# Time-resolved spectroscopy on the heartbeat state of GRS 1915+105 using
AstroSat

Divya Rawat,1,2 Ranjeev Misra,2 Pankaj Jain1 and J. S. Yadav1
1Department of physics, IIT Kanpur, Kanpur Nagar, Uttar Pradesh 208016, India
2Inter-University Center for Astronomy and Astrophysics, Ganeshkhind, Pune
411007, India
E-mail<EMAIL_ADDRESS>(DR)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

AstroSat spectra of the black hole system GRS 1915+105 during the heartbeat
state (with a varying oscillation period from 150 to 100 secs) were analysed
using a truncated relativistic disc model along with a Comptonization
component. Spectra were fitted for segments of length $\sim 24$ secs. The
oscillation can be described as coordinated variations of the accretion rate,
Comptonised flux, and the inner disc radius, with the latter ranging from
$1.235$-$5$ gravitational radii. Comparison with results from the $\chi$ and
Intermediate states shows that while the accretion rate and the high energy
photon index were similar, the inner disc radius and the fraction of
Comptonised photons were larger for these states than for the heartbeat one.
The coronal efficiency $\eta\equiv L_{ac}/\dot{M}c^{2}$, where $L_{ac}$ is the
radiative luminosity generated in the corona is found to be approximately
$\propto\dot{M}^{-2/3}$ for all the observations. The efficiency decreases
with inner radii for the heartbeat state but has similar values for the $\chi$
and Intermediate states where the inner radii is larger. The implications of
these results are discussed.

###### keywords:

accretion, accretion discs — black hole physics — X-rays: binaries — X-rays:
individual: GRS 1915+105

††pagerange: Time-resolved spectroscopy on the heartbeat state of GRS 1915+105
using AstroSat–References

##  1 Introduction

GRS 1915+105 is a galactic black hole binary system with luminosity close to
the Eddington limit. A qualitative behaviour of the transient system in the
Hardness Intensity Diagram (HID) (Muñoz-Darias et al., 2014) shows that this
source executes transitions between hard and soft state via an intermediate
state. This source was in outburst since it was discovered in 1992 and went to
the obscured state in 2019 with interrupted X-ray flares (Neilsen et al.,
2020). The binary system contains a black hole of 12.4 solar mass (Reid et
al., 2014), placed at a distance of 8.6 kpc (Reid et al., 2014) in the
Sagittarius arm of the Milky Way galaxy. The companion is a regular K-type
star (Vilhu, 2002) with a binary orbital period of 34 days (Greiner et al.,
2001). In the binary system, the compact object is a rapidly rotating Kerr
black hole with a dimensionless spin parameter $\geq$0.98 (McClintock et al.,
2006; Blum et al., 2009; Misra et al., 2020; Sreehari et al., 2020; Liu et
al., 2020).

Greiner et al. (1996) first reported a drastic change in X-ray variability on
time scale of secs to days. This source shows complex, but nearly periodic and
repetitive X-ray variability. The source is divided into 14 different X-ray
classes on the basis of its X-ray flux and Color-Color Diagram (CCD) (Belloni
et al., 2000; Klein-Wolt et al., 2002; Hannikainen et al., 2003). The long
RXTE observations reveal that the source spends most of the time in the hard
state, which belongs to $\chi$ class (Belloni et al., 2000). In this state,
the source does not show much variability. It also spends a significant amount
of time in a highly variable $\rho$ class, also known as the heartbeat class
(heartbeat class hereafter), which has been widely studied in the literature
(Taam et al., 1997; Paul et al., 1998; Yadav et al., 1999; Neilsen et al.,
2011, 2012; Pahari et al., 2014; Yan et al., 2018). The phase-resolved
spectroscopy for the heartbeat class was done by Neilsen et al. (2012) using
RXTE/PCA data in 3.3-45.0 keV band. They have explained the heartbeat
oscillation as a combination of a thermal–viscous radiation pressure
instability and Radiation pressure-driven evaporation or ejection event in the
inner accretion disc.

Migliari & Belloni (2003) performed time-resolved spectroscopy (with each
segment of length $\sim$16 sec) on the $\beta$ class using RXTE data in
3.0-25.0 keV band. They studied the variation of $\Gamma$ and $R_{in}$ with
time and reported variations of local accretion rate and mass loss. With
AstroSat data, Rawat et al. (2019) have studied the temporal properties of GRS
1915+105 and observed a state transition from $\chi$ class to Heartbeat state
via intermediate class (IMS onwards) for 28th March 2017 observation. These
transitions occur at unabsorbed 0.1-80.0 keV luminosity $\sim 0.1L_{edd}$.
Next, Misra et al. (2020) studied the spectro-temporal properties for both
28th March and 1st April 2017 observations and identified the origin of C-type
QPO frequency as dynamical frequency. Using the same set of observations as
analysed previously by Rawat et al. (2019); Misra et al. (2020), in this work,
we have undertaken time-resolved spectroscopy in the 1.0-50.0 keV band using
AstroSat observation for a heartbeat oscillation of GRS 1915+105, to quantify
the evolution of the spectral properties during the oscillation. This is in
contrast to earlier analysis of RXTE data where a phase resolved spectroscopy
method was used (Neilsen et al., 2011, 2012). The advantage of time-resolved
spectroscopy is that the analysis is not affected by the selection of the peak
of an individual cycle, averaging over cycles with different time-periods and
variation of the profile for different cycles. Thus, the technique provides a
more direct approach to understanding spectral evolution. However, the
disadvantage of time-resolved spectroscopy is that the length of the time
segment used for spectral fitting (in this case $\sim 24$ seconds) has to be
large enough to obtain statistically significant data. Hence, faster spectral
variations are not captured.
We aim to study the spectral transition of the source and correlation between
spectral parameters with simultaneous Soft X-ray Telescope (SXT) and Large
Area X-ray Proportional Counters (LAXPCs) data. Because of the low energy
coverage of SXT (0.3-8.0 KeV) in which the disc emission dominates, it enables
us to get a reasonable estimate of inner disc properties (like inner radii)
which was earlier not possible with RXTE. We have discussed the data analysis
techniques in Section 2 followed by results in Section 3. The discussion and
summary of interpretations are given in Section 4 and 5.

##  2 Observation and Data Reduction

The galactic micro-quasar source GRS 1915+105 was observed from 28th March
18:03:19 till 29th March 2017 19:54:07 (hereafter epoch 1) and 1st April
00:03:21 till 1st April 14:38:01 2017 (hereafter epoch 2) with SXT (Singh et
al., 2016, 2017) $\&$ LAXPC (Yadav et al., 2016b) onboard AstroSat (Agrawal,
2006; Singh et al., 2014) satellite. SXT is an imaging instrument that works
in 0.3-8.0 keV with a spectral resolution of 90 eV at 1.5 keV. LAXPC is a
photon-counting array and has an effective area of about 6100 cm2 (combined 3
units LX10, LX20 and LX30) at 10.0 keV (Yadav et al., 2016a). We used LAXPC
and SXT instruments for spectral analysis to cover the broad energy range of
1.0-50.0 keV. The lower energy range $<$1.0 keV is ignored because of
uncertainty in effective area and response in the SXT instrument. The high
energy range i.e $>$50.0 keV is ignored because of low signal-to-noise ratio.
LaxpcSoft 111http://astrosat-ssc.iucaa.in/?q=laxpcData software is used to
extract the LAXPC spectra in 4.0-50.0 keV band. We have used recent response
and background files provided by LAXPC POC
team222https://www.tifr.res.in/~astrosat_laxpc/LaxpcSoft.html. We have used
only LX10 $\&$ LX20 for the analysis as there was a gas leakage and
consequently instability in the gain of LX30. The SXT spectra in 1.0-5.0 keV
energy band were generated in XSELECT V2.4j using a merged event file
extracted using SXT Event Merger
Tool333https://www.tifr.res.in/~astrosat_sxt/dataanalysis.html taking a source
region of 12’. The corresponding auxiliary response file $\&$ response matrix
file were generated using ARF generation tool3 for each epoch. The
simultaneous SXT and LAXPC spectra were fitted in XSPEC version 12.11.0.

Figure 1: The left and right panel shows a full 50 sec binned simultaneous SXT
and LAXPC light curve of GRS 1915+105 for epoch 1 and epoch 2, respectively.
Light curves are plotted in 0.3-8.0 keV (using SXT) and 3.0-80.0 keV
(combining LXP10, LXP20) energy range.

###  2.1 Spectral Analysis

Misra et al. (2020) divided epoch 1 Data into 10 segments and epoch 2 Data
into 6 segments (spectral details of each segment is given in Table 1 of Misra
et al. (2020)). Details of the response and background spectra generation for
LAXPC can be found in Antia et al. (2017). To show the extent of the overlap
of SXT and LAXPC data for epoch 1 and epoch 2 we have plotted the full SXT and
LAXPC light curve in figure 1 while Rawat et al. (2019) has shown LAXPC light
curve for epoch 1 only. The zoomed view of epoch 1 $\&$ 2 light curves are
shown in figure 3, 5 and 7 of Rawat et al. (2019) and figure 1 of Misra et al.
(2020). The soft count rate in both LAXPC and SXT increases with time as the
source transits from $\chi$ class to Heartbeat state. The time-averaged fitted
spectra for $\chi$ class, IMS, and Heartbeat state are shown in figure 3 of
Misra et al. (2020). In the heartbeat state, the X-ray flux shows periodic
variation. To study the variation of the spectral parameters over the
oscillatory period, we have performed time-resolved spectroscopy on this
state.

Figure 2: The top left panel shows a $\sim$1000 sec light curve, variation of
$\dot{M}$ ($10^{18}gm$ $s^{-1}$), $r_{in}$, $\Gamma$, and $f_{s}$ with time
for epoch 1. Here light curve is plotted in 3.0-50.0 keV energy band. The
right panel shows the variation of $r_{in}$, $\Gamma$, and $f_{s}$ with
$\dot{M}$ for $\chi$ (black coloured data points), IMS (red coloured data
points) and heartbeat state (blue coloured data points) of epoch 1. Here, IMS
and HS stands for Intermediate state and Heartbeat state respectively

###  2.2 Time-resolved Spectroscopy for Heartbeat state

We have divided the heartbeat state data into different segments, with each
segment equal to 23.775 secs, which is ten times the time resolution of the
SXT instrument. The LAXPC and SXT spectra for each segment was extracted. The
same model that was used by Misra et al. (2020) was used to fit the spectra,
which is an absorbed thermal comptonization of a relativistic disc represented
by the XSPEC model tbabs*(simpl*kerrd). Following Misra et al. (2020), the
absorption column density, the relative constant between the two instruments,
and the gain offset for the SXT spectrum were kept fixed to the values
obtained from the time-averaged spectra. For all the segments, the reduced
$\chi^{2}$ ranged from $0.6-1.4$ with an average value of $\sim$0.75. The
resultant best fit spectral parameters variation as a function of time were
considered time series with a time bin of 23.775 secs. To check for a possible
time lag between the spectral parameter variation, the Cross-correlation
function (CCF) between the time series were computed using the HEASOFT
function crosscor. The time series were divided into 28 intervals of 16 bins,
i.e. the length of each interval was 380.4 secs. Then for each pair of
intervals, the cross-correlation was computed and averaged over the 28
intervals. crosscor provides the standard deviation of the averaged cross-
correlation as the error. However, to further check the robustness of the
observed cross-correlation to noise in the data, we implemented the pair
bootstrapping simulation. We simulated 10,000 pairs of time series using the
Random Subset Selection (RSS) technique (Peterson et al., 1998) and extracted
the Cross-correlation function (CCF) for each pair. The 90 $\%$ confidence
interval was estimated using the simulated CCF distribution at each lag.

##  3 Results

The left panels of Figure 2 show the variation of the count rate and spectral
parameters with time for a $\sim 1000$ sec segment of the heartbeat state. The
top panel shows the count rate in a finer 2.3775 secs time-bin, while the
spectral parameters are measured over 23.775 secs. The near sinusoidal
variation is evident for all the parameters. The right panels of Figure 2 show
the variation of the inner truncated radius, $r_{in}$, the power-law index
$\Gamma$ and the scattering fraction $f_{s}$ with the accretion rate,
$\dot{M}$, for the heartbeat state as well for the $\chi$ and intermediate
ones. For the heartbeat state, we checked the significance of any correlation
(or anti-correlation) by using a Monte Carlo simulation technique. For each
data point and its associated standard deviation in the time series, we
simulated data points considering a random normal distribution for the error.
The Pearson correlation coefficient, $r$ and the corresponding null hypothesis
probability $p$ were computed for 10,000 pairs of the simulated time-series
and averaged. The correlations of $r_{in}$ and $f_{s}$ with $\dot{M}$ are
highly significant ($r=0.680\pm 0.001$, $p=4.7\times 10^{-58}$ and
$r=-0.361\pm 0.002$, $p=6.3\times 10^{-15}$ respectively), while for $\Gamma$
with $\dot{M}$ is less so ($r=0.053\pm 0.003$, $p=1.5\times 10^{-3}$). For the
$\chi$ and Intermediate states, the variation of $r_{in}$ and $f_{s}$ with
$\dot{M}$ is clearly different from that of the heartbeat, while the $\Gamma$
values are in the same range. To verify whether there is any time (or phase
lag) between the spectral parameter variation for the heart beat state, we
computed the cross-correlation for different pairs and show them in Figure 3.
While there are hints of possible time-lags of $\sim 100$ secs between $f_{s}$
and $\dot{M}$ and $\sim 25$ secs between $\Gamma$ and $\dot{M}$, the
significance is not sufficient to make robust confident statements. Note that
this analysis is limited by the time-bin of the time-resolved spectroscopy of
23.775 seconds and hence does not capture the more rapid variability seen,
especially in the peak of the oscillations. These faster variability may have
different spectral parameter correlations than that reported here.

Figure 3: The top left and right panel shows the Cross-Correlation functions
of $r_{in}$, and $\Gamma$ with $\dot{M}$, respectively. The bottom left and
right panel shows Cross-Correlation functions of $f_{s}$ with $\dot{M}$, and
$\Gamma$, respectively.

##  4 Discussion

Our observations reveal correlated variation of spectral parameters in the
heartbeat state. We find that the total flux, accretion rate $\dot{M}$, inner
radius $r_{in}$ show periodic variation, nearly in phase with one another.
Here we discuss possible physical mechanism which may explain these
observations.

###  4.1 Viscous time-scale

The solution for the standard accretion disc model has been derived by Shakura
& Sunyaev (1973) and later relativistic corrections were included by Novikov &
Thorne (1973). In the radiation pressure dominated region where the opacity is
mainly due to electron-electron scattering, we have calculated the viscous
time-scale for heartbeat state using equation (5.9.10) from Novikov & Thorne
(1973). For $\dot{M}=0.72\pm{0.02}$ $\times$1018 gm/s and $\alpha$=0.1 the
material falling from the outer part of the disc $\sim$20 Rg takes about
100-150 secs to reach the innermost stable circular orbit (ISCO). We have used
a high value of the dimensionless spin parameter $a=0.98$ as found by
(McClintock et al., 2006; Blum et al., 2009; Misra et al., 2020; Sreehari et
al., 2020; Liu et al., 2020) for the viscous time-scale calculation.

Figure 4: The variation of the total flux with inner disc radii for $\chi$
class, IMS and HS.

Figure 5: The variation of coronal efficiency ($\eta_{coronal}$) for $\chi$
class, IMS and HS with inner disc radii and accretion rate are shown in left
and right panel respectively. The dotted line in the right panel represents
the best fit curve $\eta_{coronal}\propto{\dot{M}}^{-b}$, with $b=0.66\pm
0.02$

###  4.2 Oscillation in $\dot{M}$ as response of inward propagation of
fluctuation

As the source undergoes transitions from $\chi$ class to Heartbeat class via
IMS, $r_{in}$ starts to oscillate in response to change in accretion rate, as
shown in the figure 2. The oscillatory nature of $\dot{M}$ has been also
observed by Migliari & Belloni (2003) in the hard-state of GRS 1915+105 and
they have associated the variation in the local accretion rate to the loss of
matter in the inner region of the disc due to instability. Neilsen et al.
(2012) argued that all the heartbeat state variability could be explained as a
result of the combination of thermal-viscous radiation pressure instability
(Lightman & Eardley, 1974; Taam et al., 1997; Nayakshin et al., 2000), a local
Eddington limit, and sudden radiation-pressure driven evaporation of inner
disc (Fukue, 2004; Heinzeller & Duschl, 2007; Lin et al., 2009). In our
observations, we have observed a correlation between total unabsorbed 0.1-80.0
keV X-ray flux and inner disc radii for full HS oscillation (shown in figure
4). This is consistent with Neilsen et al. (2011, 2012) , who also observe a
steady increase in the disc radii during the phase in which the flux shows a
slow rise. This is followed by a sharp fall in radii over a very short time
interval as luminosity approaches its peak value. In our analysis, we have not
tested the behaviour at such short time intervals. We point out that the
luminosity in Neilsen et al. (2011, 2012) was close to the Eddington limit and
in our case it is approximately $0.1L_{edd}$. Furthermore, our oscillation
period is twice as large as their period.

###  4.3 Correlation between coronal efficiency and accretion rate

The heating rate of the corona should be equal to excess luminosity emitted by
it, $L_{ac}$ which would be the difference in the luminosity of the corona
$L_{c}$ and the luminosity of the seed photons entering it. Hence
$L_{ac}=L_{c}-f_{s}L_{d}$, where $f_{s}$ is the fraction of the disc
luminosity $L_{d}$ that enters the corona. We can then define a coronal
radiative efficiency as $\eta_{coronal}\equiv L_{ac}/\dot{M}c^{2}$. In the
spectral model used, the total unabsorbed flux $F_{tot}=(1-f_{s})F_{d}+F_{c}$,
where $F_{d}$ and $F_{c}$ are the disc and coronal fluxes. For isotropic
emission, the corona luminosity $L_{c}=4\pi D^{2}F_{c}$, where $D$ is the
distance to the source. The disc luminosity $L_{d}=2\pi D^{2}F_{d}/cos(i)\sim
4\pi D^{2}F_{d}$ for an inclination angle $i\sim 60^{\circ}$. From this we can
compute the excess coronal luminosity as $L_{ac}=4\pi D^{2}(F_{tot}-F_{d})$.
We estimate the total unabsorbed flux and the disc flux using the Xspec model
“cflux". For a distance of $8.5$ kpc and using the best fit values of the
fluxes and the accretion rate $\dot{M}$ for each spectral fitting, we estimate
the coronal efficiency and study its correlation with disc accretion rate and
inner disc radii.

Figure 5 shows the variation of coronal efficiency $\eta_{coronal}$ with inner
disc radii and accretion rate. The variation of the coronal efficiency with
accretion rate is relatively similar for all the states (Right panel of Figure
5). For the same accretion rate, the $\chi$ and IMS states seem to have
slightly higher efficiencies as compared to the HS state, but it is within the
dispersion seen in the HS state. The dependence can be represented as
$\eta_{coronal}\propto\dot{M}^{-2/3}$ shown as a dashed line in the Figure 5.
For the HS state, time resolved spectral results show that $\eta_{coronal}$
also varies inversely with inner radii (left panel of Figure 5). However, for
the $\chi$ and IMS states, while the efficiencies have similar values as some
of the HS segments, the inner disc radii are significantly different. In other
words, the system can exhibit nearly the same coronal efficiency for different
values of the inner radius depending on the state.

The coronal emission may arise from a hot radiatively inefficient flow at
radii less than the inner radius of the standard disc. However, for an
advection-dominated accretion flow (ADAF) the efficiency is proportional to
the accretion rate (e.g. Narayan & Yi, 1994). While a pure self-similar ADAF
solution may not be strictly valid when the standard disc is truncated at
small radii, the measured efficiency being $\propto\dot{M}^{-2/3}$ argues
against the emission being from an inner hot flow. Alternatively, the coronal
emission could be from a corona located above and below the standard disc
(sandwich geometry), where a fraction of the gravitational energy released is
dissipated in the corona (e.g. Haardt & Maraschi, 1993). For such a geometry
the coronal efficiency should decrease with inner disc radii and indeed for
the HS observations this is what is see in Figure 5. Thus, for the HS state
the results seem to indicate that the hard X-ray emission arises from a corona
above and below the disc and the inner regions at radii less then inner disc
radius does not have a significant contribution.
For the $\chi$ and Intermediate states, the coronal efficiency is similar to
the HS but the inner radii are larger. This could imply that basic geometry of
the system is different for these states and perhaps here the hard X-ray
emission arises from the inner hot flow. If one insists and assumes that the
geometry of all the states are qualitatively same, then the results imply that
the coronal efficiency is $\propto\dot{M}^{-2/3}$ generically and the
efficiency does not depend on the inner radii and the variation with inner
radii seen for the HS would then be an indirect one. In such an assumption,
neither a hot inner disc nor a corona on top and bottom of the disc would be a
favoured and a speculative interpretation could be if the corona is being
powered by the spin of the black hole i.e. by the Blandford-Znajek (BZ)
mechanism (Blandford & Znajek, 1977). The BZ mechanism has typically been
invoked to explain the powerful jets seen in black hole systems including GRS
1915+105. It has been proposed that the base of these jets is the corona which
is the source of hard X-ray emission (Markoff et al., 2001). Note that even if
the energy source of the corona is the same as that of the jet (i.e. the BZ
mechanism), the dominant radiative process for the corona could still be
thermal Comptonization instead of synchrotron by which the larger scale jet
radiates. In the BZ mechanism the energy is extracted from the spin of the
black hole and the extraction rate is proportional to $B_{\phi}^{2}$, where
$B_{\phi}^{2}$ is the poloidal magnetic field treading the black hole. If we
assume that $B_{\phi}^{2}$ is proportional to the cube root of the accretion
rate falling onto the black hole and the radiative efficiency is independent,
then the coronal flux would be $F_{coronal}\propto\dot{M}^{1/3}$ (i.e.
$\eta_{coronal}\propto\dot{M}^{-2/3}$) as reported here. Moreover, since the
process may depend only on the amount of matter falling into the black hole,
it may be independent of the inner radius of the standard disc. The details of
the BZ mechanism are complex and uncertain and in particular, the dependence
of $B_{\phi}$ on the accretion rate is theoretically not well understood.
Hence a quantitative study as to whether the observed correlation is
consistent with the mechanism is beyond the scope of this paper. However, for
the states considered here the source does not exhibit powerful jets and hence
it is not clear that the BZ mechanism would be active.
A similar analysis for the other states of GRS 1915+105 (and other black hole
systems) need to be undertaken to ascertain the scope and universality of the
coronal efficiency and accretion rate correlation, which in turn has the high
possibility of distinguishing the different mechanisms by which the corona is
heated.

##  5 Summary and Conclusions

  * •

We have performed time-resolved spectroscopy for the heartbeat state of GRS
1915+105 and found that the heartbeat state oscillation could be described as
coordinated variation of accretion rate, inner disc radii and comptonised flux
occurring at 0.1$L_{edd}$.

  * •

A correlation between coronal efficiency and accretion rate is found for
$\chi$, IMS and heartbeat state which is approximately
$\eta_{coronal}\propto\dot{M}^{-2/3}$. For the HS state the efficiency
decreases with inner disc radii. However, for the IMS and $\chi$ states the
efficiency is similar to that of the HS despite the inner radii being larger.

## Acknowledgements

We thank the referee for constructive suggestions, which were very helpful in
improving the article. We thank the members of LAXPC $\&$ SXT instrument team
for their contribution to the development of the instrument. We also
acknowledge the contributions of the AstroSat project team at ISAC. This
research work makes use of data from the AstroSat mission of the Indian Space
Research Organisation (ISRO), archived at the Indian Space Science Data Centre
(ISSDC). DR acknowledges Samuzal Barua, and Vaishak Prasad for enlightening
discussions.

## Data Availability

The datasets were derived from sources in the public domain: AstroSat Archive.
The software and tools used to process the data are available at LAXPC
Software, SXT Software.

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# Mixed-coordinate Node-link Visualization for Co-authorship Hypergraph
Networks

Mohsen Nafar
Bielefeld University
<EMAIL_ADDRESS>
&Hamed Azami Zenouzagh
<EMAIL_ADDRESS>

###### Abstract

We present an algorithmic technique for visualizing the co-authorship networks
and other networks modeled with hypergraphs (set systems). As more than two
researchers can co-author a paper, a direct representation of the interaction
of researchers through their joint works cannot be adequately modeled with
direct links between the author-nodes. A hypergraph representation of a co-
authorship network treats researchers/authors as nodes and papers as
hyperedges (sets of authors). The visualization algorithm that we propose is
based on one of the well-studied approaches representing both authors and
papers as nodes of different classes. Our approach resembles some known ones
like anchored maps but introduces some special techniques for optimizing the
vertex positioning. The algorithm involves both continuous (force-directed)
optimization and discrete optimization for determining the node coordinates.
Moreover, one of the novelties of this work is classifying nodes and links
using different colors. This usage has a meaningful purpose that helps the
viewer to obtain valuable information from the visualization and increases the
readability of the layout. The algorithm is tuned to enable the viewer to
answer questions specific to co-authorship network studies.

_K_ eywords Hypergraph visualization $\cdot$ Graph drawing $\cdot$ Co-
authorship network visualization

##  1 Introduction

The theory of Complex networks is a well-established branch of Computer
Science and Mathematics, typically utilizing graphs as models of the real-
world systems formed by interacting entities. Those interactions are not
always limited to pairwise interactions; thus, graphs, that naturally encode
just pairwise interactions, are not always the best way to model these
systems. In the case of interactions involving three and more entities,
_hypergraphs_ seem to be a more adequate mathematical abstraction. Indeed,
lately, hypergraphs have found their way into the publication stream in
network science [11, 4, 7, 24, 12]. As real-life systems vary significantly by
the number of subjects in the groups and other parameters, hypergraphs, that
correspond to such systems, are also highly diverse. Thus, it cannot be
expected that a single visual presentation or a visualization algorithm could
be equally suitable for all the use-cases. Therefore, in the current project
we restrict ourselves to specific networks, namely the _co-authorship
networks_ , in scientific disciplines where typical papers involve no more
than a dozen authors.

The dataset we use in our example is the second-largest connected component of
the Mathematics directory on arXiv (https://arxiv.org/), modeled by a
hypergraph with $33$ vertices corresponding to authors, and $48$ hyperedges
corresponding to papers. The sizes of the hyperedges range from $1$ to $4$. We
believe that our proposed algorithm generally produces competitive results for
networks with larger number of smaller hyperedges, i.e. tens to hundreds of
nodes with hyperedges of sizes one to ten, not necessarily co-authorship
networks.

In this paper, we visualize the hypergraph representing the mentioned dataset.
For this specific case, we do not have any constraints for placing the
hypergraph’s vertices. However, there are several objectives that we seek to
reach in our visualization. Among the things we would like to be able to
visually track in the network are: authors with the largest number of
publications, the most actively collaborating authors, the size of the largest
team of authors of a single paper, the most frequent size of a team of authors
of a single paper, the most frequent publication number for an author, the
connection between different authors and papers, the number of authors who
have a specific number of papers, and the appearance of the most active author
in different papers. However, we believe that for other networks with similar
parameters (modeled with a hypergraph and/or a bipartite graph), our technique
is able to address similar questions regarding the relations between different
types of entities in the network.

In this work, each paper-nodes and its adjacent edges are colored according to
the number of co-authors of a paper-node. Using different classes of colors
for nodes with similar properties assists the viewer in distinguishing some of
the graph theoretical properties of the nodes.

###  1.1 Approaches to hypergraph visualization

As hypergraphs are essentially synonymous to set systems, any set
visualization approaches can be applied for visualizing hypergraphs. According
to the survey [2], set visualization approaches fall into five main categories
but in what follows, we combine the first two categories into one.

####  1.1.1 Overlays and Euler diagrams

This approach treats hyperedges as closed curves that contains vertices. This
class of techniques is similar to that of Euler and Venn diagrams which are
vastly being used to visualize sets and their relations and are the most
popular members of set visualization techniques [6]. An example of this
approach is Bubble Sets visualization in which Collins et al., 2009, [8], used
isocontours to reveal the relation between hyperedges. Similar to those of
overlays are Euler diagrams where a set is represented by a closed curve and
set relations can be depicted by relation between curves [20]. In 2009,
Simonetto et al., [25], developed an algorithm that layouts an Euler-like
diagram that is suitable for hypergraphs with medium number of hyperedges of
medium size. Dinkla et al., 2012, [9], proposed a new visualization approach
of the class of Kelp diagrams which they call Point Set Membership. They
developed this approach to visualize hypergraphs in which the position of
their nodes are pre-defined. It seems that this approach can be a good
candidate also for hypergraphs with large hyperedges. SimpleHypergraphs.jl is
a software library that was developed in Julia programming language. It was
designed and built by Antelmi et al., 2020, [4], so as to be used for high-
performance computing on hypergraphs. MetroSets is the name of an online tool
for set systems visualization that is based on a metro map metaphor. This
approach was developed and introduced in 2020 by Jacobsen et al., [17], that
is also compatible to visualize hypergraphs with large hyperedges. Wallinger
et al., 2021, [29], in a study that targeted to compare LineSets, EulerView,
and MetroSets as three of the suitable approaches to visualize medium-sized
datasets, quoted that "Our results include statistically significant
differences, suggesting that MetroSets performs and scales better".

####  1.1.2 Node-link based diagrams:

Both hyperedges and nodes of the hypergraph are represented as two levels of a
bipartite graph vertices where an edge between two vertices stands for the set
membership in the hypergraph. An algorithm to visualize a bipartite
representation of a hypergraph using anchored maps, is proposed by Misue,
2006, [21]. Anchored maps techniques are a class of visualization methods in
which some of the vertices are restricted to be positioned on some pre-defined
places while the rest of vertices have freedom to move; the former vertices
are called anchors and the latter ones free vertices. In this visualization
approach, that is the technique for which we have presented an algorithm, one
level of vertices are positioned on a circle while the vertices belonging to
the other level have freedom to move. A so called extra-node representation of
the hypergraph was used by Ouvrard et al., 2017, [22], when they were trying
to show the improvements it can make for visualising hypergraphs. In fact, it
is a visualization of the star expansion of the hypergraph using an algorithm
called ForceAtlas2. This algorithm is a force-directed algorithm that was
developed by Jacomy et al., 2014, [18]. Another approach is SetCoLa which is a
domain-specific language that was designed, created, and contributed by
Hoffswell et al., 2018, [15], to layout graphs using constraints. According to
the authors "constraints enable flexible graph layout by combining the ease of
automatic layout with customizations for a particular domain". Py3plex, a
Python library that was implemented and introduced by Škrlj et al., 2019,
[26]. The main purpose of this library was to visualize and analyze networks
of multilayer nature. Huang et al., 2020, [16], developed an algorithm called
PLANET that creates a radial layout of the network. One objective of this
algorithm is to minimize the edge crossings while trying to distribute
vertices uniformly.

The layout of our dataset was represented as a bipartite graph according to a
free coordinate layout (force-directed algorithm) that is the most basic kind
of layout is shown in 1. In this picture, the paper-nodes are colored by
yellow and the author-nodes are the purple ones. Figure 1 shows our dataset in
anchored map view where the paper-nodes are placed on the oval (pre-defined
positions) and the author-nodes are inside the oval. The use of the anchored
maps technique improves the picture as it trivializes the distinction of types
of the nodes. Our approach is based on the _anchored maps_ technique, [21],
which we call a mixed-coordinate node-link diagram. Notably, the similarities
between the two techniques that use a curve for placing the nodes
corresponding to hyperedges, having semi-fixed coordinate, and free coordinate
nodes. A free-coordinate type of a node is restricted within a two-dimensional
region on the plane, whereas semi-fixed nodes are restricted to a one-
dimensional curve or some reasonably large discrete set of positions. On the
other hand, we have fixed coordinate nodes, which are not subject to
optimization. Moreover, we apply the bundling method in our work. Furthermore,
the algorithms developed in the two techniques are totally different.

Figure 1: a. Free-coordinate and b. anchored maps layout on our dataset

####  1.1.3 Matrix-based diagrams

These types of diagrams are basically the visualization of the incidence
matrix for a set system. Also some would refer them as matrix metaphors to
show sets and their elements. Rows and columns of the matrix represent sets
and elements, or vice versa, and entries of the matrix depict set membership
relations. UpSet is a highly interactive tool that can be used to analyze sets
from quantitative point of view. It has various abilities and properties such
as showing intersections between sets and group based and query based
aggregations [19]. As another matrix-based approach, Bertifier, a web
application, was developed and introduced by Perin et al., 2014, [23]. Its
developers quoted in their paper that their web application uses "Jacques
Bertin’s matrix analysis method, whose goal was to “simplify without
destroying” by encoding cell values visually and grouping similar rows and
columns". Valdivia et al., 2017, [27], introduced Hypenet which they have
designed for dynamic hypergraphs visualization. This technique can also be
used for pattern and inconsistency detection. HYPER-MATRIX is a visual
analytic tool for temporal hypergraph model exploration that was presented by
Fischer et al., 2020, in [13]. The technique contains various features such as
a geometric deep learning model and different interactions to be used as
combinations.

####  1.1.4 Aggregation-based techniques

In these techniques, some elements can contain multiple data-elements.
Approaches of this category strive to demonstrate the cardinality of sets
through representing the frequency of elements in set-typed data. Radial Sets
is an aggregation-based visualization presented by Alsallakh et al., 2013,
[1]. The authors designed it for visually analysis of a dataset with many
elements for their set membership relations. Wang et al., 2015, [30],
developed a software package named SuperExactTest that contains a theoretical
framework and a visualization technique for multi-set interactions in
programming language R. "A Comprehensive Visualization of Set Intersections"
is the title of a work by Alsallakh and Ren, 2016, [3], in which they
presented an approach named PowerSet. In the paper, it is pointed out that
distribution of elements among set intersections can be evaluated. Moreover,
it is suitable for exploration and comparison of elements’ attributes. PAOH,
stands for "Parallel Aggregated Ordered Hypergraph", is another visualization
approach to layout dynamic hypergraphs. In this technique, vertices are
modeled by parallel horizontal bars and vertical lines represent the
hyperedges. Valdivia et al., 2019, [28], fill the entries corresponding to a
vertex and a hyperedge that it is belong to by dots.

##  2 Proposed visualization technique: mixed-coordinate node-link diagram

The approach we are using in the current study belongs to the class of node-
link diagrams and deals with the _star expansion_ of the hypergraph. This
suits our case well as for hypergraphs with relatively small hyperedges (in
our sample dataset, the hyperedges cardinalities range from $1$ to $4$), the
incidence relations and many graph-theoretic parameters (such as graph
distances and centralities) are observable on a node-link diagram. We thus are
facing the problem of finding a good layout for the star expansion of a
hypergraph.

In our use-case, it is pretty common that the same group of researchers
publish several papers during an extended period of time. In this case, in our
hypergraph representation, we might have many hyperedges with high
multiplicity (identical as sets). Thus a natural idea is to _bundle_ such
hyperedges and represent them with a single visual glyph. As an extra visual
attribute to encode the hyperedge multiplicity, we use the size of the glyph.

We mentioned that the network contains $33$ authors and $48$ papers, which
means that before bundling the hyperedges, we would have a bipartite graph
with $81$ nodes in total. After hyperedge bundling, $48$ paper-nodes are
identified and bundled into $30$ nodes that are displayed. Therefore, the
total number of the nodes which was $81$ is reduced to $63$. The number of
links to be visualized also decreases.

Algorithm 1 Mixed-coordinate layout

Input: A bipartite graph
Output: Mixed-coordinate layout

1: Initial positioning

2: while ($E_{T}\geq\text{Threshold}\text{ }\&\text{
}i\leq\text{Max}_{iteration}$) do

3: Compute repulsion and attraction forces using a modification of force-
directed algorithm

4: Update positions of non-pendant author-nodes

5: end while

6: Compute crossings

7: while ($i\leq\text{iteration}_{1}$) do

8: Mixed-Discrete-Continuous

9: end while

10: positioning pendant author-nodes

11: return layout.

We propose restricting the coordinates of the nodes according to the three
coordinate types: _free_ , _fixed_ , and _semi-fixed_ coordinates, which we
call _mixed-coordinates_ (see Introduction). After hyperedge bundling, some
author-nodes may become pendant in addition to those that had already been
pendant which we call them pendant author-nodes. The rest of the author-nodes
are called non-pendant author-nodes. We restrict the coordinates of paper-
nodes to be semi-fixed and we place them on an oval, whereas the coordinates
of non-pendant author-nodes are of the free-coordinate type. The coordinates
of the pendant author-nodes are of the fixed-coordinate type; we will position
these nodes near the paper-node they belong to and outside of the oval. Note
that by referring to these nodes as fixed-coordinate nodes, we mean that their
positions are fixed relatively to paper-nodes, i.e. only depend on the
position of the paper-node they are connected to. The positions of the pendant
author-nodes are computed at the end of the algorithm when the positions of
all the other nodes are not subject to any change. Furthermore, this new
approach enables us to use continuous and discrete optimization methods to
compute the layout that simulates the system of forces to find the best
configuration.

Moreover, we classify the paper-nodes by the number of their co-authors
(cardinality of the original hyperedge) and to make them distinguishable in
the picture we encode the number of co-authors of a paper-node with node
color. This visual attribute enables the viewer to read out the node degrees
with more immediacy, therefore, improves the readability of the layout. To
improve the layout further, we color links by the same color as the paper-node
they are incident to.

As for the semi-fixed nodes, the primary vehicle for improving the layout with
respect to these nodes is changing the order (permutation) of these nodes on
the curve they belong to. This is where the discrete optimization of the
algorithm comes in. We have two objectives for the optimization here:
minimizing the system’s energy and minimizing the number of crossings. A
continuous optimization algorithm is used to compute the coordinates of the
free nodes. The objective of this part is to minimize the energy of the
system. The designed algorithm, which employs the discussed approach, can be
found in Algorithm 1. The result of the Algorithm 1 without lines 7-9 (without
discrete part of the algorithm) is shown in 2(a). All the paper-nodes are now
placed on the oval, all non-pendant author-nodes are inside of the oval, and
every pendant author-node is placed outside of the oval and is connected to
the paper-node, which it belongs to, with a short link. The position of every
pendant author-node is on an imaginary circle where its center is the
corresponding paper-node, but it is important that the author-node be placed
on the part of the circle that is outside of the oval that contains the paper-
nodes. All author-nodes are purple. In our example, yellow/light-green/dark-
green/blue stand for cardinalities $4,3,2,1$, respectively. The sizes of the
paper-nodes correspond to different numbers of hyperedges bundled into the
particular paper-node; the more hyperedges are in the bundle, the larger is
the corresponding paper-node.

Below you can find a brief description of the procedures of Algorithm 1.

  * •

Initial positioning. Position the nodes on two concentric ovals (pendant
author-nodes are excluded).

  * •

While loop line 2. Check the energy decrease ($E_{T}$) against a threshold and
check if the number of iterations is below the maximum.

  * •

Compute repulsion and attraction forces using a modification of force-directed
algorithm. Apply an iteration of a simple force-directed algorithm which has
the complexity of $O(n^{2})$ per iteration (e.g. Fruchterman-Reingold
algorithm [14]). Note that in this part, forces are computed between three
different pairs of nodes: attraction force between a pair of nodes that are
connected by an edge, repulsion force between all pairs of non-pendant author-
nodes, and repulsion force between pairs of non-pendant author-nodes and
paper-nodes. Since all the pendant author-nodes are excluded from this part,
we need to modify the force-directed algorithm. However, this part can be done
using Barnes-Hut algorithm [5] that has complexity of $O(n\cdot\log n)$ per
iteration that needs to be modified for our purpose.

  * •

Update positions of non-pendant author-nodes. Update positions of non-pendant
author-nodes based on total forces acting on them.

  * •

Compute crossings. Compute the number of total and per paper-node crossings
that can be accomplished, e.g., in time complexity of $O(n^{2}\cdot\log n)$,
according to an algorithm in [10].

  * •

Mixed-Discrete-Continuous. This algorithm combines discrete and continuous
optimization to improve the layout and is shown in Algorithm 2.

  * •

Pendant positioning. Add pendant author-nodes back to the graph and place them
close to their paper-nodes outside of the outer oval, no force acts on these
nodes in the whole algorithm. To do this part, we consider an imaginary circle
around the corresponding paper-node for which the center is the paper-node and
place the pendant author-node on its curve in way that it does not lie inside
of the oval on which the paper-nodes are placed.

Algorithm 2 Mixed-Discrete-Continuous

Input: Current state of the system
Output: Improved mixed-coordinate layout using Mixed-Discrete-Continuous
algorithm

1: while ($cr^{{}^{\prime}}<cr^{*}\text{ }\&\text{
}i\leq\text{iterations}_{2}$) do

2: Choose nodes

3: Check number of crossings

4: if (pair is a good candidate) then

5: Swap pair

6: while ($E_{T}\geq\text{Threshold}\text{ }\&\text{
}i\leq\text{iterations}_{3}$) do

7: Compute repulsion and attraction forces using a modification of force-
directed algorithm like in algorithm 1

8: end while

9: end if

10: end while

As previously discussed, the main idea of Mixed-Discrete-Continuous algorithm
is to add another objective (minimizing the number of crossings) to the
problem and solve it by combining discrete and continuous optimization. The
reader may compare the quality of the layouts of 2(a) (before improvement)
with 2(b) that is the result of the improvement made by our algorithm. Note
the decrease in the number of crossings when comparing the two layouts. We
will briefly describe Algorithm 2 in the following.

  * •

While loop line 1. Makes sure that the loop runs while the crossing number has
not decreased ($cr^{{}^{\prime}}$ refers to the number of crossings before the
loop and $cr^{*}$ refers to the number of crossings after one iteration) _and_
the limit on the maximum number of iterations has not been reached.

  * •

Choose nodes. Select two candidates from paper-nodes to swap them.

  * •

Check number of crossings. Check to see whether swapping the chosen paper-
nodes will reduce the number of crossings, we can simply do it in linear time
($O(n)$); since we already know the number of crossings every paper-node
participates in, we need to compute the number of crossings that these nodes
cause in case we replace them with each other.

  * •

Swap pair. Exchange the position of the chosen paper-nodes.

(a) before improvement

(b) after improvement

Figure 2: Mixed-coordinate layout with/without Mixed-Discrete-Continuous
algorithm

##  3 Discussion and results

Assigning different colors to paper-nodes with different cardinalities makes
it easier for the user to visually approximate the number of the papers with
certain number of authors, which was one of the aims we introduced in
introduction of this work as an objective. Since the color of every edge is
the same as the paper-node it is incident to, by looking at an author-node the
user can easily track what the sizes of the collaboration groups (papers) that
an author participated in are, which is another aim that we set for our
technique. Moreover, looking at author-nodes, the user can visually notice
those that have published the largest number of papers. Although bundling the
paper-nodes decreases the degree of the author-nodes it is connected to,
different size of the bundled paper-nodes that depends on the number of the
articles in the bundle solves this problem, therefore, the user is able to
find the most active authors. Furthermore, the user is able to understand the
relation between any two authors by tracking their incident links and since
the colors of the links have a particular meaning the user does not need to
track the edges that have different colors (e.g. two authors under
consideration do not have links with the same color, then they surly have no
common paper). In addition to what we discussed so far, the user is able to
find the most frequent size of a team of authors of a single paper by just
visually tracking the number of paper-nodes that have the same color and
comparing for different colors. Therefore, our technique enables the user not
only to see the relations between vertices of the original hypergraph (in our
case author-nodes), but also the relations between hyperedges and vertices of
the original hypergraph (paper-nodes and author-nodes). We believe that the
information that our technique is able to reveal can not be revealed by other
visualization approaches, unless a combination of approaches get in use. The
color coding of the paper-nodes and links also helps in situations when a node
is placed close to a link which is not incident to.

In current section, we compare our technique with four different visualization
methods which belong to different categories of hypergraph visualization
techniques. Some of them are the-states-of-the-art in the visualization
literature, i.e. SimpleHypergraphs.jl, ForceAtlas2, Bertifier, and Parallel
Aggregated Ordered Hypergraph Visualization (PAOH). Figure 3 shows the output
of these visualization techniques on the dataset we used in this paper. We
evaluated the methods with each other according to these questions: Q1. Which
authors do have the largest number of publications? Q2. Who is the most
actively collaborating author? Q3. Who are the most actively collaborating
team of authors? Q4. What is the size of the largest team of authors of a
single paper? Q5. What is the most frequent size of a team of authors of a
single paper? Q6. What is the most frequent publication number for an author?
Q7. What are the connection between different authors and papers? Q8. What is
the number of authors who have a specific number of papers? Q9. what is the
appearance of the most active author in different papers?

(a) SimpleHypergraphs.jl

(b) ForceAtlas2

(c) Bertifier Matrix

(d) PAOH

Figure 3: Four visualization methods output for a data set with 33 authors and
48 articles

We summarized the results of the comparison of the techniques in answering the
questions in table 1. In this table, a method obtains check-mark for a
question, if a viewer answers the question faster by only looking at the
layout and without doing any calculation except comparing values. In other
words, additional calculations such as summation of values are not allowed and
leads to cross-sign. If the number of an author’s publication is much greater
than the others, a viewer can answer Q1 by using all of these five methods.
However, if the difference is small, PAOH and our method act better than the
others. The reason that PAOH and our technique are able to lead the viewer to
answer the Q1 faster is that we encode the degree of paper-nodes with specific
colors and PAOH possesses additional information in row and column. According
to our method properties such as nodes positioning and using colors in a
meaningful way, the method enables the viewer to answer Q2 and Q3 which cannot
be answered by using other methods. SimpleHypergraph.jl and ForceAtlas2 failed
in finding the size of the largest team of authors of a single paper. In
Bertin’s matrix method, the viewer should add up the values of each column to
obtain the number of authors for each paper and then answers Q4. By comparing
the additional information in PAOH, a viewer can answer Q4. Finding the
largest number is even easier with our layout because the problem is changed
into checking the colors assigned to specific degree of paper-nodes which is a
very easy visual task. For Q5, Q6, and Q8, the description is the same as that
one which was stated for Q4. In these cases, the viewer focuses on the
repetition number of specific colors (as they reveal very specific information
about the paper-nodes). In our technique, the connection between authors and
articles are shown by meaningful colors. As a result, these connections are
clear using our method which the others lack this property. Additional
information of PAOH and the color-using property of our method enables the
viewer to answer Q9. Although, Mixed-coordinate Node-link Visualization method
visualizes complicated networks, the trustworthy of the method are not as good
as that ones which are obtained for networks with larger number of smaller
hyperedges.

Qs |  SimpleHypergraphs.jl |  ForceAtlas2 |  Bertin’s matrix |  PAOH |  Our technique
---|---|---|---|---|---
1 |  ✓ |  ✓ |  ✓ |  ✓ |  ✓
2 |  $\times$ |  $\times$ |  $\times$ |  $\times$ |  ✓
3 |  $\times$ |  $\times$ |  $\times$ |  $\times$ |  ✓
4 |  $\times$ |  $\times$ |  $\times$ |  ✓ |  ✓
5 |  $\times$ |  $\times$ |  $\times$ |  ✓ |  ✓
6 |  $\times$ |  $\times$ |  $\times$ |  ✓ |  ✓
7 |  $\times$ |  $\times$ |  $\times$ |  $\times$ |  ✓
8 |  $\times$ |  $\times$ |  $\times$ |  ✓ |  ✓
9 |  $\times$ |  $\times$ |  $\times$ |  ✓ |  ✓
Table 1: Comparing our method with 4 different visualizations methods

##  4 Conclusion

The algorithm that we have proposed has several novelties that benefits the
readability of the resulting layout. We combined both freely positioned nodes
and nodes attached to the fixed circle, thus providing clear visual difference
between the nodes corresponding to the authors and the papers. Moreover, we
employ node bundling and special positioning of the pendant nodes to maximally
de-clutter the visualization in center of the canvas. We use color coding to
make it easy to visually estimate node degrees, also additional use of color
coding on the links allows for estimation of the degrees of the paper-nodes
that neighbour a given author-node. Furthermore, we have combined continuous
and discrete optimization in a single algorithm for best visual results. Below
we outline some areas for improvement on the proposed technique and topics for
further investigation.

Firstly, in Algorithm 2, one can have a multitude of strategies for choosing
the two paper-nodes as the candidates for swapping in the discrete
optimization stage of the algorithm. One such strategy is to choose the nodes
one after another, and while doing it, we can have different criteria for
these choices. For instance, we can choose the first paper-node randomly or
greedily; take the paper-node with the highest energy or the paper-node with
the highest share in the crossings. Note that we can make multiple copies of
the system after choosing the first paper-node and compute different cases for
different candidate choices of the second paper-node in parallel. Moreover,
the choice of the two paper-nodes can be dependent on each other or
independent. Furthermore, it is possible to choose a pair of paper-nodes at
once rather than choose the two vertices in consecutive steps. However,
currently in this part of the algorithm we choose the two paper-nodes
consecutively, randomly, independently of each other, and according to their
share in the number of crossings. The other mentioned cases remain for further
investigation.

Secondly, the cardinality of the discrete set of positions for placing semi-
fixed coordinate nodes (paper-nodes) is equal to the number of paper-nodes
after bundling. This means that paper-nodes are equally distanced on the oval.
We believe that the layout may benefit from uneven distribution of the paper-
nodes on the oval in some cases, although we do not have the corresponding
implementation for it right now.

Lastly, the approach that we have presented is adequate for the visualization
of hypergraphs sharing similar distributions of node degrees and hyperedge
cardinalities as ours. The question remaining for further investigation is how
far our approach scales for other types of networks. Moreover, in our
approach, we have a circular area and a circumference on which the nodes of
some particular type are placed. We are currently investigating a novel
visualization type where nodes corresponding to hyperedges of different
cardinalities are placed on different concentric circles surrounding the area
of author-nodes. The corresponding work, which is an extension of the current
paper, is under preparation.

## Acknowledgements

We gratefully acknowledge the assistance of professor Elena Bazanova for her
helpful contributions in the editing phase of the paper.

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