# ‘Just because you are right, doesn’t mean I am wrong’:
Overcoming a Bottleneck in the Development and Evaluation of
Open-Ended Visual Question Answering (VQA) Tasks

Man Luo, Shailaja Keyur Sampat, Riley Tallman, Yankai Zeng,
Manuha Vancha, Akarshan Sajja, Chitta Baral
Arizona State University, Tempe, AZ, USA
<EMAIL_ADDRESS>

###### Abstract

GQA (Hudson and Manning, 2019) is a dataset for real-world visual reasoning
and compositional question answering. We found that many answers predicted by
the best vision-language models on the GQA dataset do not match the ground-
truth answer but still are semantically meaningful and correct in the given
context. In fact, this is the case with most existing visual question
answering (VQA) datasets where they assume only one ground-truth answer for
each question. We propose Alternative Answer Sets (AAS) of ground-truth
answers to address this limitation, which is created automatically using off-
the-shelf NLP tools. We introduce a semantic metric based on AAS and modify
top VQA solvers to support multiple plausible answers for a question. We
implement this approach on the GQA dataset and show the performance
improvements.

##  1 Introduction

One important style of visual question answering (VQA) task involves open-
ended responses such as free-form answers or fill-in-the-blanks. The
possibility of multiple correct answers and multi-word responses makes the
evaluation of open-ended tasks harder, which has forced VQA datasets to
restrict answers to be a single word or a short phrase. Despite enforcing
these constraints, from our analysis of the GQA dataset (Hudson and Manning,
2019), we noticed that a significant portion of the visual questions have
issues. For example, a question  “Who is holding the bat?” has only one ground
truth answer “batter” while other reasonable answers like “batsman”, “hitter”
are not credited. We identified six different types of issues with the dataset
and illustrated them in Table 1.

A large-scale human-study conducted by (Gurari and Grauman, 2017) on VQA
(Antol et al., 2015) and VizWiz (Gurari et al., 2019) found that almost 50%
questions in these datasets have multiple possible answers. datasets had
similar observations. The above evidence suggests that it is unfair to
penalize models if their predicted answer is correct in a given context but
does not match the ground truth answer.

With this motivation, we leverage existing knowledge bases and word embeddings
to generate Alternative Answer Sets (AAS) instead of considering visual
questions to have fixed responses. Since initially obtained AAS are generated
from multiple sources and observed to be noisy, we use textual entailment to
verify semantic viability of plausible answers to make alternative answer sets
more robust. We justify the correctness and quality of the generated AAS by
human evaluation. We introduce a semantic metric based on AAS and train two
vision-language models LXMERT (Tan and Bansal, 2019) and ViLBERT (Lu et al.,
2019) on two datasets. The experimental results show that the AAS metric
evaluates models’ performances more reasonably than the old metric. Lastly, we
incorporate AAS in the training phase and show that it further improves on the
proposed metric. Figure 2 gives an overview of our work.

Issue Type | Definition | %
---|---|---
[1] Synonym and Hypernym |  | Synonym or hypernym of the ground-truth can also be
---
considered as a correct answer for a given question-image pair.
9.1
[2] Singular/Plural |  | Singular or plural of the ground-truth can also be considered
---
as a correct answer for a given question-image pair.
1.0
[3] Ambiguous Objects |  | Question refers to an object but the image contains multiple
---
such objects that can lead to different possible answers.
5.8
[4] Multiple Correct Answers |  | If a given image-question pair is not precise, annotators might
---
have different opinion which leads to multiple correct answers
7.0
[5] Missing Object(s) | Object referred in the question is not clearly visible in image. | 4.3
[6] Wrong Label | The ground-truth answer to a question-image pair is incorrect. | 6.7
Table 1: Six types of issues observed in the GQA dataset, their definition and
their distribution observed in manual review of 600 samples from testdev
balanced split. For example of each issue type, refer Figure 1.

[1] Synonym and Hypernym  Question: Who is holding the bat? Ground-truth:
batter Explanation: ‘batsman’ is a synonym of ’batter’ SU-AAS: batter,
batsman, hitter, ballplayer, player

[2] Singular/Plural  Question: Who is wearing the dress? Ground-truth: women
Explanation: singular form ‘woman’ is also correct (there is only one woman in
the picture) SU-AAS: women, female, woman, people, adult female

[3] Ambiguous Objects

Question: Does the person in front of the cabinets
---
have brunette color?
Ground-truth: Yes
Explanation: there are two people in front of the cabinet
and it is not clear which person is being referred to
SU-AAS: yes

[4] Multiple Correct Answers  Question: Which place is it? Ground-truth: road
Explanation: some person might answer ‘road’ and some might answer ‘street’
SU-AAS: road, street, roadway, paved

[5] Missing Object(s)  Question: Does the marker look narrow? Ground-truth:
Yes Explanation: the ‘marker’ is missing from the image SU-AAS: yes

[6] Wrong Label  Question: Do the door and the logo have the same color?
Ground-truth: Yes Explanation: the correct answer is ‘no’ as the door is white
and logo is green. SU-AAS: yes

Figure 1: Examples from GQA dataset for each issue type and SU-AAS i.e. AAS of
ground-truth based on semantic union approach. SU-AAS can resolve Synonym and
Hypernym, Singular/Plural, and Multiple Correct Answers for a given problem.

##  2 Related Works

We discuss related works from two aspects, dataset creation and evaluation.

#### Dataset Creation-Level

Large-scale VQA datasets are often curated through crowd-sourcing, where open-
ended ground-truths are determined by majority voting or annotator agreement.
The subjectivity in crowd-sourced datasets is well-studied in human-computer
interaction literature- (Gurari and Grauman, 2016, 2017; Yang et al., 2018)
etc. Ray et al. (2018) suggested creating a semantically-grounded set of
questions for consistent answer predictions. (Bhattacharya et al., 2019)
analyzed VQA and VizWiz datasets to present 9-class taxonomy of visual
questions that suffer from subjectivity and ambiguity. Our analysis on GQA
partially overlaps with this study. GQA dataset only provides one ground truth
for each question; thus, we propose AAS to extend answers by phrases with
close semantic meaning as the ground-truth answer.

#### Evaluation-Level

For open-ended VQA tasks, the standard accuracy metric can be too stringent as
it requires a predicted answer to exactly match the ground-truth answer. To
deal with different interpretations of words and multiple correct answers,
(Malinowski and Fritz, 2014) defined a WUPS scoring from lexical databases
with Wu-Palmer similarity (Wu and Palmer, 1994). (Abdelkarim et al., 2020)
proposed a soft matching metric based on wordNet (Miller, 1998) and word2vec
(Mikolov et al., 2013). Different from them, we incorporate more advanced NLP
resources tools to generate answer sets and rely on textural entailment to
validate semantics for robustness. We propose a new metric to evaluate a
system’s response.

##  3 Analysis of GQA Dataset

GQA is a dataset for real-world visual reasoning and compositional question
answering. Instead of human annotation, answers to the questions in GQA are
generated from the scene graphs of images. We found that automatic creation
leads to flaws in the dataset; thus, we manually analyze 600 questions from
the testdev balanced split of GQA dataset, and identify six issues shown in
Table 1. Figure 1 shows examples of each type of issue.

These issues are caused by (not limited to) three reasons. First, the dataset
assumes only one ground truth so that other answers with semantic closed
meaning are ignored. We propose AAS to address this issue to some extent and
describe AAS in the next section. Second, some questions referring to multiple
objects cause ambiguous meaning. We leverage scene graphs to address this
issue and found 2.92% and 2.94% ambiguous questions in balanced training split
and balanced validation split, respectively. These ambiguous questions can be
removed from the dataset. Third, there are incorrect scene graph detections so
that some questions and/or labels do not match with the given images. We plan
to address these issues in our future work.

##  4 Alternative Answer Set

To credit answers with semantically close meaning as the ground-truth, we
propose a workflow that can be visualized from Figure 2. Each item in VQA
dataset consists of $<$I, Q, GT$>$, where I is an image, Q is a question, and
GT is a ground-truth answer. We define an Alternative Answer Set (AAS) as a
collection of phrases {A1, A2, A3,.., An} such that Ai replaced with GT is
still a valid answer to the given Image-Question pair. We construct AAS for
each unique ground-truth automatically from two knowledge bases: Wordnet
(Miller, 1998) and ConcpetNet (Liu and Singh, 2004), two word embeddings: BERT
(Devlin et al., 2018) and counter-fitting (Mrkšić et al., 2016). We assign a
semantic score to each alternative answer by textural entailment and introduce
the AAS metric.

###  4.1 Semantic Union AAS

We take a union of four methods to find all alternative answers. For example,
“stuffed animal” is semantic similar to “teddy bear”, which appears in the AAS
based on BERT but not in WordNet. However, the union might include phrases
that we want to distinguish from the label like “man” is in the AAS of “woman”
when using the BERT-based approach. For this reason, we employ the textural
entailment technique to compute a semantic score of each alternative answer.
For each label, we first obtain 50 sentences containing the ground-truth label
from GQA dataset. We take each sentence as a premise, replace the label in
this sentence with a phrase in its AAS as a hypothesis to generate an
entailment score between 0-1. Specifically, we use publicly available RoBERTa
(Liu et al., 2019) model trained on SNLI (Stanford Natural Language Inference)
(Bowman et al., 2015) dataset for entailment computation. The semantic score
of the alternative answer is the average of 50 entailment scores. If the
semantic score is lower than the threshold of 0.5, then this alternative
answer is thrown out. We choose 0.5 since it is the middle of 0 and 1.

Lastly, we sort the AAS by semantic score and keep the top K in the semantic
union AAS, annotated by SU-AAS. We experiment with different values of K from
2 to 10, and decide K to be 6, a trade-off between accuracy and robustness.
Note that the performance of textual entailment model is a contributing factor
in obtaining quality AAS. Therefore, we recommend using the state-of-the-art
entailment model when our proposed method is applied on other VQA datasets.

###  4.2 Evaluation Metric Based on AAS

We propose AAS metric and semantic score: given a question Qi, an image Ii,
the alternative answer set of GTi denoted by S${}_{\text{GT}_{i}}$, the
prediction of model Pi is correct if and only if it is found in
S${}_{\text{GT}_{i}}$, and the score of Pi is S${}_{\text{GT}_{i}}$(Pi), where
S${}_{\text{GT}_{i}}$(Pi) is the semantic score of Pi. Mathematically,

$\text{Acc(Q${}_{i}$, I${}_{i}$, S${}_{\text{GT}_{i}}$,
{P}${}_{i}$)}=\begin{cases}\text{S}_{\text{GT}_{i}}(\text{P}_{i})&\text{if
P}_{i}\in\text{S}_{\text{GT}_{i}}\\\ 0&\text{else}\end{cases}$

##  5 Experiments

In this section, we first show that the performance of vision-language models
on two datasets is improved based on the AAS metric. Then, we describe our
experiment to incorporate AAS with one model on GQA dataset. Last, we verify
the correctness of AAS by human evaluation.

Figure 2: (top) The workflow for generating Alternative Answer Set (AAS) for
VQA datasets (bottom) An example from GQA dataset showing semantically valid
AAS for the answer ‘batter’ generated using above workflow

###  5.1 Baseline Methods

We select two top Vision-and-Language models, ViLBERT (Lu et al., 2019) and
LXMERT (Tan and Bansal, 2019) and evaluate their performances based on the AAS
metric. From Table 2, we see that for the GQA dataset, LXMERT and ViLBERT have
4.49%, 4.26% improvements on union AAS metric separately. For VQA2.0 dataset,
LXMERT and ViLBERT have 0.82%, 0.53% improvements on union AAS metric
separately. It is expected that the improvement on VQA2.0 dataset is less than
GQA since the former dataset already provides multiple correct answers. Figure
3 shows the impacts of the value K of Union AAS on the scores. From the
figure, we see that when K increases from 2 to 6, the score gets increased
significantly, and slightly when k increases from 6 to 9, but not increases
more after K is 9. Since values 7 and 8 do not significantly improve the
score, and the value 9 introduces noise, we take the top 6 as the SU-AAS.

Dataset | Model | Original Metric | WordNet | BERT | CounterFit | ConceptNet | Union
---|---|---|---|---|---|---|---
GQA | LXMERT | 60.06 | 61.79 | 62.69 | 62.75 | 63.58 | 64.55
(testdev) | ViLBERT | 60.13 | 61.90 | 62.69 | 62.74 | 63.67 | 64.39
VQA | LXMERT | 69.98 | 70.21 | 70.54 | 70.33 | 70.52 | 70.80
(valid) | ViLBERT | 77.65 | 77.82 | 78.10 | 77.93 | 78.06 | 78.28

Table 2: The evaluation of two models on GQA and VQA with original metric and
AAS based metrics.  Figure 3: Union AAS score of different value of K

###  5.2 Training with AAS

We incorporate SU-AAS of ground truth in training phase, so the model learns
that more than one answer for a given example can be correct. We train LXMERT
on GQA dataset with this objective.

Table 3 shows the results of LXMERT trained with AAS compared with the
baseline. Not surprisingly, the performance evaluated on the original method
drops because the model has a higher chance to predict answers in AAS, which
are different from the ground truth, and thus the performance evaluated on SU-
AAS metric increases.

Dataset | Exact Matching Accuracy | SU-AAS Accuracy
---|---|---
LXMERT | LXMERTAAS | LXMERT | LXMERTAAS
GQA(testdev) | 60.06 | 59.02 | 64.55 | 65.22

Table 3: Incorporate AAS in the training phase of LXMERT (LMXERTAAS) on GQA
dataset.

###  5.3 Evaluation of AAS

To validate the correctness of AAS, we measure the correlation between human
judgment and AAS. Specifically, for each label of GQA, we take the SU-AAS and
ask three annotators to justify if alternative answers in AAS can replace the
label. If the majority of annotators agree upon, we keep the answer in the
AAS, remove otherwise. In this way, we collect the human-annotated AAS. We
compare the human-annotated AAS with each automatically generated AAS. We take
the intersection over union (IoU) score to evaluate the correlation between
automatic approach and human annotation: a higher IoU score means stronger
alignment.

Method | WordNet | BERT | CounterFit | ConceptNet | Union
---|---|---|---|---|---
IoU% | 48.25 | 56.18 | 58.95 | 58.39 | 80.5

Table 4: The IoU scores between human annotations and AAS based on five
approaches.

##  6 Discussion and Conclusion

To evaluate a model from a semantic point of view, we define an alternative
answer set (AAS). We develop a workflow to automatically create robust AAS for
ground truth answers in the dataset using Textual Entailment. Additionally, we
did human verification to assess the quality of automatically generated AAS.
The high agreement score indicates that entailment model is doing a careful
job of filtering relevant answers. From experiments on two models and two VQA
datasets, we show the effectiveness of AAS-based evaluation using our proposed
metric.

AAS can be applied to other tasks, for example, machine translation.
BLEU(Papineni et al., 2002) score used to evaluate machine translation models
incorporates an average of n-gram precision but does not consider the
synonymy. Therefore, METEOR (Banerjee and Lavie, 2005) was proposed to
overcome this problem. However, METEOR only relies on the synset of WordNet to
get the synonyms. Our proposed AAS has the advantage of both knowledge base
and word embeddings, which would help better evaluate translation tasks.

## Acknowledgments

We are thankful to Tejas Gokhale for useful discussions and feedback on this
work. We also thank anonymous reviewers for their thoughtful feedback. This
work is partially supported by the National Science Foundation grant
IIS-1816039.

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# Complex network view for $\delta$ Scuti stars

Elham Ziaali1, Nasibe Alipour1, and Hossein Safari1,2 E-mail<EMAIL_ADDRESS>
(1Department of Physics, Faculty of Science, University of Zanjan, University
Blvd., Zanjan, 45371-38791, Zanjan, Iran
2Observatory, Faculty of Science, University of Zanjan, University Blvd.,
Zanjan, 45371-38791, Zanjan, Iran
)

##  1 Abstract

Extraction of characteristics of the complex light curve of pulsating stars is
essential for stellar physics. We investigate the complex network (natural and
horizontal visibility graphs) properties of the $\delta$ Sct stars light
curves observed by TESS. We find that the average shortest path length of
$\delta$ Sct light curves is a linear function of the logarithm of the network
sizes, indicating the small-world and non-random properties. The small-world
property signifies the connection of significant peaks of the light curve with
small nearing peaks and other important peaks. The lognormal behavior of
nodes’ degree is evidence for non-random processes for stellar pulsations.
This may be rooted in the different mechanisms of stellar dynamics, such as
rotation, excitation of modes, and magnetic activity. The PageRank and nodes’
degree distributions of $\delta$ Sct stars collect in two HADS and non-HADS
groups. The lower clustering for HADS than non-HADS indicates a more
straightforward light curve (containing one or two independent modes) than a
more complex light curve (including various oscillation modes) that might be a
signature of surface gravity as an indication of stellar evolution. We show
that reducing the window size of a light curve to about 5% of the original one
based on the network ranking preserves most of the star modes information. In
this case, we also observe that the amplitude of most natural modes amplifies
compared to the noise background in the power spectrum. These results indicate
the capability of the network approach for interpreting pulsating stars’ light
curves.

##  2 Introduction

Asteroseismology is a fundamental approach for identifying and modeling
stellar light curves to determine the internal structure and properties of
pulsating stars (e.g. Aerts et al.,, 2003; Christensen-Dalsgaard,, 2004;
Baglin,, 2003; Chaplin et al.,, 2009; Aerts et al.,, 2010; Handler,, 2013;
Basu and Chaplin,, 2017; Bowman,, 2017; Aerts,, 2021; Kurtz,, 2022). Space
telescopes such as CoRoT (Baglin et al.,, 2009), Kepler (Gilliland et al.,,
2010) and TESS (Ricker et al.,, 2015) provide the most accurate light curves
for astrophysical and exoplanets investigations. The complex network methods
for pulsating stars may extract meaningful statistics and hidden
characteristics of the system. In complex network theory, a real-world time
series maps to a graph to extract the topological features of the system that
do not occur in the simple analysis due to the complex behavior of the system.

Several analyses have been employed, e.g., on internal structure, mode
identification, and classification for $\delta$ Sct stars (e.g., Balona and
Dziembowski,, 2011; Uytterhoeven et al.,, 2011; Chang et al.,, 2013; Kahraman
Aliçavuş et al.,, 2017; Niemczura et al.,, 2015; Pakštienė et al.,, 2018;
Antoci et al.,, 2019; Jayasinghe et al.,, 2020; Bedding et al.,, 2020; Murphy
et al.,, 2020; Chaplin et al.,, 2020; Audenaert et al.,, 2021; Bowman et al.,
2021a, ; Barbara et al.,, 2022; Kirmizitas et al.,, 2022). The $\delta$ Sct
stars are multi-periodic pulsating stars with intermediate masses. In the HR
diagram, the $\delta$ Sct stars are at the transition region between low-mass
and high-mass stars. The low-mass stars ($\leq 1M_{\odot}$) have thick
convective envelopes, and high-mass stars ($\geq 2M_{\odot}$) have large
convective cores and radiative envelopes. These stars have spectral types
A0-F5 and pulsate in low-order pressure modes. They are in the lower part of
the classical instability strip, within or just above the main sequence stars.
Effective temperatures of $\delta$ Sct stars are approximately between 6500 K
and 9500 K. The $\delta$ Sct stars represent various cepheid-like significant
amplitude radial pulsations and the non-radial multi-periodic pulsations
within the classical instability strip with dominant pulsation frequencies in
the range of 5-80 d-1 (e.g., Breger,, 2000; Pamyatnykh,, 2000; Bowman et al.,,
2016; Michel et al.,, 2017; Balona,, 2018; Bowman and Kurtz,, 2018; Pakštienė
et al.,, 2018; Qian et al.,, 2018; Ziaali et al.,, 2019; Jayasinghe et al.,,
2020; Bedding et al.,, 2020; Hasanzadeh et al.,, 2021; Barac et al.,, 2022).

The high amplitude $\delta$ Sct stars (called HADS), are classified in
population $\it{I}$ of $\delta$ Sct stars with V band amplitude $\geq 0.3$ mag
and also $v\sin i\leq\ 30\ kms^{-1}$. However, the SX Phoenicis type of large
amplitude $\delta$ Sct stars is metal-poor population II stars. The frequency
spectrum of HADS stars mostly shows only one or two independent modes, which
are probably radial (Breger,, 2000; Poretti,, 2003; McNamara et al.,, 2007;
McNamara,, 2011; Balona et al.,, 2012; Balona,, 2016; Bowman et al., 2021b, ;
Yang et al.,, 2022). In $\delta$ Sct stars, the well-known $\kappa$ mechanism,
which operates in zones of partial ionization of hydrogen and helium, can
drive low-order radial and nonradial modes of the low spherical degree to
measurable amplitudes (opacity-driven unstable modes). In low temperature
$\delta$ Sct stars, near the red edge of the instability strip with
substantial outer convection zones, the selection mechanism of modes with
observable amplitudes could be affected by induced fluctuations of the
turbulent convection. Some studies (e.g., Houdek et al.,, 1999; Samadi et
al.,, 2002; Antoci et al.,, 2011; Antoci,, 2014) suggested the opacity-driven
unstable p modes, in which nonlinear processes limit their amplitudes, and
intrinsically stable stochastically driven (solar-like) p modes can be excited
simultaneously in the same $\delta$ Sct star.

de Franciscis et al., (2018) investigated the fractal property of $\delta$ Sct
stars via the rescaled range analysis, multifractal spectra analysis, and
coarse-graining spectral analysis. They inferred the scale invariances of
light curve of $\delta$ Sct stars may be due to the turbulent. The complex
system methods have been widely applied for distinguishing individual and
collective features in many scientific fields, such as economics (Souma et
al.,, 2003), biology (Barabasi and Oltvai,, 2004), earthquakes (Baiesi and
Paczuski,, 2004; Pastén et al.,, 2018; Vogel et al.,, 2020), and Solar physics
(Daei et al.,, 2017; Gheibi et al.,, 2017; Lotfi et al.,, 2020; Mohammadi et
al.,, 2021). The complex network approach could classify features with the
same characteristics (e.g., the degree distribution, average clustering
coefficient, transitivity, and PageRank) and quantify the complexities of
dynamic systems. Graph theory is a powerful mathematical tool for exploring
the characteristics of complex systems. Time series analysis based on the
complex network provides information about feature structure. By mapping the
time series into a natural and horizontal visibility graph, we can capture the
essential characteristics of the features into distinct individual categories.

Due to small-amplitude nonradial modes in light curves, categorizing the HADS
and non-HADS (low amplitude multi-mode $\delta$ Sct stars) stars is
challenging. Amplitude modulation, non-linear driving mechanisms of modes,
binarity, and planets’ effects are complex features of stars’ light curves.
Here, we study the characteristics of HADS and non-HADS stars by mapping the
light curves into the graph using the horizontal visibility graph (HVG) and
natural visibility graph (NVG) approaches. We study the networks’ local and
global characteristics (nodes’ degree distribution, clustering coefficients,
path length, PageRank, etc.) for HADS and non-HADS stars to show the
capability of complex network approaches to identifying HADS and non-HADS
stars.

Section 3 explains the TESS data set for HADS and non-HADS stars. Section 4
discuss the methods, including the HVG and NVG algorithms, to map the stellar
light curves to the network. Section 5 gives results and discussions. Section
6 remarks on the important findings of this study.

##  3 Data

TESS is a NASA mission and a high-precision photometric instrument (Sullivan
et al.,, 2015; Barclay et al.,, 2018) and scans the sky in several sectors for
millions of stars at 600-1000 nm bands (Ricker et al.,, 2015). TESS observed
hundreds of thousands of stars with a short cadence of 2 minutes and a long
cadence of 30 minutes (Campante et al.,, 2016; Stassun et al.,, 2018;
Feinstein et al.,, 2019) that are collected into
MAST111https://archive.stsci.edu in both target pixels and light curve files.
Due to fewer data points, the complex network (Section 4.1) did not form for
long cadence TESS observation targets. So, we queried the information and
short cadence light curves of 33 (Table 1) and 40 (Table 2) HADS and non-HADS
stars, respectively, by using the Python Lightkurve package (Lightkurve
Collaboration et al.,, 2018).

##  4 Methods

###  4.1 Network representation of a stellar light curve

A complex network is a helpful approach to studying the characteristics of
natural complex systems that evolve via complex dynamics. A complex network
extracts a complex system’s behavior, composed of individual parts or
interacting components. Then show emergent collective characteristics of the
system. Natural visibility graph (NVG) and horizontal visibility graph (HVG)
are among the several algorithms for building a network for a time series to
study the system’s characteristics (e.g., Newman,, 2003; Luque et al.,, 2009;
Newman,, 2010; Gheibi et al.,, 2017; Daei et al.,, 2017; Najafi et al.,, 2020;
Lotfi et al.,, 2020; Mohammadi et al.,, 2021; Taran et al.,, 2022).

NVG and HVG are algorithms that focus on the interactions of system elements.
To map a time series like a stellar light curve to a network, we consider each
data point as a node (vertex), and if two elements interact, they are then
connected in pairs by lines (edges or links). Figure 1 illustrates the
connection of nodes (data points) for HD 112063 (TIC 9591460) light curve via
the NVG and HVG algorithms.

Figure 1: Horizontal visibility (a) and natural visibility (b) for a point of
the light curve of HD 112063 $\delta$ Sct star. For HVG and NVG algorithms,
the nodes are connected based on Equations (1) and (2), respectively.

For the HVG network for HD 112063 star, node $(t_{a},y_{a})$ connects to
$(t_{b},y_{b})$ if an arbitrary point $(t_{c},y_{c})$ such that
$t_{a}<t_{c}<t_{b}$ satisfies the condition (Figure 1 top panel)

$y_{a},y_{b}>y_{c}.$ (1)

In the NVG algorithm for HD 112063 star, we map the light curve into a network
based on the visibility criterion. For two data points $(t_{a},y_{a})$ and
$(t_{b},y_{b})$ in the light curve have connected nodes in the graph via the
criteria:

$y_{c}<y_{b}+(y_{a}-y_{b})\frac{t_{b}-t_{c}}{t_{b}-t_{a}}$ (2)

in which $(t_{c},y_{c})$ is an arbitrary data point such that
$t_{a}<t_{c}<t_{b}$.

In the network approach, a set of local and global metrics reflect some
particular features of the system. Local metrics describe individual nodes or
edges, and global metrics interpret the graph as a whole. We briefly describe
the nodes degree, clustering coefficient, shortest path length, and
transitivity properties of an HVG network.

The node’s degree of a network is one of the local metrics which measures the
centrality. The node’s degree is the number of edges linked to the node.
Centrality shows the most influential nodes with effective connectivity
through the network (Acosta-Tripailao et al.,, 2021). Local and average
clustering coefficients are local metrics in a graph that indicate the node’s
tendency to cluster together and their neighbors. Transitivity is a global
clustering coefficient that determines the density of triangles in a complex
network. Transitivity measures the fraction of triples with their third edge
served to complete the triangle. Average shortest-path length is another
global metric for complex networks that define the average number of links as
the shortest paths for all pairs of nodes. The Google founders developed the
PageRank metric that measures the importance of every single node. The
assumption is that important nodes have many in-links from essential nodes.

###  4.2 Power-law and lognormal distribution fits

We applied the maximum likelihood estimation in the Bayesian framework
(Clauset et al.,, 2009; Farhang et al.,, 2018; Alipour et al.,, 2022) to
obtain the fit parameters of both power-law and lognormal distributions to
data. Also, applying the bootstrapping sampling, we computed the parameters’
uncertainty and average (true) value for both the power law and lognormal
distributions. We examined a hypothesis test based on K-S (Kolmogorov–Smirnov)
statistic. The null hypothesis supposes no significant difference between the
nodes’ degree distribution and the lognormal/power law model. But, the
alternative hypothesis assumes a substantial difference between the nodes’
degree distribution and the model. We calculate a p-value to decide whether or
not the lognormal/power law distribution hypothesis is plausible for our
nodes’ degree. A p-value more petite than the threshold of 0.1 refutes the
null hypothesis showing that the lognormal/power law distribution is ruled
out. We cannot deny the null for a p-value more remarkable than the threshold
of 0.1.

The power-law distribution function is given by

${\rm PDF}(x,x_{\rm min},\alpha)=\frac{\alpha-1}{x_{\rm
min}}\left(\frac{x}{x_{\rm min}}\right)^{-\alpha},$ (3)

where $x_{\rm min}$ and $\alpha$ are threshold and power index, respectively.
This power-law or scale-free property is an essential characteristic of a
self-oscillatory process that can be considered as the underlying mechanism of
self-similar, self-organized, or self-organized criticality systems.

The lognormal distribution function is introduced by

${\rm PDF}(x,\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(\log
x-\mu)^{2}}{2\sigma^{2}}\right),$ (4)

where $\mu$ and $\sigma$ are the scale and shape parameters, respectively. A
lognormal distribution relates to a multiplicative mechanism that shows the
effect of the system’s independent varying parameters (Bazarghan et al.,,
2008; Farhang et al.,, 2022).

##  5 Results and discussions

In the current work, we analyzed the complex networks of $\delta$ Sct stars
observed by TESS. The HADS stars are mostly radial mono-periodic or double-
mode pulsators, but the other $\delta$ Sct stars show several oscillations
frequencies (multi modes). We provided a list of HADS stars, including 20
mono-periodic and 13 double-mode HADS stars. We also have 40 non-HADS stars,
typically multi-periodic pulsators. We mapped the TESS stellar light curves of
HADS and non-HADS stars into the individual networks via the HVG and NVG
algorithms.

We applied the HVG and NVG algorithms to investigate the characteristics of 33
and 40 HADS and non-HADS stars, respectively. Due to different algorithms of
network structure for HVG and NVG, we represent the results for key metrics of
each algorithm. Some network properties in both HVG and NVG algorithms for
HADS and non-HADS stars show similar behavior, so we presented the results for
one of the algorithms. Also, for some metrics without simple interpretable
behavior, we ignored the analysis for those metrics.

Figure 2 represents the average shortest path length for the HVG network of GP
And (HADS) and HD 113211 (non-HADS) that compared with the equivalent random
network (Luque et al.,, 2009). As shown in the figure, the average shortest
path length network for both stars has deviated from the random network. The
linear dependency of the average shortest path length to the logarithm of
network size for both cases indicated the small-world behavior of networks. We
observed similar behavior for all target stars of Tables 1 and 2. The small
world behavior for $\delta$ Sct stars network shows that the high peaks at the
light curve are connected to several neighboring small peaks and the other
high peaks at the light curve. In the context of complex networks, reported a
similar behavior for other small-world networks (Watts and Strogatz,, 1998;
Mathias and Gopal,, 2001; Latora and Marchiori,, 2001).

Figure 2: The relation between average shortest path length of the HVG network
with the network sizes for HD 113211 (blue circles) as a non-HADS star and GP
And star (red triangles) as a HADS star. The fitted line for HD 113211 (blue
line) and GP And star (red line) deviated from the random network (olive dash-
dotted line).

Figure 3 depicts the scattering of the average clustering coefficients and
transitivity of HVG networks for HADS (triangles) and non-HADS (circles)
stars. The color bar indicates surface gravity, $3\leq\log g\leq 3.5$ for
Terminal-age main sequence (TAMS), $3.5<\log g\leq 4.0$ for Mid-age main
sequence (MAMS), and $4.0<\log g\leq 4.5$ for Zero-age main sequence (ZAMS)
stars (Bowman and Kurtz,, 2018). The HADS stars have high transitivity and low
clustering coefficients, but the other $\delta$ Sct stars have different
values. So, the HADS and non-HADS stars are clustered in groups with different
slopes (linear fits). The RX Cae (HD 28837) star (indicated by a black arrow)
is defined as a HADS candidate. As shown in the figure, the HADS are mostly
MAMS samples that consider the old stars comparing the non-HADS with large
surface gravity. The low clustering coefficient indicates the simple light
curve (containing one or two independent modes) for HADS compared with the
more complicated light curve (including various oscillation modes) for non-
HADS.

Figure 3: Scatter plot of the transitivity versus the average local clustering
coefficient for HVG network of non-HADS (circles) and HADS (triangles) stars.
The linear fits represent the dependency of the clustering coefficient and
transitivity. The age of stars indicated by color bars of surface gravity. The
black arrow indicates $\delta$ Sct RX Cae star.

Figure 4 shows PDF and CCDF for nodes’ degree of the HVG network for six HADS
stars. As shown in the figure, the distribution of nodes degree is heavy-tail,
so we fitted the power-law and lognormal distribution via the maximum
likelihood estimation in the Bayesian framework (Section 4.2).

Figure 4: PDF and CCDF for the nodes degree of six HADS stars. Power laws
(power law fit: blue dashed lines), lognormal functions (lognormal fit: red
line) and random functions (random fit: olive dash-dotted) are fitted to each
distribution.

We observe that the peak of the nodes’ degree distribution (Figure 4) for HADS
(mono-periodic: GP And (TIC 436546358), CY Aqr (TIC 422412568), AD Cmi (HD
64191), and V367 Cam (TIC 354872568); double mode: V1384 Tau (TIC 415333069)
and HD 146953) are precisely three or four. We analyzed the HVG
characteristics of a single-mode artificial sinusoidal light curve to address
this essential property of HADS stars. Figure 5 displays the horizontal
visibility required for the HVG network for a part of a mono-periodic
artificial time series. We observe that the nodes with three links are more
frequent than the others, with the number of connections differing from three.
In other words, we follow that the nodes (black points) connect to three other
nodes due to the behavior of a sinusoidal mono-periodic artificial time
series—this property of a fully sinusoidal time series with various
frequencies/periods. However, most mono-periodic stellar light curves have the
HVG node distribution peaks of about three or four, similar to the artificial
time series. However, the range of degrees of nodes is different for regular
artificial time series and mono-periodic stellar light curves. These
differences may indicate the diverse nature of light curves of pulsating stars
as $\kappa$, stochastic, or other complex generative mechanisms (Houdek et
al.,, 1999; Samadi et al.,, 2002; Antoci et al.,, 2011; Antoci,, 2014) for
pulsations from a regular sinusoidal oscillation. We find the p-value for both
power-law fit and random model (Luque et al.,, 2009) is less than 0.01, which
rejects both models for nodes’ degree distribution of $\delta$ Sct stars. The
rejection of the random model implies that the $\delta$ Sct stars light curve
is not random. The p-values for lognormal fits are more than 0.1 (Table 1),
which indicates we can not refute the lognormal model for nodes’ degree
distribution of HADS stars. We obtain the lognormal distribution parameters in
the range of 3.05 to 3.86 and 0.2 to 0.6 for scale and shape, respectively.
The lognormal distribution is an essential feature of systems handled with
multiplicative independent varying parameters (Mitzenmacher,, 2004; Bazarghan
et al.,, 2008; Tajfirouze and Safari,, 2012; Tokovinin,, 2014; Ruocco et al.,,
2017; Alipour et al.,, 2022; Farhang et al.,, 2022). The lognormal
distribution of the network parameter for $\delta$ Sct stars might originate
in the pulsating driving mechanisms. The different mechanisms of stellar
dynamics control the pulsating driving mechanisms, including rotation,
excitation of modes, and magnetic activity (de Franciscis et al.,, 2018).

Figure 5: Schematic representation of a mono-periodic sinusoidal time series
(light curve). The red double head arrows show the links of nodes via the HVG
approach. The nodes with the color black, blue, and red have a degree of 3, 4,
and 2, respectively. We ignored the network for analyzing two ends of the time
series. Nodes with a degree of three are the most frequent.

Figure 6 shows PDF and CCDF for nodes’ degree of the HVG network for six non-
HADS stars, HD 77914, HD 112063, HD 86312, HD 44930, HD 56843, and V479 Tau
(HD 24550). We obtain the p-values more than 0.1 (Table 2) for lognormal fits,
which shows we can not reject the lognormal model for nodes’ degree
distribution of non-HADS stars. We find the lognormal distribution parameters
in the range of 2.4 to 3.9 and 0.33 to 0.6 for scale and shape, respectively.
A lognormal distribution is due to a multiplicative process (e.g., Hubble,,
1934; Kolmogorov,, 1962; Miller and Scalo,, 1979; McBreen et al.,, 1994; Mouri
et al.,, 2009). So, the lognormal behavior of nodes degree for stellar flux
network may indicate the flux of $\delta$ Sct stars forming via some
multiplicative process. Pauluhn and Solanki, (2007) and Bazarghan et al.,
(2008) showed that the small-scale solar brightening features distribution
follows a lognormal distribution.

Figure 6: PDF and CCDF for the nodes degree of six non-HADS stars. Power laws
(power law fit: blue dashed lines), lognormal functions (lognormal fit: red
line) and random functions (random fit: olive dashdot) are fitted to each
distribution.

Figure 7 depicts the CCDF of nodes’ degree and PageRank of NVG network for
$\delta$ Sct stars. We observe that both nodes’ degree and PageRank
distributions for HADS stars (red lines) are grouped above levels than the
non-HADS (blue lines) stars. The high nodes’ degree shows the significant
number of connections of nodes with others. However, the high PageRank
indicates the important nodes or data points in the light curve with many in-
links from other nodes. Since HADS are mostly mono-periodic or double mode
stars, so high PageRank nodes (importance nodes) show a significant
probability over the other nodes. However, due to the superposition of
amplitudes of several modes for non-HADS stars, the high PageRanks (tail of
the distribution) have low probability than the low PageRank.

Analysis for the network properties (e.g., PageRank and nodes’ degree
distributions) indicates that the V353 Vel, TIC 448892817, and GW Dra stars
are in the interface of HADS and non-HADS stars. Lv et al., (2022) showed that
the TIC 448892817 is a HADS star, confirmed by the present network analysis.
Along with the clustering coefficient analysis (Figure 3), the PageRank and
degree distribution for RX Cae also suggest considering this star in the HADS
subclass. Also, the dominant period (${\rm P}=0.15$ d) and luminosity (${\rm
M_{V}}=1.04$ mag) for RX Cae satisfy the revised P-L relation ${\rm
M_{V}}=(-2.94\pm 0.06)\log{\rm P}-(1.34\pm 0.06)$ (Ziaali et al.,, 2019).

Figure 7: CCDF of nodes’ degree (top panel) and PageRank (bottom panel) of NVG
network for HADS (red lines) and non-HADS (blue lines) stars. The CCDF of
nodes’ degree and PageRank for RX Cae, V353 Vel, TIC 448892817, and GW Dra are
samples that show different behavior of multi-mode $\delta$ Sct stars.

###  5.1 Effect of the noise on networks’ parameters

We investigate the effect of noise on the networks’ parameters for a synthetic
time series with several frequencies and amplitudes (in the range of $\delta$
Sct stars) with harmonics, including the white noises. Figure 8 shows the
synthetic time series and its power spectral density (PSD) calculated with
fast Fourier transformation. We reconstruct the related time series via
inverse Fourier transformation for different threshold values as a fraction of
PSD subtracted from the primary PSD. For each time series, we build the HVG
network. Figure 9 displays the standard deviation (std) of network parameters
(nodes degree, clustering coefficient, and PageRank) versus the threshold. We
observe that the thresholds of decreasing steps for the std of network
parameters corresponded to the PSD of frequencies. Therefore, the number of
significant steps corresponds to the frequencies made in the time series.
Also, the fluctuations of the small thresholds might correspond to the
different noise levels.

###  5.2 Reducing the size of the window for light curves

Reducing the size of the window of a time series without significantly
decreasing the essential characteristics of the time series is very important
in machine learning applications (Moradkhani et al.,, 2015). The modes’
information, such as frequencies, amplitudes, and phases, are the main
characteristics of pulsating star light curves (Aerts et al.,, 2010). We apply
the complex network parameter (e.g., PageRank) to reduce a stellar light
curve’s data points. To find the key data points of light curves, we first
ranked the nodes of a light curve according to their HVG PageRanks. Then we
selected a given percentile of data points with higher PageRanks. We extract
the characteristics (frequencies and amplitudes) for the resultant light
curve.

Figure 10 represents the frequency power spectrum for 5% (top panel) and 15%
(bottom panel) of TIC 448892817 light curve chosen by the large HVG PageRanks
(blue) and random sample (red) that compared with the power spectrum of the
full light curve in sector 5 (green). We observe that compared with the random
sampling, the modes (frequencies, amplitudes, and phases) obtained from light
curves reduced by the network approach are significantly satisfied with the
full light curves (black circles) reported by Lv et al., (2022). In this case,
we also observe that the most amplitudes for identified frequencies were
amplified from the noise background. This result implies that reducing the
size of the window to e.g., 5% of the original light curve via the network
approach preserves the main characteristics of the light curve. However, some
characteristics of the light curve were missed by the random sampling.

Figure 8: Synthetic time series (top panel) and its PSD clean and noisy
(bottom panel) signal.

Figure 9: Std of network parameters including of nodes degree (top panel),
clustering coefficient (middle panel), and PageRank (bottom panel) versus the
threshold. Threshed of corresponds to noise and frequencies steps are
distinguished.

Figure 10: Power spectrum of TIC 448892817 for 5% (top panel) and 15%(bottom
panel) of light curve selected by the large HVG PageRanks (blue) and randomly
chosen (red). The power spectrum of full light curve in sector 5 (green) is
added for diagrams. The well-known frequencies are indicated by black circles.

##  6 Conclusions

We investigate the network characteristics $\delta$ Sct stars via the HVG and
NVG algorithms, which both algorithms study the interaction of data points
along the light curves (Figure 1). We focus on the network metrics to discuss
the property of HADS and non-HADS stars.

  * •

The linear dependency of the average shortest path length (Figure 2) for HVG
network size (in log space) of $\delta$ Sct stars indicates that the networks
are small-world complex networks with non-random properties. This small-world
property implies that the peaks of the stellar flux light curves connect with
some small close peaks and are then linked to other significant peaks along
the light curves.

  * •

Scattering of transitivity and average clustering coefficient of HADS and non-
HADS stars are clustered in two groups overlapping at some deals (Figure 3).
The low average clustering for RX Cae (HD 28837) star is defined as a HADS
sample. The lower clustering coefficient for most HADS stars indicates the
simple light curve (containing one or two independent modes). In contrast, the
higher clustering for non-HADS cases is a signature of the more complicated
light curve (including various oscillation modes). These results may be due to
HADS ages (low surface gravity) as older than the non-HADS stars. In other
words, the older stars may have simpler light curves than the younger stars.

  * •

The nodes’ degree distribution of HVG networks (Figure 4 and 6) of $\delta$
Sct stars obey the lognormal distribution. The p-value of K-S statistics
rejects both power-law and random network distribution. A lognormal behavior
suggests a multiplicative generative mechanism for oscillations of $\delta$
Sct stars.

  * •

We find that the nodes’ degree and PageRank distributions for NVG networks of
HADS and non-HADS stars are grouped in two classes for most cases (Figure 7).

  * •

Based on the network ranking for modes, we reduced the window size of a light
curve to about 5% of the original one, which preserves most of the star
oscillation modes characteristics (frequencies, amplitudes, and phases). We
also observe that most natural modes amplify the amplitude compared to the
power spectrum’s noise background.

Finally, we observe that the network approach can recognize HADS in the
$\delta$ Sct stars as a subgroup of pulsating stars. We conclude that the
network analysis of light curves helps interpret pulsating stars and their
originating mechanisms via stellar dynamics.

Table 1: Parameters and p-value of fitted lognormal distribution for HADS stars. Star ID | Lognormal
---|---
Name | HD | TIC | $\mu$ | $\sigma$ | p-value
AD Cmi | 64191 | 266328148 | $1.25\pm 0.08$ | $0.29\pm 0.06$ | 0.14
BE Lyn | 79889 | 56914404 | $1.15\pm 0.07$ | $0.45\pm 0.05$ | 0.16
CY Aqr | – | 422412568 | $1.19\pm 0.07$ | $0.44\pm 0.06$ | 0.18
DX Cet | 16189 | 278962831 | $1.35\pm 0.06$ | $0.33\pm 0.06$ | 0.14
GP And | – | 436546358 | $1.17\pm 0.07$ | $0.35\pm 0.05$ | 0.18
GW Uma | – | 150276417 | $1.12\pm 0.07$ | $0.44\pm 0.06$ | 0.21
PT Com | – | 335826251 | $1.19\pm 0.07$ | $0.38\pm 0.06$ | 0.28
RS Gru | 206379 | 139845816 | $1.17\pm 0.06$ | $0.46\pm 0.06$ | 0.19
V 524 And | – | 196562983 | $1.23\pm 0.08$ | $0.33\pm 0.06$ | 0.15
V367 Cam | – | 354872568 | $1.27\pm 0.06$ | $0.34\pm 0.04$ | 0.19
V1162 Ori | – | 34512862 | $1.30\pm 0.07$ | $0.33\pm 0.04$ | 0.21
V2455 Cyg | 204615 | 266794067 | $1.25\pm 0.06$ | $0.33\pm 0.06$ | 0.17
YZ Boo | – | 233465540 | $1.21\pm 0.07$ | $0.37\pm 0.05$ | 0.16
AN Lyn | – | 56882581 | $1.32\pm 0.06$ | $0.36\pm 0.05$ | 0.20
– | 146953 | 210548440 | $1.26\pm 0.06$ | $0.35\pm 0.06$ | 0.23
V1384 Tau | – | 415333069 | $1.35\pm 0.06$ | $0.32\pm 0.04$ | 0.26
V1393 Cen | 121517 | 241787384 | $1.24\pm 0.06$ | $0.34\pm 0.05$ | 0.20
V2855 Ori | 254061 | 166979292 | $1.32\pm 0.07$ | $0.32\pm 0.05$ | 0.22
ZZ Mic | 199757 | 126659093 | $1.35\pm 0.06$ | $0.27\pm 0.03$ | 0.06
AE Uma | – | 357132618 | $1.12\pm 0.07$ | $0.43\pm 0.07$ | 0.20
$\rho$ Pup | 67523 | 154360594 | $1.35\pm 0.06$ | $0.21\pm 0.06$ | 0.02
BL Cam | – | 392774261 | $1.32\pm 0.06$ | $0.37\pm 0.05$ | 0.24
BS Aqr | 223338 | 9632550 | $1.15\pm 0.08$ | $0.39\pm 0.06$ | 0.17
DE Lac | – | 119486942 | $1.26\pm 0.07$ | $0.38\pm 0.05$ | 0.27
EH Lib | – | 157861023 | $1.04\pm 0.08$ | $0.42\pm 0.06$ | 0.20
KZ Hya | 94033 | 188209486 | $1.15\pm 0.07$ | $0.39\pm 0.06$ | 0.19
RY Lep | 38882 | 93441696 | $1.28\pm 0.04$ | $0.29\pm 0.02$ | 0.08
SS Psc | – | 456857185 | $1.36\pm 0.06$ | $0.33\pm 0.05$ | 0.32
SX Phe | 223065 | 224285325 | $1.26\pm 0.06$ | $0.35\pm 0.06$ | 0.20
V1719 Cyg | 200925 | 290277380 | $1.25\pm 0.04$ | $0.30\pm 0.03$ | 0.30
VX Hya | – | 289711518 | $1.29\pm 0.05$ | $0.40\pm 0.06$ | 0.30
VZ Cnc | 73857 | 366632312 | $1.17\pm 0.06$ | $0.52\pm 0.06$ | 0.21
XX Cyg | – | 233310793 | $1.07\pm 0.07$ | $0.45\pm 0.07$ | 0.29
Table 2:  Parameters and p-value of fitted lognormal distribution for non-HADS stars. Star ID | Lognormal
---|---
Name | HD | TIC | $\mu$ | $\sigma$ | p-value
– | 129831 | 81003 | $0.93\pm 0.09$ | $0.47\pm 0.07$ | 0.35
– | 77914 | 975071 | $0.92\pm 0.09$ | $0.47\pm 0.07$ | 0.29
– | 112063 | 9591460 | $0.89\pm 0.09$ | $0.48\pm 0.07$ | 0.29
RX Cae | 28837 | 7808834 | $1.11\pm 0.07$ | $0.34\pm 0.05$ | 0.19
CV Phe | 13755 | 7245720 | $1.17\pm 0.07$ | $0.35\pm 0.05$ | 0.32
– | 21295 | 12524129 | $0.92\pm 0.09$ | $0.58\pm 0.07$ | 0.38
– | 79111 | 18658256 | $0.88\pm 0.08$ | $0.48\pm 0.06$ | 0.35
– | 86312 | 26957587 | $0.91\pm 0.08$ | $0.52\pm 0.07$ | 0.35
– | 181280 | 30624832 | $0.89\pm 0.09$ | $0.57\pm 0.07$ | 0.38
– | 44930 | 34737955 | $1.16\pm 0.07$ | $0.45\pm 0.07$ | 0.40
– | 185729 | 79659787 | $0.91\pm 0.09$ | $0.50\pm 0.08$ | 0.33
– | 25674 | 34197596 | $1.28\pm 0.07$ | $0.41\pm 0.06$ | 0.24
– | 113221 | 102192161 | $0.97\pm 0.08$ | $0.57\pm 0.08$ | 0.42
– | 180349 | 121729614 | $0.97\pm 0.09$ | $0.57\pm 0.09$ | 0.43
– | 216728 | 137796620 | $0.91\pm 0.09$ | $0.59\pm 0.08$ | 0.38
– | 113211 | 253296458 | $0.92\pm 0.08$ | $0.58\pm 0.08$ | 0.43
– | 31322 | 246902545 | $1.31\pm 0.06$ | $0.37\pm 0.05$ | 0.28
– | 32433 | 348792358 | $1.03\pm 0.08$ | $0.48\pm 0.07$ | 0.38
– | 56843 | 387235455 | $0.97\pm 0.08$ | $0.50\pm 0.07$ | 0.38
– | 38597 | 100531058 | $1.22\pm 0.07$ | $0.41\pm 0.06$ | 0.25
– | – | 448892817 | $0.88\pm 0.08$ | $0.49\pm 0.07$ | 0.29
– | 99302 | 458689740 | $0.97\pm 0.08$ | $0.50\pm 0.07$ | 0.41
V479 Tau | 24550 | 459908110 | $0.98\pm 0.08$ | $0.51\pm 0.07$ | 0.39
V353 Vel | 93298 | 106886169 | $1.21\pm 0.07$ | $0.38\pm 0.06$ | 0.29
– | 17341 | 122615966 | $1.26\pm 0.06$ | $0.41\pm 0.06$ | 0.29
– | 183281 | 137341551 | $1.13\pm 0.07$ | $0.51\pm 0.07$ | 0.32
– | 182895 | 159647185 | $1.37\pm 0.07$ | $0.42\pm 0.06$ | 0.36
– | 46722 | 172193026 | $1.28\pm 0.06$ | $0.34\pm 0.04$ | 0.24
– | 8043 | 196921106 | $1.3\pm 0.06$ | $0.46\pm 0.03$ | 0.38
CC Gru | 214441 | 161172103 | $0.87\pm 0.06$ | $0.45\pm 0.06$ | 0.16
– | 24572 | 242944780 | $1.32\pm 0.06$ | $0.41\pm 0.07$ | 0.31
– | – | 274038922 | $1.37\pm 0.06$ | $0.34\pm 0.05$ | 0.29
IN Dra | 191804 | 269697721 | $1.29\pm 0.06$ | $0.35\pm 0.05$ | 0.31
GW Dra | – | 329153513 | $1.31\pm 0.05$ | $0.35\pm 0.05$ | 0.27
CP Oct | 21190 | 348772511 | $1.32\pm 0.06$ | $0.33\pm 0.05$ | 0.27
IO Dra | 193138 | 403114672 | $1.34\pm 0.06$ | $0.34\pm 0.05$ | 0.31
– | 42005 | 408906554 | $1.17\pm 0.07$ | $0.46\pm 0.05$ | 0.33
DE Cmi | 67852 | 452982723 | $1.32\pm 0.06$ | $0.34\pm 0.04$ | 0.28
V435 Car | 44958 | 255548143 | $1.32\pm 0.06$ | $0.35\pm 0.05$ | 0.32
V1790 Ori | 290799 | 11361473 | $1.23\pm 0.07$ | $0.45\pm 0.06$ | 0.33

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