# Front-End Diarization for Percussion Separation in Taniavartanam of Carnatic
Music Concerts

###### Abstract

Instrument separation in an ensemble is a challenging task. In this work, we
address the problem of separating the percussive voices in the taniavartanam
segments of Carnatic music. In taniavartanam, a number of percussive
instruments play together or in tandem. Separation of instruments in regions
where only one percussion is present leads to interference and artifacts at
the output, as source separation algorithms assume the presence of multiple
percussive voices throughout the audio segment. We prevent this by first
subjecting the taniavartanam to diarization. This process results in
homogeneous clusters consisting of segments of either a single voice or
multiple voices. A cluster of segments with multiple voices is identified
using the Gaussian mixture model (GMM), which is then subjected to source
separation. A deep recurrent neural network (DRNN) based approach is used to
separate the multiple instrument segments. The effectiveness of the proposed
system is evaluated on a standard Carnatic music dataset. The proposed
approach provides close-to-oracle performance for non-overlapping segments and
a significant improvement over traditional separation schemes.

Index Terms—  Diarization, percussion separation, taniavartanam, Carnatic
music

##  1 Introduction

Carnatic music is one of the oldest music forms in the world. It is a
classical music form popular in India. Carnatic music attributes its origin to
Samveda [1]. Carnatic music has large repositories and is of great interest in
music information retrieval (MIR) [2, 3, 4, 5, 6, 7, 8, 9]. A concert in
Carnatic music comprises a lead vocal artist, a violinist, and percussion
instrument artists. The solo instrument performances attempt to mimic the
vocalist. Mridangam is the lead percussion instrument. A concert is made up of
a sequence of items known as alapana, composition, and taniavarnatnam
segments. The melody of different compositions has different rhythm patterns.
Taniavartanam refers to the solo performance by the percussion artists. It is
a “call and response” section where the percussion artists communicate via
different complex phrases. During taniavartanam, a percussion artist
challenges the other percussion artist with a particular complex phrase. The
other artist gives a fitting response to the challenge. Taniavartanam ends
with the overlapped section, where all the percussion artists play certain
phrases together. The instrument diarization system annotates the input
taniavartanam with relative instrument labels [10]. The duration of the
phrases used in taniavartanam is dependent on the duration of a cycle, and a
segment can be as short as 500 ms. This makes the diarization of taniavartanam
a challenging task.

Source separation deals with separating various sources from a mixed audio
signal. Percussion instrument separation has applications in different
research areas as follows; (i) analysis of musical signals to identify the
artist [5], (ii) to detect the percussive onsets [6], (iii) Akshara
transcription [9], and (iv) to perform other rhythmic [11] and melodic
analyses [12, 13]. Hence, percussive source separation in taniavartanam is an
important problem. Recently, methods based on deep neural networks for source
separation have been used [14, 15]. The approaches to source separation are
performed on the entire recording [16, 17]. The approaches in literature treat
the whole recording as a mixed signal and attempt to separate the sources in
the whole signal. Recently, authors in [18] use diarization and source
separation modules in tandem on the speech conversation recording that handles
mixtures with more speakers than microphones. The percussive voice111The words
“percussive voice” and “voice” are used interchangeably. separation system can
introduce artifacts in the separated output channels. Hence, the output
sources from the actual single source regions (i.e., the solo instrument
regions) get distorted due to artifacts introduced by the source separating
system.

A recording of a taniavartanam can be as long as 5 mins to 30 mins. The
overlapping segments account for 15-20% of the total duration, and the
remaining parts are percussion solo with instrument-homogeneous segments. The
objective of this work is to avoid performing source separation in the single
source regions, thereby retaining the original quality for the single source
regions. This work focuses on two majorly used percussion instruments in
taniavartanam; the mridangam and the ghatam. Our proposed approach is
motivated by the ideas in the speech domain [18]. To the best of our
knowledge, this paper presents for the first time a percussion separation
system using diarization as a front-end module. The proposed system has the
following stages. (i) Diarization: Obtain clusters of solo and mixed audio
segments. (ii) Identification: Obtain the actual identity for each cluster.
(iii) Percussion separation: Perform separation only for mixed audio segments.
Since the proposed system performs separation only in the mixed audio
segments, it retains the original signal quality for non-overlapped regions.

The paper is organized as follows. Section 2 describes the proposed
methodology. In Section 3, we describe the multi-channel dataset used to
demonstrate the effectiveness of the proposed system. Section 4 describes the
experimental setup, results, and analysis of the proposed system. Finally,
Section 5 concludes the paper.

Fig. 1: Block diagram of the proposed system.

##  2 Proposed Methodology

Fig. 1 shows the block diagram of the proposed system. The stages involved and
the individual modules used in the proposed system are explained in detail in
the following subsections.

###  2.1 Diarization of Taniavartanam

An agglomerative information bottleneck approach to diarization is a bottom-up
clustering approach based on information bottleneck (IB) principle [19]. It is
popular in the speaker diarization literature owing to its low runtime [20,
21, 22, 23]. Recently, we proposed a varying length information bottleneck
(VarIB) based diarization system [10]. This approach has low runtime and has
shown good performance in diarizing taniavartanam. We diarize the
taniavartanam to obtain the clusters for solo and mixed audio segments. The
VarIB approach initializes the clusters by dividing the whole audio recording
into short segments such that each segment contains an equal number of
strokes. This distributes instrument information uniformly across the
segments. Let $\mathbf{X_{v}}$ represent the short segments. A GMM is trained
using these segments. The components of the GMM form the relevant variables
represented by $\mathbf{Y}$. The VarIB algorithm arranges segments
$\mathbf{X_{v}}$ into clusters $\mathbf{C}$ using a bottom-up agglomerative
approach such that the relevant information $\mathbf{Y}$ is preserved. The
optimization equation is given as follows,

$\mathcal{F}=\mathrm{I}(\mathbf{Y},\mathbf{C})-\mathbf{\frac{1}{\beta}}\mathrm{I}(\mathbf{C},\mathbf{X_{v}})$
(1)

where $\beta$ is a Lagrange multiplier and $\mathrm{I(\cdot)}$ is the mutual
information between the variables. The bottom-up clustering is stopped when
the threshold on normalized mutual information (NMI) given by
$\frac{\mathrm{I}(\mathbf{Y,C})}{\mathrm{I}(\mathbf{X_{v},Y})}$ is reached
[20]. Finally, the segment boundaries are realigned to obtain the final
diarized output [24].

###  2.2 Cluster Identification

We classify the clusters obtained from VarIB based diarization to get their
actual identities. The VarIB system clusters the segments based on the timbre.
Since the timbre of the overlapped segments differs from the respective solo
instruments, the VarIB system forms a separate cluster for the overlapped
segments. We propose a two-level classification approach to obtain the actual
identities for each of the clusters. (i) First, the cluster with overlapping
segments is detected based on the cluster likelihood using a GMM trained on
separate overlapped data. (ii) We then remove the overlapped cluster222Cluster
with overlapping segments is referred to as an overlapped cluster. and
identify the cluster corresponding to ghatam using a GMM trained on separate
data consisting of segments from ghatam. We do not use the trained mridangam
model as it is the most dominating instrument and gets confused easily with
the overlapped segments.

It is important to note that the likelihood of the whole cluster is considered
during identification instead of segment-wise likelihoods. Mridangam is a
dominant instrument played for around 60-70% of the total time (45-50% in
solo, 15-20% in overlapped). It dominates even more in the overlapped
sections. Hence, it is difficult for a segment-wise classifier to detect the
mridangam segments. Thus, it is essential to first diarize to get the clusters
of audio segments and then classify them to get the actual identities. These
identities are required for obtaining final separated channel outputs. Only
the segments belonging to the overlapped cluster are fed to the source
separation stage.

###  2.3 Percussion Instrument Separation

The separation between the percussive sources is very challenging as the time-
frequency (T-F) structure shows significant similarity, and the strokes are
well-overlapped at certain rhythmic locations. We employ a deep recurrent
neural network (DRNN) with source-specific mask prediction for performing the
separation of two percussion instruments [6]. This framework was originally
proposed for singing voice separation [17]. Recently, this framework has shown
good performance on separating percussion from musical mixtures in [6] on the
Carnatic music dataset. The architecture consists of an input layer, hidden
layers with recurrent connections, and two output layers corresponding to the
mridangam and the ghatam. Magnitude spectrogram is used as the feature, and
the estimated mask is mixed with a mixture spectrogram for obtaining the
magnitude spectrogram estimates of the sources. This is then mixed with the
noisy phase to generate the individual channels.

The loss function is regularised not only to minimize the difference between
the estimated and the corresponding target percussive voices but also to
maximize the difference between the estimated and the other percussive voice,
owing to a very small timbre difference. We use a single model to learn masks
for both of the sources. The objective function for this discriminative
training is given by,

$||\widehat{y}_{m}-y_{m}||^{2}+||\widehat{y}_{g}-y_{g}||^{2}-\gamma(||\widehat{y}_{m}-y_{g}||^{2}+||\widehat{y}_{g}-y_{m}||^{2})$
(2)

where $\widehat{y}$ and $y$ are the estimated and the original magnitude
spectra, and $m$ and $g$ denote mridangam and ghatam, respectively. We want
the small timbre difference to be captured. Hence, the discriminative term is
added to the loss function. The magnitude spectrogram prediction is formulated
as a soft mask estimation problem where each T-F bin is classified as either
percussive or non-percussive voice with some probability. After percussion
separation, an instrument channel is formed in the non-overlapping part by
concatenating the segments of the instrument when it is present, and by
replacing the temporal positions by zeros when it is absent.

The proposed system is modular; hence each module can be replaced by other
suitable algorithms. In this paper, we use modules that have shown good
performance on the Carnatic music dataset.

##  3 Dataset description

We use live multi-channel audio recordings of the Carnatic music concerts from
the standard MusicBrainz [25] repository developed as a part of CompMusic
Project333CompMusic [accessed 04-Mar-2021]: http://compmusic.upf.edu/. Six
live concert recordings are chosen randomly that have mridangam and ghatam
played in the taniavartanam section. This section is manually extracted, and
the dataset is created by annotating the instrument change points and the
overlapped sections. For each of the taniavartanam, the channels corresponding
to mridangam and ghatam are mixed to form one audio in the test phase. The
details of the dataset are given in Table 1. We use one concert (AK) to train
the source separation model, and the rest are used for testing purpose.

Table 1: Details of the dataset. Durations of Mridangam (Mri), Ghatam (Ghat),
and overlapped part in each of the recordings in the dataset are mentioned.

Concert ID | Segments (mm:ss) |  Total Length (mm:ss)
---|---|---
Mri | Ghat | Overlap
AK | 06:13 | 03:52 | 01:36 | 11:41
DR | 08:29 | 04:26 | 02:09 | 15:04
MD | 03:13 | 02:06 | 01:28 | 06:47
RM | 06:22 | 05:16 | 01:57 | 13:35
RT | 03:47 | 03:28 | 01:47 | 09:02
SR | 09:18 | 07:47 | 04:47 | 21:52
Total | 37:22 | 26:55 | 13:44 | 01:18:01

We use the Tani-Dev dataset that was introduced in [10] to tune the
hyperparameters for diarization systems. This dataset contains ten
taniavartanam mono-channel recordings, which are also a part of the standard
MusicBrainz repository. The models for the cluster identification is also
trained on this dataset. All recordings are sampled at 44.1 kHz.

##  4 Experimental Studies

In this section, we demonstrate the effectiveness of the proposed approach on
standard Carnatic datasets.

###  4.1 Experimental Setup

The input to the diarization module is 19-dimensional MFCC features. The
threshold on NMI and $\beta$ for both IB444Code:
https://github.com/idiap/IBDiarization [20] and VarIB [10] systems were set to
0.4 and 10, respectively. The fixed duration segment length for the IB system
was set to 2 seconds, and the minimum number of strokes for the VarIB system
was set to 15. As there are two percussion instruments, the maximum number of
clusters for IB and VarIB was set to three. These hyperparameters are also the
same as used in [10]. The number of GMM components for the cluster
identification was set to three. For the source separation module, a 1024
point short-time Fourier transform is used as the feature. The DRNN
architecture555Code:
https://github.com/posenhuang/deeplearningsourceseparation has 500 nodes at
three hidden layers with recurrent connections (stacked RNN) and 513 nodes at
the output layer. ReLU non-linearity is applied at the output of each layer.
Mean squared error is used as the loss function, and the hyper-parameter
$\gamma$ is set at 0.08. All the hyperparameters were tuned to obtain the best
results on the development dataset.

Percussion instrument separation on the entire taniavartanam segment without
diarization is considered as the traditional system. The final output of the
proposed system consists of the concatenation of separated channels from the
overlapped clusters and corresponding segments from the non-overlapped
clusters. Note that all these segments are obtained from diarization.
Recordings of the individual channels form the oracle system for non-
overlapped segments.

###  4.2 Results and Discussion

We not only report the final separation performance but also report the
intermediate performances of each module in the proposed system. The results
are discussed in the following subsections.

Table 2: Diarization Error Rate (DER) for IB and VarIB systems on different
datasets.

Sys. |  Dev Set | Test Set
---|---|---
AK | DR | MD | RM | RT | SR | Avg.
IB | 17.9 | 7.2 | 33.4 | 30.1 | 5.4 | 3.8 | 29.3 | 20.0
VarIB | 13 | 6.9 | 13.5 | 18.9 | 6.5 | 3.8 | 13.3 | 10.6

####  4.2.1 Diarization Module

We use a standard tool from
NIST666https://github.com/nryant/dscore/blob/master/scorelib/md-eval-22.pl to
evaluate the diarization error rate (DER). DER is the sum of missed segments,
false alarms, and instrument errors. Since the silence is negligible in the
taniavartanam, voice activity detection is not needed. Hence, the missed
segments and false alarms are almost zero. The lower the DER, the better is
the clustering solution. We use a strict forgiveness collar of 0.15 sec in
evaluating DER [10]. The forgiveness collar is essential as it ignores both,
the errors made by the system and also the human errors made during ground
truth annotations within a specified window.

Table 2 shows the DER obtained on development and test datasets. The DER
reported for the development set is averaged over all the ten recordings,
while for the test set, we report individual DER for each recording. It can be
seen that the VarIB system outperforms the traditional IB system for most
recordings. The average DER for VarIB is absolute 9.4% (47% relative) better
than the baseline system on test data. This also confirms that varying length
segmentation yields a significantly better clustering solution [10].

Table 3: Average cluster purity and identification accuracy (Acc.) for VarIB
diarization followed by cluster identification.

Metric | Dev Set | Test Set
---|---|---
AK | DR | MD | RM | RT | SR | Avg.
Purity | 0.89 | 0.91 | 0.86 | 0.80 | 0.93 | 0.95 | 0.85 | 0.88
Acc. (%) | 86.0 | 90.7 | 85.5 | 79.8 | 92.7 | 95.3 | 85.2 | 88.2

####  4.2.2 Cluster Identification

The performance of the system after VarIB based diarization and cluster
identification is measured in terms of averaged cluster purity and
identification accuracy. Since the segments in a cluster can be partially
overlapped/pure, the purity and accuracy are reported in terms of time
duration using an open-source tool [26]. Purity is calculated as the ratio of
the sum of the duration of the most frequent segments in a cluster to the
duration of that cluster. Hence, a cluster with purity 1.0 is a perfectly pure
cluster. The accuracy is calculated in terms of percentage of the time
duration of the correctly identified taniavartanam.

Table 3 shows the average cluster purity and accuracy for each concert for the
cluster identification stage. The observed average cluster purity is 0.89 and
0.88 on development and test data, respectively. The GMM classifier shows a
good average accuracy of 86.0% and 88.2% on the train and test data,
respectively. This shows that the proposed GMM classifier detects all the
clusters obtained from the diarization module correctly. However, the reported
frame-level accuracy is not 100% because of a few short segments that get
wrongly clustered during the diarization stage. The exact value of these
errors can be obtained as 1-purity from Table 3.

####  4.2.3 Percussion Separation

We use blind source separation evaluation (BSS Eval) metric [27] to measure
the quality of the source separation. The results on the length-weighted
signal-to-distortion ratio (SDR) averaged across the segments (Global SDR) are
reported. This overall measure accounts for both the artifacts introduced and
the interfering source present in the separated channel. SDR considers the
overall quality, which includes the other BSS measures, namely, SAR (Signal to
Artifact Ratio) and SIR (Signal to Interference Ratio). Hence, these measures
are not reported separately in the paper. This evaluation measure requires the
knowledge of original sources for computing the separation quality. Since both
the channels are active for the overlapped audio, we can measure the SDR
value. The oracle system consists of source separation on the ground truth
segments, and the proposed system contains the same separation on the segments
with boundaries coming from the diarization system.

Table 4: Global SDR for overlapped segments of Proposed (diarized segments)
and Oracle (ground truth segments) systems. The GSDR for the traditional
system is same as oracle for overlapped segments.

System | Test Set
---|---
DR | MD | RM | RT | SR | Avg.
Proposed | 1.475 | 0.914 | 3.953 | 4.037 | 3.867 | 2.849
Oracle | 1.138 | 0.896 | 4.076 | 4.032 | 3.406 | 2.710

The GSDR of the proposed approach and the oracle system for overlapped
segments are shown in Table 4. Since the traditional system blindly separates
the whole recording, the GSDR values are the same as the oracle system for
overlapped segments. The GSDR values are inferior to other musical separation
tasks as these are live concert recordings, and there is a high similarity
between the T-F structures of the percussion instruments considered. Moreover,
the timbre for the same instrument varies across the concerts (based on the
base frequency, i.e., tonic). Note that the proposed method has better GSDR
values than the oracle system for some test sets. This is because few
overlapping parts are classified as non-overlapping, and for these segments,
the GSDR is lesser than the oracle average. As they are not included in the
GSDR computation only for the proposed system, the resulting measure will be
higher than the oracle.

We have to evaluate the system for non-overlapped segments separately. This is
because though the ground truth should ideally have zero amplitudes in the
inactive channel, it has traces of other channel and microphone noises. The
input mixture is the sum of this inactive channel and the instrument channel.
Providing a zero amplitude signal or a small noisy signal (the estimated
inactive output) causes problems for calculating BSS evaluation metrics as
they are computed by projecting the estimated source energy to the original
source sub-spaces [27]. Nevertheless, the GSDR can be computed for the
traditional system since both channels are active for non-overlapped segments
(reported later in this section).

Fig. 2: Subjective evaluation scores on non-overlapped segments.

For the non-overlapping segments, we perform qualitative analysis to compare
the performances of (i) the oracle system (ground truth segments and no
separation), (ii) the proposed system (diarization-based segments and no
separation), and (iii) the traditional system (ground truth segments with
separation). Subjective mean opinion score (MOS) [28] test is performed with
15 subjects listening to four randomly selected examples (2 from mridangam and
ghatam each) from non-overlapped clusters in each of the concerts. For each
example, having a length of 20 sec, three systems are evaluated after hiding
their identity. All the examples were randomly shuffled before presenting them
to the subject. The subject was asked to listen to the audio clips from all
the three different systems and then rate them between 1 to 5 (5 being the
highest) based on the perceived quality of audio. The final evaluation scores
averaged over all subjects are given in Fig. 2.

The average GSDR value of the traditional system on non-overlapping segments
is 2.727 (DR=1.776, MD=1.708, RM=3.855, RT=3.432, SR=2.862). GSDR for non-
overlapped clusters is very high (ideally infinity) for the proposed approach
as the output signal is similar to the respective individual channels. This is
also observed for most of the concerts in the subjective evaluation in Fig. 2.
It can be seen from the Fig. 2 that the scores of the proposed system are
close to that of the oracle system. The small average reduction in the
evaluation score for the proposed approach with respect to the oracle system
(0.27) is mainly due to the noise captured in the inactive channel, which
makes the sum of two channels different from the actual voices (active
channel). Hence, a significant improvement in performance over a traditional
separation system is achieved for the non-overlapping segments by
incorporating diarization and subsequent cluster identification.

Fig. 3: An example excerpt from non-overlapped mridangam segment. Outputs of
the proposed (left), ground truth (middle), and traditional separation (right)
systems. Observe that the artifacts are introduced in the output of
traditional system.

To observe the system artifact, we also plot the spectrogram for a non-
overlapped segment for all the three systems, as shown in Fig. 3. It can be
observed clearly in Fig. 3 that the artifacts are introduced by the
traditional system in the separated segment. Also, the traditional source
separation system would consider the whole non-overlapped section and blindly
separate it, generating the percussive voices for the inactive channel as
well. The proposed system does not separate the non-overlapped segments. The
performance gain achieved by the proposed approach is huge since it is
improper to separate a non-overlapping segment.

##  5 Conclusion

We propose a diarization-driven percussion separation pipeline for
taniavartanam. The advantage of this approach is that the systems in each of
the modules can be easily replaced with another algorithm. The proposed system
is able to reduce the artifacts arising in the traditional separation systems
owing to the separation performed in the non-overlapping segments. The
performance of the proposed system on the non-overlapping segments is close to
oracle. This suggests the importance of diarization as a front-end to reduce
the deterioration effects (artifacts and interferences) of traditional source
separation systems.

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# Heat flow and concentration of measure on
directed graphs with a lower Ricci curvature bound

Ryunosuke Ozawa  Department of Mathematics, National Defense Academy of Japan,
1-10-20 Hashirimizu, Yokosuka, 239-8686 Japan<EMAIL_ADDRESS>,  Yohei
Sakurai  Department of Mathematics, Saitama University, 255 Shimo-Okubo,
Sakura-ku, Saitama-City, Saitama, 338-8570, Japan
<EMAIL_ADDRESS>and  Taiki Yamada  Interdisciplinary Faculty
of Science and Engineering, Shimane University, 1060 Nishikawatsu-cho, Matsue,
Shimane, 690-8504, Japan<EMAIL_ADDRESS>
(Date: February 24, 2022)

###### Abstract.

In a previous work [13], the authors introduced a Lin-Lu-Yau type Ricci
curvature for directed graphs referring to the formulation of the Chung
Laplacian. The aim of this note is to provide a von Renesse-Sturm type
characterization of our lower Ricci curvature bound via a gradient estimate
for the heat semigroup, and a transportation inequality along the heat flow.
As an application, we will conclude a concentration of measure inequality for
directed graphs of positive Ricci curvature.

###### Key words and phrases:

Directed graph; Ricci curvature; Gradient estimate; Heat flow; Concentration
of measure; Functional inequality

###### 2010 Mathematics Subject Classification:

Primary 05C20, 05C12, 05C81, 53C21, 53C23

##  1\. Introduction

###  1.1. Main results

On a smooth Riemannian manifold, von Renesse-Sturm [14] have characterized a
lower Ricci curvature bound in terms of a gradient estimate for the heat
semigroup, and a transportation inequality along the heat flow. Münch-
Wojciechowski [11] have studied its discrete analogue, and produced a
characterization of a lower bound of the Ricci curvature introduced by Lin-Lu-
Yau [10] on (undirected) graphs (see also [12]).

In [13], the authors have introduced a Lin-Lu-Yau type Ricci curvature for
directed graphs inspired by the formulation of the Chung Laplacian ([3], [4]).
In the present note, we aim to extend the results of Münch-Wojciechowski [11]
to our directed setting. Let us introduce our main results (see Section 2 for
the precise meaning of the notations). Let $(V,\mu)$ be a simple, strongly
connected, finite weighted directed graph, where $V$ is the vertex set, and
$\mu:V\times V\to[0,\infty)$ is the (non-symmetric) edge weight. We denote by
$d:V\times V\to[0,\infty)$ the (non-symmetric) graph distance function on $V$,
and by $W$ the Wasserstein distance function. For $x,y\in V$ with $x\neq y$,
let $\kappa(x,y)$ stand for the Ricci curvature introduced in [13]. For
$f:V\to\mathbb{R}$, let $\mathrm{Lip}f$ denote its Lipschitz constant. We
denote by $\mathcal{L}$ the Chung Laplacian, and by $P_{t}$ the heat semigroup

(1.1) $P_{t}:=e^{-t\mathcal{L}}.$

Furthermore, let $p^{t}_{x}$ be the associated heat kernel measure at $x$.

In [13], the authors have derived a representation formula for the Ricci
curvature in terms of the Chung Laplacian (see Theorem 2.4 below). With the
help of such a formula, we first prove the following characterization theorem
in our directed setting:

######  Theorem 1.1.

Let $(V,\mu)$ denote a simple, strongly connected, finite weighted directed
graph. For $K\in\mathbb{R}$, the following are equivalent:

  1. (1)

$\inf_{x\neq y}\kappa(x,y)\geq K$;

  2. (2)

for all $f:V\to\mathbb{R}$ and $t>0$,

(1.2) $\mathrm{Lip}\,P_{t}f\leq e^{-Kt}\,\mathrm{Lip}\,f;$

  3. (3)

for all $x,y\in V$ and $t>0$,

(1.3) $W(p^{t}_{x},p^{t}_{y})\leq e^{-Kt}d(x,y).$

Theorem 1.1 has been obtained by Münch-Wojciechowski [11] in the undirected
case (see [11, Theorem 3.8]). Notice that in the undirected case, they have
proven it not only for finite graphs but also for infinite graphs.

Having Theorem 1.1 at hand, we will investigate the concentration of measure
phenomena with respect to the Perron measure $\mathfrak{m}$. Due to the lack
of symmetry of the distance function, we need to introduce the following
notion: For $x,y\in V$, we set

$\mathcal{D}(x,y):=\max\\{d(x,y),d(y,x)\\},\quad\mathcal{D}_{x}:=\sup_{y\in\mathcal{N}_{x}}\mathcal{D}(x,y),$

where $\mathcal{N}_{x}$ is the neighborhood of $x$. Note that
$\mathcal{D}_{x}\geq 1$ in general. In the undirected case, the distance
function $d$ is symmetric, and hence we possess $\mathcal{D}_{x}=1$. Taking
the asymmetry into account, we conclude the following concentration inequality
for directed graphs:

######  Theorem 1.2.

Let $(V,\mu)$ denote a simple, strongly connected, finite weighted directed
graph. For $K>0$ and $\Lambda\geq 1$, we assume $\inf_{x\neq y}\kappa(x,y)\geq
K$ and $\sup_{x\in V}\mathcal{D}_{x}\leq\Lambda$. Then for every $1$-Lipschitz
function $f:V\to\mathbb{R}$ we have

$\mathfrak{m}(\\{f\geq\mathfrak{m}(f)+r\\})\leq
e^{-\frac{Kr^{2}}{\Lambda^{2}}},$

where $\mathfrak{m}(f)$ is the mean of $f$ defined as

$\mathfrak{m}(f):=\sum_{x\in V}f(x)\mathfrak{m}(x).$

In the undirected case, Jost-Münch-Rose [8] have shown Theorem 1.2 based on
the method of [15] (see [8, Theorem 3.1]). Also, Fathi-Shu [6] have
(implicitly) shown a similar result for reversible Markov chains via
functional inequalities (see [6, Theorems 1.13 and 2.4], and cf. [5]).

###  1.2. Organization

In Section 2, we will review basics of directed graphs. In Section 3, we prove
Theorem 1.1. In Sections 4 and 5, we prove Theorem 1.2 in two different ways.
In Section 4, we do it by following the argument of [15] as in [8]. In Section
5, we examine functional inequalities such as transportation-information
inequality and transportation-entropy inequality, and apply them to another
proof of Theorem 1.2 as in [6].

##  2\. Preliminaries

We review basics on directed graphs. We refer to [13].

###  2.1. Directed graphs

Let $(G,\mu)$ be a finite weighted directed graph, namely, $G=(V,E)$ is a
finite directed graph, and $\mu:V\times V\to[0,\infty)$ is a function such
that $\mu(x,y)>0$ if and only if $x\rightarrow y$, where $x\rightarrow y$
means $(x,y)\in E$. The function $\mu$ is called the edge weight, and we write
$\mu(x,y)$ by $\mu_{xy}$. Note that $(G,\mu)$ is undirected if and only if
$\mu_{xy}=\mu_{yx}$ for all $x,y\in V$, and simple if and only if $\mu_{xx}=0$
for all $x\in V$. It is also called unweighted if $\mu_{xy}=1$ whenever
$x\rightarrow y$. The weighted directed graph can be written as $(V,\mu)$
since the full information of $E$ is included in $\mu$.

For $x\in V$, its outer neighborhood $N_{x}$, inner one
$\overleftarrow{N}_{x}$, and neighborhood $\mathcal{N}_{x}$ are defined as

$N_{x}:=\left\\{y\in V\mid x\rightarrow
y\right\\},\quad\overleftarrow{N}_{x}:=\left\\{y\in V\mid y\rightarrow
x\right\\},\quad\mathcal{N}_{x}:=N_{x}\cup\overleftarrow{N}_{x},$

respectively.

A sequence $\left\\{x_{i}\right\\}_{i=0}^{l}$ of vertices is called a directed
path from $x$ to $y$ if $x_{0}=x,\,x_{l}=y$ and $x_{i}\rightarrow x_{i+1}$ for
all $i=0,\dots,l-1$, where $l$ is called its length. $(V,\mu)$ is said to be
strongly connected if for any $x,y\in V$, there is a directed path from $x$ to
$y$. For strongly connected $(V,\mu)$, the (non-symmetric) distance function
$d:V\times V\to[0,\infty)$ is defined as follows: $d(x,y)$ is defined to be
the minimum of the lengths of directed paths from $x$ to $y$.

For $f:V\to\mathbb{R}$, its Lipschitz constant is defined by

$\mathrm{Lip}f:=\sup_{x\neq y}\nabla_{xy}f,$

where $\nabla_{xy}$ is the gradient operator defined as

$\nabla_{xy}f:=\frac{f(y)-f(x)}{d(x,y)}.$

For $L>0$, $f$ is said to be $L$-Lipschitz if

$\mathrm{Lip}f\leq L.$

Let $\mathrm{Lip}_{L}(V)$ stand for the set of all $L$-Lipschitz functions.

###  2.2. Chung Laplacian

Hereafter, let $(V,\mu)$ be a simple, strongly connected, finite weighted
directed graph. In this subsection, we review the formulation of the Chung
Laplacian introduced in [3], [4]. The transition probability kernel $P:V\times
V\to[0,1]$ is defined as

$P(x,y):=\frac{\mu_{xy}}{\mu(x)},$

where

$\mu(x):=\sum_{y\in V}\mu_{xy}.$

Since $(V,\mu)$ is finite and strongly connected, the Perron-Frobenius theorem
ensures that there is a unique (up to scaling) positive function
$m:V\to(0,\infty)$ such that

(2.1) $m(x)=\sum_{y\in V}m(y)P(y,x).$

A probability measure $\mathfrak{m}:V\to(0,1]$ satisfying (2.1) is called the
Perron measure (or the stationary probability measure).

Let $\mathfrak{m}$ be the Perron measure. For a non-empty subset
$\Omega\subset V$, we set

$\mathfrak{m}(\Omega):=\sum_{x\in\Omega}\mathfrak{m}(x).$

The reverse transition probability kernel $\overleftarrow{P}:V\times
V\to[0,1]$, and the mean transition probability kernel $\mathcal{P}:V\times
V\to[0,1]$ are defined as

$\overleftarrow{P}(x,y):=\frac{\mathfrak{m}(y)}{\mathfrak{m}(x)}P(y,x),\quad\mathcal{P}:=\frac{1}{2}(P+\overleftarrow{P}).$

Let $\mathcal{F}$ be the set of all functions on $V$. The Chung Laplacian
$\mathcal{L}:\mathcal{F}\to\mathcal{F}$ is given by

$\mathcal{L}f(x):=f(x)-\sum_{y\in V}\mathcal{P}(x,y)f(y).$

We will also use the negative Laplacian $\Delta:\mathcal{F}\to\mathcal{F}$
defined by

$\Delta:=-\mathcal{L}.$

We define a function $\mathfrak{m}:V\times V\to[0,\infty)$ by

$\mathfrak{m}(x,y):=\frac{1}{2}(\mathfrak{m}(x)P(x,y)+\mathfrak{m}(y)P(y,x))=\mathfrak{m}(x)\mathcal{P}(x,y).$

We write $\mathfrak{m}(x,y)$ by $\mathfrak{m}_{xy}$. The following properties
hold: (1) $\mathfrak{m}_{xy}=\mathfrak{m}_{yx}$; (2) $\mathfrak{m}_{xy}>0$ if
and only if $y\in\mathcal{N}_{x}$ (or equivalently, $x\in\mathcal{N}_{y}$);
(3) $\mathcal{P}(x,y)=\mathfrak{m}_{xy}/\mathfrak{m}(x)$.

The inner product on $\mathcal{F}$ is defined by

$(f_{0},f_{1}):=\sum_{x\in
V}f_{0}(x)f_{1}(x)\mathfrak{m}(x)=\mathfrak{m}(f_{0}f_{1}).$

Also, the $\Gamma$-operator is defined as

$\Gamma(f_{0},f_{1}):=\frac{1}{2}\left(\Delta(f_{0}\,f_{1})-f_{0}\,\Delta
f_{1}-f_{1}\,\Delta f_{0}\right),\quad\Gamma(f):=\Gamma(f,f).$

By direct computations, we see the following (cf. [13, Proposition 7.1]):

$\Gamma(f_{0},f_{1})(x)=\frac{1}{2}\sum_{y\in
V}(f_{0}(y)-f_{0}(x))(f_{1}(y)-f_{1}(x))\mathcal{P}(x,y).$

We possess the following integration by parts formula (see e.g., [13,
Proposition 2.4]):

######  Proposition 2.1.

Let $\Omega\subset V$ be a non-empty subset. Then for all
$f_{0},f_{1}:V\to\mathbb{R}$,

$\displaystyle\sum_{x\in\Omega}\mathcal{L}f_{0}(x)f_{1}(x)\mathfrak{m}(x)$
$\displaystyle=\frac{1}{2}\sum_{x,y\in\Omega}(f_{0}(y)-f_{0}(x))(f_{1}(y)-f_{1}(x))\mathfrak{m}_{xy}$
$\displaystyle\quad-\sum_{x\in\Omega}\sum_{y\in
V\setminus\Omega}(f_{0}(y)-f_{0}(x))f_{1}(x)\mathfrak{m}_{xy}.$

In particular,

$(\mathcal{L}f_{0},f_{1})=\mathfrak{m}(\Gamma(f_{0},f_{1}))=(f_{0},\mathcal{L}f_{1}).$

Let $P_{t}$ be the heat semigroup defined as (1.1) such that $P_{t}f$ solves

$\begin{cases}\partial_{t}u=\Delta u,\\\ u|_{t=0}=f.\end{cases}$

From Proposition 2.1 we can derive

(2.2) $(P_{t}f_{0},f_{1})=(f_{0},P_{t}f_{1}).$

The heat kernel measure $p^{t}_{x}$ is defined as

$p^{t}_{x}:=\frac{\mathfrak{m}}{\mathfrak{m}(x)}P_{t}\delta_{x}$

for the Dirac measure $\delta_{x}$ at $x$. The equality (2.2) yields

(2.3) $P_{t}f(x)=\sum_{y\in V}p^{t}_{x}(y)f(y).$

Furthermore, we see $P_{t}1_{V}=1_{V}$ by (2.2), and hence $p^{t}_{x}$ is a
probability measure.

###  2.3. Optimal transport theory

We next recall the basics of the optimal transport theory (cf. [16]). For two
probability measures $\nu_{0},\nu_{1}$ on $V$, a probability measure
$\pi:V\times V\to[0,\infty)$ is called a coupling of $(\nu_{0},\nu_{1})$ if

$\sum_{y\in V}\pi(x,y)=\nu_{0}(x),\quad\sum_{x\in V}\pi(x,y)=\nu_{1}(y).$

Let $\Pi(\nu_{0},\nu_{1})$ stand for the set of all couplings of
$(\nu_{0},\nu_{1})$. The Wasserstein distance from $\nu_{0}$ to $\nu_{1}$ is
defined as

(2.4) $W(\nu_{0},\nu_{1}):=\inf_{\pi\in\Pi(\nu_{0},\nu_{1})}\sum_{x,y\in
V}d(x,y)\pi(x,y),$

which is a (non-symmetric) distance function on the set of all probability
measures on $V$.

The following Kantorovich-Rubinstein duality formula is well-known (cf. [16,
Theorem 5.10 and Particular Cases 5.4 and 5.16]):

######  Proposition 2.2.

For any two probability measures $\nu_{0},\nu_{1}$ on $V$, we have

$W(\nu_{0},\nu_{1})=\sup_{f\in\mathrm{Lip}_{1}(V)}\sum_{x\in
V}f(x)\left(\nu_{1}(x)-\nu_{0}(x)\right).$

######  Remark 2.3.

In the discussion of [16, Particular Cases 5.4], the symmetry for cost
functions is not required, and hence Proposition 2.2 holds in our directed
setting.

###  2.4. Ricci curvature

In this subsection, we recall the formulation of the Ricci curvature
introduced in [13]. For $\varepsilon\in[0,1]$, and for $x,y\in V$ with $x\neq
y$, we set

$\kappa_{\varepsilon}(x,y):=1-\frac{W(\nu^{\varepsilon}_{x},\nu^{\varepsilon}_{y})}{d(x,y)},$

where $\nu^{\varepsilon}_{x}:V\to[0,1]$ is a probability measure defined by

$\nu^{\varepsilon}_{x}(z)=(1-\varepsilon)\delta_{x}(z)+\varepsilon\,\mathcal{P}(x,z).$

The authors [13] have introduced the Ricci curvature as follows (see [13,
Definition 3.6]):

$\kappa(x,y):=\lim_{\varepsilon\to
0}\frac{\kappa_{\varepsilon}(x,y)}{\varepsilon},$

which is well-defined (see [13, Lemmas 3.2 and 3.4, and Definition 3.6]). In
the undirected case, this is nothing but the Lin-Lu-Yau Ricci curvature in
[10].

We have the following representation formula, which has been established by
Münch-Wojciechowski [11] in the undirected case (see [13, Theorem 3.10] and
[11, Theorem 2.1]):

######  Theorem 2.4 ([11], [13]).

$\kappa(x,y)=\inf_{f\in\mathcal{F}_{xy}}\nabla_{xy}\mathcal{L}f,$

where

$\mathcal{F}_{xy}:=\\{f\in\mathrm{Lip}_{1}(V)\mid\nabla_{xy}f=1\\}.$

##  3\. Heat flow

In this section, we give a proof of Theorem 1.1.

###  3.1. Curvature bound and gradient estimate

In this subsection, we show the equivalence of (1) and (1.2) in Theorem 1.1.

######  Proposition 3.1.

For $K\in\mathbb{R}$, the following are equivalent:

  1. (1)

$\inf_{x\neq y}\kappa(x,y)\geq K$;

  2. (2)

for all $f:V\to\mathbb{R}$ and $t>0$, we have (1.2).

###### Proof.

We begin with the implication from (1) to (2). For a fixed $x,y\in V$, and
$f:V\to\mathbb{R}$, it is enough to prove

$\nabla_{xy}P_{t}f\leq e^{-Kt}\mathrm{Lip}f$

for every $t>0$. If $f$ is constant, then this immediately follows from
$P_{t}1_{V}=1_{V}$. Thus we may assume that $f$ is non-constant; in
particular, $\mathrm{Lip}f>0$. We will prove it by contradiction. Suppose that
there exists $t>0$ such that

$e^{Kt}\nabla_{xy}P_{t}f>\mathrm{Lip}f.$

We here notice that

$e^{Kt}\nabla_{xy}P_{t}f|_{t=0}=\nabla_{xy}f\leq\mathrm{Lip}f.$

In this case, there is $t_{0}>0$ such that

$e^{Kt_{0}}\nabla_{xy}P_{t_{0}}f>\mathrm{Lip}f,\quad\partial_{t}(e^{Kt}\nabla_{xy}P_{t}f)|_{t=t_{0}}>0.$

Due to Theorem 2.4,

$0<\frac{e^{-Kt_{0}}}{\nabla_{xy}P_{t_{0}}f}\partial_{t}(e^{Kt}\nabla_{xy}P_{t}f)|_{t=t_{0}}=\left(K-\nabla_{xy}\mathcal{L}\frac{P_{t_{0}}f}{\nabla_{xy}P_{t_{0}}f}\right)\leq\left(K-\kappa(x,y)\right),$

and hence $\kappa(x,y)<K$. This contradicts with the curvature bound.

We next show the opposite direction. By (1.2), for any $f\in\mathcal{F}_{xy}$,

$\nabla_{xy}\mathcal{L}f=-\partial_{t}\nabla_{xy}P_{t}f|_{t=0}=\lim_{t\to
0}\frac{1}{t}(\nabla_{xy}f-\nabla_{xy}P_{t}f)\geq\lim_{t\to
0}\frac{1}{t}(1-e^{-Kt})=K,$

here we used $\mathrm{Lip}f=1$. Theorem 2.4 leads us to the desired
conclusion. $\Box$

###  3.2. Gradient estimate and transportation inequality

Next, we prove the equivalence of (1.2) and (1.3), and conclude Theorem 1.1.

######  Proposition 3.2.

For $K\in\mathbb{R}$, the following are equivalent:

  1. (1)

For all $f:V\to\mathbb{R}$ and $t>0$, we have (1.2);

  2. (2)

for all $x,y\in V$ and $t>0$, we have (1.3).

###### Proof.

We start with the implication from (1) to (2). By (2.3) and (1.2), it holds
that

$\displaystyle\sum_{z\in V}f(z)(p^{t}_{y}(z)-p^{t}_{x}(z))$
$\displaystyle=P_{t}f(y)-P_{t}f(x)$
$\displaystyle\leq\mathrm{Lip}P_{t}f\,d(x,y)\leq
e^{-Kt}\mathrm{Lip}f\,d(x,y)\leq e^{-Kt}\,d(x,y)$

for every $f\in\mathrm{Lip}_{1}(V)$. With the help of Proposition 2.2, we
arrive at (1.3).

We now consider the opposite one. We fix $f:V\to\mathbb{R}$. If $f$ is
constant, then the desired assertion is trivial by $P_{t}1_{V}=1_{V}$. If $f$
is non-constant, then $\mathrm{Lip}f$ is positive, and hence we can define
$g\in\mathrm{Lip}_{1}(V)$ by

$g:=\frac{f}{\mathrm{Lip}f}.$

Therefore, (2.3), Proposition 2.2 and (1.3) imply

$\displaystyle P_{t}f(y)-P_{t}f(x)$
$\displaystyle=\mathrm{Lip}f(P_{t}g(y)-P_{t}g(x))=\mathrm{Lip}f\sum_{z\in
V}g(z)(p^{t}_{y}(z)-p^{t}_{x}(z))$
$\displaystyle\leq\mathrm{Lip}f\,W(p^{t}_{x},p^{t}_{y})\leq
e^{-Kt}\mathrm{Lip}fd(x,y).$

By dividing the both sides by $d(x,y)$, we complete the proof. $\Box$

We are now in a position to conclude Theorem 1.1.

###### Proof of Theorem 1.1.

Theorem 1.1 is a direct consequence of Propositions 3.1 and 3.2. $\Box$

###  3.3. Characterization via heat flow

In the undirected case, Münch-Wojciechowski [11] have formulated not only the
characterization of lower Ricci curvature bound but also that of Ricci
curvature itself (see [11, Theorem 5.8]). In our directed setting, we also
have the following characterization of Münch-Wojciechowski type:

######  Theorem 3.3.

$\kappa(x,y)=\lim_{t\to
0}\frac{1}{t}\left(1-\frac{W(p^{t}_{x},p^{t}_{y})}{d(x,y)}\right).$

We can prove Theorem 3.3 by the same argument as in the proof of [11, Theorem
5.8]. We omit the proof.

##  4\. Concentration of measure

In this section, we will prove Theorem 1.2 along the line of the proof of [8,
Theorem 3.1].

###  4.1. Laplace functional estimates

For $\lambda\geq 0$, the Laplace functional (or moment generating functional)
is defined as follows (see e.g., [9, Subsection 1.6], [2, Subsection 2.1]):

$E(\lambda):=\sup_{f\in\mathrm{Lip}_{1}(V)}\mathfrak{m}(e^{\lambda f}),$

where the supremum is taken over all $f\in\mathrm{Lip}_{1}(V)$ with
$\mathfrak{m}(f)=0$. In this subsection, we give an upper bound of the Laplace
functional under the same setting as in Theorem 1.2. To do so, we prepare the
following lemma (cf. [15]):

######  Lemma 4.1.

For all $\lambda\geq 0$ and $f:V\to\mathbb{R}$ we have

$\mathfrak{m}(\Gamma(f,e^{\lambda f}))\leq\lambda(e^{\lambda f},\Gamma(f)).$

###### Proof.

We see

$\displaystyle\mathfrak{m}(\Gamma(f,e^{\lambda f}))$
$\displaystyle=\frac{1}{2}\sum_{x,y\in V}(f(y)-f(x))(e^{\lambda
f(y)}-e^{\lambda f(x)})\mathfrak{m}_{xy}$
$\displaystyle=\sum_{f(y)>f(x)}(f(y)-f(x))(e^{\lambda f(y)}-e^{\lambda
f(x)})\mathfrak{m}_{xy}.$

We now recall the following elementary inequality (see e.g., [9, Corollary
5.8]): For all $s>t$,

$\frac{e^{s}-e^{t}}{s-t}\leq\frac{e^{s}+e^{t}}{2}.$

If $f(y)>f(x)$, then

$\frac{e^{\lambda f(y)}-e^{\lambda
f(x)}}{f(y)-f(x)}\leq\lambda\,\frac{e^{\lambda f(y)}+e^{\lambda f(x)}}{2}.$

It follows that

$\displaystyle\mathfrak{m}(\Gamma(f,e^{f}))$
$\displaystyle\leq\frac{\lambda}{2}\sum_{f(y)>f(x)}\left(e^{\lambda
f(y)}+e^{\lambda f(x)}\right)\,(f(y)-f(x))^{2}\mathfrak{m}_{xy}$
$\displaystyle=\frac{\lambda}{2}\sum_{x,y\in V}e^{\lambda
f(y)}\,(f(y)-f(x))^{2}\mathfrak{m}_{xy}=\lambda(e^{\lambda f},\Gamma(f)).$

This proves the lemma. $\Box$

We now state the desired assertion:

######  Proposition 4.2.

For $K>0$ and $\Lambda\geq 1$, we assume $\inf_{x\neq y}\kappa(x,y)\geq K$ and
$\sup_{x\in V}\mathcal{D}_{x}\leq\Lambda$. Then for every $\lambda\geq 0$,

$E(\lambda)\leq e^{\frac{\lambda^{2}\Lambda^{2}}{4K}}.$

###### Proof.

Let $f\in\mathrm{Lip}_{1}(V)$ with $\mathfrak{m}(f)=0$. Using Proposition 2.1
and Lemma 4.1, we obtain

$\partial_{t}\,\mathfrak{m}(e^{\lambda
P_{t}f})=-\lambda\,(\mathcal{L}P_{t}f,e^{\lambda
P_{t}f})=-\lambda\,\mathfrak{m}(\Gamma(P_{t}f,e^{\lambda
P_{t}f}))\geq-\lambda^{2}(e^{\lambda P_{t}f},\Gamma(P_{t}f)).$

By virtue of Theorem 1.1,

$\displaystyle\Gamma(P_{t}f)$ $\displaystyle=\frac{1}{2}\sum_{x,y\in
V}(P_{t}f(y)-P_{t}f(x))^{2}\mathfrak{m}_{xy}$
$\displaystyle\leq\frac{1}{2}(\mathrm{Lip}P_{t}f)^{2}\sum_{x,y\in
V}\mathcal{D}(x,y)^{2}\mathfrak{m}_{xy}\leq\frac{1}{2}\Lambda^{2}\,e^{-2Kt}(\mathrm{Lip}f)^{2}\leq\frac{1}{2}\Lambda^{2}\,e^{-2Kt}.$

By combining the above inequalities,

$\partial_{t}\,\mathfrak{m}(e^{\lambda
P_{t}f})\geq-\frac{\lambda^{2}\Lambda^{2}}{2}e^{-2Kt}\mathfrak{m}(e^{\lambda
P_{t}f}),$

and hence

$\log\mathfrak{m}(e^{\lambda P_{t}f})-\log\mathfrak{m}(e^{\lambda
f})\geq\frac{\lambda^{2}\Lambda^{2}}{4K}\left(e^{-2Kt}-1\right).$

By letting $t\to\infty$ we arrive at

$\mathfrak{m}(e^{\lambda f})\leq e^{\frac{\lambda^{2}\Lambda^{2}}{4K}}.$

Here we used $\mathfrak{m}(e^{\lambda P_{t}f})\to 1$ as $t\to\infty$, which is
a consequence of $\mathfrak{m}(f)=0$ and the fact that $\mathfrak{m}$ is a
probability measure. This completes the proof. $\Box$

###  4.2. Concentration inequalities

Let us recall the following Chernoff bounding method (see e.g., [9,
Proposition 1.14], [2, Subsection 2.1]):

######  Proposition 4.3.

Let $c>0$. If

$E(\lambda)\leq e^{\frac{\lambda^{2}}{2c}}$

for all $\lambda\geq 0$, then

$\mathfrak{m}(\\{f\geq\mathfrak{m}(f)+r\\})\leq e^{-\frac{cr^{2}}{2}}.$

Now, one can now derive Theorem 1.2.

###### Proof of Theorem 1.2.

Proposition 4.2 together with Proposition 4.3 with $c=2K/\Lambda^{2}$ implies
Theorem 1.2. $\Box$

##  5\. Functional inequalities

Here we discuss several functional inequalities, and give another proof of
Theorem 1.2.

###  5.1. Transportation-information inequality

In this subsection, we examine a transportation-information inequality. We
first show the following lemma (cf. [6, Lemma 5.1]):

######  Lemma 5.1.

For $K>0$ and $\Lambda\geq 1$, we assume $\inf_{x\neq y}\kappa(x,y)\geq K$ and
$\sup_{x\in V}\mathcal{D}_{x}\leq\Lambda$. Then for every probability density
$\rho:V\to[0,\infty)$ $($i.e., $\mathfrak{m}(\rho)=1$$)$ we have

$W(\mathfrak{m},\rho\mathfrak{m})\leq\frac{\Lambda}{2K}\sum_{x,y\in
V}|\rho(y)-\rho(x)|\mathfrak{m}_{xy}.$

###### Proof.

Proposition 2.2 can be written as

$W(\mathfrak{m},\rho\mathfrak{m})=\sup_{f\in\mathrm{Lip}_{1}(V)}(f,\rho),$

where the supremum is taken over all $f\in\mathrm{Lip}_{1}(V)$ with
$\mathfrak{m}(f)=0$. By Proposition 2.1,

$W(\mathfrak{m},\rho\mathfrak{m})=-\sup_{f\in\mathrm{Lip}_{1}(V)}\int^{\infty}_{0}\,\frac{d}{dt}(P_{t}f,\rho)dt=\sup_{f\in\mathrm{Lip}_{1}(V)}\int_{0}^{\infty}\,\mathfrak{m}(\Gamma(P_{t}f,\rho))\,dt.$

Let us fix $f\in\mathrm{Lip}_{1}(V)$ with $\mathfrak{m}(f)=0$. Theorem 1.1
tells us that

$\displaystyle\int_{0}^{\infty}\,\mathfrak{m}(\Gamma(P_{t}f,\rho))\,dt$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\,\sum_{x,y\in
V}(P_{t}f(y)-P_{t}f(x))(\rho(y)-\rho(x))\mathfrak{m}_{xy}\,dt$
$\displaystyle\leq\frac{1}{2}\int_{0}^{\infty}{\mathrm{Lip}P_{t}f\,dt\sum_{x,y\in
V}\mathcal{D}(x,y)|\rho(y)-\rho(x)|\mathfrak{m}_{xy}}$
$\displaystyle\leq\frac{\Lambda}{2K}\sum_{x,y\in
V}|\rho(y)-\rho(x)|\mathfrak{m}_{xy}.$

This proves the lemma. $\Box$

For a probability density $\rho:V\to[0,\infty)$, the Fisher information is
defined by

$\mathcal{I}(\rho):=4\mathfrak{m}(\Gamma(\sqrt{\rho}))=2\sum_{x,y\in
V}\left(\sqrt{\rho(y)}-\sqrt{\rho(x)}\right)^{2}\mathfrak{m}_{xy}.$

We now state our desired inequality (cf. [6, Theorem 1.13]).

######  Theorem 5.2.

For $K>0$ and $\Lambda\geq 1$, we assume $\inf_{x\neq y}\kappa(x,y)\geq K$ and
$\sup_{x\in V}\mathcal{D}_{x}\leq\Lambda$. Then for every probability density
$\rho:V\to[0,\infty)$ we have

$W(\mathfrak{m},\rho\mathfrak{m})^{2}\leq\frac{\Lambda^{2}}{2K^{2}}\mathcal{I}(\rho)\left(1-\frac{1}{8}\mathcal{I}(\rho)\right)\leq\frac{\Lambda^{2}}{2K^{2}}\mathcal{I}(\rho).$

###### Proof.

Fathi-Shu [6] have proved a similar result for reversible Markov chains (see
[6, Theorem 1.13]). We will prove it along the line of their argument. Since
$\rho$ is a probability density, we see

$\displaystyle\sum_{x,y\in
V}\left(\sqrt{\rho(y)}+\sqrt{\rho(x)}\right)^{2}\mathfrak{m}_{xy}$
$\displaystyle=\sum_{x,y\in
V}\left(2\rho(y)+2\rho(x)-\left(\sqrt{\rho(y)}-\sqrt{\rho(x)}\right)^{2}\right)\mathfrak{m}_{xy}$
$\displaystyle=4-\frac{1}{2}\mathcal{I}(\rho).$

From Lemma 5.1 we deduce

$\displaystyle W(\mathfrak{m},\rho\mathfrak{m})$
$\displaystyle\leq\frac{\Lambda}{2K}\sum_{x,y\in
V}|\rho(y)-\rho(x)|\mathfrak{m}_{xy}$
$\displaystyle=\frac{\Lambda}{2K}\sum_{x,y\in
V}\left|\sqrt{\rho(y)}-\sqrt{\rho(x)}\right|\left(\sqrt{\rho(y)}+\sqrt{\rho(x)}\right)\mathfrak{m}_{xy}$
$\displaystyle\leq\frac{\Lambda}{2K}\sqrt{\mathcal{I}(\rho)}\sqrt{\frac{1}{2}\underset{x,y\in
V}{\sum}\hskip
2.84526pt\left(\sqrt{\rho(y)}+\sqrt{\rho(x)}\right)^{2}\mathfrak{m}_{xy}}\leq\frac{\Lambda}{2K}\sqrt{\mathcal{I}(\rho)}\sqrt{2-\frac{1}{4}\mathcal{I}(\rho)}.$

This completes the proof. $\Box$

###  5.2. Transportation-entropy inequality

We next investigate a transportation-entropy inequality. For a probability
density $\rho:V\to[0,\infty)$, its relative entropy is defined by

$\mathcal{E}(\rho):=\mathfrak{m}(\rho\log\rho).$

We notice the following characterization (see e.g., [9, (5.13)]):

(5.1) $\mathcal{E}(\rho)=\sup_{g}(g,\rho),$

where the supremum is taken over all $g:V\to\mathbb{R}$ with
$\mathfrak{m}(e^{g})\leq 1$.

We verify the following Bobkov-Götze type criterion due to the lack of
symmetry of the distance function (cf. [1, Theorem 1.3], [9, Proposition
6.1]):

######  Lemma 5.3.

For $c>0$, the following are equivalent:

  1. (1)

For every probability density $\rho:V\to[0,\infty)$ we have

(5.2) $W(\mathfrak{m},\rho\mathfrak{m})^{2}\leq\frac{2}{c}\mathcal{E}(\rho);$

  2. (2)

for all $\lambda\geq 0$,

(5.3) $E(\lambda)\leq e^{\frac{\lambda^{2}}{2c}}.$

###### Proof.

Let us show the implication from (5.2) to (5.3). Fix $f\in\mathrm{Lip}_{1}(V)$
with $\mathfrak{m}(f)=0$, and set

(5.4) $g_{f}:=\lambda
f-\frac{\lambda^{2}}{2c},\quad\rho_{f}:=\frac{e^{g_{f}}}{\mathfrak{m}(e^{g_{f}})}.$

By Proposition 2.2,

$(f,\rho_{f})\leq
W(\mathfrak{m},\rho_{f}\mathfrak{m})\leq\sqrt{\frac{2}{c}\mathcal{E}(\rho_{f})}\leq\frac{\lambda}{2c}+\frac{1}{\lambda}\mathcal{E}(\rho_{f}),$

and hence $(g_{f},\rho_{f})\leq\mathcal{E}(\rho_{f})$. On the other hand,
straightforward computations imply

$\mathcal{E}(\rho_{f})=(g_{f},\rho_{f})-\log\mathfrak{m}(e^{g_{f}}).$

Therefore, $\mathfrak{m}(e^{g_{f}})\leq 1$, which is equivalent to

$\mathfrak{m}(e^{\lambda f})\leq e^{\frac{\lambda^{2}}{2c}}.$

We have shown the desired estimate.

We prove the opposite one. Fix a probability density $\rho$. Proposition 2.2
can be written as

(5.5)
$W(\mathfrak{m},\rho\mathfrak{m})=\sup_{f\in\mathrm{Lip}_{1}(V)}(f,\rho),$

where the supremum is taken over all $f\in\mathrm{Lip}_{1}(V)$ with
$\mathfrak{m}(f)=0$. We also fix such a $f$, and define $g_{f}$ as (5.4). From
(5.3) we derive $\mathfrak{m}(e^{g_{f}})\leq 1$. In view of (5.1),

$\lambda(f,\rho)-\frac{\lambda^{2}}{2c}=(g_{f},\rho)\leq\mathcal{E}(\rho)$

By letting $\lambda\to\sqrt{2c\mathcal{E}(\rho)}$, we arrive at

(5.6) $(f,\rho)\leq\sqrt{\frac{2}{c}\mathcal{E}(\rho)}.$

Combining (5.5) and (5.6), we obtain (5.2). We complete the proof. $\Box$

Proposition 4.2 together with Lemma 5.3 yields the following transportation-
entropy inequality:

######  Theorem 5.4.

For $K>0$ and $\Lambda\geq 1$, we assume $\inf_{x\neq y}\kappa(x,y)\geq K$ and
$\sup_{x\in V}\mathcal{D}_{x}\leq\Lambda$. Then for every probability density
$\rho:V\to[0,\infty)$ we have

$W(\mathfrak{m},\rho\mathfrak{m})^{2}\leq\frac{2\Lambda^{2}}{K}\mathcal{E}(\rho).$

###  5.3. Relation between functional inequalities

We finally mention the relation between the transportation-information
inequality and the transportation-entropy inequality. In order to do so, we
prepare the following (cf. [6, Lemma 2.3]):

######  Lemma 5.5.

For every $f:V\to\mathbb{R}$ we have

$\mathfrak{m}(\Gamma(e^{f}))\leq(e^{2f},\Gamma(f)).$

###### Proof.

It holds that

$\mathfrak{m}(\Gamma(e^{f}))=\frac{1}{2}\sum_{x,y\in
V}(e^{f(y)}-e^{f(x)})^{2}\,\mathfrak{m}_{xy}=\sum_{f(y)>f(x)}(e^{f(y)}-e^{f(x)})^{2}\,\mathfrak{m}_{xy}.$

If $f(y)>f(x)$, then we see

$\frac{e^{f(y)}-e^{f(x)}}{f(y)-f(x)}\leq\frac{e^{f(y)}+e^{f(x)}}{2}.$

We also notice that

$\left(\frac{e^{f(y)}+e^{f(x)}}{2}\right)^{2}=\frac{e^{2f(y)}+e^{2f(x)}}{2}-\left(\frac{e^{f(y)}-e^{f(x)}}{2}\right)^{2}\leq\frac{e^{2f(y)}+e^{2f(x)}}{2}.$

It follows that

$\displaystyle\mathfrak{m}(\Gamma(e^{f}))$
$\displaystyle\leq\sum_{f(y)>f(x)}\left(\frac{e^{2f(y)}+e^{2f(x)}}{2}\right)\,(f(y)-f(x))^{2}\mathfrak{m}_{xy}$
$\displaystyle=\frac{1}{2}\sum_{x,y\in
V}e^{2f(y)}\,(f(y)-f(x))^{2}\mathfrak{m}_{xy}=(e^{2f},\Gamma(f)).$

This proves the lemma. $\Box$

We possess the following relation (cf. [6, Theorem 2.4], [7, Theorem 2.1]):

######  Proposition 5.6.

For $\Lambda\geq 1$, we assume $\sup_{x\in V}\mathcal{D}_{x}\leq\Lambda$. Let
$c>0$, and let $\rho:V\to[0,\infty)$ be a probability density. If

(5.7)
$W(\mathfrak{m},\rho\mathfrak{m})^{2}\leq\frac{1}{c^{2}}\mathcal{I}(\rho),$

then

$W(\mathfrak{m},\rho\mathfrak{m})^{2}\leq\frac{\sqrt{2}\Lambda}{c}\mathcal{E}(\rho).$

###### Proof.

Fix $f\in\mathrm{Lip}_{1}(V)$ with $\mathfrak{m}(f)=0$, and set

$\rho_{f,\lambda}:=\frac{e^{\lambda f}}{\mathfrak{m}(e^{\lambda f})}.$

From Proposition 2.2, (5.7), and Lemma 5.5, it follows that

$\displaystyle\frac{d}{d\lambda}\log\mathfrak{m}(e^{\lambda f})$
$\displaystyle=(f,\rho_{f,\lambda})\leq
W(\mathfrak{m},\rho_{f,\lambda}\mathfrak{m})\leq\sqrt{\frac{4}{c^{2}}\frac{1}{\mathfrak{m}(e^{\lambda
f})}\mathfrak{m}(\Gamma(e^{\frac{\lambda f}{2}}))}$
$\displaystyle\leq\sqrt{\frac{4}{c^{2}}\frac{1}{\mathfrak{m}(e^{\lambda
f})}\left(e^{\lambda f},\Gamma\left(\frac{\lambda
f}{2}\right)\right)}=\sqrt{\frac{\lambda^{2}}{c^{2}}\frac{1}{\mathfrak{m}(e^{\lambda
f})}\left(e^{\lambda f},\Gamma\left(f\right)\right)}.$

Now, we have

$2\Gamma(f)(x)=\sum_{y\in V}(f(y)-f(x))^{2}\mathcal{P}(x,y)\leq\sum_{y\in
V}\mathcal{D}(x,y)^{2}\mathcal{P}(x,y)\leq\Lambda^{2},$

and hence

$\frac{d}{d\lambda}\log\mathfrak{m}(e^{\lambda
f})\leq\frac{\Lambda\lambda}{\sqrt{2}c}.$

Integrating the both sides, we obtain

$E(\lambda)\leq e^{\frac{\Lambda\lambda^{2}}{2\sqrt{2}c}}.$

Thanks to Lemma 5.3, we arrive at the desired inequality. $\Box$

We are now in a position to provide another proof of Theorem 1.2.

###### Another proof of Theorem 1.2.

Theorem 5.2 together with Lemma 5.3 and Proposition 5.6 with
$c=\sqrt{2}K/\Lambda$ implies the same conclusion as in Proposition 4.2. Thus,
we conclude Theorem 1.2 due to Proposition 4.3. $\Box$

### Acknowledgements

The authors are grateful to the anonymous referees for valuable comments. The
first named author was supported in part by JSPS KAKENHI (19K14532). The first
and second named authors were supported in part by JSPS Grant-in-Aid for
Scientific Research on Innovative Areas “Discrete Geometric Analysis for
Materials Design” (17H06460). The third named author was supported in part by
JSPS KAKENHI (19K23411).

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REDISCUSSION OF ECLIPSING BINARIES. PAPER XVIII.

THE F-TYPE SYSTEM OO PEGASI

By John Southworth

Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK

OO Peg is a detached eclipsing binary system containing two late-A-type stars
in a circular orbit with a period of 2.985 d. Using published spectroscopic
results and a light curve from the Transiting Exoplanet Survey Satellite
(TESS) we determine their masses to be $1.69\pm 0.09$ and $1.74\pm
0.06$$~{}{\rm M}_{\odot}$ and their radii to be $2.12\pm 0.03$ and $1.91\pm
0.03$$~{}{\rm R}_{\odot}$. The TESS data are of high quality, but
discrepancies in the radial velocities from two sources prevent a precise mass
measurement. The primary star is definitively hotter, larger and more luminous
than its companion, but its mass is lower (albeit to a significance of only
1.1$\sigma$). Using published apparent magnitudes and temperatures, we find a
distance of $238.8\pm 6.1$ pc, in agreement with the Gaia DR3 parallax.
Although both components are in the $\delta$ Scuti instability strip, we find
no evidence of pulsations. More extensive spectroscopy is needed to improve
our understanding of the system.

## Introduction

In this series of papers 1 we have been systematically reanalysing known
detached eclipsing binaries (dEBs) in order to determine their physical
properties to high precision. The main improvements versus previous work stem
from the availability of high-quality light curves from space missions such as
Kepler 2 and TESS (Transiting Exoplanet Survey Satellite 3) – see ref. 4 for a
review.

This work is important because dEBs are our primary source of direct
measurements of the basic properties (mass and radius) of normal stars 5; 6.
They are widely used to calibrate physical processes included in theoretical
models of stellar evolution 7; 8, such as atomic diffusion 9, convective core
overshooting 10 and the size of the core 11. A high precision in the
measurements of the stellar properties is vital for reliable results 12 and
can approach 0.2% precision in mass and radius in the best cases 13.

## OO Pegasi

Table I: Basic information on OO Pegasi.  Property | Value | Reference
---|---|---
Right ascension (J2000) | 21:41:37.70 | 14
Declination (J2000) | +14:39:30.8 | 14
Henry Draper designation | HD 206417 | 15
Hipparcos designation | HIP 107099 | 16
Gaia DR3 designation | 1770729907069675392 | 14
Gaia DR3 parallax |  $4.2534\pm 0.0245$ mas | 14
TESS Input Catalog designation | TIC 314847177 | 17
$U$ magnitude | $8.650\pm 0.010$ | 18
$B$ magnitude | $8.635\pm 0.021$ | 19
$V$ magnitude | $8.354\pm 0.018$ | 19
$J$ magnitude | $7.676\pm 0.023$ | 20
$H$ magnitude | $7.633\pm 0.027$ | 20
$K_{s}$ magnitude | $7.555\pm 0.018$ | 20
Spectral type | A7 V + A8 V | 21

In this work we present an analysis of OO Pegasi (Table I) based on published
spectroscopy and new space-based photometry. The eclipsing nature of OO Peg
was found using data from the Hipparcos satellite 16; 22. A first detailed
analysis was presented by Munari et al. 23 (hereafter M01) with the aim of
assessing the expected quality of results from the then-forthcoming Gaia
mission; of the three systems in that paper V505 Per and V570 Per have already
been revisited by the current author using the new TESS data 24; 25. M01 used
only Hipparcos photometry and radial velocity (RV) measurements from ground-
based spectroscopy in the 850–875 nm region, to represent the type of
observations that Gaia was expected to obtain. Due to these limitations they
were only able to obtain masses to 2% and radii to 4% precision.

A subsequent analysis of OO Peg was presented by Çakırlı 21 (hereafter C15)
who added new spectroscopic RV measurements and a more extensive light curve
from the All Sky Automated Survey (ASAS 26) to determine the properties of the
components more precisely. C15 also measured the atmospheric parameters of the
components, their projected rotational velocities, and the reddening and
distance of the system. He searched for but found no evidence of pulsations in
the light curves from Hipparcos and ASAS, despite both components being in the
$\delta$ Scuti instability strip 27.

The apparent mangitudes in Table I come from a variety of sources. The $U$
magnitude is from Oja 18 and relies on just two observations so may not
reflect the brightness of the system outside eclipse. This number is not used
in our analysis, but the consistency between the distances measured in the
various passbands (see below) suggest it does represent an out-of-eclipse
measurement. The $BV$ magnitudes are from the Tycho experiment 19 on the
Hipparcos satellite and each comprise the average of 55 measurements well-
distributed in orbital phase. The $JHK_{s}$ magnitudes are from 2MASS 20 and
were obtained at a single epoch corresponding to orbital phase 0.615, which is
not within an eclipse.

## Photometric observations

FIG. 1:  TESS short-cadence SAP photometry of OO Peg. The flux measurements
have been converted to magnitude units then rectified to zero magnitude by
subtraction of the median. The data rejected from the analysis are shown using
open circles, and the corresponding cutoff times indicated with vertical
dashed lines.

OO Peg has been observed just once by TESS, in sector 55, beginning on
2022/08/05 and concluding on 2022/09/01. A second set of observations is
scheduled for sector 82 and will occur in 2024 August if the spacecraft
remains healthy. The observations from sector 55 were obtained with a cadence
of 600 s, which is lower than desired and decreases the information content of
the data.

The available light curve from TESS was downloaded from the NASA Mikulski
Archive for Space Telescopes
(MAST***https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html) using
the lightkurve package 28. We used the simple aperture photometry (SAP) data
from the TESS-SPOC data reduction 29. A quality flag of “hard” yielded a total
of 3412 datapoints (Fig. 1). We rejected the data in the time interval BJDTDB
2459804.25 to 2459811.70 to avoid a stretch of data with the eclipses either
partially covered or not observed at all, leaving 2831 datapoints for further
analysis. These were normalised using lightkurve, converted to differential
magnitude, and the median magnitude of the sector subtracted.

A query of the Gaia DR3 database†††https://vizier.cds.unistra.fr/viz-
bin/VizieR-3?-source=I/355/gaiadr3 returned a total of 56 objects within 2
arcmin of OO Peg. Of these, the brightest is fainter by 4.23 mag in the $G$
band and 3.98 mag in the $G_{\rm RP}$ band. A small amount of third light is
therefore expected to contaminate the TESS light curve of OO Peg, at the level
of approximately 1%.

## Light curve analysis

FIG. 2:  The 600-s cadence TESS light curves of OO Peg (filled circles) and
its best fit from jktebop (white-on-black line) versus orbital phase. The
residuals are shown on an enlarged scale in the lower panel.

The components of OO Peg are small compared to their orbital separation, so
the system is suitable for analysis with the
jktebop‡‡‡http://www.astro.keele.ac.uk/jkt/codes/jktebop.html code 30; 31, for
which we used version 43. We defined the primary eclipse to be the deeper of
the two eclipses, star A to be the component eclipsed at primary eclipse, star
B to be its companion, and the primary eclipse to occur at orbital phase zero.
In the case of OO Peg star A is both hotter and larger than star B, by small
but significant amounts.

The jktebop fitted parameters included the fractional radii of the stars
($r_{\rm A}$ and $r_{\rm B}$), expressed as their sum ($r_{\rm A}+r_{\rm B}$)
and ratio ($k={r_{\rm B}}/{r_{\rm A}}$), the central surface brightness ratio
($J$), orbital inclination ($i$), orbital period ($P$), and a reference time
of primary minimum ($T_{0}$). A circular orbit was assumed, after confirming
that allowing for orbital eccentricity has a negligible effect on the values
of the fitted parameters.

Initial attempts to fit for third light returned values that were very small
and slightly less than zero. An experiment with it fixed to a value of 2%, to
account for the nearby stars discussed above, yielded a solution with
significantly larger residuals and a noticably poorer fit to the eclipses. We
therefore fixed third light to zero.

Limb darkening was included using the power-2 law 32; 33; 34 defined according
to

$\frac{F(\mu)}{F(1)}=1-c(1-\mu^{\alpha})$ (1)

where $\mu=\cos\gamma$, $\gamma$ is the angle between the observer’s line of
sight and the surface normal, $F(\mu)$ is the surface brightness at position
$\mu$ on the stellar disc, $c$ is the linear coefficient and $\alpha$ is the
nonlinear coefficient. As the two stars are very similar we assumed their
limb-darkening behaviours to be identical. Initial fits showed that we were
able to fit for one but not both of the limb darkening coefficients, so we
fitted for $c$ and left $\alpha$ fixed at a theoretical value 35; 36.

The relatively low 600 s sampling rate of the TESS data was accounted for by
numerically integrating the model to match 37. In effect we calculated the
model at five points, each spaced by 120 s, and averaged the results before
comparing to an observed datapoint. We found that this had a negligible effect
on the results, but continued to do so as the increase in computation time was
not a problem. The coefficients of two quadratic functions, one for each half
of the TESS sector, were also included to precisely normalise the light curve
to zero differential magnitude.

We found no evidence for changes in the orbital period for OO Peg, in
agreement with the results of C15. We therefore included the observed time of
primary minimum from the Hipparcos light curve calculated by M01
($2448499.1545\pm 0.0020$) to help constrain the orbital ephemeris more
precisely. This step lowered the uncertainty in $P$ by approximately a factor
of 3.

Table II:  Parameters of OO Peg, with their 1$\sigma$ uncertainties, measured from the TESS sector 55 light curves using the jktebop code. Parameter | Value
---|---
Fitted parameters: |
Primary eclipse time (BJDTDB) | $2459813.984151\pm 0.000007$
Orbital period (d) | $2.98465593\pm 0.00000049$
Orbital inclination (∘) | $83.629\pm 0.013$
Sum of the fractional radii | $0.30576\pm 0.00020$
Ratio of the radii | $0.8983\pm 0.0038$
Central surface brightness ratio | $0.96661\pm 0.00016$
LD coefficient $c$ | $0.709\pm 0.014$
LD coefficient $\alpha$ | 0.431 (fixed)
Derived parameters: |
Fractional radius of star A | $0.16107\pm 0.00028$
Fractional radius of star B | $0.14489\pm 0.00039$
Light ratio $\ell_{\rm B}/\ell_{\rm A}$ | $0.7794\pm 0.0065$

The resulting best fit is shown in Fig. 2 and the parameters are given in
Table II. Uncertainties in the fitted parameters were calculated using both
Monte Carlo and residual-permutation simulations 38; 39, and the larger of the
two options chosen for each parameter. The two error estimation algorithms
were in good agreement for all parameters, as expected because there is no
obvious systematic noise present in the data. The uncertainties in the all-
important fractional radii are encouragingly low at 0.17% and 0.27%, despite
the relatively poor sampling rate of the TESS photometry.

## Radial velocity analysis

M01 published a set of 21 RVs for each component of OO Peg, which are
tabulated in the paper. The spectra on which they were based were deliberately
obtained at quasi-random times in order to simulate a dataset that might be
expected from Gaia. As a result, two spectra are too blended to give precise
RVs and there is only one spectrum near second quadrature. We reanalysed the
RVs from M01 to confirm their results, and followed these authors in omitting
the RVs from the two most blended spectra.

To fit the RVs we used jktebop and the orbital ephemeris from Table II. The
fitted parameters were the velocity amplitudes of the two stars, $K_{\rm A}$
and $K_{\rm B}$, and the systemic velocity ($V_{\gamma}$) of the two stars. We
also allowed for a change in $T_{0}$ to insure against ephemeris drift or
period changes, but all solutions were consistent with the ephemeris given in
Table II. Separate fits were obtained with $V_{\gamma}$ either assumed to be
the same for the two stars or allowed to be different. The errorbars of the
RVs for each star were scaled to give a reduced $\chi^{2}$ of
$\chi_{\nu}^{\,2}=1.0$ versus the best fit. Parameter uncertainties were
calculated using the Monte Carlo procedure 24. The results are given in Table
III.

It was immediately clear that our star A is the _secondary_ component for M01,
something that can happen easily when the primary and secondary eclipses are
of similar depth and the photometric data are quite scattered. We accounted
for this in our analysis. We note that this is also apparent in C15 (see his
fig. 4) but not commented on by that author. Our $K_{\rm A}$ agrees with M01,
but our $K_{\rm B}$ and $V_{\gamma}$ do not agree within the uncertainties. We
also notice that there is a counterintuitive result that the r.m.s. residuals
are lower for star A when the $V_{\gamma}$ values of the two stars are
required to be the same – this occurs because of the rescaling of the RV
uncertainties combined with the RV measurements having a range of
uncertainties.

Table III:  Spectroscopic orbits for OO Peg from the literature and from the reanalysis of the RVs in the current work. $K_{\rm A}$ and $K_{\rm B}$ values were not given by M01, so we have calculated the values that would reproduce their mass measurements. All quantities are in km s-1. Source | $K_{\rm A}$ | $K_{\rm B}$ | ${V_{\gamma}}$ | ${V_{\gamma}}_{\rm,A}$ | ${V_{\gamma}}_{\rm,B}$ | r.m.s. residual
---|---|---|---|---|---|---
|  |  |  |  |  | star A | star B
M01 (our calculation) | 112.0 | 110.1 | $8.47\pm 0.46$ |  |  |  |
C15 | $111\pm 2$ | $114\pm 3$ | $6.7\pm 0.1$ |  |  |  |
M01 (our fit, $V_{\gamma}$ same) | $112.1\pm 1.6$ | $107.6\pm 1.0$ | $10.8\pm 0.7$ |  |  | 5.1 | 6.7
M01 (our fit, $V_{\gamma}$ different) | $112.4\pm 1.7$ | $107.6\pm 1.0$ |  | $11.4\pm 1.2$ | $10.5\pm 0.8$ | 5.4 | 6.6
C15 (our fit, $V_{\gamma}$ same) | $112.9\pm 1.6$ | $111.8\pm 1.2$ | $5.5\pm 0.9$ |  |  | 5.1 | 3.2
C15 (our fit, $V_{\gamma}$ different) | $112.3\pm 1.5$ | $111.5\pm 1.3$ |  | $7.6\pm 1.2$ | $3.9\pm 1.2$ | 4.2 | 3.1
Combined ($V_{\gamma}$ same) | $112.7\pm 1.1$ | $109.1\pm 0.9$ | $0.2\pm 0.7$ |  |  | 5.1 | 5.9
Combined ($V_{\gamma}$ different) | $112.5\pm 1.0$ | $109.0\pm 0.9$ |  | $0.3\pm 0.8$ | $-0.4\pm 0.7$ | 5.1 | 5.6
Adopted values | $112.6\pm 1.2$ | $109.1\pm 2.8$ |  |  |  |  |

C15 obtained 15 spectra of OO Peg from which 15 RVs were obtained for star A
and 14 for star B. C15 fitted spectroscopic orbits to the spectra from his own
observations together with those from M01. We first modelled the RVs from C15
separately. Four of the RVs for star A and three for star B are close to
conjunction so suffer from blending and contribute little to pinning down
$K_{\rm A}$ and $K_{\rm B}$, so we ran solutions with these omitted. The
results were similar to those for all RVs, and we adopted these as our
standard datasets for the RVs from C15.

FIG. 3:  RVs of OO Peg from M01 and C15 (filled circles for star A and open
circles for star B), compared to the best fit from jktebop (solid lines) with
a separate $V_{\gamma}$ value for each star. The residuals are given in the
lower panels separately for the two components.

Table III shows that there are significant discrepancies between the solutions
of different RV datasets, both calculated in this work and versus the
literature. We also consistently find that $K_{\rm A}$ is larger than $K_{\rm
B}$, thus star A is less massive than star B (although the difference is of
similar size to the uncertainties). Some of these discrepancies are driven by
small-number statistics, and some are likely due to differences in
$V_{\gamma}$ from the differing RV measurement processes used by M01 and C15.
The biggest discrepancy is in the $K_{\rm B}$ values from the two sources of
RVs, which differ by 4 km s-1.

We made the choice to fit the RVs from M01 and C15 simultaneously, both with
the combined and independent $V_{\gamma}$ values (Fig. 3). In each case we
scaled the errorbars of the individual datasets to give $\chi_{\nu}^{\,2}=1$
and subtracted the best-fitting $V_{\gamma}$ before fitting the combined data.
Our results are in between those for the two RV sources separately, as
expected. The $K_{\rm A}$ values are consistent over all solutions so we adopt
a value of $112.6\pm 1.2$ km s-1. The errorbar is the quadrature addition of
the 1.1 km s-1 uncertainty in Table III and the 0.5 km s-1 which is the
largest difference between the adopted $K_{\rm A}$ and the other fitted
values. For $K_{\rm B}$ we adopt a value of $109.1\pm 2.8$ km s-1, where the
uncertainty is the quadrature addition of 0.9 km s-1 and 2.7 km s-1 following
the same argument. This $K_{\rm B}$ is unfortunately rather uncertain, which
prevents the measurement of the masses of the stars to high precision.

## Physical properties and distance to OO Peg

Table IV: Physical properties of OO Peg defined using the nominal solar units given by IAU 2015 Resolution B3 (ref. 40). The $T_{\rm eff}$ values are from C15.  Parameter | Star A | Star B
---|---|---
Mass ratio $M_{\rm B}/M_{\rm A}$ | $1.032\pm 0.029$
Semimajor axis of relative orbit ($\mathcal{R}^{\rm N}_{\odot}$) | $13.16\pm 0.18$
Mass ($\mathcal{M}^{\rm N}_{\odot}$) | 1.689 $\pm$ | 0.088 | 1.744 $\pm$ | 0.058
Radius ($\mathcal{R}^{\rm N}_{\odot}$) | 2.120 $\pm$ | 0.029 | 1.907 $\pm$ | 0.027
Surface gravity ($\log$[cgs]) | 4.013 $\pm$ | 0.011 | 4.119 $\pm$ | 0.005
Density ($~{}\rho_{\odot}$) | 0.1774 $\pm$ | 0.0027 | 0.2515 $\pm$ | 0.0040
Synchronous rotational velocity ( km s-1) | 35.93 $\pm$ | 0.50 | 32.32 $\pm$ | 0.45
Effective temperature (K) | 7850 $\pm$ | 350 | 7600 $\pm$ | 450
Luminosity $\log(L/\hbox{$\mathcal{L}^{\rm N}_{\odot}$})$ | 1.19 $\pm$ | 0.08 | 1.04 $\pm$ | 0.10
$M_{\rm bol}$ (mag) | 1.77 $\pm$ | 0.20 | 2.14 $\pm$ | 0.26
Interstellar reddening $E(B\\!-\\!V)$ (mag) | $0.09\pm 0.02$
Distance (pc) | $238.8\pm 6.1$

Using the photometric and spectroscopic results from Tables I, II and III, we
have determined the physical properties of the OO Peg system using the
jktabsdim code 41. The results are given in Table IV and show that the masses
are measured to 5.2% (star A) and 3.3% (star B), and the radii to 1.4% (both
stars). This is not the desired 2% precision 5; 42 due to the uncertainty in
the value of $K_{\rm B}$. The mass measurements agree well with those of M01
but not C15; the radius measurements disagree with both. In particular, the
$R_{\rm B}$ value from M01 ($1.37\pm 0.05$$~{}{\rm R}_{\odot}$) is extremely
low. Our results are based on a careful analysis of the available RVs and much
higher-quality light curves from TESS, so should be preferred to previous
values.

M01 determined the effective temperatures ($T_{\rm eff}$s) of the stars from a
comparison between the observed and synthetic spectra, finding $T_{\rm
eff,A}=8770\pm 150$ K and $T_{\rm eff,B}=8683\pm 180$ K. The ratio of these
values agrees well with the surface brightness ratio measured from the light
curve (Table II). Using these $T_{\rm eff}$s and the apparent magnitudes of
the system (Table I), we determined the distance to OO Peg using the $K$-band
surface brightness method 41 and calibrations from Kervella et al. 43. The
interstellar reddening was determined by requiring the $UBV$ and $JHK$
distances to agree, via manual iteration, resulting in $E(B\\!-\\!V)=0.21\pm
0.03$ mag and a distance of $245.2\pm 4.9$ pc. This reddening is rather larger
than expected – the stilism§§§https://stilism.obspm.fr online tool 44; 45
gives a value of $0.037\pm 0.018$ mag – and the distance is also $2\sigma$
beyond the value Gaia DR3 14 value of $234.1\pm 1.3$ pc.

C15 determined rather smaller temperatures of $T_{\rm eff,A}=7850\pm 350$ K
and $T_{\rm eff,B}=7600\pm 450$ K, via comparison with reference-star spectra.
Using these values instead of the ones from M01, we obtain
$E(B\\!-\\!V)=0.09\pm 0.02$ mag and a distance of $238.8\pm 6.1$ pc. This
$E(B\\!-\\!V)$ is in much better agreement with the stilism value, and the
distance is also consistent with the Gaia DR3 parallax at the 0.8$\sigma$
level. We therefore adopt these $T_{\rm eff}$s and $E(B\\!-\\!V)$ as our final
values in Table IV. Supporting evidence for these lower temperatures are the
catalogue $T_{\rm eff}$s of $7476\pm 149$ K given in v8 of the TESS Input
Catalog 17 and $7347\pm 17$ K from the Gaia DR3 APSIS pipeline 46; 47. Both
catalogues treat point sources as single stars, but in the case of OO Peg this
is a reasonable approximation due to the similarity of the two components.

C15 measured $E(B\\!-\\!V)=0.29\pm 0.01$ from the strength of the interstellar
Na D lines; such a large reddening is highly inconsistent with our results and
would require the stars to have $T_{\rm eff}$s in the region of 10,000 K for
the distances measured in the optical to match those measured in the IR.

## Summary and final points

OO Peg is a dEB containing two component with late-A spectral types, on a
circular orbit with a period of 2.98 d, whose eclipsing nature was discovered
thanks to the Hipparcos satellite. We have presented a reanalysis of the
system based on a space-based light curve from the TESS mission and published
spectroscopic parameters. The TESS light curve is of high quality and allows
the fractional radii of the stars to be determined to 0.2% precision; star A
is clearly larger, hotter and more luminous than its companion. However, our
reanalysis of published RVs from two sources yields both a disagreement in the
value of $K_{\rm B}$ and the measurement of a lower mass for star A than star
B. This discrepancy would be problematic for stellar evolutionary theory, but
is thankfully not significant due to the uncertainty in the measured masses.
Using published apparent magnitudes of the system and $T_{\rm eff}$ values of
the stars, we have determined a distance to the system in agreement with the
Gaia DR3 parallax.

Both components of OO Peg are in a region of the luminosity versus $T_{\rm
eff}$ diagram where a high fraction of stars show $\delta$ Scuti pulsations
27, but are not known to pulsate. We therefore calculated a Fourier transform
of the residuals of the jktebop fit using version 1.2.0 of the period04 code
48. No significant periodicity was found up to the Nyquist frequency of 72
d-1, with a noise level of approximately 0.01 mmag from 1 d-1 to the Nyquist
frequency. Lower-amplitude pulsations may be present but would require
significantly more photometry to measure.

We made a brief comparison of the masses, radii and $T_{\rm eff}$s of the
stars to the parsec 1.2S theoretical stellar evolutionary models 49; 50.
Agreement was found for a solar chemical composition and an age of $1.0\pm
0.3$ Gyr, which supports the lower $T_{\rm eff}$ values found by C15 versus
those obtained by M01.

The quality of our results has been limited by the imprecision of the
spectroscopic parameters measured for the system: both the RVs and the $T_{\rm
eff}$s are quite uncertain. Conversely, the TESS data allow high-quality
measurements of its photometric parameters. Further work should therefore
concentrate on performing a more extensive spectroscopic analysis of OO Peg.
Forthcoming data releases from the Gaia satellite may well help.

## Acknowledgements

We thank the anonymous referee for a helpful report. This paper includes data
collected by the TESS mission and obtained from the MAST data archive at the
Space Telescope Science Institute (STScI). Funding for the TESS mission is
provided by the NASA’s Science Mission Directorate. STScI is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5–26555. 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. The following
resources were used in the course of this work: the NASA Astrophysics Data
System; the SIMBAD database operated at CDS, Strasbourg, France; and the
ar$\chi$iv scientific paper preprint service operated by Cornell University.

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