# Rethinking the Memory Staleness Problem in Dynamics GNN

Mor Ventura, Hadas Ben Atya, Dekel Brav  MSc
Technion - IIT
Haifa, Israel
{mor.ventura, hds<EMAIL_ADDRESS>
###### Abstract

The staleness problem is a well-known problem when working with dynamic data,
due to the absence of events for a long time. Since the memory of the node is
updated only when the node is involved in an event, the i’s memory becomes
stale. Usually it refer to lack of events such as a temporal deactivating of a
social account [1]. To overcome the memory staleness problem [1] aggregate
information from the node’s neighbors memory in addition to the node’s memory.
Inspired by that, we design an updated embedding module that inserts the most
similar node in addition to the node’s neighbors. Our method achieved similar
results to the TGN, with a slight improvement. This could indicate a potential
improvement after fine-tuning our hyper-parameters, especially the time
threshold, and using a learnable similarity metric.

###### Index Terms:

GNN, Staleness problem, Social networks.

##  I Introduction

Many fields have turned to Graph Neural Networks (GNNs) for modeling systems
of relations and interactions, such as social sciences and biology. While most
of the models assume that the underlying graph is static, real-life systems of
interactions are dynamic [1].

Recently dynamic graphs have been researched and can be divided into two
groups, discrete-time dynamic graphs (DTDG) and continuous-time dynamic graphs
(CTDG). However, discrete-time dynamic graphs are unsuitable for several real-
world settings such as social networks in which the dynamic graph are
continuous and evolving.

On the other hand, continuous-time dynamic graph my suffer from a memory
staleness problem, due to a lack of events.

To handle the staleness of representations, [2] propose a method to learn how
the representation for a node i evolves when no observation involving the node
is made [3].

The TGN model [1] mitigates the staleness problem by aggregating information
from a node’s temporal neighbor’s memory. According to their article, when a
node has been inactive for a while, it is likely that some of its temporal
neighbors have been recently active, and by aggregating their memories, TGN
can compute an up-to-date embedding for the node. They used temporal graph
attention to select which neighbors are more important based on both features
and timing information.

We propose a new method, for nodes that were stale for relative a long period
$\Delta T$, using the most similar nodes’ information as well as the temporal
neighbors.

Our code is available on GitHub
https://github.com/venturamor/dynamic_graph.git.

##  II Related Work

To represent accurately dynamic systems using graphs, there is a must to
represent the temporal evolution within them. While most of the works focus on
discrete-time dynamic graphs [4, 5], It has also been suggested to represent
the continuous-time changes using a sequence of time-stamped graph nodes’
embeddings [1]. One of the main problems is the memory staleness problem.[6]
When the update of node embedding is based on its memory, and it was not
updated for a while due to lack of usage, for example when a user stops using
a social network platform it causes stale memory, it may lead to worse
performance, especially when the end task is recommendation system based on
the edges. The solution that was suggested in the baseline paper to mitigate
this problem is to apply a TGN, graph embedding module with a temporal
attention mechanism that is updated by aggregating information from a node’s
neighbors’ memory [1]. It is based on the assumption that though it, i.e the
user, was not active recently, its neighbors were. This temporal graph
attention takes into account the last N temporal neighbors. For instance, in
the Wikipedia dataset, a temporal neighbor is one item of the 10 recent items
that were edited by this user. Solutions that lack the consideration of a
node’s neighbors when computing its embedding are susceptible to the staleness
problem.

##  III Datasets

We used two datasets in our experiments, Wikipedia and Reddit. The Reddit
dataset is a graph dataset from Reddit posts made in the month of September
2014. The nodes are the community, or “subreddit”, to which a post belongs to.
An interaction occurs when a user writes a post to the subreddit[7, 1]. In the
Wikipedia dataset, users and items (pages) are the nodes, and an interaction
represents a user editing an item. In both aforementioned datasets, the
interactions are represented by text features, and labels represent whether a
user is banned. Both interactions and labels are time-stamped, and each event,
user, and item has a unique id [1].

##  IV Method

”Tell me who your friends are, and I will tell you who you are”. This is the
guiding saying of our method. We look for the most similar source nodes
(users), based on their memory features and use theirs embedding when
computing the current relative staled node.

Figure 1: The updated flow of operations of TGN. The embedding is computed
through staleness check based on relative time difference and KNN algorithm to
compute similarity to other nodes memory. (based on Figure 2 in [1])

Our method differentiate between short-time stale nodes and long-time stale.
When a node was stale for a period time longer than our threshold, $\Delta
T>\Delta T_{th}$, We update the node’s embedding with the embedding from it’s
$K$ most similar nodes, in additional to it’s $N$ temporal neighbors.

The new node’s embedding can be written as the following equation:

$Emd(S_{i})=Emd(S_{i})+\sum_{k=1}^{K}Emb(S_{k})+GAT(\\{S_{1}^{i},..,S_{N}^{i}\\})$

(1)

Where $Emd(S_{i})$ is the i’s node embedding representation, $Emb(S_{k})$ is
the embedding of the k’th most similar node to i, and
$GAT(\\{S_{1}^{i},..,S_{N}^{i}\\})$ is the Temporal Graph Attention, the
aggregating information from the i’s L-hop temporal neighborhood as described
in [1].

It’s important to note that we searched for other nodes that already have kind
of history, which means that already have memory features that are not
initialized i.e, equal to zeros.

For simplicity, we decided to use $K=1$, only the most similar node.

###  IV-A Define events-time difference

For each node in the current batch, we calculate $\Delta T_{th}$, the events-
time difference between the node’s last updated event’s timestamp and the
current event’s timestamp.

Using the $\alpha=0.025$ fraction of those events-time difference of the nodes
in the current batch, we calculated the threshold value for $\Delta T$, as in
equation 2 For $p=1-\alpha=0.975$.
It means we look for the relative staled nodes in the current batch. When we
use the quantile we make sure that the threshold is conditioned with the
current batch and we define staleness as relative to the other nodes, or
users:

$\Delta T_{th}=Q(p)=\inf\\{t\in\Re:p(t)\\}$

(2)

Thus, only nodes that their last event occurred before $\Delta T_{th}$ or
longer will be updated with the similar nodes information.

##  V Similarity Matrices

In order to find the most similar nodes with test to matrices of K-Nearest-
Neighbors (KNN), using the ’sklearn’ python package. First, we used the KNN
ball-tree algorithm, for fast generalized N-point problems. The ball-tree
algorithm divides the data points into two clusters. Then, the process of
dividing the data points into two clusters is repeated within each cluster
until a defined depth is reached [8], as shown in “Fig. 2.”

The second KNN algorithm we used is the brute-force search.

Figure 2: Visualization of the Ball Tree Algorithm, [8]

##  VI Results and Discussion

For two different similarity matrices and the TGN method as a baseline, we
carried out three experiments. Our experiments were performed using two
datasets: Wikipedia and Reddit.

The results are drawn from the test set, which includes new nodes that the
network was not exposed to during training or validation. We also checked the
impact of the time-quantile threshold on the network performance, by
decreasing the qunatile $q=1-\alpha$ for applying our similarity method on
larger amount of nodes.

Table [I] shows the networks performance while using a fixed quantile of
$q=1-\alpha=0.975$, and Table [II] shows the networks performance with
different quantiles $q={0.975,0.80,0.70}$.

model | dataset | AUC | precision
---|---|---|---
TGN | wikipedia | 0.963 | 0.967
Ball-Tree | wikipedia | 0.961 | 0.966
Brute-force | wikipedia | 0.963 | 0.967
TGN | reddit | 0.953 | 0.958
Ball-Tree | reddit | 0.957 | 0.960
Brute-force | reddit | 0.953 | 0.957
TABLE I: area under curve (AUC) and precision results for quantile $1-\alpha=0.975$ model | quantile | AUC | precision
---|---|---|---
TGN | - | 0.963 | 0.967
Ball-Tree | 0.975 | 0.961 | 0.966
Ball-Tree | 0.8 | 0.966 | 0.970
Ball-Tree | 0.7 | 0.958 | 0.963
TABLE II: different time quantile thresholds on Wikipedia dataset.

Model weights are available on request.

##  VII Conclusion and Future-Work

In the work, we study the influence of information aggregation from similar
nodes on stale nodes. temporal deactivation of a node might effect the quality
of recommendation system and edge prediction.

Nodes may share similar or mutual information, so it can be exploit to express
similarity when node becomes relative stale. Though our solution has not
showed significant improvement, there are future steps that may mitigate this
issue by better effect on the stale nodes.

Using a larger number of similar nodes may increase their effectiveness and
improve network performance. Also, decreasing the time-quantile threshold may
increase the impact of our new method, by apply it on larger number of nodes.
In addition, applying attention mechanism or learnable nodes similarity
network to find the most similar combination of nodes instead of KNN also may
improve the staleness performance. Moreover, there is future work on an
evaluation metric that will be specific for the stale nodes.

## References

  * [1] E. Rossi, B. Chamberlain, T. Fabrizio, F. Twitter, D. E. Twitter, F. Monti, and M. Bronstein Twitter, “TEMPORAL GRAPH NETWORKS FOR DEEP LEARNING ON DYNAMIC GRAPHS.”
  * [2] S. Kumar, X. Zhang, and J. Leskovec, “Learning Dynamic Embeddings from Temporal Interaction Networks.” [Online]. Available: https://doi.org/10.1145/nnnnnnn.nnnnnnn
  * [3] S. M. Kazemi, R. Goel, K. Jain, I. Kobyzev, A. Sethi, P. Forsyth, and P. Poupart, “Representation Learning for Dynamic Graphs: A Survey,” _Journal of Machine Learning Research_ , vol. 21, pp. 1–73, 2020. [Online]. Available: http://jmlr.org/papers/v21/19-447.html.
  * [4] A. Sankar, Y. Wu, L. Gou, W. Zhang, and H. Yang, “DYNAMIC GRAPH REPRESENTATION LEARNING VIA SELF-ATTENTION NETWORKS.”
  * [5] B. Yu, M. Li, J. Zhang, and Z. Zhu, “3D Graph Convolutional Networks with Temporal Graphs: A Spatial Information Free Framework For Traffic Forecasting,” 3 2019.
  * [6] S. M. Kazemi, R. Goel, K. Jain, I. Kobyzev, A. Sethi, P. Forsyth, and P. Poupart, “Representation Learning for Dynamic Graphs: A Survey,” _Journal of Machine Learning Research_ , vol. 21, pp. 1–73, 2020. [Online]. Available: http://jmlr.org/papers/v21/19-447.html.
  * [7] “Reddit Dataset — Papers With Code.” [Online]. Available: https://paperswithcode.com/dataset/reddit
  * [8] “Tree algorithms explained: Ball Tree Algorithm vs. KD Tree vs. Brute Force — by Hucker Marius — Towards Data Science.” [Online]. Available: https://towardsdatascience.com/tree-algorithms-explained-ball-tree-algorithm-vs-kd-tree-vs-brute-force-9746debcd940


# An Assessment of the In-Situ Growth of the Intracluster Light in the High
Redshift Galaxy Cluster SpARCS1049+56

Capucine Barfety Félix-Antoine Valin  Tracy M.A. Webb Department of Physics,
McGill Space Institute, McGill University, 3600 rue University, Montréal,
Québec, Canada, H3A 2T8  Min Yun Department of Astronomy, University of
Massachusetts Amherst, Lederle Graduate Research Tower, 619-E, 710 N Pleasant
Street, Amherst, MA 01003-9305  Heath Shipley  Department of Physics, McGill
Space Institute, McGill University, 3600 rue University, Montréal, Québec,
Canada, H3A 2T8  Kyle Boone DiRAC Institute, Department of Astronomy,
University of Washington, 3910 15th Ave NE, Seattle, WA, 98195, USA  Brian
Hayden  Space Telescope Science Institute, 3700 San Martin Drive Baltimore, MD
21218, USA  Julie Hlavacek-Larrondo Département de physique, Université de
Montréal, Complexe des Sciences, Case postale 6128, succursale Centre-ville,
Montréal, Québec, H3C 3J7, Canada  Adam Muzzin Department of Physics and
Astronomy, York University, 4700 Keele St., Toronto, Ontario, Canada, MJ3 1P3
Allison G. Noble ASU School of Earth and Space Exploration, PO Box 871404,
Tempe, AZ 85287, USA  Saul Perlmutter Department of Physics, University of
California, 366 Physics North, MC 7300, Berkeley, CA, 94720-7300  Carter Rhea
Département de physique, Université de Montréal, Complexe des Sciences, Case
postale 6128, succursale Centre-ville, Montréal, Québec, H3C 3J7, Canada
Gillian Wilson  Department of Physics and Astronomy, University of California-
Riverside, 900 University Avenue, Riverside, CA 92521, USA  H.K.C. Yee The
David A. Dunlap Department of Astronomy and Astrophysics, University of
Toronto, 50 St George St., Toronto, ON M5S 3H4, Canada

###### Abstract

The formation of the stellar mass within galaxy cluster cores is a poorly
understood process. It features the complicated physics of cooling flows, AGN
feedback, star formation and more. Here, we study the growth of the stellar
mass in the vicinity of the Brightest Cluster Galaxy (BCG) in a $z=1.7$
cluster, SpARCS1049+56. We synthesize a reanalysis of existing HST imaging, a
previously published measurement of the star formation rate, and the results
of new radio molecular gas spectroscopy. These analyses represent the past,
present and future star formation respectively within this system. We show
that a large amount of stellar mass – between $(2.2\pm 0.5)\times
10^{10}\>M_{\odot}$ and $(6.6\pm 1.2)\times 10^{10}\>M_{\odot}$ depending on
the data processing – exists in a long and clumpy tail-like structure that
lies roughly 12 kpc off the BCG. Spatially coincident with this stellar mass
is a similarly massive reservoir (1.0$\pm$0.7$\times$1011M⊙) of molecular gas
that we suggest is the fuel for the immense star formation rate of 860 $\pm$
130 M⊙yr-1, as measured by infrared observations. Hlavacek-Larrondo et al.
(2020) surmised that massive, runaway cooling of the hot intracluster X-ray
gas was feeding this star formation, a process that had not been observed
before at high-redshift. We conclude, based on the amount of fuel and current
stars, that this event may be rare in the lifetime of a cluster, producing
roughly 15 to 21% of the Intracluster Light (ICL) mass in one go, though
perhaps a common event for all galaxy clusters.

Galaxy clusters — High-redshift galaxy clusters — Intracluster medium

††journal: ApJ††facilities: HST(WFC3), JVLA††software: Astropy (Astropy
Collaboration et al., 2013), CASA (McMullin et al., 2007), SExtractor (Bertin
& Arnouts, 1996)

tablenum

##  1 Introduction

The centres of galaxy clusters remain among the most violent yet least
understood regions of the universe, at all redshifts. Indeed, their complex
nature - which includes the physics of large scale gas cooling, star
formation, Active Galactic Nuclei (AGN) feedback, and galaxy mergers and
interactions - make them key, if complicated, laboratories for hierarchical
processes of structure growth. The observational study of galaxy cluster cores
is especially difficult at cosmic noon ($\sim z=1-2$) where many of the
aforementioned processes peak. Here, the challenge of identifying clusters is
coupled with the difficult interpretation of imaging and spectroscopy of high-
redshift sources, which have both lower S/N and lower angular resolution.
Nevertheless, progress is being made through wide-field cluster surveys, that
are simultaneously able to reveal unique clusters (providing windows onto
particular physical processes) and facilitate global statistical studies of
young clusters.

A cluster core at high redshift contains several important components. Most
clusters contain a single large galaxy sitting at the bottom of the
gravitational potential well, known as the Brightest Cluster Galaxy (BCG).
This object is surrounded by a hot intracluster medium, which, if AGN feedback
is lacking or insufficient, may cool onto the BCG (McNamara & Nulsen, 2012;
McDonald et al., 2019). Over time, cluster galaxies are accreted into the core
of the cluster, resulting in galaxy-galaxy interactions and mergers with the
central BCG. This latter process is further believed (e.g., Dressler (1979))
to feed a halo of stars, unbound to any galaxy, around the BCG which
themselves produce the Intracluster Light (ICL).

The formation and evolution of BCGs and the ICL, and the role of cooling flows
and mergers, are not well understood. Semi-analytic models (SAMs) (e.g.,  De
Lucia & Blaizot (2007); Tonini et al. (2012)) find that below $z\lesssim$1 the
BCG alone grows by a factor of 2-3. This appears to be supported by
observational evidence (Lidman et al., 2012), which indicates that roughly 50%
of tidal remnants from mergers add to the growth of the BCG, whereas the
remaining 50% ends up in the ICL, though see also Lin et al. (2013); Oliva-
Altamirano et al. (2014); Bellstedt et al. (2016). Some observational studies,
however, have claimed that significant amounts of in-situ star formation
occurs within the BCG (Webb et al., 2015; McDonald et al., 2016), while some
simulations (Puchwein et al., 2010) indicate that a significant fraction of
the ICL could possibly form in-situ as well. Indeed, the work of Hlavacek-
Larrondo et al. (2020) surmised that the growth of the ICL could be fed by
runaway cooling flows, in the absence of AGN heating. The relative importance
of in-situ (i.e. the stars formed where we observe them) vs ex-situ (i.e. the
stars formed elsewhere, and were accreted via galaxy interactions) star
formation for BCGs and the ICL is likely a function of redshift, as is the
process driving each (e.g. McDonald et al. (2016)).

###  1.1 The SpARCS1049 Cluster

In this paper, we present the study of the stellar mass build-up in the core
of a single system, SpARCSJ-104922.6+564032.5 (hereafter SpARCS1049), located
at redshift $z=1.71$. This object was first detected as a cluster in the
Spitzer Adaptation of the Red-sequence Cluster Survey (SpARCS) (Wilson et al.,
2009; Muzzin et al., 2009; Demarco et al., 2010) and published in Webb et al.
(2015). This work presented both the cluster itself (through the spectroscopic
detection of 27 cluster member galaxies) as well as the high star forming core
(860$\pm$130 M⊙yr-1) which, at the time, was associated with the BCG. The
cluster has a lensing mass of 4$\times$1014M⊙ (Finner et al., 2020) and has
been detected in the X-ray by Hlavacek-Larrondo et al. (2020). The latter
work, as mentioned in the previous section, furthermore argued that SpARCS1049
contains a cool core and that the intense star formation is fed by the large
scale, runaway cooling of intracluster gas, which has been allowed to cool in
the cluster core due to the lack of feedback from the central AGN (Trudeau et
al., 2019). They surmised that this cooling flow-fed star formation might be
forming the ICL in-situ.

Here we present a new analysis of archival Hubble Space Telescope (HST)
observations, to assess the current stellar mass present in the region; and
the complete analysis of CO(1-0) observations (briefly shown in Hlavacek-
Larrondo et al. (2020)), that trace the molecular gas mass of the system.
Finally, we add to this previously published infrared data that target the on-
going star formation in the core of the cluster. Using these three legs we
present a census of the past, future and present star formation in the central
regions of SpARCS1049 and, we posit, in agreement with Hlavacek-Larrondo et
al. (2020), the in-situ build-up of the ICL.

In §2 we outline the data, including the new Jansky Very Large Array CO(1-0)
observations, and we recap the existing Hubble Space Telescope optical imaging
and multi-facility infrared imaging and spectroscopy. §3 outlines our
methodology, including the determination of the stellar mass in the vicinity
of the BCG and measurement of the molecular gas mass in the same region. We
then present our main results in §4 and discuss their implications in §5.
Finally, we conclude and summarize in §6. Throughout the paper we use a
standard $\Lambda$ CDM cosmology with H∘ = 70 km s-1 Mpc-1; and the kiloparsec
to arcsec conversion at the redshift of interest is 1kpc $\sim\>0.11"$.

##  2  Observations and Data Reduction

###  2.1 Previous Infrared Imaging and Spectroscopy

The first detection of the BCG was made by Lonsdale et al. (2003) with
$Spitzer$-MIPS 24$\mu$m, as part of the SWIRE survey of the Lockman field. As
a bright $z\sim 2$ ULIRG candidate, it was followed up by Farrah et al. (2008)
with $Spitzer$-IRS which determined its spectroscopic redshift to be $z=1.7$.
Finally, Webb et al. (2015) presented evidence that this object was at the
heart of a massive galaxy cluster. This latter paper also presented new SCUBA2
imaging at 450 and 850$\mu$m; and synthesized the infrared Spectral Energy
Distribution (SED), including limits and detections derived from the Herschel
Space Observatory. Using this extensive data set, they estimated an infrared
luminosity of $\sim$ 7 $\times$1012 L☉ for the BCG, and a corresponding AGN-
corrected Star Formation Rate (SFR) (Kennicutt, 1998; Pope et al., 2008) of
860$\pm$130 M☉yr-1.

Unfortunately, the PSFs of the infrared instruments are large, with the
smallest beam corresponding to 6′′ for $Spitzer$-MIPS at 24$\mu$m. This made
the source of the SFR hard to determine, and lead to associating the emission
with the BCG (Farrah et al. (2008)). As we will argue in this paper (also in
Trudeau et al. (2019) and Hlavacek-Larrondo et al. (2020)), the location of
the bright infrared emission actually lies 1.5′′ ($\sim 13$ kpc) off the BCG,
on top of what appears to be galactic debris.

###  2.2 Archival Multi-wavelength HST Optical Imaging

The HST data, first published in Webb et al. (2015), were retrieved from the
Hubble Legacy Archives, having been targeted by the Hubble Space Telescope’s
(HST) Wide Field Camera 3 (WFC3) in 2014 and 2015 through Cycle 22
programs111Based on observations made with the NASA/ESA Hubble Space
Telescope, obtained [from the Data Archive] at the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are
associated with program # GO-13677 and # GO-13747.. The object was observed
through the F160W, F105W, and F814W filters with integration times of 9237s,
8543s, and 2846s respectively. The data were drizzled into images with 0.06”
pixel scale using the STScI software,
AstroDrizzle222http://www.stsci.edu/scientific-
community/software/drizzlepac.html.

###  2.3 New Jansky Very Large Array Observations

We obtained new radio observations using the Jansky Very Large Array (JVLA),
in Socorro, New Mexico. We observed SpARCS1049 for a total of 11 hours
combining two different observing sessions from January 5 and 12 2019
(Proposal ID: 18B-177, PI: T. Webb) during optimal observing conditions with
an optical depth $\tau_{$42.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$}\sim
0.06$. We used the Q-Band in the C-Configuration which provided a synthesized
beam width of $0.47\text{\,}\mathrm{\SIUnitSymbolArcsecond}$.

All of the interferometry data were reduced using the pipeline reduction
routine provided by the NRAO center in Socorro. For the CO(1-0) line
($\nu_{rest}=$115.3\text{\,}\mathrm{GHz}$$), we expected an observed line
frequency of $\sim$42.5\text{\,}\mathrm{GHz}$$ considering a previous Large
Millimetre Telescope redshift measurement of $z=1.709$ (Webb et al., 2017).

Figure 1 shows the CO(1-0) line detection from the JVLA. The spectrum was
extracted in the following iterative manner. First, using the CASA tclean
task, we created a cube composed of
$32\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$-wide slices centered around our
target frequency $\nu_{obs}=$42.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$. We
then visually searched the image slices for emission. Extended emission was
identified at the expected frequency and we then applied an aperture around
said emission to extract the spectrum shown in Figure 1.

We used a regular Gaussian fitting routine over the previously mentioned
channels and measured a peak emission at
$42.54781\pm$0.000\,09\text{\,}\mathrm{GHz}$$, which corresponds to a redshift
of $z=1.708\pm 0.001$. This agrees well with the previously determined value
of $z=1.709\pm 0.001$ (Webb et al., 2017). The FWHM, which corresponds to the
measured velocity dispersion of the line, is
$630\pm$270\text{\,}\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$$.

Figure 1: CO line detection from both the 1-0 and previously detected 2-1
(Webb et al., 2017) transition. The black dotted line represents the CO(2-1)
transition line and the solid blue line represents the CO(1-0) transition
line. Each have had a Gaussian curve fit to them in order to estimate the
redshift. The CO(2-1) line has been scaled in the frequency domain by a factor
of 2 to match the CO(1-0) and the CO(2-1) flux density also has been scaled
down by a factor of 4 to compensate for the $\nu^{2}$ SLED effect. (Narayanan
& Krumholz, 2014)

.

To create an integrated intensity map of the molecular gas reservoir, we
generate a single image from multiple channels. We combine 2 spectral windows
each with 64 channels using CASA’s tclean task with a resolution of
$0.2\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ per pixels for a 1024 pixels
square map. We also use a $1\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ taper in
uv-space such that longer baselines have a much more limited weight on the
final image. This usually reduces the overall flux sensitivity, and decreases
the resolution. We show this emission overlaid atop the HST optical image in
Figure 2.

Figure 2: HST optical image of the SpARCS1049 cluster (colour image, using
F160W, F140W, and F814W). We identify the BCG and two foreground objects
within this map. The white contours show the CO(1-0) map contours starting at
2$\sigma$ and increasing in 0.5$\sigma$ steps. The red cross marks the peak of
the MIPS-24$\mu$m detection, and the cyan cross denote the center of the soft
X-ray emission. The size of the crosses represent their respective positional
uncertainties. The white circle in the upper right corner denotes the radio
beam.

##  3 Methods and Analysis

Here we describe the new analyses using the data outlined in the previous
section. First we present the extraction of the stellar mass in the region
surrounding the BCG, from the HST optical data, followed by a detailed summary
of the CO characterization from the JVLA data. The analysis on the infrared
star formation is described thoroughly in Webb et al. (2015), where the
infrared luminosity was converted to SFR using the Kennicutt (1998) law, which
was then corrected for AGN contribution. No new analysis was done.

###  3.1 Measuring the Stellar Mass in the Core

####  3.1.1 Method for Removing Galaxies

Imaging observations of clusters poses many challenges due to the large amount
of light that blends together from different sources in the cluster and
foreground/background. As we are interested in the central intracluster mass
we need to model and subtract all the nearby galaxies contributing
significantly to the light in the region. We chose the method demonstrated by
Shipley et al. (2018) for an iterative processing method to remove galaxies in
the Hubble Frontier Fields. We summarize the method here and refer the reader
to Shipley et al. (2018) for a detailed discussion.

The procedure measures the isophotal parameters of a galaxy and removes the
resulting model as described by Ferrarese et al. (2006). The modeling method
has been shown to work well for elliptical galaxies with significant isophotal
twisting, which are the predominant type of bright galaxies in the cluster
environment. The method utilizes the IRAF tasks ELLIPSE to measure the
isophotal parameters, which are then passed to the IRAF task BMODEL to create
the best fit model for each selected galaxy with statistical finessing of the
data to achieve the best fit model even out to large radii (by splining and
interpolating the isophotal parameters). Also, a local background is applied
to create a smooth profile at large radii, and the curve of growth is measured
to mitigate extra light being modeled that should be attributed to the sky.
Furthermore, the method minimizes effects from poor modeling of a nearby
galaxy by masking the image except for the galaxy being modeled. This is
important in the initial stages of modeling in crowded regions.

Figure 3: (a) Original drizzled image of SpARCSJ-104922.6+564032.5 through the
F160W filter. The position of the central galaxy is indicated by the red
cross. (b) Residual image after the galaxies were fit and removed (see
§3.1.1).

To achieve improved modeling of the galaxies being removed, the method
utilizes an iterative processing step. After the initial run (described
above), the code runs through 10 more iterations of each selected galaxy for
11 total iterations adding in one selected galaxy at a time and modeling it
again for each iteration. The result of this improves the model and the
residual for each galaxy selected without having contamination from all other
selected galaxies that hindered the initial models. Furthermore, the method
averages over the iterative models to create a better model overall that is
more resistant to spurious model results from any one iteration. The IRAF task
IMCOMBINE is used with the following parameters (combine=“average”,
reject=“minmax”, nlow=“4”, nhigh=“2”) for the modeled galaxies to achieve
this.

We chose to model and subtract 17 galaxies (10 of which are visible on the
left panel of Figure 3) that were bright, nearby and/or improved the modeling
that allows for the best measurement of the intracluster mass near the BCG.
Finally, the residual patterns left by the subtracted galaxies (primarily in
the core of each model) are masked.

###  3.2 Estimating the Stellar Mass

Our goal was to derive mass-to-light ratios for each individual pixel of the
residual map. To do this, we exploited the relation between optical and near-
infrared (NIR) colors and stellar M/L as presented in Bell et al. (2003),
computed using a Salpeter IMF (Salpeter (1955)) modified to have only 70% of
the mass. We used the following equation and parameters:

$\log_{10}\left(\frac{M}{L}\right)=a_{r}+(b_{r}*color).$ (1)

where $a_{r}=-0.223$ and $b_{r}=0.299$ are empirically determined parameters,
and the $color$ is the difference in ST magnitudes between the F105W filter
and the F160W. Prior to the color estimation, both F105W and F160W magnitudes
are k-corrected to map the rest frame magnitudes of Sloan Digital Sky Survey
(SDSS) u and r filters - which are the filters used in determining Equation 1
-, as prescribed in Hogg et al. (2002). The k-correction is computed using the
Bruzual & Charlot (2003) models, with Chabrier IMF (Chabrier, 2003) and solar
metallicities.

With a mass-to-light ratio in hand we can then convert the observed F160W flux
density to a stellar mass, taking into account cosmological dimming. We also
measure the mass in the three brightest clumps of the tail, to determine if
they have stellar masses comparable to dwarf galaxies.

This method is reproduced twice, once on the data as presented in the right
panel of Figure 3, and once on a reprocessed version of the data, presented in
Figure 5, re-binned using the Cappellari & Copin (2003) 2D implementation of
the Voronoi binning technique. We reiterate the mass measurement using the
second method as Zibetti et al. (2009) estimates that a signal-to-noise ratio
(S/N) $\geq$ 20 is necessary to obtain M/L values accurate within 30%. The
repetition of the measurement therefore allows us to obtain a better sense of
the variation in the measured stellar mass depending on the methodology.

####  3.2.1 Using the Original Data

For the first estimate of the stellar mass, the mass-per-pixel map obtained
using the un-binned data is shown in Figure 4. This map traces, but is not
identical to, the F160W light map; in which some of the clumps are more
defined.

We note that sky subtraction - one step of data reduction through AstroDrizzle
- returned negative values for some of the background pixels in both HST
images. This leads to errors when converting the data to magnitudes before
getting the color. To overcome this issue, the steps described in the
paragraph above were repeated twice, once on the absolute value of all the
pixels in the image, and once on the absolute value of the pixels flagged as
negative. The final mass value is then the difference between the sum of all
the pixels in the mass pixel map (Figure 4) and the sum of the“negative”
pixels.

Figure 4: Mass-per-pixel map of the area shown in Figure 3. The red apertures
circle the three most massive clumps in the region. The noisy pixels in the
background are due to pixels with un-physical colours and the galaxy
subtraction. These fluctuations are taken into account in the errors budget
(see §3.2.3). Here we see the mass distribution is compressed into the tail
which extends roughly 65 kpc, and is composed of several large mass clumps.

In addition, noise in the colour values lead to high M/L values for some
pixels, which then transferred to high mass values. To estimate the magnitude
of this effect, we set a threshold value above which the M/L of the pixel is
deemed un-physical, and replaced by a background value. By varying the value
of the threshold from the lowest (using the minimum threshold below which
physical mass in the tail is cut-off) to infinity (without a threshold), we
observe that the mass varies around $6.6\times 10^{10}\>M_{\odot}$, with a 10%
scatter on this value, which is used as the uncertainty on the mass from this
effect. It should be noted that Figure 4 displays the pixel map of the mass
for a threshold value of 10.

Finally, despite the sky subtraction step in AstroDrizzle, we included
background fluctuations in our error analysis. To do so, we divide the area
around the tail of SpARCSJ-1049 into 64 squares of 20x20 pixels; in which we
replace the pixel values associated to light sources by randomly chosen sky
values. We proceed to compute the mean values of the pixels in each of the
squares, and use the standard deviation of those 64 values as the uncertainty
in the pixel values. This uncertainty can then be propagated to both the color
and luminosity, and finally to the mass, using error propagation formulas.
This amounts to an error of $<1\%$ in the final mass measurement.

####  3.2.2 Using the Binned Data

For the second estimate of the stellar mass, both the F160W and F105W images
are re-processed using the 2D implementation of the Voronoi binning technique
by Cappellari & Copin (2003). The results are shown in Figure 5 for the F160W
data, which displays the average flux density per bin using two different
target S/N. While the number of bins varies with the S/N, the overall shape of
the structure is well preserved. We also re-process the F105W data using the
binning map of the F160W image.

We then proceeded to reproduce the analysis presented in §3.2 on the new data,
to obtain another estimate of the stellar mass. Propagating the errors from
each bin to the mass only adds a 1% uncertainty on the mass measurement.

Figure 5: Resulting F160W image after reprocessing the data using the Voronoi
binning, showing the average flux density value per bin, where (a) had a
target S/N of 10, and (b) had a target S/N of 20. The mass measurements were
made using the S/N = 20 images, but we note that both returned values in
agreement with each other. The red annuli circle the same clumps as in Figure
4.

####  3.2.3 Additional Uncertainties in the Mass Estimate

There are several uncertainties inherent or introduced in both of our
measurements by the method outlined in §3.2.

  * •

Varying the input model used to compute the k-correction adds an additional 4%
error in the mass measurement.

  * •

Other studies have found varying values for $a_{\lambda}$ and $b_{\lambda}$ in
Equation 1. We repeated the mass measurement using values from Zibetti et al.
(2009) and García-Benito et al. (2019), and found varying the parameters
introduces a 14% error in the stellar mass from the original data, and a 20%
error in the stellar mass from the binned data.

###  3.3 Estimating the Molecular Gas Mass

We now discuss the methodology of measuring the molecular gas mass using the
JVLA radio data. We use the spectral tool from the casaviewer on the cube
image mentioned in §2.3 to generate a data file containing the spectrum of a
6${}^{\prime\prime}\times$6′′ circular aperture around our source peak.

To estimate the flux in the line, with an associated error we adopt a Monte
Carlo approach. We generate 20 000 new realizations of the line using the rms
uncertainties in each channel. We can then fit and integrate a Gaussian curve
on each of these generated emission lines and compute the mean and standard
deviation of the integrated line flux:
$S_{\text{CO}}\Delta\nu=1.0\pm$0.6\text{\,}\mathrm{Jy}\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$.

We compute the CO integrated source brightness temperature using equation 2,
where $D_{L}$ is the luminosity distance, $z$ is the redshift, $\nu_{obs}$ is
the observed frequency mentioned above.

$L_{\text{CO}}^{\prime}=$3.25\text{\times}{10}^{7}\text{\,}\times$\
S_{\text{CO}}\Delta\nu\frac{D_{L}^{2}}{(1+z)^{3}\nu_{obs}^{2}}$\mathrm{K}\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{pc}}^{2}$$
(2)

We find $L_{\text{CO}}^{\prime}=(1.3\pm 0.9)\times
10^{11}$\mathrm{K}\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{pc}}^{2}$$.

The integrated source brightness temperature can be converted into a H_2
molecular gas mass using a conversion factor $\alpha_{\text{CO}}$. Here we
employ $\alpha_{\text{CO}}=0.8$, which is typical for ultra-luminous infrared
galaxies (ULIRG) and also allows for a conservative mass estimate (Webb et
al., 2017). Using equation 3 we compute a mass
$M_{\ce{H_{2}}}=1.0\pm$0.7\text{\times}{10}^{11}\text{\,}\mathrm{\text{M}_{\odot}}$$.

$\frac{M_{\ce{H_{2}}}}{\text{M}_{\odot}}=\alpha_{\text{CO}}\frac{L_{\text{CO}}^{\prime}}{$\mathrm{K}\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{pc}}^{2}$}$
(3)

##  4 Results

In this paper we set out to synthesize three important measurements: (1) the
stellar mass in the vicinity of the BCG (but not within the BCG proper) -
representing the history of the star formation within the core; (2) the
molecular gas mass around the BCG - representing the future star formation
within the core; and (3) the total current star formation rate occurring on or
near the BCG. We now discuss each in turn.

The total stellar mass in the $12.9^{\prime\prime}\times 11.7^{\prime\prime}$
cutout is measured using the methods described in §3.2, and is dominated by
the mass within the clumpy tidal tail. This returns final values for the
stellar mass in the BCG’s vicinity of $M_{near}=(6.6\pm 1.2)\times
10^{10}\>M_{\odot}$ for the original data and $M_{near}=(2.2\pm 0.5)\times
10^{10}\>M_{\odot}$ for the re-binned data. We note that the BCG-proper has
previously been estimated to contain (3.0$\pm 0.4)\times$1011 M⊙ (Webb et al.,
2015). The totals of these measurements ($(3.7\pm 0.4)\times 10^{11}M_{\odot}$
and $(3.2\pm 0.4)\times 10^{11}M_{\odot}$) sit below the measurement of the
mass of the ICL (within 100kpc radius)+BCG undertaken by DeMaio et al. (2020)
for SpARCS1049 of (5$\pm$0.5)$\times$1011M⊙.

The three brightest clumps in the tail (circled in red, Figure 4) have stellar
masses (from top to bottom) $1.1\times 10^{9}\>M_{\odot}$, $1.3\times
10^{9}\>M_{\odot}$, and $1.2\times 10^{9}\>M_{\odot}$; which indicates that
they might be dwarf galaxies formed by the intense star formation. They only
amount to $\sim 6\%$ of the total mass measured in the area.

The molecular gas mass is measured in §3.3 to be 1.0$\pm$0.7$\times$1011M⊙.
The high spatial resolution of the JVLA clearly spatially isolates this mass
to an area roughly 12kpc distant from the BCG, and coincident with the clumpy
tail that contains the bulk of the stellar mass. This mass is in agreement
with the previous large beam (23′′) CO(2-1) mass estimate of Webb et al.
(2017) of (1.1$\pm$0.1)$\times$1011M⊙. The good agreement between these two
measurements indicates all of the molecular gas is located within the 12 kpc
(radius) clump resolved by the JVLA, and we are not missing extended flux in
the interferometric measurement.

Finally, we reiterate the measurement of the star formation rate in a large -
$\sim$ 50 kpc - aperture (that is, with a 6′′ beam). The positional peak of
this flux is coincident with the peak of the molecular gas reservoir. After
removing flux attributed to a weak AGN (Pope et al., 2008), Webb et al. (2015)
estimate a star formation rate of SFR = 860$\pm$130M⊙yr-1. To place this value
in the same context as the two above masses we note that for a duty cycle of
1Gyr this star formation would produce $\sim$ 9$\times$1011M⊙.

##  5 Discussion

The results summarized in the previous section indicate that a large amount of
stellar mass is being assembled in the core (inner $\sim$100 kpc) of
SpARCS1049. Building on the work from Webb et al. (2015) and Hlavacek-Larrondo
et al. (2020) \- the latter of which surmised that the cooling-flow was
feeding the in-situ formation of the ICL - these results add to the picture
that a significant amount of mass has already been built up and the fuel is in
place to continue to feed the star formation for some time ($\sim$1Gyr).
Moreover, the HST and JVLA imaging indicate that this mass is not being formed
within the BCG itself, but rather off-axis by $\sim$ 12 kpc and extended along
a long filamentary structure that stretches for $\sim$65 kpc. Although no
dynamical information is yet available it is unlikely, given the projected
distance (tens of kpc) that this material is bound to the BCG and we instead
suggest that it will be incorporated into the ICL.

Intergalactic star formation in intracluster space has been reported in many
low-redshift systems (Sun et al., 2007, 2010; Lee et al., 2000; Gerhard et
al., 2002; Cortese et al., 2004, 2007; Yoshida et al., 2008; Smith et al.,
2010; Sivanandam et al., 2010; Gullieuszik et al., 2020), and explored by
simulations (Schulz & Struck, 2001; Vollmer et al., 2001; Kronberger et al.,
2008; Kapferer et al., 2009; Puchwein et al., 2010). This star formation
occurs within long galactic tails of material that has been stripped out of
infalling galaxies and is generally seen to be low levels of activity (c.f.
Cortese et al. (2007) who report LIRG-levels of star formation of $\sim$50
M⊙yr-1), contrasting with the very high level of SF activity seen here.
Indeed, the large star formation rate in the centre of SpARCS1049 is
reminiscant of the excessive cooling seen in many hydrodynamical simulations
without AGN feedback (e.g., Puchwein et al. (2008, 2010); Sijacki et al.
(2007); McCarthy et al. (2010)). These unhindered cooling flows over-produce
the stars in BCGs and make the stellar population too blue. Most current
hydrodynamical simulations thus assume strong AGN feedback and disallow
intergalactic star formation.

Hlavacek-Larrondo et al. (2020) however, showed that the SpARCS1049 cluster
contains a cool-core and suggested that the star formation was being fueled by
just such a large-scale runaway cooling flow of the hot intracluster X-ray
gas. This was based on deep Chandra X-ray observations, which shows that the
coolest intracluster X-ray gas (0.7-1.0 keV) was remarkably co-spatial with
the stellar mass and molecular gas that we see here. Hlavacek-Larrondo et al.
(2020) further conjectured that the sloshing off-axis of the BCG (which
contains a weak AGN, Trudeau et al. (2019)) has allowed this cooling to occur
in the absence of central AGN heating. We expect this combination to support
the star formation already present in the cluster, and transform most of the
molecular gas reservoir into intra-cluster stars. While it is possible that
feedback from supernovae blows away part of the gas, recent studies indicate
that it cannot by itself quench star formation in high-mass systems (Henriques
et al., 2019; Bluck et al., 2019), and without feedback from the AGN; it is
unlikely that the SF in this clusters stops before running out of gas.

Coupled with the simulations this would suggest that a dynamically unrelaxed
cluster with suppressed AGN heating is necessary for such in-situ ICL growth
and that the duty cycle for such an event would be roughly the relaxation time
of the sloshed BCG. This chain of events is similar to that of a $z=0.485$
cluster observed in the Clusters Hiding in Plain Sight (CHiPS) survey
(Somboonpanyakul et al., 2021); in which galaxy interactions (in this case a
major merger) triggered off-axis cooling of low-entropy gas, itself leading to
active star formation.

Exactly what fraction of the ICL is forming is difficult to assess, given the
uncertain constraints on the total mass of the ICL at these redshifts.
Observational measurements (Furnell et al., 2021; Gonzalez et al., 2005;
González et al., 2010; Sand et al., 2011; Gal-Yam et al., 2003; Zibetti et
al., 2005; Krick & Bernstein, 2007) indicate that for a halo of the mass of
SpARCS1049 (M200 = 4$\times$1014M⊙) approximately 10-20% (inside roughy r200)
of the stellar mass will be contained in the ICL at low redshift. If the mass
of the ICL and cluster galaxies co-evolves then we can apply this ratio to
high redshift. A back of the envelope calculation can thus be done: SpARCS1049
would have stellar mass in galaxies of roughly 8$\times$1012 (using
$f_{500}^{stars}\sim 0.02$, Giodini et al. (2009)), and thus the expected ICL
mass would be $\sim$8$\times$1011 M⊙. Given that we see
$\sim$2.2$\times$1010M⊙ to $\sim$6.6$\times$1010M⊙ of stellar mass in place,
and another $\sim$1$\times$1011M⊙ of fuel, with ample star formation to use up
this fuel in a short amount of time, SpARCS1049 will have amassed a
significant portion ($\sim 15$ to $21$%, depending on the data processing
method) of its ICL through this mechanism. Such an event can be roughly
encompassed by the constraints on the ICL placed by studies of the Virgo
cluster. Williams et al. (2007) find that 20-30% of the intracluster stellar
population in Virgo is younger than 10Gyr and conclude that Virgo’s
intracluster stars must have multiple origin mechanisms. Indeed, Edwards et
al. (2020) estimate an average age of the ICL of 9.2$\pm$3.5 Gyr.

This is in sharp contrast to what is prescribed in semi-analytical models
however, where the ICL forms through the stripped stellar mass of cluster
galaxies, during interactions with the BCG or between themselves (e.g.,
Contini et al. (2018), see also Contini (2021) for a review of the ICL). In
these simulations, AGN feedback is invoked to prohibit excessive cooling, and
intergalactic star formation is, in general, not allowed to occur. Thus, the
lack of any in-situ star formation is not a direct prediction of the models,
but is written into the physics.

We note also, that a recent publication has presented an alternative view of
the SpARCS1049 core. Castignani et al. (2020) suggest that the immense amount
of gas seen in the central region of the cluster was delivered, not by a
cooling flow, but through gas clouds stripped from an infalling galaxy as seen
in many clusters at lower redshift. This distant single Jelly Fish galaxy is
destined, in their view, to merge with the BCG and contribute to its stellar
mass growth. However, this explanation for the cluster is unlikely, as the
X-ray emission is too massive and bright to be attributed to a jellyfish
galaxy; and can only be associated with a massive cool core. Moreover, the
fact that the cool 0.7-1.0 keV gas and the optical tail are remarkably co-
spatial makes it all the more probable that the two emissions are related.
Further observations with better spatial resolution would be needed to further
confirm this postulate, in particular surrounding the star forming region.
Additional information regarding the dynamics at play in that same region
would also allow to distinguish between the two hypotheses mentioned in this
discussion.

##  6 Conclusions

Here we present and summarize evidence that, taken together, suggests that we
are witnessing the in-situ buildup a substantial amount of stellar mass in the
core of SpARCS1049. A summary of our specific conclusions is as follows:

  * •

We present two estimates of the sum of the established stellar mass in the
clumpy tidal tail within the core of the SpARCS1049 cluster, one made using
the original HST imaging, and one made using re-processed data. We find that,
over a $12.9^{\prime\prime}\times 11.7^{\prime\prime}$ region, but dominated
by the tail, there exist a substantial amount of stellar mass: $(6.6\pm
1.2)\times 10^{10}\>M_{\odot}$ using the first method, and $(2.2\pm 0.5)\times
10^{10}\>M_{\odot}$ using the second one. The scatter in Equation 1 also
introduces an average systematic error of $\sim 40\%$ (Bell et al., 2003).

  * •

We present a new CO(1-0) measurement of the total molecular gas mass within a
single large clump in the core of SpARCS1049. We measure this to be
1.0$\pm$0.7$\times$1011M⊙.

  * •

Both the large amount of stellar mass and the vast reservoir of molecular gas
lie roughly 12 kpc off the BCG. We therefore suggest, in line with Hlavacek-
Larrondo et al. (2020), that we are not seeing the BCG stellar mass assembly,
but rather the in-situ growth of the ICL.

  * •

Simple arguments indicate that if this is indeed the in-situ growth of the
ICL, it may be forming a significant fraction ($\sim 15$ to $21$%) of the ICL
in this cluster.

C.B. and T.M.A.W. acknowledges support from NSERC via the Discovery and from
the Fonds de Recherche du Québec - Nature et Technologies (FRQNT). J.H.-L.
acknowledges support from NSERC via the Discovery grant program, the
Accelerator grant program, as well as the Canada Research Chair program. GW
gratefully acknowledges support from the National Science Foundation through
grant AST-1517863, from HST program number GO-15294, and from grant number
80NSSC17K0019 issued through the NASA Astrophysics Data Analysis Program
(ADAP). Support for program number GO-15294 was provided by NASA through a
grant from the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Incorporated, under
NASA contract NAS5-26555. We thank Vivian Tan and Visal Sok for their help
with the Voronoi binning. We thank the JVLA staff with their immense help
reducing and analysing the data. The National Radio Astronomy Observatory is a
facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. This research is based on
observations made with the NASA/ESA Hubble Space Telescope obtained from the
Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.
These observations are associated with program(s) # GO-13677 and # GO-13747.

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