# Non-tensorial Gravitational Wave Background in NANOGrav 12.5-Year Data Set

Zu-Cheng Chen<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical
Physics, Institute of Theoretical Physics, Chinese Academy of Sciences,
Beijing 100190, China  School of Physical Sciences, University of Chinese
Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China  Chen Yuan
<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, No.
19A Yuquan Road, Beijing 100049, China  Qing-Guo Huang  Corresponding author:
<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, No.
19A Yuquan Road, Beijing 100049, China  School of Fundamental Physics and
Mathematical Sciences Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China  Center for Gravitation and Cosmology, College of Physical
Science and Technology, Yangzhou University, Yangzhou 225009, China

###### Abstract

We perform the first search for an isotropic non-tensorial gravitational-wave
background (GWB) allowed in general metric theories of gravity in the North
American Nanohertz Observatory for Gravitational Waves (NANOGrav) 12.5-year
data set. By modeling the GWB as a power-law spectrum, we find strong Bayesian
indication for a spatially correlated process with scalar transverse (ST)
correlations whose Bayes factor versus the spatially uncorrelated common-
spectrum process is $107\pm 7$, but no statistically significant evidence for
the tensor transverse, vector longitudinal and scalar longitudinal
polarization modes. The median and the $90\%$ equal-tail amplitudes of ST mode
are $\mathcal{A}_{\mathrm{ST}}=1.06^{+0.35}_{-0.28}\times 10^{-15}$, or
equivalently the energy density parameter per logarithm frequency is
$\Omega_{\mathrm{GW}}^{\mathrm{ST}}=1.54^{+1.21}_{-0.71}\times 10^{-9}$, at
frequency of 1/year.

Introduction. The direct detection of gravitational waves (GWs) from compact
binary coalescences Abbott  _et al._ (2016a, 2019a, 2020a) has marked the
beginning of a new era of GW astronomy and provides a powerful tool to test
gravitational physics in the strong-field regime Abbott  _et al._ (2019b,
2020b). The current ground-based GW detectors are sensitive to GWs at
frequencies of $10\sim 10^{4}$ Hz Abbott  _et al._ (2016b). As a complementary
tool, the stable millisecond pulsars are natural galactic scale GW detectors
that are sensitive in nano-Hertz frequency band, opening a new window to
explore the Universe. By monitoring the spatially correlated fluctuations
induced by GWs on the time of arrivals (TOAs) of radio pulses from an array of
pulsars Sazhin (1978); Detweiler (1979); Foster and Backer (1990), a pulsar
timing array (PTA) seeks to detect the very low frequency GWs which might be
sourced by the inspiral of supermassive black hole binaries (SMBHBs) Jaffe and
Backer (2003); Sesana  _et al._ (2008, 2009), the first-order phase transition
Witten (1984); Hogan (1986), the scalar-induced GWs Saito and Yokoyama (2009);
Yuan  _et al._ (2019a, b), etc. The null-detection of GWs by PTAs has
successfully constrained various astrophysical scenarios, such as cosmic
strings Lentati  _et al._ (2015); Arzoumanian  _et al._ (2018); Yonemaru  _et
al._ (2020), continuous GWs from individual SMBHBs Zhu  _et al._ (2014); Babak
_et al._ (2016); Aggarwal  _et al._ (2018), GW memory effects Wang  _et al._
(2015); Aggarwal  _et al._ (2019), primordial black holes Chen  _et al._
(2020), and stochastic GW backgrounds (GWBs) of a power-law spectrum Lentati
_et al._ (2015); Shannon  _et al._ (2015); Arzoumanian  _et al._ (2018).
However, the direct detection of GWs by PTAs remains a key task in
astrophysical experiments, and is hopefully achieved in the next few years
Siemens  _et al._ (2013); Taylor  _et al._ (2016).

Recently, the North American Nanohertz Observatory for Gravitational Waves
(NANOGrav) collaboration has reported strong evidence for a stochastic common-
spectrum process, which is significantly preferred over an independent red-
noise process in each pulsar Arzoumanian  _et al._ (2020). The characteristic
strain of this process is described by a power-law model, $h_{c}(f)\propto
f^{-2/3}$, corresponding to the GW emission from inspiraling SMBHBs. NANOGrav
announced there was no statistically significant evidence for quadrupolar
spatial correlations. Moreover, this process shows moderately negative
evidence for monopolar and dipolar correlations, which may come from the
reference clock and solar system ephemeris (SSE) anomalies, respectively.
Lacking definitive evidence for quadrupolar spatial correlations Arzoumanian
_et al._ (2020), NANOGrav argued that it is inconclusive to claim a detection
of GWB consistent with general relativity (GR), and the origin of this process
remains controversial.

Even though there is no definitive evidence for tensor transverse (TT)
correlations predicted by GR in the NANOGrav 12.5-year data set, it does not
exclude the possibility of other GW polarization modes allowed in general
metric theories of gravity. In fact, a most general metric gravity theory can
allow two vector modes and two scalar modes besides the two tensor modes, and
these different modes have distinct correlation patterns Lee  _et al._ (2008);
Chamberlin and Siemens (2012); Gair  _et al._ (2015); Boîtier  _et al._
(2020), allowing the GW detectors to explore them separately. To figure out
whether the signal originates from a GWB or not, it is necessary to fit the
data with all possible correlation patterns. In this letter, we perform the
first Bayesian search for the stochastic GWB signal modeled by a power-law
spectrum with all the six polarization modes in the NANOGrav 12.5-year data
set. Such a power-law spectrum of GWB can be produced by the inspiraling
SMBHBs by assuming circular orbits whose decays are dominated by GWs and
neglecting higher moments Cornish  _et al._ (2018). We find the Bayes factor
in favor of a spatially correlated common-spectrum process with the scalar
transverse (ST) correlations versus the spatially uncorrelated common-spectrum
process (UCP) is $107\pm 7$ which indicates that strong Bayesian indication
for the ST correlations in the NANOGrav 12.5-year data set.

Detecting GWB Polarizations with a PTA. The radio pulses from pulsars,
especially millisecond pulsars, arrive at the Earth at extremely steady rates,
and pulsar timing experiments exploit this regularity. The geodesics of the
radio waves can be perturbed by GWs, inducing the fluctuations in the TOAs of
radio pulses Sazhin (1978); Detweiler (1979). The presence of a GW will
manifest as the unexplained residuals in the TOAs after subtracting a
deterministic timing model that accounts for the pulsar spin behavior and the
geometric effects due to the motion of the pulsar and the Earth Sazhin (1978);
Detweiler (1979). By regularly monitoring TOAs of pulsars from an array of the
ultra rotational stable millisecond pulsars Foster and Backer (1990) and using
the expected form for cross correlations of a signal between pulsars in the
array, it is feasible to discriminate the GW signal from other systematic
effects, such as clock or SSE errors.

For any two pulsars ($a$ and $b$) in a PTA, the cross-power spectral density
of the timing residuals induced by a GWB at frequency $f$ will be Lee  _et
al._ (2008); Chamberlin and Siemens (2012); Gair  _et al._ (2015)

$S_{ab}(f)=\sum_{P}\frac{h_{c,P}^{2}}{12\pi^{2}f^{3}}\Gamma^{P}_{ab}(f),$ (1)

where $h_{c}^{P}(f)$ is the characteristic strain and the sum is over all the
six possible GW polarizations which may be presented in a general metric
gravity theory, namely $P=+,\times,x,y,l,b$. Here, “$+$” and “$\times$” denote
the two different spin-2 transverse traceless polarization modes; “$x$” and
“$y$” denote the two spin-1 shear modes; “$l$” denotes the spin-0 longitudinal
mode; and “$b$” denotes the spin-0 breathing mode. The overlap function
$\Gamma^{P}_{ab}$ for two pulsars is given by Lee  _et al._ (2008); Chamberlin
and Siemens (2012)

$\displaystyle\Gamma^{P}_{ab}(f)=$ $\displaystyle\frac{3}{8\pi}\int
d\hat{\Omega}\left(e^{2\pi
ifL_{a}(1+\hat{\Omega}\cdot\hat{p}_{a})}-1\right)\times$ (2)
$\displaystyle\left(e^{2\pi
ifL_{b}(1+\hat{\Omega}\cdot\hat{p}_{b})}-1\right)F^{P}_{a}(\hat{\Omega})F^{P}_{b}(\hat{\Omega}),$

where $L_{a}$ and $L_{b}$ are the distance from the Earth to the pulsar $a$
and $b$ respectively, $\hat{\Omega}$ is the propagating direction of the GW,
and $\hat{p}$ is the direction of the pulsar with respect to the Earth. The
antenna patterns $F^{P}(\hat{\Omega})$ are given by

$F^{P}(\hat{\Omega})=e^{P}_{ij}(\hat{\Omega})\frac{\hat{p}^{i}\hat{p}^{j}}{2(1+\hat{\Omega}\cdot\hat{p})},$
(3)

where $e^{P}_{ij}$ is the polarization tensor for polarization mode $P$ Lee
_et al._ (2008); Chamberlin and Siemens (2012). Following Cornish  _et al._
(2018), we define

$\displaystyle\Gamma^{\mathrm{TT}}_{ab}(f)$ $\displaystyle=$
$\displaystyle\Gamma^{+}_{ab}(f)+\Gamma^{\times}_{ab}(f),$ (4)
$\displaystyle\Gamma^{\mathrm{ST}}_{ab}(f)$ $\displaystyle=$
$\displaystyle\Gamma^{b}_{ab}(f),$ (5)
$\displaystyle\Gamma^{\mathrm{VL}}_{ab}(f)$ $\displaystyle=$
$\displaystyle\Gamma^{x}_{ab}(f)+\Gamma^{y}_{ab}(f),$ (6)
$\displaystyle\Gamma^{\mathrm{SL}}_{ab}(f)$ $\displaystyle=$
$\displaystyle\Gamma^{l}_{ab}(f).$ (7)

For the $\mathrm{TT}$ and $\mathrm{ST}$ polarization modes, the overlap
functions are approximately independent of the distance and frequency and can
be analytically calculated by Hellings and Downs (1983); Lee  _et al._ (2008)

$\displaystyle\Gamma^{\mathrm{TT}}_{ab}(f)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\delta_{ab})+\frac{3}{2}k_{ab}\left(\ln
k_{ab}-\frac{1}{6}\right),$ (8) $\displaystyle\Gamma^{\mathrm{ST}}_{ab}(f)$
$\displaystyle=$
$\displaystyle\frac{1}{8}\left(3+4\delta_{ab}+\cos\zeta_{ab}\right),$ (9)

where $\delta_{ab}$ is the Kronecker delta symbol, $\zeta_{ab}$ is the angle
between pulsars $a$ and $b$, and $k_{ab}\equiv(1-\cos\zeta_{ab})/2$. Note that
$\Gamma^{\mathrm{TT}}_{ab}$ is known as the Hellings & Downs (HD) Hellings and
Downs (1983) or quadrupolar correlations. However, there exist no analytical
expressions for the vector longitudinal ($\mathrm{VL}$) and scalar
longitudinal ($\mathrm{SL}$) polarization modes, and we calculate them
numerically.

PTAs are sensitive to the GWs at frequencies of approximately $10^{-9}\sim
10^{-7}$ Hz, and it is expected that the GWB from a population of inspiraling
SMBHBs will be the dominant source in this frequency band Jaffe and Backer
(2003); Sesana  _et al._ (2008, 2009). Assuming the binaries are in circular
orbits and the orbital decay is dominated by the GW emission, the cross-power
spectral density of Eq. (1) can be approximately estimated by Cornish  _et
al._ (2018)

$S_{ab}(f)=\sum_{I={\mathrm{TT},\mathrm{ST},\mathrm{VL},\mathrm{SL}}}\Gamma^{I}_{ab}\frac{\mathcal{A}_{I}^{2}}{12\pi^{2}}\left(\frac{f}{f_{\mathrm{yr}}}\right)^{-\gamma_{I}}f_{\mathrm{yr}}^{-3},$
(10)

where $\mathcal{A}_{I}$ is the GWB amplitude of polarization mode $I$, and
$f_{\mathrm{yr}}=1/\mathrm{year}$. The power-law index $\gamma_{I}$ for the TT
polarization is $\gamma_{\mathrm{TT}}=13/3$, and
$\gamma_{\mathrm{ST}}=\gamma_{\mathrm{VL}}=\gamma_{\mathrm{SL}}=5$ for other
polarizations. The dimensionless GW energy density parameter per logarithm
frequency for the polarization mode $I$ is related to $\mathcal{A}_{I}$ by,
Thrane and Romano (2013),

$\Omega_{\mathrm{GW}}^{I}(f)=\frac{2\pi^{2}}{3H_{0}^{2}}f^{2}h_{c,I}^{2}=\frac{2\pi^{2}f_{\mathrm{yr}}^{2}}{3H_{0}^{2}}\mathcal{A}_{I}^{2}\left(\frac{f}{f_{\mathrm{yr}}}\right)^{5-\gamma_{I}},$
(11)

where $H_{0}$ is the Hubble constant and we take
$H_{0}=67.4\,\mathrm{km}\sec^{-1}\mathrm{Mpc}^{-1}$ from Planck 2018 (Aghanim
_et al._ , 2020).

PTA data analysis. The NANOGrav collaboration has searched the isotropic GWB
in their 12.5-year timing data set Alam  _et al._ (2021) and found strong
evidence for a stochastic common-spectrum process but without statistically
significant evidence for the TT spatial correlations Arzoumanian  _et al._
(2020). In this letter, we perform the first search for the GWB from the non-
tensorial polarization modes in the NANOGrav 12.5-year data set.

Table 1: Parameters and their prior distributions used in the analyses. parameter | description | prior | comments
---|---|---|---
White Noise
$E_{k}$ | EFAC per backend/receiver system |  Uniform $[0,10]$ | single-pulsar analysis only
$Q_{k}$[s] | EQUAD per backend/receiver system |  log-Uniform $[-8.5,-5]$ | single-pulsar analysis only
$J_{k}$[s] | ECORR per backend/receiver system |  log-Uniform $[-8.5,-5]$ | single-pulsar analysis only
Red Noise
$A_{\rm{RN}}$ | red-noise power-law amplitude |  log-Uniform $[-20,-11]$ | one parameter per pulsar
$\gamma_{\rm{RN}}$ | red-noise power-law spectral index |  Uniform $[0,7]$ | one parameter per pulsar
Uncorrelated Common-spectrum Process (UCP)
$\mathcal{A}_{\mathrm{UCP}}$ | UCP power-law amplitude |  log-Uniform $[-18,-14]$ | one parameter for PTA
$\gamma_{\mathrm{UCP}}$ | UCP power-law spectral index |  delta function ($\gamma_{\mathrm{UCP}}=13/3$) | fixed
GWB Process
$\mathcal{A}_{\mathrm{TT}}$ | GWB amplitude of TT polarization |  log-Uniform $[-18,-14]$ | one parameter for PTA
$\mathcal{A}_{\mathrm{ST}}$ | GWB amplitude of ST polarization |  log-Uniform $[-18,-14]$ | one parameter for PTA
$\mathcal{A}_{\mathrm{VL}}$ | GWB amplitude of VL polarization |  log-Uniform $[-19,-15]$ | one parameter for PTA
$\mathcal{A}_{\mathrm{SL}}$ | GWB amplitude of SL polarization |  log-Uniform $[-20,-16]$ | one parameter for PTA
BayesEphem
$z_{\rm drift}$ [rad/yr] |  drift-rate of Earth’s orbit about ecliptic $z$-axis |  Uniform [$-10^{-9},10^{-9}$] | one parameter for PTA
$\Delta M_{\rm jupiter}$ [$M_{\odot}$] | perturbation to Jupiter’s mass | $\mathcal{N}(0,1.55\times 10^{-11})$ | one parameter for PTA
$\Delta M_{\rm saturn}$ [$M_{\odot}$] | perturbation to Saturn’s mass | $\mathcal{N}(0,8.17\times 10^{-12})$ | one parameter for PTA
$\Delta M_{\rm uranus}$ [$M_{\odot}$] | perturbation to Uranus’ mass | $\mathcal{N}(0,5.72\times 10^{-11})$ | one parameter for PTA
$\Delta M_{\rm neptune}$ [$M_{\odot}$] | perturbation to Neptune’s mass | $\mathcal{N}(0,7.96\times 10^{-11})$ | one parameter for PTA
PCAi | principal components of Jupiter’s orbit |  Uniform $[-0.05,0.05]$ | six parameters for PTA

Following NANOGrav Arzoumanian  _et al._ (2020), in our analyses, we use 45
pulsars whose timing baseline is greater than three years. To calculate the
longitudinal response functions $\Gamma^{\mathrm{VL}}_{ab}$ and
$\Gamma^{\mathrm{SL}}_{ab}$ that are dependent on the pulsar distance from the
Earth, we adopt the distance information from the Australia Telescope National
Facility (ATNF) pulsar
database111https://www.atnf.csiro.au/research/pulsar/psrcat/ Manchester  _et
al._ (2005). Due to the uncertainty in the pulsar distance measurement, the
estimation uncertainty of the overlap function can be $\lesssim 3\%$ for VL
mode, and $\lesssim 20\%$ for SL mode. The timing residuals of each single
pulsar after subtracting the timing model from the TOAs can be decomposed as
Arzoumanian  _et al._ (2016)

$\delta\bm{t}=M\bm{\epsilon}+F\bm{a}+\bm{n}.$ (12)

The term $M\bm{\epsilon}$ accounts for the inaccuracies in the subtraction of
timing model, where $M$ is the timing model design matrix and $\bm{\epsilon}$
is a vector denoting small offsets for the timing model parameters. The timing
model design matrix is obtained through
libstempo222https://vallis.github.io/libstempo package which is a python
interface to TEMPO2 333https://bitbucket.org/psrsoft/tempo2.git Hobbs  _et
al._ (2006); Edwards  _et al._ (2006) timing software. The term $F\bm{a}$
describes all low-frequency signals, including both the red noise intrinsic to
each pulsar and the common red noise signal common to all pulsars (such as a
GWB), where $F$ is the Fourier design matrix with components of alternating
sine and cosine functions and $\bm{a}$ is a vector giving the amplitude of the
Fourier basis functions at the frequencies of
$\\{1/T,2/T,\cdots,{N_{\text{mode}}}/T\\}$ with $T$ the span between the
minimum and maximum TOA in the PTA van Haasteren and Vallisneri (2014).
Similar to NANOGrav Arzoumanian  _et al._ (2020), we use $30$ frequency
components (${N_{\text{mode}}}=30$) for the pulsar intrinsic red noise with a
power-law spectrum while using $5$ frequency components
(${N_{\text{mode}}}=5$) for the common-spectrum process to mitigate the effect
of potentially coupling between the higher-frequency components of common red
noise process and the white noise Arzoumanian  _et al._ (2020). The last term
$\bm{n}$ describes the timing residuals induced by white noise, including a
scale parameter on the TOA uncertainties (EFAC), an added variance (EQUAD),
and a per-epoch variance (ECORR) for each backend/receiver system Arzoumanian
_et al._ (2016).

Similar to NANOGrav Arzoumanian  _et al._ (2020), we use the latest JPL SSE,
DE438 Folkner and Park (2018), as the fiducial SSE. For verification, we also
allow for the BayesEphem Vallisneri  _et al._ (2020) corrections to DE438 to
model the SSE uncertainties. However, one should bear in mind that introducing
BayesEphem would subtract the power from the putative GWB process and suppress
the evidence of the GWB process Vallisneri  _et al._ (2020); Arzoumanian  _et
al._ (2020); Pol  _et al._ (2020). To extract information from the data, we
perform the Bayesian parameter inferences by closely following the procedure
in Arzoumanian  _et al._ (2018, 2020). The parameters of our models and their
prior distributions are summarized in Table 1. To reduce the computational
costs, in our analyses, we fixed the white noise parameters to their max
likelihood values from results released by
NANOGrav444https://github.com/nanograv/12p5yr_stochastic_analysis. We use
enterprise Ellis  _et al._ (2020) and
enterprise_extension555https://github.com/nanograv/enterprise_extensions
software packages to calculate the likelihood and Bayes factors and use
PTMCMCSampler Ellis and van Haasteren (2017) package to do the Markov chain
Monte Carlo sampling. To reduce the number of samples needed for the chains to
burn in, we use draws from empirical distributions to sample the pulsars’ red
noise parameters as was done in Aggarwal  _et al._ (2018); Arzoumanian  _et
al._ (2020), with the distributions based on the posteriors obtained from an
initial Bayesian analysis that includes only the pulsars’ red noise (i.e.
excluding any common red noise process).

Our analysis is mainly based on the Bayesian inference in which the Bayes
factor
$\mathcal{B}_{10}\equiv\rm{Pr}(\mathcal{D}|\mathcal{M}_{1})/\rm{Pr}(\mathcal{D}|\mathcal{M}_{0})$
is used to quantify the model selection, where
$\rm{Pr}(\mathcal{D}|\mathcal{M})$ denotes the probability that the data
$\mathcal{D}$ are produced under the assumption of model $\mathcal{M}$. In
Kass and Raftery (1995), $\mathcal{B}_{10}\in[20,150]$ and
$\mathcal{B}_{10}>150$ respectively correspond to strong and very strong
evidence for $\mathcal{M}_{1}$. More optimistically,
$\mathcal{B}_{10}\in[10,30]$, $\mathcal{B}_{10}\in[30,100]$, and
$\mathcal{B}_{10}>100$ correspond to strong, very strong and extreme evidence
for $\mathcal{M}_{1}$ in Lee and Wagenmakers (2014). NANOGrav found strong
evidence for a common-spectrum process in the 12.5-year data set and reported
the Bayes factors of UCP model versus the pulsar-intrinsic red noise only
model to be $10^{4.5}$ with DE438, and $10^{2.4}$ with BayesEphem Arzoumanian
_et al._ (2020). In this letter, the UCP model with fixed spectral index
$\gamma_{\mathrm{UCP}}=13/3$ is taken as the fiducial model $\mathcal{M}_{0}$,
and the model $\mathcal{M}_{1}$ with $\mathcal{B}_{10}\gg 1$ is supposed to be
significantly preferred over the UCP model. We perform analyses on various
models by considering different correlation combinations as presented in Eq.
(10).

ephemeris | TT | ST | VL | SL
---|---|---|---|---
DE438 | $4.96(9)$ | $107(7)$ | $1.94(3)$ | $0.373(5)$
BayesEphem | $2.35(3)$ | $18.4(7)$ | $1.31(2)$ | $0.555(7)$
Table 2: The Bayes factors for various models compared to the UCP model with
$\gamma=13/3$. The digit in the parentheses gives the uncertainty on the last
quoted digit.

Results and discussion. Our results are summarized in Table 2 in which we list
the Bayes factors for different models with respect to the UCP model. The
Bayes factor of the TT model compared to the UCP model is $4.96\pm 0.09$ with
DE438, and $2.35\pm 0.03$ with BayesEphem, indicating no statistically
significant evidence for the TT correlations in the data, which is consistent
with the results from NANOGrav Arzoumanian  _et al._ (2020). The Bayes factors
of VL and SL models compared to the UCP model are smaller than $3$, implying
the VL and SL signals are “not worth more than a bare mention” Kass and
Raftery (1995). However, the Bayes factor for the ST model versus the UCP
model is $107\pm 7$ with DE438, implying strong indication for the ST
correlations Kass and Raftery (1995); Lee and Wagenmakers (2014), and we
obtain the median and the $90\%$ equal-tail amplitudes as
$\mathcal{A}_{\mathrm{ST}}=1.06^{+0.35}_{-0.28}\times 10^{-15}$ or
equivalently $\Omega_{\mathrm{GW}}^{\mathrm{ST}}=1.54^{+1.21}_{-0.71}\times
10^{-9}$, at frequency of 1/year. It is known that BayesEphem may absorb a
common-spectrum process and weaken the evidence of the GWB process if it
exists in the data Vallisneri  _et al._ (2020); Arzoumanian  _et al._ (2020);
Pol  _et al._ (2020). Nevertheless, even in the case of BayesEphem, the Bayes
factor for the ST model is $18.4\pm 0.7$ which is still significant in the
sense of statistics. See the Bayesian posteriors for the ST amplitude
$\mathcal{A}_{\mathrm{ST}}$ obtained in the ST model in Fig. 1. Although
NANOGrav reported the UCP is more consistent with $\gamma=5.5$ Arzoumanian
_et al._ (2020), we found that such a large Bayes factor for ST model versus
the UCP model cannot be explained by the ST spectral index
$\gamma_{\rm{ST}}=5$ because the Bayes factor for the ST model versus the UCP
model with $\gamma=5.5$ is $96\pm 9$ with DE438. It implies that the preferred
ST model is likely attributed to the cross-correlations. Furthermore, we also
consider a model that includes a common-spectrum process and an off-diagonal
ST-correlated process where all auto-correlation terms are set to zero. The
Bayesian amplitude posteriors are shown in Fig. 2 in which the amplitude
posterior of the off-diagonal ST-correlated process is significant and
comparable to the amplitude posterior of the common-spectrum process,
indicating that the large Bayes factor for the ST model should be attributed
to the cross-correlations in the NANOGrav 12.5-year data set.

Figure 1:  Bayesian posteriors for the ST amplitude
$\mathcal{A}_{\mathrm{ST}}$ obtained in the ST model under DE438 and
BayesEphem ephemeris schemes, respectively. Figure 2:  Bayesian amplitude
posteriors in a model (with DE438) that includes a common-spectrum process and
an off-diagonal ST-correlated process where all auto-correlation terms are set
to zero. The posteriors shown here are marginalized to each other.

In addition, we also consider an ST+TT model in which we simultaneously take
into account both the ST and TT correlations. The contour plot and the
posterior distributions of the ST and TT amplitudes in the ST+TT model are
shown in Fig. 3, which implies that the presence of ST correlations is
preferred even using BayesEphem, but no significant evidence for additional TT
correlations. The amplitude of ST mode from this model with DE438 is
$\mathcal{A}_{\mathrm{ST}}=1.02^{+0.36}_{-0.44}\times 10^{-15}$ or
equivalently $\Omega_{\mathrm{GW}}^{\mathrm{ST}}=1.45^{+1.21}_{-0.98}\times
10^{-9}$, at frequency of 1/year. This result is consistent with the former
one in the ST model.

Figure 3:  One and two-dimensional marginalized posteriors of ST and TT
amplitudes obtained from the ST+TT model under DE438 and BayesEphem ephemeris
schemes, respectively. We show both the $1\sigma$ and $2\sigma$ contours in
the two-dimensional plot.

To summarize, we find strong Bayesian indication for the ST correlations but
no statistically significant evidence for the TT, VL, and SL correlations in
the NANOGrav 12.5-year data set. We hope that the future PTA data sets growing
in timespan and number of pulsars continue to confirm our results presented in
this letter.

Acknowledgments.  We would like to thank the anonymous referee for the useful
suggestions and comments. We also acknowledge the use of HPC Cluster of ITP-
CAS and HPC Cluster of Tianhe II in National Supercomputing Center in
Guangzhou. This work is supported by the National Key Research and Development
Program of China Grant No.2020YFC2201502, grants from NSFC (grant No.
11975019, 11690021, 11991052, 12047503), Strategic Priority Research Program
of Chinese Academy of Sciences (Grant No. XDB23000000, XDA15020701), and Key
Research Program of Frontier Sciences, CAS, Grant NO. ZDBS-LY-7009.

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# Identifying Competition and Mutualism Between Online Groups

Nathan TeBlunthuis 1 2, Benjamin Mako Hill 1

###### Abstract

Platforms often host multiple online groups with overlapping topics and
members. How can researchers and designers understand how related groups
affect each other? Inspired by population ecology, prior research in social
computing and human-computer interaction has studied related groups by
correlating group size with degrees of overlap in content and membership, but
has produced puzzling results: overlap is associated with competition in some
contexts but with mutualism in others. We suggest that this inconsistency
results from aggregating intergroup relationships into an overall
environmental effect that obscures the diversity of competition and mutualism
among related groups. Drawing on the framework of community ecology, we
introduce a time-series method for inferring competition and mutualism. We
then use this framework to inform a large-scale analysis of clusters of
subreddits that all have high user overlap. We find that mutualism is more
common than competition.

##  1 Introduction

Online groups do not exist in isolation.111We use the term “online group”
instead of “online community” to help avoid confusion with our term “community
ecology” which plays an important conceptual and analytic role in our paper.
Analyzing interdependence between online groups can be complex (Hill and Shaw
2019), but recent social computing research has sought to quantify how online
groups share users or topics (Datta, Phelan, and Adar 2017), and how such
interactions relate to outcomes like the emergence of new groups (Tan 2018),
the spread of hate speech (Chandrasekharan et al. 2017) and contributions to
peer-produced knowledge (Vincent, Johnson, and Hecht 2018). While this work
has demonstrated that intergroup interactions matter, very little intergroup
research has tackled questions of group success—i.e., why some online groups
succeed in maintaining active and long-lived participation while most do not.
Can intergroup relationships explain whether online groups will grow or
decline?

We seek to answer this question in terms of _ecological relationships_
following prior studies in social computing (Wang, Butler, and Ren 2012; Zhu,
Kraut, and Kittur 2014; Zhu et al. 2014), we take inspiration from
organizational ecology, an influential body of theory in sociology, and
analyze _competition_ and _mutualism_ between online groups (Hannan and
Freeman 1989; Baum and Shipilov 2006). Prior ecological studies of online
groups have yielded inconsistent results that differ both from one context to
another and from theoretical predictions. For example, wikis whose memberships
overlap with other wikis survived longer (Zhu et al. 2014), but Usenet groups
with overlapping memberships failed more quickly (Wang, Butler, and Ren 2012).

We propose that limitations of the _population ecology_ framework used by
these studies give rise to these inconsistencies. We therefore introduce an
alternative framework inspired by _community ecology_ that seeks to directly
study competitive and mutualistic interactions between groups. Population
ecology models how overlapping resources among groups affect their subsequent
growth, decline, or survival (Freeman and Audia 2006), but it does not
directly study interactions. By contrast, community ecology models related
groups as an “ecological community” structured by a network of competitive and
mutualistic relationships.

We introduce our community ecology approach and compare it to the population
ecology approach from prior work in a two-part empirical study of 641 clusters
of online groups among the 10,000 communities on Reddit with the most
contributors. In Study A, we demonstrate a prototypical population ecology
analysis by testing density dependence theory. Prior studies would interpret
the results of this analysis as suggesting that high degrees of user overlap
are associated with competition.

In Study B, we introduce our method for inferring networks of ecological
relationships among related online groups based on clustering analysis and
vector autoregression (VAR) models of group size over time (Ives et al. 2003).
We illustrate the method in four case studies and present a large-scale
analysis showing that mutualistic interactions are far more common than
competitive ones. To validate our approach, we show that including ecological
interactions in our VAR models improves time series forecasting.

Our findings illuminate the different contributions of population ecology and
community ecology. While Study A suggests that competition is strongest when
user overlap is high, Study B finds widespread mutualism among groups with
highly overlapping memberships. Although these findings might seem
contradictory, population and community ecology analyses provide complementary
views. Population ecology points to favorable or unfavorable conditions for
building online groups—conditions that may or may not involve competition and
mutualism. A community ecology analysis can infer local networks of
competition and mutualism to explain how specific ecological relationships
contribute to growth or decline. By demonstrating that ecological
relationships within clusters of highly related groups are important—and by
describing how to measure them—this paper lays the groundwork for future
investigations into interdependent online groups and designs that support
ecological communities.

##  2 Related Work

Online groups are important sites for social support (De Choudhury and De
2014), entertainment (Ducheneaut et al. 2006), information sharing (Benkler
2006), and political mobilization of disinformation campaigns and protest
movements (Choudhury et al. 2016; Benkler et al. 2013; Krafft and Donovan
2020). Although an online group’s ability to achieve its goals depends on
attracting and retaining contributors, few develop a sizable group of
participants (Kraut, Resnick, and Kiesler 2012). Many attempts to explain
growth and decline of online groups look to properties of individual groups
like characteristics of founders and designs for regulating behavior (Kraut,
Resnick, and Kiesler 2012; Halfaker et al. 2013).

By contrast, recent research shows the importance of interdependence among
online groups (Kairam, Wang, and Leskovec 2012; Tan 2018; Waller and Anderson
2019). For example, banning hate subreddits reduced hate speech in related
subreddits (Chandrasekharan et al. 2017), Reddit and Stack Overflow receive
substantial benefits from activity on Wikipedia (Vincent, Johnson, and Hecht
2018), and editors make valuable and qualitatively different contributions
across different languages of Wikipedia (Hale 2015). Our work contributes to
this literature by providing a new conceptual lens and statistical method for
studying intergroup connections.

###  2.1 Ecological Interdependence

Ecological approaches to online groups see online groups as depending on
resources. Our conceptual approach, like prior ecological research in social
computing and information systems, builds on resource dependence theory (RDT)
(Butler 2001; Wang, Butler, and Ren 2012). According to RDT, members of online
groups contribute resources such as content, information, attention or social
interactions that sustain the group. Ecological approaches observe that
interrelated online groups may share resources with one another and effect
each other’s growth and survival as a result. Rival resources like
participants’ time, attention, and efforts become unavailable to others when
used by one group (Benkler 2006; Romer 1990), and competition over important
rival resources can explain declines participation (Wang, Butler, and Ren
2012).

On the other hand, the value of a nonrival resource does not decrease (and may
even increase) when it is used. Nonrival resources that “spill over” can
result in mutualism that promotes growth in related groups (Zhu, Kraut, and
Kittur 2014). For example, the usefulness of a communication network increases
as more people join it (Fulk et al. 1996). Similarly, the usefulness of an
information good can increase as more people come to know, refer to, and
depend upon it. Ecological approaches seek to understand how different types
of resources will limit or promote growth.

###  2.2 Population Ecology, Density Dependence and Overlapping Resources

Our work builds on a tradition rooted in organizational ecology. First
developed in the late 1970s by sociologists studying interactions between
firms, organizational ecology was inspired by, and has drawn closely from,
ecological studies in biology (Hannan and Freeman 1989). Organizational
ecology has inspired at least three high-quality empirical studies of how
resources shared by online groups shared shape their growth, decline, or
survival (Wang, Butler, and Ren 2012; Zhu, Kraut, and Kittur 2014; Zhu et al.
2014). These studies draw from the population ecology strand of organizational
ecology, and specifically enage with density dependence theory (DDT).

DDT concieves of competitive or mutualistic forces as a function of population
density. In the earliest and most influential studies of DDT, density is
simply the size of the population, which defines a homogenous set of
organizations of groups facing the same competitive and mutualistic pressures
(Aldrich and Ruef 2006). However, online groups sharing a platform have
diverse topics (Kairam, Wang, and Leskovec 2012), norms (Chandrasekharan et
al. 2018), and user bases (Tan 2018). To account for this diversity,
ecological studies of online groups have modeled density dependence based on
the concept of _overlap density_ (Baum and Shipilov 2006; Wang, Butler, and
Ren 2012; Zhu, Kraut, and Kittur 2014; Zhu et al. 2014). Overlap density
measures the extent to which one group’s members or topics overlap with all
other groups’. Overlap density thus characterizes a group’s _niche_ or local
_resource environment_ defined by its distinctive topic and membership.

DDT proposes a model for the growth of organizational populations in which
mutualism drives a virtuous cycle of population growth (Carroll and Hannan
1989; Hannan and Freeman 1989). For example, a population of online groups,
such as those sharing a platform, may grow in size as their platform gains in
popularity, as established groups spin off new ones, and as useful knowledge
develops that can be shared between groups (Tan 2018; Zhu, Kraut, and Kittur
2014). On the other hand, when density is high, competition among population
members over rival resources limits growth (Hannan and Freeman 1989). DDT thus
proposes a trade-off in which low density reflects limited opportunities for
mutualistic contributions of nonrival resources, while high density reflects
competition over rival resources. Therefore, DDT predicts that the
relationship between density and positive outcomes like growth or survival is
$\cap$-shaped (inverse-U-shaped) (Baum and Shipilov 2006; Carroll and Hannan
1989).

Tests of DDT in populations of online groups yield inconsistent results. In
Wang, Butler, and Ren (2012), user overlap in Usenet newsgroups is associated
with decreasing numbers of participants. Similarly, TeBlunthuis, Shaw, and
Hill (2020) find that topical overlaps between online petitions are negatively
associated with participation. By contrast, Zhu, Kraut, and Kittur (2014) find
that membership overlap is positively associated with increasing survival of
new Wikia wikis. Only Zhu et al. (2014) find support for the $\cap$-shaped
relationship predicted by DDT in an enterprise social media platform.

In Study A, we provide a test of DDT using data from Reddit. The classical
logic of DDT appears reasonable in the context of Reddit because low overlap
density likely reflects an impoverished environment lacking in non-rival
resources like skills and knowledge of experienced users, while a group with
high overlap density likely faces competition over its members (Zhu et al.
2014; Zhu, Kraut, and Kittur 2014): (H1) The relationship between overlap
density and the growth of online groups is $\cap$-shaped (inverse-U-shaped).

###  2.3 Introducing Community Ecology

The distinction between population ecology and community ecology theories is
in where they locate ecological dynamics like competition and mutualism. In
population ecology, competition and mutualism are properties of an
environmental niche; but in community ecology, they are relations in networks
of interdependent groups called _ecological communities_ (Freeman and Audia
2006). While most community ecology studies of classical organizations analyze
ecological communities of different organizational forms, some, like our
study, analyze communities of related organizations (Freeman and Audia 2006).

Community ecology focuses on _ecological interactions_ (Aldrich and Ruef
2006). Mutualism is an ecological interaction where one group has a positive
influence on the second such that growth in the first group leads to growth in
the second. Competition is when one group has a negative effect on the second
such that growth in the first group leads to decline in the second. These
relationships are modeled as edges of a directed network. As a result,
ecological interactions can be mutualistic in one direction and competitive in
the other and mutualism (or competition) from one group to another group may
(or may not) be returned in kind. The goal of many community ecology analyses
in both biology and organization science is to infer and analyze the
_community matrix_ , which quantifies competitive and mutualistic interactions
(Ives et al. 2003; Aldrich and Ruef 2006).

In Study B, we demonstrate community ecology by inferring networks of
ecological interactions in ecological communities on Reddit to determine
whether mutualism or competition among subreddits is more common. We then
present case studies to illustrate different types of ecological communities.
Finally, we evaluate whether modeling ecological interactions is useful for
making time series forecasts of participation in online groups: (H2) The
addition of ecological interactions to a baseline time series model improves
the forecasting performance.

##  3 Materials & Methods

We analyze data from the publicly available Pushshift archive of Reddit
submissions and comments from December 5th 2005 to April 13th 2020
(Baumgartner et al. 2020). Within this dataset, we limit our analysis to
submissions and comments from the 10,000 subreddits with the highest number of
comments. There are 702 subreddits larger than the smallest subreddit included
in our dataset having a majority of submissions marked “NSFW,” which typically
indicates pornographic material. As others have done in large-scale studies of
Reddit (e.g., Datta, Phelan, and Adar 2017), we exclude these subreddits to
avoid asking members of our research team to inspect clusters including
pornography. The top 10,000 subreddits provide a sufficiently large number of
ecological communities for our statistical analysis.

###  3.1 Study A: Density Dependence Theory

User overlap $o_{i,j}$ quantifies the degree to which two subreddits ($i$ and
$j$) share users. Zhu, Kraut, and Kittur (2014) and Wang, Butler, and Ren
(2012) both measure user overlap between two groups by counting the number of
users contributing to both groups at least once and exclude users who appear
in more than 10 groups. In our preliminary analysis, we found that this
measure led to similarity measures and clusters with poor face validity. These
issues may have stemmed from how Reddit users often peripherally participate
in many groups while participating heavily in few (Zhang et al. 2017).
Therefore, our measure of user overlap follows Datta, Phelan, and Adar (2017)
by using the number of comments each user makes in each pair of groups.

To measure user overlap between subreddits, we first build user frequency
vectors by counting the number of times each user comments in each subreddit.
We prevent giving undue weight to subreddits with higher overall activity
levels by normalizing the comment counts for each subreddit by the maximum
number of comments by a single author in the subreddit:

$f_{u,j}=\frac{n_{\mathrm{u,j}}}{max_{v\in\mathrm{J}}n_{v,j}}$ (1)

where $n_{u,j}$, the user frequency, is the number of times that user $u$
authors a comment in subreddit $j$. This results in a user frequency vector
$F_{j}$ for each subreddit that is sparse and high-dimensional, having one
element for each user account that comments in any subreddit in our dataset.
Next, we use latent semantic analysis (LSA) to reduce the dimensionality of
the user frequency vectors:

$\displaystyle\mathbf{F}$ $\displaystyle=\mathbf{U\Sigma V}^{T}$ (2)
$\displaystyle\mathbin{\stackrel{{\scriptstyle\sim}}{{\smash{F_{j}}\rule{0.0pt}{4.95134pt}}}}$
$\displaystyle=\mathbf{U_{k}}^{T}F_{j}$

where $\mathbf{F}$ is the matrix where columns are author frequency vectors
$F_{j}$ and $\mathbf{U\Sigma V}^{T}$ is its singular value decomposition.
Truncating the singular value decomposition to use only the first $k$ left-
singular vectors gives $\mathbf{U_{k}}$. Left-multiplying a subreddit’s author
frequency vector by $\mathbf{U_{k}}$ transforms the high-dimensional author
frequencies into
$\mathbin{\stackrel{{\scriptstyle\sim}}{{\smash{F_{j}}\rule{0.0pt}{4.95134pt}}}}$,
their approximation in the $k$-dimensional space. Our measure of user overlap
($o_{i,j}$) is the cosine similarity between these vectors:

$o_{i,j}=\frac{\mathbin{\stackrel{{\scriptstyle\sim}}{{\smash{F_{j}}\rule{0.0pt}{4.95134pt}}}}\cdot\mathbin{\stackrel{{\scriptstyle\sim}}{{\smash{F_{i}}\rule{0.0pt}{4.95134pt}}}}}{\lVert\mathbin{\stackrel{{\scriptstyle\sim}}{{\smash{F_{i}}\rule{0.0pt}{4.95134pt}}}}\rVert\lVert\mathbin{\stackrel{{\scriptstyle\sim}}{{\smash{F_{j}}\rule{0.0pt}{4.95134pt}}}}\rVert}$
(3)

where
$\lVert\mathbin{\stackrel{{\scriptstyle\sim}}{{\smash{F_{i}}\rule{0.0pt}{4.95134pt}}}}\rVert$
is the euclidean norm of the transformed user frequencies for subreddit $i$.

Growth $Y_{i}$, the dependent variable in our density dependence model testing
H1 is measured as the change in the (log-transformed) size of a subreddit over
the final 24 weeks of our data, from to November 4th 2019 to April 13th 2020.

Overlap density $d_{i}$, the normalized average user overlap for a given
subreddit is the independent variable in our density dependence model testing
H1:

$\displaystyle d^{*}_{i}$ $\displaystyle=\frac{1}{\left|S\right|-1}\sum_{j\in
R;j\neq i}\mathrm{o}_{i,j}$ $\displaystyle d_{i}$
$\displaystyle=\frac{d_{i}^{*}}{\mathrm{max}_{j}d_{j}^{*}}$ (4)

where $S$ is the set of groups in our dataset.

#### Regression model for H1

To test H1, we fit Model 1:

$\displaystyle\mathrm{Model~{}1}$ $\displaystyle
Y_{i}=B_{0}+B_{1}d_{i}+B_{2}d^{2}_{i}$ (5)

where $Y_{i}$ is the growth of subreddit $i$ and $d_{i}$ is its overlap
density. The model has first and second-order terms for overlap density to
allow for a curvilinear relationship between _overlap density_ and _growth_.

###  3.2 Study B: Introducing Community Ecology

Analyzing networks of ecological interactions is the key difference between
community ecology and population ecology. To identify ecological communities
of related subreddits, we use a clustering procedure based on user overlap. We
selected a clustering model using grid search to obtain a high silhouette
coefficient. The silhouette coefficient captures the degree to which a
clustering creates groups of subreddits with high within-cluster similarity.

We ran the affinity propagation, HDBSCAN and k-means clustering algorithms and
selected the algorithm, hyperparameters, and LSA dimensions $k$ that resulted
in the clustering with a high silhouette coefficient having less than 5,000
isolated subreddits, and at least 50 clusters. We limit the number of isolated
subreddits because some choices of hyperparamters for the HDBSCAN algorithm
could improve the silhouette coefficient, but at the cost of greatly
increasing numbers of isolated subreddits. Choosing a high limit to the number
of isolates helps ensure that our clusters contain highly related communities.
We chose an HDBSCAN clustering with 731 clusters, 4964 isolated subreddits,
$k=600$ LSI dimensions, and a silhouette score of 0.48. We exclude the
isolated subreddits from our analysis. More details about our clustering
selection process are found in the online supplement.

We evaluate the external validity of the chosen clustering using the purity
evaluation criterion. To do so, an undergraduate research assistant examined a
random sample of 100 clusters including 744 subreddits. By visiting the
subreddits and using their own judgment, the assistant flagged subreddits that
did not seem like a good fit for their assigned cluster. Using these labels
and excluding 25 subreddits that have been deleted, made private, or banned,
we calculated the purity of our clustering as 0.92. Therefore, we estimate
that 92% of subreddits belong to their assigned cluster.

Group size is the dependent variable of the models we use to infer ecological
interactions. Measured as the number of distinct users commenting in a
subreddit each week, group size quantifies the number of people who
participate in a subreddit over time. Typical of social media participation
data, group size is highly skewed—i.e., we transform it by adding 1 and taking
the natural logarithm.

#### Inferring ecological interactions

The community matrix $\mathbf{\Phi}$ of ecological interactions can be
inferred from time series data using vector autoregression models (VAR
models). VAR models are a workhorse in biological ecology because VAR(1)
models (i.e., VAR models with a single autoregressive term) have a close
relationship with the Gompertz of population growth, a common theoretical
model in ecology (Ives et al. 2003).

VAR(1) models generalize auto-regressive AR(1) models in time series analysis.
Where AR(1) models predict the state of a single time series as a function of
its previous values, VAR(1) models predict several time series as a function
of each other’s previous values (Ives et al. 2003):

$Y_{t}=B_{0}+B_{1}t+\sum_{k\in K}A_{k}x_{k,t}+\sum_{j\in
M}\Phi_{j}y_{j,t-1}+\epsilon_{t}$ (6)

where $Y_{t}$ is a vector containing the sizes of a set of online groups ($M$)
at time $t$. $B_{0}$ is the vector of intercept terms and $B_{1}$ is the
vector of linear time trends ($b_{1,j}$) for each community ($j$). $\Phi_{j}$
represents the influence of $y_{j,t-1}$, the size of the $j^{\mathrm{th}}$
online group at time $t-1$ on $Y_{t}$. $\Phi_{j}$ is a column of
$\mathbf{\Phi}$, a matrix of coefficients in which the diagonal elements
correspond to intrinsic growth rates (marginal to the trend) for each online
group and the off-diagonal elements are intergroup influences, and
$\epsilon_{t}$ is the vector of error terms

Additional time-dependent predictors ($x_{k,t}$) can be included in the
vectors $X_{k}$ with coefficients $a_{k}$. Because subreddits are created at
different times, growth trends must begin only after the subreddit is created.
We use $X_{k}$ to introduce a counter-trend during the period prior to the
creation of subreddits so that each group’s growth trend begins in the period
the group is created. For each group $j$ created at time $t^{0}_{j}$ we fill
$X_{j}$ with the sequence $[1,2,3,\ldots\ ,t^{0}_{j}-1,0,0,0,\ldots\ ]$. In
other words, $X_{j}$ adds a counter-trend only during the period prior to the
first comment in subreddit $j$. We fix the elements $a_{j,i}$ of $A_{j}$ equal
to 0 unless $i=j$, so the counter trend only influences subreddit $j$. This
effectively sets $a_{j,j}$ approximately equal to $-b_{1,j}$.

We fit VAR(1) models using ordinary least squares as implemented in the vars R
package to predict the group size each week over the history of each subreddit
prior to November 4th 2019\. We hold out 24 weeks of data for forecast
evaluation and fit our models on the remainder. To ensure that sufficient data
is available for fitting the models, we exclude 946 subreddits and 89 clusters
having less than 156 weeks of activity.

#### Characterizing ecological communities

In Study B, we interpret the community matrix $\mathbf{\Phi}$ as a directed
network of ecological interactions, a _competition-mutualism network_ (Ives et
al. 2003). Although the elements of $\mathbf{\Phi}$ correspond to direct
associations between group sizes, ecological interactions can also be
indirect. Consider 3 one-directional interactions between three groups ($a$,
$b$, $c$) such that growth in $a$ predicts decreased growth in $b$
($\phi_{a,b}<0$), growth in $b$ predicts decreased growth in $c$
($\phi_{b,c}<0$), but $a$ and $c$ do not directly interact ($\phi_{a,c}\approx
0$).

This does not necessarily mean that groups A and C are independent. Rather, an
exogenous increase in A predicts a decrease in B and thereby an eventual
increase in C. Such indirect relationships are analyzed by using impulse
response functions (IRFs) to interpret a VAR model. In large VAR models
containing many groups, the great number of parameters can mean that few
specific elements of $\mathbf{\Phi}$ will be statistically significant, even
as many weak direct relationships can combine into statistically significant
impulse response functions (IRFs) (Ives et al. 2003).

An IRF predicts how much each group’s size would change in response to a
sudden increase in the size of each other group:

$\mathbf{\Theta_{t}}=\mathbf{\Theta_{t-1}}\mathbf{\Phi},t=1,2,...$ (7)

where $\mathbf{\Theta_{t}}$ is the impulse response function at time $t$.
$\mathbf{\Theta_{0}}$ is an $M$-by-$M$ identity matrix so our impulses
represent a log-unit increase of 1 to each group. $\mathbf{\Theta_{t}}$ is a
matrix with elements $\theta^{t}_{i,j}$ corresponding to the response of group
$j$ to the impulse of group $i$.

We use IRFs of our VAR(1) models to make our visualizations of example
competition-mutualism networks in §4.2. We compute the IRFs with bootstrapped
confidence intervals (CI) based on 1,000 samples using the vars R package. We
draw an edge $i\rightarrow j$ in the competition-mutualism network if the 95%
CI of $\theta^{t}_{i,j}$ does not include zero at any time $10>=t>0$. If
$\theta^{t}_{i,j}>0$, the edge indicates mutualism and if $\theta^{t}_{i,j}<0$
the edge indicates competition.

Average ecological interaction $\overline{m}$ measures the extent to which an
overall ecological community is mutualistic or competitive by taking the mean
point estimate of the off-diagonal coefficients of $\mathbf{\Phi}$:

$\overline{m}=\frac{1}{\left|M\right|-1}\sum_{i\in M}\sum_{j\in M;j\neq
i}\phi_{i,j}$ (8)

If $\overline{m}>0$ then mutualistic interactions within the ecological
community are stronger than competitive ones, and if $\overline{m}<0$ then
competitive interactions are stronger then mutualistic ones.

Ecological interaction strength $\kappa$ quantifies the overall strength of
ecological interactions in an ecological community as the mean absolute value
of the point estimates of the off-diagonal coefficients of $\mathbf{\Phi}$:

$\kappa=\frac{1}{\left|M\right|-1}\sum_{i\in M}\sum_{j\in M;j\neq
i}\left|\phi_{i,j}\right|$ (9)

where $\left|\phi_{i,j}\right|$ is the absolute value of the coefficient
$\phi_{i,j}$. The average ecological interaction can be close to 0 if
ecological interaction strength is low or if the ecological interaction
strength is high and results from a mixture of competitive and mutualistic
interactions that cancel one another out when averaged.

#### Forecasting growth

To test H2, we evaluate whether modeling ecological interactions improves time
series forecasting of future participation in online groups by comparing the
model in Equation 6 to a baseline model with off-diagonal elements of
$\mathbf{\Phi}$ fixed to 0. This baseline model is equivalent to our VAR
model, but excludes ecological interactions.

We compare our VAR model to the baseline in terms of two forecasting metrics
with differing assumptions: root-mean-square-error (RMSE) and the continuous
ranked probability score (CRPS). RMSE is commonly used, non-parametric, and
intuitive, but does not take differing scales of the predicted variable or
forecast uncertainty into account. Thus, it may place excessive weight on
larger subreddits having greater variation in size. The CRPS accounts for
variance in the data and rewards forecasts for both accuracy and precision and
is thus a “proper scoring rule”” for evaluating probabilistic forecasts
(Gneiting and Raftery 2007). Our CRPS calculations assume that the predictive
forecast distribution for each community is normal with standard deviations
given by the 68.2% forecast confidence interval. We calculate CRPS using the
scoringRules R package.

##  4 Results

###  4.1 Study A: Density Dependence Theory

We test density dependence theory as formulated in H1 using Model 1 which has
first- and second-order terms for the effect of overlap density on growth. As
described in §2.2, H1 hypothesizes that overlap density will have a
curvilinear $\cap$-shaped (inverse-U-shaped) relationship with growth
indicated by a positive first-order regression coefficient and a negative
second-order coefficient.

Figure 1: A 2D histogram of subreddits with overlap density (log-transformed)
on the X-axis and the change in the logarithm of the number of distinct
commenting users on the Y-axis. The black line shows the marginal effect of
overlap density on growth as predicted by Model 2. The gray region shows the
95% confidence interval of the marginal effect.

We observe this predicted relationship between overlap density and growth.
Figure 1 plots the marginal effects of overlap density on growth for the
median subreddit laid over a scatterplot of the data. The point where
increasing density ceases to predict increasing growth and begins to predict
decreasing growth is at the 49th percentile. Prototypical subreddits at this
overlap density grew slightly (95% CI:[0.001,0.06]). Yet subreddits at the
lower and upper extremes of overlap density slightly declined on average.
Typical groups at the 20th percentile of overlap density decline by 1.1
members (95% CI:[-1.1,-1.15]) and typical groups at the 80th percentile
decline by 1.2 members (95% CI:[-1.1,-1.28]).

###  4.2 Study B: Introducing Community Ecology

Figure 2 visualizes the distribution of average ecological interaction and
ecological interaction strength over the 641 ecological communities we
identify. We observe ecological communities characterized by strong forms of
both mutualism and competition, others having mixtures of the two, and some
with few significant ecological interactions. Mutualism is more common than
competition, with the mean community having an average ecological interaction
of 0.03 ($t=14.5$, $p<0.001$). We find that 524 clusters (81.7%) are
mutualistic. Not only are most ecological communities mutualistic, but more
mutualistic ecological communities have greater ecological interaction
strength (Spearman’s $\rho=0.58$, $p<0.001$). Therefore, our community ecology
analysis suggests that among groups with similar users, mutualistic ecological
interactions are more common than competitive ones.

Figure 2: Two-dimensional histogram showing ecological communities on Reddit
in our typology. The X-axis shows the overall degree of mutualism or
competition in clusters of subreddits with high user overlap based on the
average ecological interaction. The Y-axis shows the ecological interaction
strength representing the overall magnitude of competition or mutualism.

#### Example ecological communities

We present four case studies to illustrate our typology of ecological
communities of online groups. Figure 2 shows that we find clusters of
subreddits characterized by mutualism, competition, a mixture of mutualism and
competition, and few ecological relationships at all. We select one case from
each of these four types using our measures of average ecological interaction
($\overline{m}$) and ecological interaction strength ($\kappa$). To allow for
more interesting network structures, we draw our cases from the 367 large
clusters having at least five subreddits.

(a) The ecological community of subreddits for supporting mental health and
survivors of abuse is dense with largely mutualistic interactions.

(b)  The subreddits about real estate and finance are relatively competitive.

(c) Subreddits about watches are dense with both mutualistic and competitive
interactions.

(d) The ecological community of subreddits about Call of Duty video games is
characterized by relatively sparse ecological interactions.

Figure 3: Network visualizations of commensal relationships in example
ecological communities of subreddits with overlapping users. Yellow indicates
competition and purple indicates mutualism.

Figure 3 presents visualizations of competition-mutualism networks
representing statistically significant impulse response functions. For each
example, we examined the terms of the vector autoregression parameter
$\mathbf{\Phi}$, the impulse response functions, and the model fits and
forecasts, all of which are available in our online supplement. We also
visited each subreddit in the clusters and read their sidebars and top posts
to support our brief qualitative descriptions.

#### Mutualism among mental health subreddits

To find a case characterized by mutualism, we selected the top 37 large
clusters with the greatest average ecological interaction. From these, we
arbitrarily chose one interesting ecological community, the mental health
cluster, which includes 11 subreddits for supporting people in struggles with
mental health, addiction, and surviving abuse. Constitutive subreddits include
those focused on specific mental health diagnoses like r/bpd (bipolar
disorder) and r/cptsd (complex post traumatic stress disorder) while others
like r/survivorsofabuse and r/adultsurvivors are support groups.

The interactions among these subreddits are dense and primarily mutualistic as
shown in Figure 3. There are a handful of competitive interactions like the
reciprocal competition detected between r/codedependence and r/bpd. We also
observe some interactions that are mutualistic in one direction and
competitive in the other. For example, growth in r/addiction predicts
increasing growth in r/cptsd even as that growth in r/cptsd predicts
decreasing growth in r/addiction. This suggests a pattern in which r/cptsd
siphons members from r/addiction. That said, the density of mutualistic
interactions shown in Figure 3 suggests that different subreddits have
complementary roles in this ecological community as people turn to different
types of groups for help with interrelated problems. While attempting to
explain why different online groups form mutualistic or competitive
interactions is left to future research, the example of mental health
subreddits demonstrates how groups with related topics and overlapping
participants can have mutualistic interactions.

#### Competition among real estate and finance subreddits

To find competitive clusters, we selected an ecological community that we
label finance from the 36 large clusters with the lowest average ecological
interaction having six subreedits. Three of them: r/realestateinvesting,
r/realestate and r/commercialrealestate, deal in different aspects of the real
estate industry while r/financialindependence and r/fatfire (the acronym
“fire” means “financial independence/retire early”) focus on building wealth
and becoming financially independent and r/financialplanning is a general
financial advice subreddit.

In contrast to the mental health ecological community, the finance cluster has
mostly competitive ties as visualized in Figure 3. We detect three reciprocal
competitive interactions among the three subreddits that focus on real estate.
The edges from r/fatfire to r/commercialrealestate and r/financialindependence
are competitive as well.

Although this cluster is among the most competitive in our data, it contains
mutualistic ties between the general finance subreddits (r/financialplanning
and r/financialindependence) and r/realestate. This reflects just how
prevalent mutualism is among subreddits with high degrees of user overlap.

#### Mixed interactions among timepiece subreddits

Next, we turn to the timepiece ecological community of 7 subreddits about
watches that has low average ecological interaction but high ecological
interaction strength. We selected the timepiece subreddits from 36 large
clusters with the average ecological interaction closest to 0 and then from
the 15 clusters with the greatest ecological interaction strength.

As shown in Figure 3, the network of timepiece subreddits is dense with
ecological interactions (although not as dense as the mental health
subreddits). We observe both reciprocated mutualistic interactions, like that
between r/rolex and r/gshock, and competitive interactions like that between
r/gshock and r/seiko. We also observe numerous unreciprocated competitive and
mutualistic relationships like the mutualism between r/watchexchange and
r/watchcirclejerk222The suffix is widely understood on Reddit to signify a
jokey, meme, or satirical subreddit. and the competition between
r/japanesewatches and r/seiko. Though the average ecological interaction among
these subreddits is near 0, our analysis reveals a complex ecological
community with a mixture of competition and mutualism.

#### Sparse interactions among Call of Duty subreddits

To find a case where ecological interactions are weak, we return to the group
of the 36 large clusters with the average ecological interaction closest to 0
but select from the 15 clusters within this group with the lowest ecological
interaction strength. From these, we chose the Call of Duty cluster containing
five groups about the popular series of video games.

The Call of Duty ecological community is sparse, having only two significant
ecological interactions among its 5 member groups. This ecological community
includes subreddits about different editions of the series such as
r/blackops3, r/infinitewarfar and r/wwii as well as one about a popular spin-
off zombie game r/codzombies and the more general r/callofduty subreddit. We
find that growth in r/blackops3 or r/codzombies predicts growth in
r/infinitewarfare, but no other ecological interactions.

The timepiece and Call of Duty ecological communities illustrate how
subreddits with overlapping users can have relatively strong or weak forms of
ecological interdependence. Although both clusters are characterized by high
degrees of user overlap and low average ecological interaction, the timepiece
cluster has a dense competition-mutualism network while the call of duty
network is sparse.

#### Forecasting accuracy

As described in §3.2, we test H2 by comparing two forecasting metrics of
whether we have improved the 24-week forecast performance for all subreddits
which were assigned to clusters: root-mean-square-error (RMSE) and continuous
ranked probability score (CRPS). We find that VAR models including ecological
interactions have forecasting performance superior to the baseline model in
terms of both. The RMSE under the baseline model (0.84) is greater than the
RMSE of the VAR models (0.75) and the CRPS of the baseline model (72,853) is
also greater than the CRPS of the VAR models (72,669).

##  5 Threats to Validity

Our work is subject to several important threats to validity. First, we study
only one platform hosting online groups and our results may not generalize to
other platforms or time periods. The method we propose for identifying
ecological interactions between online groups has limitations common to all
time series analysis of observational data. While our community ecology
approach assumes that ecological interactions drive dynamics in the size of
groups over time and cause groups to grow or decline, drawing causal
conclusions using our method would depend on several untestable assumptions.
For example, groups we do not consider—including groups on other
platforms—could play a role in an ecological community accounted for in our
models. Potential omitted variables might also include additional time lags of
group size. For simplicity we chose to use VAR(1) with a single time lag, but
we hope future work will model more complex dynamics with additional lags. Our
vector autoregression models assumes that the error terms are trend
stationary. This is a common assumption in time series analysis and is
difficult to evaluate empirically (Ives et al. 2003).

Additional threats to validity stem from our use of algorithmic clustering to
identify ecological communities. While we choose clusters based on high
degrees of user overlap and validate our clustering, we might have obtained
different results if we had clustered in a different way. Additionally, our
efforts to obtain clusters with a high silhouette coefficient led us to remove
subreddits from our analysis. Thus, our results are not representative of
Reddit overall, but only of those subreddits that were included in our
analysis.

##  6 Discussion

In the final chapter of their book on Building Successful Online Communities,
Kraut, Resnick, and Kiesler (2012) advise managers of online groups to select
an effective niche and beware of competition. However, these recommendations
are based on little direct evidence from studies of online groups and offer
almost no concrete steps that designer or group should take based on either
piece of advice. Although further research into ecological interactions is
needed to derive design principles, we provide a novel framework for online
group managers to think about ecological constraints on group size. While
intuition suggests that online group managers might seek out mutualistic
relationships and avoid competitive ones, it is not clear whether another
group is a competitor or mutualist. Our method provides a way for group
managers to know.

We presented two studies with the purpose of introducing our community ecology
framework and comparing it to past work using population ecology. In Study A,
we found support for H1 showing—as predicted by density dependence theory—that
overlap density has an $\cap$-shaped association with subreddit growth.
Subreddits with moderate overlap density in our data declined less than
subreddits with either very low or very high overlap density. According to
population ecology theory, this suggests that high-density environments are
competitive and as a result are less conducive to growth than medium-density
environments.

Surprisingly, this seems to contrast with our results in Study B. We studied
the diversity of ecological communities using vector autoregression models of
group size over time to infer networks of ecological interactions. We found
that ecological communities of subreddits are typically mutualistic and that
these mutualistic interactions are stronger on average than competitive ones.
These findings corroborate recent qualitative studies arguing that multiple
online communities about the same topic exist because they provide different
and complementary benefits (TeBlunthuis et al. 2022), such as those provided
by combinations of small and large subreddits (Hwang and Foote 2021).

We also found a diversity of ecological dynamics among clusters of subreddits
with high degrees of user overlap. We found ecological communities that are
mutualistic, competitive, that mix the two, or that have few significant
ecological interactions at all. This explains the puzzling set of empirical
results from prior work about the relationship between overlap density and
outcomes like growth, decline and survival (Wang, Butler, and Ren 2012; Zhu,
Kraut, and Kittur 2014; Zhu et al. 2014). These studies have measured the
density of an online group’s niche in terms of its overlap in participants or
topics. Although we find support for the relationship between density and
growth predicted by density dependence theory, our analysis of ecological
communities suggests that degrees of overlap may have little to do with
whether two groups are mutualists or competitors. Instead, we argue that
density can be associated with growth when a platform provides a hospitable
environment to build online communities sharing certain topics or membership
bases. Yet, when conditions change and these topics lose popularity or
membership bases migrate off a platform (Fiesler and Dym 2020), this density
may lead to decline.

For example, differing environmental conditions of Wikis and Usenet groups
might explain why user overlap was associated with the survival of wikis (Zhu,
Kraut, and Kittur 2014) but with the decline of Usenet groups (Wang, Butler,
and Ren 2012). Wikia was a young and growing platform during Zhu, Kraut, and
Kittur’s data collection period when the growth of groups may have been
limited by knowledge of how organize and build a wiki; perhaps this knowledge
was provided by overlapping experienced members. Usenet was in decline during
Wang, Butler, and Ren’s study period and this may have created competition
over increasingly scarce members.

Future work should seek to explain when two online groups will be mutualists
or competitors. Long-held understandings of ecological interactions in
evolutionary theory suggest that, as we find, mutualism will be more common
than competition (Aldrich and Ruef 2006). Competition is unlikely to persist
because it decreases survival, but mutualistic relationships are likely to
endure because they increase the parties’ survival (Kropotkin 2012). In this
line of theory, groups might avoid competition by adopting specialized roles
in their ecological communities, a dynamic known as resource partitioning
(Carroll 1985). For example, the competition among real estate subreddits
observed in Figure 3 may happen because of insufficient specialization. By
contrast, mental health support groups like those observed in Figure §3 appear
to have specialized purposes or roles.

Online groups may use multiple platforms with distinctive affordances for
different purposes (Kiene, Jiang, and Hill 2019). Since our VAR method relies
only on time series data to infer ecological interactions, it can be applied
to study ecological communities spanning social media platforms. While we
focus on relationships between groups sharing a platform, one can apply our
concepts and methods to how understand higher levels of social organization
emerge from interdependent systems of technologies and users on social media
platforms.

##  7 Conclusion

An ecological explanation for the success of online groups looks beyond
internal mechanisms to understand how different groups influence each other’s
growth or decline. Prior research has investigated competition and mutualism
among online groups with overlapping users and topics using the population
ecology framework (Wang, Butler, and Ren 2012; Zhu, Kraut, and Kittur 2014;
Zhu et al. 2014), yet has not provided a way to infer competitive or
mutualistic interactions among related groups. We introduce the community
ecology framework as a complementary perspective to population ecology. By
inferring competition-mutualism networks directly from time-series data, our
community ecology approach helps resolve the empirical tensions raised by
prior work and reveals that most interactions within clusters of highly
overlapping subreddits are mutualistic. Our methods provide a foundation for
future work investigating related online groups.

##  8 Ethics, Impact and Competing Interests

The intended broader impact of this work is to improve the design and
management of how online communities by advancing understanding of overlapping
communities. It is conceivable that this work may contribute to potential
harms, such as if it is used to organize socially harmful online communities.
We hope and believe any negative consequences of this work will be outweighed
by positive and productive ones. We are considerate of the ethics of studies
such as ours, a large-scale analysis of publicly available social media
communication and behavior in that the individuals whose traces we analyze are
unlikely to anticipate their data will be used in such a way. That said,
because our analysis aggregates these activities to such a degree that no
individual is exposed to scrutiny, so we believe that the resulting harms are
minimal. We have no competing financial interests in this work.

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