars would be estimated more
accurately by means of $FUV$ and $NUV$ magnitudes which are observed outside
of the Earth atmosphere. Transformation equations are derived as a function of
(only) the luminosity class, and as a function of both the luminosity class
and metallicity. In both cases the statistical results promise accurate $U-V$
colours for the main-sequence and sub-giant stars, estimated by using the
$FUV$ and $NUV$ magnitudes. However, the same case does not hold for the
giants.

We used the inverted parallaxes as a distance estimate when calculating the
interstellar absorption. However, Schönrich, McMillan & Eyer (2019) have shown
that the Gaia DR2 parallaxes can be biased. We compared the distances
estimated via Gaia DR2 trigonometric parallaxes and the ones of Schönrich et
al. (2019) which are obtained by a statistical method to see the impact of the
distances of the stars on the analysis, as explained in the following. The
mean of the differences between the distances estimated by two procedures and
the corresponding standard deviation are -0.10 pc and 0.66 pc, respectively.
As seen Fig. 10, almost all distances fit with the one-to-one straight line.
Hence, we do not expect any systematic uncertainty for our results.

Figure 10: Comparison of the distances for the sample stars estimated via Gaia
DR2 trigonometric parallaxes and statistical method of Schönrich et al.
(2019). The distances calculated by two different methods are quite compatible
with one-to-one line.

We compared the spectroscopic atmospheric model parameters taken from 14
different papers in the literature to investigate the confirmation of their
homogeneity. Figs. A1, A2 and A3 show that the three parameters, $T_{eff}$,
$\log g$, and [Fe/H] lie on the one-to-one line in the corresponding figure.
Hence, we can argue that the atmospheric model parameters taken from different
studies are in an agreeable homogeneity.

The transformation equations are applied to the F-G type main-sequence stars
Tunçel Güçtekin et al. (2016) which are provided with accurate photometric
data. The sample of stars in the cited study reduced from 168 to 70 due to the
absence of $FUV$ and $NUV$ magnitudes in GALEX DR7 (Bianchi et al., 2017)
database. We used the corresponding coefficients in Table 4 and Table 5, and
estimated the $(U-V)_{0}$ colours of 70 stars in question. Residuals (Fig. 11)
and the statistical results (Table 6) show that combination of the luminosity
class and metallicity provides more accurate $(U-V)_{0}$ colours relative to
the ones estimated by considering only the luminosity class. We should
emphasize that the results corresponding only to the luminosity class are also
consistent.

Table 6: Statistical results based on the comparison of the observed and calculated $(U-V)_{0}$ colours according to the coefficients in Tables 4 and 5 (sum of differences ($\Sigma(\Delta(U-V)_{0})$), means of differences ($\Sigma(\Delta(U-V)_{0})/N$), and standard deviations of differences $\sigma_{\Sigma(\Delta(U-V)_{0})/N}$) for 70 main-sequence stars. |  | Coefficients in Table 4 | Coefficients in Table 5
---|---|---|---
${\rm[Fe/H]}$ (dex) | $N$ | $\Sigma(\Delta(U-V)_{0})$ | $\Sigma(\Delta(U-V)_{0})/N$ | $\sigma_{\Sigma(\Delta(U-V)_{0})/N}$ | $\Sigma(\Delta(U-V)_{0})$ | $\Sigma(\Delta(U-V)_{0})/N$ | $\sigma_{\Sigma(\Delta(U-V)_{0})/N}$
$0<{\rm[Fe/H]}\leq 0.5$ | 7 | 0.204 | 0.029 | 0.035 | -0.159 | -0.023 | 0.031
$-0.5<{\rm[Fe/H]}\leq 0$ | 38 | -0.501 | -0.013 | 0.044 | 0.059 | 0.002 | 0.041
$-1<{\rm[Fe/H]}\leq-0.5$ | 25 | -0.837 | -0.033 | 0.051 | 0.422 | 0.017 | 0.042

The transformation equations between the GALEX and UBV colours would be used
for estimation of the $U$ magnitude of stars for which this magnitude cannot
be observed accurately. This is important for the intermediate spectral type
main-sequence stars. Because, the $U$ magnitude thus obtained, i.e. $U_{est}$,
would be combined with the $B$ magnitude of UBV photometric system and the
$U_{est}-B$ colour would be used in the (photometric) metallicity estimation
which is important in studying the chemical structure and evolution of our
Galaxy.

Figure 11: Colour residuals for 70 main-sequence stars taken from Tunçel
Güçtekin et al. (2016) in terms of $(FUV-NUV)_{0}$ (left column) and
$(B-V)_{0}$ (right column). Residuals for stars with different metallicities
are indicated in the panels. Dashed lines show $\pm 1\sigma$ prediction
levels.

##  5 Acknowledgments

The authors thank the anonymous referee for her/his suggestions that helped
improve the quality of the paper. This research has made use of NASA’s
(National Aeronautics and Space Administration) Astrophysics Data System and
the SIMBAD Astronomical Database, operated at CDS, Strasbourg, France and
NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration. This work has made use of data
from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

Figure 12: Corner plot showing a comparison of $T_{eff}$ for overlapping stars
in 14 research groups. The numbers indicate the research groups: (1) Boesgaard
et al. (2011), (2) Nissen & Schuster (2011), (3) Ishigaki et al. (2012), (4)
Mishenina et al. (2013), (5) Molenda-Zakowicz et al. (2013), (6) Bensby et al.
(2014), (7) da Silva et al. (2015), (8) Sitnova et al. (2015), (9) Jofré et
al. (2015), (10) Brewer et al. (2016), (11) Kim et al. (2016), (12) Maldonado
& Villaver (2016), (13) Luck (2017), (14) Delgado Mena et al. (2017). Figure
13: Corner plot showing a comparison of $\log g$ for overlapping stars in 14
research groups. The numbers indicate the the research groups as in Fig. A1.
Figure 14: Corner plot showing a comparison of [Fe/H] for overlapping stars in
14 research groups. The numbers indicate the research groups as in Fig. A1.

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