# Implicit degree bias in the link prediction task

Rachith Aiyappa1, Xin Wang3, Munjung Kim1, Ozgur Can Seckin1, Jisung Yoon2,
Yong-Yeol Ahn1, Sadamori Kojaku3
1Center for Complex Networks and Systems Research, Luddy School of
Informatics, Computing, and Engineering, Indiana University, Bloomington, IN,
USA &
2KDI School of Public Policy and Management, Sejong City, South Korea &
3Department of Systems Science and Industrial Engineering, Binghamton
University, Binghamton, NY, USA &
<EMAIL_ADDRESS>jisung.yoon92<EMAIL_ADDRESS>
###### Abstract

Link prediction—a task of distinguishing actual hidden edges from random
unconnected node pairs—is one of the quintessential tasks in graph machine
learning. Despite being widely accepted as a universal benchmark and a
downstream task for representation learning, the validity of the link
prediction benchmark itself has been rarely questioned. Here, we show that the
common edge sampling procedure in the link prediction task has an implicit
bias toward high-degree nodes and produces a highly skewed evaluation that
favors methods overly dependent on node degree, to the extent that a “null”
link prediction method based solely on node degree can yield nearly optimal
performance. We propose a degree-corrected link prediction task that offers a
more reasonable assessment that aligns better with the performance in the
recommendation task. Finally, we demonstrate that the degree-corrected
benchmark can more effectively train graph machine-learning models by reducing
overfitting to node degrees and facilitating the learning of relevant
structures in graphs.

##  1 Introduction

Standardized benchmarks such as ImageNet [1, 2] and SQuAD [3, 4] play a
pivotal role in guiding the evolution of machine learning, creating a
competitive environment that propels innovation by setting clear, measurable
goals. A core benchmark for graph machine learning is the link prediction
task, the task of identifying missing edges in a graph, with diverse
applications including the recommendations of friends and contents [5, 6, 7,
8], knowledge discoveries [9, 10], and drug development [11, 12, 13, 14, 15,
16, 17]. The link prediction benchmarks have been serving as a crucial
benchmark that facilitates quantitative evaluations and drives the advancement
of graph machine learning techniques [18, 19, 20, 12, 16, 17, 21].

Despite its significant role in graph machine learning, the link prediction
benchmark itself has rarely been examined critically for its effectiveness,
reliability, and potential biases. A widely-used link prediction benchmark
evaluates methods by their ability to classify pairs of nodes as either
connected or unconnected [5, 18, 20]. The connected node pairs (i.e., edges)
are randomly sampled from existing edges as the hidden positive set, and an
equal number of node pairs are randomly chosen from unconnected node pairs.
Existing criticisms mainly consider the disconnect from real-world scenarios.
For instance, unconnected node pairs are far more common than connected node
pairs because of the sparsity of edges in graphs [22, 23], which can lead to
biased performance evaluations [24, 25, 26, 27, 28]. Additionally, while the
link prediction benchmark tests methods by classifying a predefined set of
potential edges, the practical link prediction task involves identifying the
potential edges from the entire graph. Despite the misalignment with real-
world scenarios, high benchmark performance is often regarded as a testament
to successful learning in graph machine learning [18, 29, 12, 15, 16, 17, 30,
31, 32, 33].

Here, we argue that the standard link prediction benchmark has a fundamental
and severe bias that favors methods that exploit node degree information
(i.e., the number of edges a node has). The degree bias arises from the
sampling of edges used for performance evaluations, i.e., when sampling an
edge from a graph uniformly at random, a node with $k$ edges is $k$ times more
likely to be selected than a node with a single edge ($k=1$). Meanwhile, the
negative set of edges is randomly sampled from unconnected node pairs, which
do not have this degree bias. As a result, we have a distinct and useful
feature (degree) for the methods that can be leveraged without understanding
any non-trivial structural features of the graph. We show that this degree
bias is so profound that a “null” link prediction method based solely on node
degree can yield nearly optimal performance, questioning the usefulness of the
link prediction benchmark even as a general objective for graph machine
learning and raising awareness on the need for being more intentional and
careful about what the evaluation tasks themselves actually evaluate.

To address this bias, we propose a degree-corrected link prediction benchmark
that samples the unconnected node pairs with the same degree bias. We
demonstrate that the degree-corrected link prediction benchmark can better
capture the performance of algorithms in the recommendation task. Moreover, we
show that the degree-corrected benchmark trains graph neural networks more
effectively by reducing overfitting to node degrees, thereby improving the
learning of community structure in graphs.

##  2 Design flaw of the link prediction benchmark

### Preliminary

We focus on the unweighted, undirected graph $G=(\mathcal{V},\mathcal{E})$,
where $\mathcal{V}$ is the set of nodes and $\mathcal{E}$ is the set of edges.
We assume that $G$ has no self-loops, no multiple edges, and is highly sparse
($|\mathcal{E}|\ll|\mathcal{V}|^{2}$), which is a common characteristic of
real-world graphs [22, 23]. Degree $k_{i}$ of a node $i\in\mathcal{V}$ is the
number of edges emanating from node $i$. We use $\sim$ to denote proportional
relationships. We exclude node attributes from data, if present, to ensure
consistency across all graph-based link prediction methods.

Figure 1:  Illustration of the degree bias in the link prediction benchmark.
A: A node with degree $k$ appears $k$ times in the edge list, making it $k$
times more likely to be sampled as a positive edge than a node with degree 1.
B: The degree distribution of the nodes in the positive and negative edges
sampled from a Price graph of $N=10^{5}$ nodes and $M=10^{6}$ edges. The
y-axis, “CCDF”, denotes the complementary cumulative distribution function,
representing the probability that a node’s degree is at least $k$. Dashed
lines illustrate the relationship described by Eq. (1). C: The AUC-ROC score
for the Preferential Attachment (PA) method on empirical graphs, with the
dashed line indicating the expected AUC-ROC based on the “null” link
prediction method that uses only the node degree information (Eq. (9)). D:
AUC-ROC of 26 methods across 90 graphs. PA outperforms 54% of the method on
average across 90 graphs and outperforms most other methods in the most
heterogeneous graphs. E: AUC-ROC of the same methods for the degree-corrected
benchmark, showing that nearly 86% of the methods perform better than PA on
average.

### Link prediction benchmark

The standard procedure for link prediction benchmarks is outlined as follows
[5, 29, 12, 15, 16, 17, 30, 31, 32, 33]. First, we randomly sample a fraction
$\beta$ of edges from the edge set $\mathcal{E}$ as _positive edges_. Second,
we randomly sample an equal number of unconnected node pairs with replacement
from the node set $\mathcal{V}$ as _negative edges_. We resample the negative
edges if they form a loop or are already in the positive or test edges. Third,
each node pair $(i,j)$ is scored by a link prediction method, where a higher
score $s_{ij}$ indicates a higher likelihood of an edge existing between the
nodes. Fourth, the effectiveness of a method is evaluated using the Area Under
the Receiver Operating Characteristic Curve (AUC-ROC), which represents the
probability that the method gives a higher score to a positive edge than a
negative edge. While alternative benchmark designs use different evaluation
metrics or sampling strategies for negative edges [24, 25, 26, 27, 28]
(Section 4), this outlined procedure has been widely used [5, 29, 12, 15, 16,
17, 30, 31, 32, 18].

### Sampling bias due to node degree

A well-known, counterintuitive fact about graphs is that a uniform random
sampling of edges results in a _biased_ selection of nodes in terms of degree
[34, 35, 36, 37, 38]. The bias arises because a node with $k$ edges appears
$k$ times in the edge list and thus $k$ times more likely to be chosen than a
node with $k^{\prime}=1$ edge (e.g., node 1 and 4 in Fig. 1A). Consequently,
for a graph with degree distribution $p(k)$, the nodes in the positive edges
have degree distribution $p_{\text{pos}}(k)$ proportional to $\sim k\cdot
p(k)$. By normalizing $k\cdot p(k)$, the degree distribution of the positive
edges is given by

$\displaystyle p_{\text{pos}}(k)=\frac{1}{\sum_{\ell}\ell p(\ell)}k\cdot
p(k)=\frac{1}{\langle k\rangle}k\cdot p(k),$ (1)

where $\langle k\rangle$ is the average degree of the graph. On the other
hand, the nodes in the negative edges are uniformly sampled from the node list
(i.e., $\mathcal{V}$) and thereby have degree distribution $p_{\text{neg}}(k)$
identical to $p(k)$ (i.e., $p_{\text{neg}}(k)=p(k)$).

We demonstrate the degree bias using the Price graph of $N=10^{5}$ nodes and
$M=10^{6}$ edges, with the power-law degree distribution approximating
$p(k)\propto k^{-3}$ (Fig. 1B). We sample $\beta=0.25$ of the edges uniformly
at random from $\mathcal{E}$ as positive edges, together with the same number
of unconnected node pairs by uniformly sampling nodes from $\mathcal{V}$. The
degree distributions for nodes in the positive and negative edges closely
follow $p_{\text{pos}}(k)$ and $p_{\text{neg}}(k)$, respectively,
demonstrating the sampling bias due to node degree. We note that the degree
bias is not specific to the Price graph but appears for any graph with non-
uniform degree distribution.

### Impact of degree bias on the link prediction benchmark

We demonstrate the impact of degree bias on link prediction benchmark
(Fig.1D). We evaluated 26 link prediction methods across 90 graphs. These
methods include 7 network topology-based methods (e.g., Common Neighbors (CN)
[19]), 13 graph embedding methods (e.g., Laplacian EigenMap (EigenMap) [39]),
2 network models (e.g., Stochastic Block Model (e.g., SBM) [40]), and 4 graph
neural networks (GNNs) (e.g., Graph Convolutional Network (GCN) [41]).
Detailed descriptions of the methods and graphs are available in SI Section 1.
We set the fraction of test edges to $\beta=0.25$ and repeat the experiment 5
times. We quantify the heterogeneity $\sigma$ of node degree by fitting a log-
normal distribution to $p(k)$ and calculating its variance parameter $\sigma$.
We will show that $\sigma$ is a good indicator of the impact of degree bias in
Section 2.

We focus on the Preferential Attachment (PA) link prediction method, which
calculates the prediction score $s_{ij}=k_{i}k_{j}$ using only node degrees.
Although PA is a crude method that neglects key predictive features such as
the number of common neighbors and shortest distance [24, 25, 42, 29, 43], it
still outperforms more than half of the advanced methods with an average AUC-
ROC of 0.83 (ranked 13th out of 26 methods; see Fig. 2A). PA performs better
as the heterogeneity of node degrees increases. The outperformance of PA is
attributed to the degree bias, where the positive edges are more likely to be
formed by nodes with high degree and thereby are easily distinguishable from
the negative edges. As a result, the current benchmark design favors methods
that make predictions based largely on node degrees.

### Theoretical analysis

Many empirical graphs exhibit heterogeneous degree distributions, with a few
nodes having an exceptionally large degree and most having a small degree. The
heterogeneous degree distributions are often characterized by power-law degree
distribution $p(k)\propto k^{-\alpha}$ with $\alpha\in(2,3]$ (i.e., scale-free
networks) [44, 45, 46, 47] or log-normal distributions [48, 49]. While the
power-law and log-normal distributions are both continuous, they are often
used to characterize the discrete degree distribution and provide
approximations [48, 49, 50, 51, 52, 53]. We show that the AUC-ROC for PA
reaches the nearly maximum under log-normal distributions with heterogeneous
node degrees. See SI Section 2.2 for the case of power-law distributions.

Let us consider a general degree distribution $p(k)$ without restricting
ourselves to log-normal distributions. The AUC-ROC has a probablistic
interpretation [54]: it is the probability that the score $s^{+}$ for the
positive edges is larger than the score $s^{-}$ for the negative edges.
Recalling that PA computes $s_{ij}=k_{i}k_{j}$, the AUC-ROC for PA is given by

$\displaystyle\text{AUC-ROC}=P(s_{i^{-},j^{-}}\leq
s_{i^{+},j^{+}})=P(k_{i^{-}}k_{j^{-}}\leq k_{i^{+}}k_{j^{+}}),$ (2)

where $i^{\pm}$ and $j^{\pm}$ represent the nodes of the positive and negative
edges, respectively. Now, let us assume that $p(k)$ follows the log-normal
distribution, $\text{LogNorm}(k\mid\mu,\sigma^{2})$, given by [54]:

$\displaystyle
p(k)=\text{LogNorm}(k\mid\mu,\sigma^{2})=\frac{1}{\sqrt{2\pi}\sigma
k}\exp\left[-\frac{(\ln k-\mu)^{2}}{2\sigma^{2}}\right],$ (3)

where $\mu$ and $\sigma$ are the parameters of the log-normal distribution.
The mean of the log-normal degree distribution is $\langle
k\rangle=\exp(\mu+\sigma^{2}/2)$ [54]. By substitutiong Eq. (3) into Eq. (1),
we derive the degree distribution of nodes in the positive edges as:

$\displaystyle p_{\text{pos}}(k)$ $\displaystyle=\frac{k}{\langle
k\rangle}\frac{1}{\sqrt{2\pi}\sigma k}\cdot\exp\left[-\frac{(\ln
k-\mu)^{2}}{2\sigma^{2}}\right]$ $\displaystyle=\frac{1}{\sqrt{2\pi}\sigma
k}\exp\left[-\frac{1}{2\sigma^{2}}\left((\ln k)^{2}-2\mu\ln
k+\mu^{2}\right)+\ln k-\mu-\sigma^{2}/2\right]$
$\displaystyle=\frac{1}{\sqrt{2\pi}\sigma
k}\exp\left[-\frac{1}{2\sigma^{2}}\left((\ln k)^{2}-2(\mu+\sigma^{2})\ln
k+\mu^{2}+2\mu\sigma^{2}+\sigma^{4}\right)\right]$
$\displaystyle=\frac{1}{\sqrt{2\pi}\sigma k}\exp\left[-\frac{(\ln
k-\mu-\sigma^{2})^{2}}{2\sigma^{2}}\right]=\text{LogNorm}(k\mid\mu+\sigma^{2},\sigma^{2}).$
(4)

Equation (4) indicates that the degree distribution for nodes in the positive
edges also follows a log-normal distribution, parameterized by
$\mu+\sigma^{2}$ and $\sigma$.

We derive the AUC-ROC for PA by leveraging a unique characteristic of log-
normal distributions, i.e., the logarithm $\ln k$ of log-normally-distributed
degree $k$ follows a normal distribution with mean $\mu$ and variance
$\sigma^{2}$, i.e.,

$\displaystyle P(\ln
k)=\text{Norm}(k\mid\mu,\sigma^{2}),\;\text{where}\;\;\text{Norm}(k\mid\mu,\sigma^{2})=\frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(k-\mu)^{2}}{2\sigma^{2}}\right].$
(5)

We assume no degree assortativity in the graph, where
$P(k^{+}_{i},k^{+}_{j})=P(k_{i})P(k_{j})$. Although empirical graphs often
exhibit degree assortativity, our results indicate that it does not
significantly impact the AUC-ROC (SI Section 2.1). For the negative edges, the
distribution for $\ln s_{i^{-},j^{-}}=\ln k^{-}_{i}+\ln k^{-}_{j}$ follows a
normal distribution with mean $2\mu$ and variance $2\sigma^{2}$, as the sum of
independent normal variables also forms a normal distribution with additive
means and variances [55], i.e.,

$\displaystyle P(\ln s_{i^{-},j^{-}})=\text{Norm}\left(\ln s_{i^{-},j^{-}}\mid
2\mu,2\sigma^{2}\right).$ (6)

For the positive edges, the degree distribution also follows a log-normal
distribution (Eq. (4)). Thus, the distribution for $\ln s_{i^{+},j^{+}}=\ln
k^{+}_{i}+\ln k^{+}_{j}$ is given by

$\displaystyle P(\ln s_{i^{+},j^{+}})=\text{Norm}\left(\ln s_{i^{+},j^{+}}\mid
2\mu+2\sigma^{2},2\sigma^{2}\right).$ (7)

By using Eqs. (2), (6), and (7), we have

AUC-ROC $\displaystyle=P(\ln s^{-}<\ln s^{+})$
$\displaystyle=\int_{-\infty}^{\infty}\text{Norm}(x^{-}\mid
2\mu,2\sigma^{2})\left[1-\int^{x^{-}}_{-\infty}\text{Norm}(x^{+}\mid
2\mu+2\sigma^{2},2\sigma^{2})\text{d}x^{+}\right]\text{d}x^{-}$
$\displaystyle=1-\int_{-\infty}^{\infty}\text{Norm}(x^{-}\mid
2\mu,2\sigma^{2})\int^{x^{-}}_{-\infty}\text{Norm}(x^{+}\mid
2\mu+2\sigma^{2},2\sigma^{2})\text{d}x^{+}\text{d}x^{-}$ (8)

We reparameterize Eq. (8) by using
$z^{\pm}=\frac{x^{\pm}-2\mu}{\sqrt{2}\sigma}$. Noting that
$\text{Norm}(x^{-}\mid
2\mu,2\sigma^{2})\cdot\sqrt{2}\sigma=\text{Norm}(z^{-}\mid 0,1)$ and ${\rm
d}x^{\pm}=(\sqrt{2}\sigma){\rm d}z^{\pm}$, we have

$\displaystyle P(\ln s^{-}<\ln s^{+})$
$\displaystyle=1-\int_{-\infty}^{\infty}(2\sigma^{2})\text{Norm}(x^{-}\mid
2\mu,2\sigma^{2})\int^{x^{-}}_{-\infty}\text{Norm}(x^{+}\mid
2\mu+2\sigma^{2},2\sigma^{2})\cdot\text{d}z^{+}\text{d}z^{-}$
$\displaystyle=1-\int_{-\infty}^{\infty}\text{Norm}\left(z^{-}\mid
0,1\right)\int^{z^{-}}_{-\infty}\text{Norm}\left(z^{+}-\sqrt{2}\sigma\mid
0,1\right)\text{d}z^{+}\text{d}z^{-}$
$\displaystyle=1-\int_{-\infty}^{\infty}\text{Norm}(z^{-}\mid
0,1)\Phi\left(z^{-}-\sqrt{2}\sigma\right)\rm{d}z,$ (9)

where $\Phi(z^{-})$ is the cumulative distribution function for the standard
normal distribution, i.e., $\Phi(z^{-})=\int_{\infty}^{z^{-}}\text{Norm}(y\mid
0,1){\rm d}y$. Equation (9) suggests the key behavior of AUC-ROC for PA. The
AUC-ROC for PA is an increasing function of the parameter $\sigma$ of the log-
normal distribution (Fig. 1C). The parameter $\sigma$ of the log-normal
distribution controls the spread of the distribution, with larger $\sigma$
resulting in a more fat-tailed distribution. While our assumptions about the
log-normal degree distribution and degree assortativity may not always align
with real-world data, Eq. (9) still effectively captures the AUC-ROC behavior
for PA (Fig. 1C). Further analysis of power-law distributions is described in
SI Section 2.2.

This theoretical result highlights the significant issue with the current link
prediction benchmark: a high benchmark performance can be achieved by only
learning node degrees, posing the question of whether the link prediction
benchmark is an effective objective of graph machine learning.

##  3 The degree-corrected link prediction benchmark

Algorithm 1 Degree-corrected link prediction benchmark

1:Input: Graph $G(\mathcal{V},\mathcal{E})$, Sampling fraction $\beta\in[0,1]$
for positive edges

2:Output: Set of negative edges $\mathcal{E}_{\text{neg}}$ and set of positive
edges $\mathcal{E}_{\text{pos}}$

3:Generate $\mathcal{E}_{\text{pos}}$ by randomly sampling $\beta$ fraction of
edges in $\mathcal{E}$.

4:Initialize $\mathcal{E}_{\text{neg}}\leftarrow\emptyset$

5:Create a node list $L$ where each node $i\in\mathcal{V}$ with degree $k_{i}$
appears $k_{i}$ times

6:while $|\mathcal{E}_{\text{neg}}|<|\mathcal{E}_{\text{pos}}|$ do

7: Randomly select two nodes $i,j$ from $L$ with replacement

8: if $(i,j)\notin\mathcal{E}$ and $(i,j)\notin\mathcal{E}_{\text{neg}}$ and
$i\neq j$ then

9: $\mathcal{E}_{\text{neg}}\leftarrow\mathcal{E}_{\text{neg}}\cup\\{(i,j)\\}$

10: end if

11:end while

12:return $\mathcal{E}_{\text{neg}}$, $\mathcal{E}_{\text{pos}}$

The link prediction benchmark yields biased evaluations due to mismatched
degree distributions between positive and negative edges, i.e.,
$p_{\text{neg}}(k)\neq p_{\text{pos}}(k)$. To mitigate the mismatch, we
introduce _the degree-corrected link prediction benchmark_ that samples
negative edges with the same degree bias as positive edges (Algorithm 1).
Specifically, we create a list of nodes where each node with degree $k$
appears $k$ times. Then, we sample negative edges by uniformly sampling two
nodes from this list with replacement until the sampled node pairs are not
connected and not in the test edge set. Crucially, nodes with degree $k$ are
$k$ times more likely to be sampled than nodes with degree 1, mirroring the
degree bias of the positive edges. Consequently, the positive and negative
edges in the degree-corrected benchmark are indistinguishable based on node
degrees. A Python package for the degree-corrected benchmark is available at
https://github.com/skojaku/degree-corrected-link-prediction-benchmark.

### Comparison of the benchmark evaluations

We reevaluated the methods with the degree-corrected link prediction benchmark
(Fig. 1E). We use the same parameters in the original link prediction
benchmark for all methods. There is some agreement between the original and
degree-corrected benchmarks (Fig. 2A). For example, they rank GAT and LRW as
top performers, whileNB, SGTAdjNeu, and SGTAdjExp are consistently ranked
lower. On the other hand, the degree-corrected benchmark ranks PA as the
lowest performer, with its average AUC-ROC dropping from 0.83 to 0.54, placing
it last out of 26 methods. The AUC-ROC near 0.5 for PA suggests that it
performs as random guessing, confirming that the degree-corrected benchmark
successfully makes the degree distributions of positive and negative edges
indistinguishable. Other methods such as LPI, GIN also experience a
substantial drop in their rankings from 2nd to 10th and 9th to 20th,
respectively (Fig. 2A). On the other hand, GCN, node2vec, DeepWalk, and
EigenMap increase their rankings substantially from 14th to 2nd, 17th to 5th,
21th to 7th, and 20th to 12th, respectively (Fig. 2A).

Figure 2:  Comparative analysis of link prediction and recommendation
benchmarks. A: Rank changes across the original and degree-corrected link
prediction benchmarks. The red and blue lines indicate the methods that change
their rankings more than 8 places from the original one. We compute the
ranking of the methods based on the AUC-ROC of the link prediction benchmark
and compare it against the ranking based on the link retrieval task. B: The
degree-corrected benchmark ranks link prediction methods more similarly with
the recommendation task than the original benchmark. The RBO (rank-biased
overlap) represents the similarity between the ranking of the link prediction
methods and that based on the recommendation task.

### The degree-corrected benchmark aligns better with recommendation tasks

Link prediction methods are often used in recommendation tasks [24, 25, 26,
27, 28]. However, there are key differences between standard link prediction
benchmarks and recommendation tasks [24, 25, 26]. In the link prediction
benchmark, a set of potential edges is provided, and the goal is to determine
the existence of these edges. In contrast, recommendation tasks require
identifying potential edges from the entire graph without a preset list of
candidate edges. This makes recommendation tasks more challenging due to the
large number of unconnected nodes that could be just as likely to have an edge
as connected nodes. For example, a link prediction task in a social network
involves predicting friendships from a given list of potential friends. On the
other hand, recommendation tasks must evaluate all potential unlisted
friendships, including absent but equally probable friendships as existing
ones. Nevertheless, given the common use of link prediction in recommendation
tasks, it is crucial that the link prediction benchmarks accurately reflect
the performance in the recommendation tasks.

We assess how well the link prediction benchmark provides coherent performance
assessment with the recommendation task as follows. In the common
recommendation task [25], the method recommends, for each node $i$, its top
$C=50$ nodes $j$ based on the highest scores $s_{ij}$. We note that our
results are consistent regardless of the chosen $C$ value (refer to SI Section
2.3). The effectiveness of the recommendations is then measured using the
vertex-centric max precision recall at $C$ (VCMPR@C) defined as the higher of
the precision and recall scores at $C$ [25]. The VCMPR@C is designed to
evaluate recommendation methods and addresses the excessive penalty on the
precision scores at $C$ for small-degree nodes, which often have low precision
due to their limited number of edges. We perform this task five times and
average the VCMPR@C scores across different runs.

For each graph, we evaluate the alignment between the rankings based on the
recommendation task and those based on the link prediction benchmarks using
Rank Biased Overlap (RBO) [56]. RBO is a ranking similarity metric with larger
weights on the top performers in the two rankings. A larger RBO score
indicates that the top performers in the two rankings are more similar. The
weights on the top performer are controlled by the parameter $p\in(0,1)$.
While we set $p=0.5$ in our experiment, we confirmed that our results are
robust to the choice of $p$ (SI Section 2.3).

Our results from 90 graphs show that the degree-corrected benchmark achieves
higher RBO scores than the original benchmark. We find consistent results for
different values of $C$ and parameter $p$ of the RBO (SI Section 2.3). These
results indicate that the degree-corrected benchmark more accurately mirrors
the performance in recommendation tasks, providing a more reliable measure of
the effectiveness of methods in practical applications.

### Degree-corrected benchmark facilitates the learning of community structure

Figure 3:  The degree-corrected benchmark improves GNNs in learning community
structure in the LFR graphs. The LFR graphs consist of $3,000$ nodes with the
average degree of $25$. A: The performance of community detection for the LFR
graphs with a power-law degree distribution with $\tau_{1}=3.0$ as a function
of $\mu$. B: The average performance (by the area under the performance curve)
shows that the methods trained with the degree-corrected training set
outperform their counterparts trained with the traditional link prediction
training set. C, D: The same plots for LFR graphs with a fatter power-law
degree distribution with $\tau_{1}=2.5$. The error bars represent the 95%
confidence interval estimated by a bootstrap of 1,000 repetitions.

The link prediction benchmark is a common unsupervised learning objective for
GNNs [57, 58, 38, 59]. The degree bias implies that GNNs trained using the
original link prediction benchmark tend to overfit to node degrees because
they can easily differentiate between positive and negative edges based on
node degrees. We show that the degree-corrected benchmark effectively prevents
overfitting to node degrees and improves the learning of salient graph
structures.

We evaluate GNNs on the common unsupervised task of community detection in
graphs [60, 40, 61]. The community detection task identifies densely connected
groups (i.e., communities) in a graph. Communities often correspond to
functional units (e.g., social circles with similar opinions and protein
complexes) in the graph, and detecting communities is a crucial task in many
graph applications [60, 40, 61, 62, 63]. Specifically, we test the GNNs by
using the Lancichinetti-Fortunato-Radicchi (LFR) community detection benchmark
[64], a standard benchmark for community detection [61, 40, 65, 66]. The LFR
benchmark generates synthetic graphs with predefined communities as follows.
Each node $i$ is assigned a degree $k_{i}$ from a power-law distribution
$p(k)\sim k^{-\tau_{1}}$, with maximum degree $k_{\text{max}}$. Nodes are
randomly grouped into $L$ communities, with community sizes (i.e., the number
of nodes in a community) following another power-law distribution $p(n)\sim
n^{-\tau_{2}}$ bounded between $n_{\text{min}}$ and $n_{\text{max}}$. Edges
are then formed such that each node $i$ connects to a fraction $1-\mu$ of its
$k_{i}$ edges within its community and the remaining fraction $\mu$ to nodes
in other communities. We generate 10 graphs for
$\mu\in\\{0.05,0.1,0.15,\ldots,0.95,1\\}$ using the following parameter
values: the number of nodes $N=3,000$, the degree exponent
$\tau_{1}\in\\{2.5,3\\}$, the average degree $\langle k\rangle=25$, the
maximum degree $k_{\text{max}}=1000$, the community-size exponent
$\tau_{2}=3$, the minimum and maximum community size $n_{\text{min}}=100$ and
$n_{\text{max}}=1000$. We obtained qualitatively similar results for different
values of the parameters of the LFR benchmark (SI Section 2.5).

We train GNNs using either the original or the degree-corrected link
prediction benchmarks to minimize binary entropy loss in classifying the
positive and negative edges. Using the trained GNNs, we generate node
embeddings and apply the $K$-means clustering algorithm to detect communities,
where $K$ is set to the number of true communities. Although the number $K$ of
communities is often unknown, we use the ground-truth number to eliminate
noise from estimating $K$ and to concentrate on evaluating the quality of the
learned node embeddings, a standard practice in benchmarking node embeddings
for the community detection task [65, 66, 67]. We measure the performance of
GNNs by comparing the detected communities against the true communities using
the adjusted element-centric similarity [66, 67, 68], where higher scores
indicate a higher similarity between the node partitions for the true and
detected communities. We observe qualitatively similar results for partition
similarities based on the normalized mutual information (SI Section 2.4).

All GNNs, except for GIN, perform well when $\mu\leq 0.5$, where communities
are distinct and easily identifiable, but their performance declines as $\mu$
increases (Fig. 3A). Across a broad range of $\mu$, degree-corrected GNNs,
particularly GIN, GCN, and GraphSAGE, outperform original GNNs in identifying
communities, as shown by the area under the performance curve (Fig. 3B). The
advantage of degree-corrected benchmarks becomes more evident with a more
heterogeneous degree distribution (Fig. 3C and D). This indicates that degree
correction effectively reduces overfitting to node degrees, enhancing the
learning of community structures in graphs.

##  4 Discussion

We showed that the common link prediction benchmark suffered from a sampling
bias arising from node degree and could mislead performance evaluations by
favoring methods that overfit to node degree. To address this issue, we
proposed a degree-corrected benchmark that corrected the discrepancy between
the degree distributions of the connected and unconnected node pairs sampled
for performance evaluation. This new benchmark provided more accurate
evaluations that aligned better with the performance for recommendation tasks
and also improved training for GNNs by reducing the overfitting to node degree
and facilitating the learning of communities in the graph.

While we focused on node degree, other features can also effectively
distinguish between positive and negative edges [24, 42, 29, 43]. Previous
studies showed that nodes in the negative edge set tend to be distant from
each other, making them easily identifiable by computing the distance between
nodes [24, 42]. Our findings add a new direction to the important examination
of the fundamental graph machine-learning task by demonstrating that node
degree, a simpler structural attribute, is sufficient, in many cases, for
differentiating the positive and negative edges. We note that the degree bias
partially accounts for the distance discrepancy in the positive and negative
edges because a high degree heterogeneity can induce loops of short length in
the graph [69]. Crucially, the degree bias stems from the degree distribution
rather than the connections between nodes and thus is present in any non-
regular graph. More broadly, the degree bias is manifested through the edge
sampling process, a general technique for evaluating and training graph
machine learning. For instance, mini-batch training [57, 70], which samples
subsets of edges for efficient GNN training, may also exhibit bias due to node
degrees, leading to skewed training sets. Given the widespread use of edge
sampling across various graph machine-learning tasks, our findings have broad
implications beyond link prediction benchmarks, extending to a range of
benchmarks and training frameworks.

Our study has several limitations. First, we employed standard hyperparameters
(e.g., embedding dimensions, GNN layers) across all link prediction methods,
achieving reasonable performance in link prediction and community detection
tasks. Given the extensive variety of methods and hyperparameters, we did not
optimize hyperparameters for specific tasks. Although fine-tuning could
potentially enhance performance [24], our primary goal was to evaluate the
effectiveness of link prediction benchmarks, not to compare the difference in
performances between different link prediction methods. Second, we did not
explore the reasons behind the varying performance of different link
prediction methods. It is important to note that there is no single link
prediction method that is universally effective for all graphs because the
performance of link prediction methods depends on the assumptions on the graph
structure to make the predictions [18]. Future studies will explore an in-
depth examination of the link prediction methods on specific graphs. Third, we
focus on community structure to test the effectiveness of the proposed
benchmark as a training framework. However, other non-trivial graph
structures, such as centrality, could be tested through network dismantling
benchmarks [71].

Even with the aforementioned limitations, our results suggest that sampling
graph data is a highly non-trivial task than commonly considered. Because
sampling edges from a graph is integral to evaluating and training graph
machine learning methods, our results underline the importance of careful
sampling to ensure the effectiveness of evaluations and training of graph
machine learning methods.

##  5 Broader Impacts

Our findings indicate that the standard benchmark promotes link prediction
algorithms that enhance the “rich-get-richer” effect, where well-connected
nodes tend to gain even more connections. This effect, known as _preferential
attachment_ , is common in various societal contexts. For instance, popular
individuals often gain more friends, widely shared content spreads further,
and frequently cited papers are more likely to attract additional citations.
While preferential attachment can benefit distributing resources and
recognition, it also presents significant disadvantages. For example,
preferential attachment can disproportionately favor well-connected
individuals in professional networks and hinder scientific progress by
undervaluing innovative papers with less citations [72]. Our analysis
indicates that the recommendation algorithms trained using the existing
benchmark may exacerbate social inequalities by reinforcing the preferential
attachment mechanism. Our new benchmark directly addresses these concerns by
negating the degree bias, promoting fairer algorithm evaluations, and
supporting the development of fairness-aware machine learning methods.

### Code and Data Availability

We provide the code to reproduce the results in this study at
https://github.com/skojaku/degree-corrected-link-prediction-benchmark. The
network data used in this study is available at
https://figshare.com/projects/Implicit_degree_bias_in_the_link_prediction_task/205432.

## Acknowledgments and Disclosure of Funding

R.A., M.K., O.C.S., S.K., and Y.Y.A. acknowledge the support by the Air Force
Office of Scientific Research under award number FA9550-19-1-0391. The authors
acknowledge the computing resources at Binghamton University and Indiana
University and thank NVIDIA Corporation for their GPU resources.

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##  Appendix A Observed parameters

###  A.1 Stellar mass

Table 3:  The stellar mass measuring methods of the observation references presented in Fig. 1 | Data |  | Method
---
Yang et al. (2008) |  | SDSS DR4
---
(NYU-VAGC)
| Photometric.
---
Petrosian r-band magnitude within the Petrosian radius is converted to the
stellar mass based on the
M/L from the SPS model of Bell et al. (2003) assuming ‘diet’ Salpeter IMF.
Yang et al. (2012) | SDSS DR7 |  | Photometric.
---
Similar to Yang et al. (2008), using Kroupa (2002) IMF.
Moster et al. (2013) | SDSS DR7 |  | Photometric.
---
Model r-band magnitude is converted to the stellar mass using Li & White
(2009) stellar mass
measurements and the M/L from the SPS model of Blanton & Roweis (2007)
assuming Chabrier IMF.
SDSS model magnitude is calculated by fitting a surface brightness profile to
the exponential or the
de Vacoulers profile, whichever better, and integrating it to the infinite
radius. It is a better estimate of
the total mass of galaxies than Petrosian
magnitudes141414http://classic.sdss.org/dr7/algorithms/photometry.html.
Kravtsov et al. (2018) | SDSS DR8 |  | Photometric.
---
Extended luminosity profiles are constructed by fitting a triple-Sérsic
profile and then integrated out to
$50\,\rm kpc$ or the infinite radius. The luminosity is converted to the
stellar mass based on the M/L from the
SPS model of Bell et al. (2003) and then adjusted to emulate Chabrier IMF.
L18 | 2MASS |  | Photometric.
---
Total K-band magnitude is calculated by fitting a single-Sérsic profile and
integrating it to the infinite
radius. The luminosity is then converted to the stellar mass based on the M/L
from the dynamical
modelling of Cappellari (2013).
van Uitert et al. (2018) | GAMA |  | SPS modelling.
---
SED templates are built based on $ugriz$\- and NIR-magnitude scaled to cover
10 effective radius of the
galaxies. SED fitting is performed using the SPS model by Bruzual & Charlot
(2003) assuming an
exponential star formation history and Chabrier IMF.
Erfanianfar et al. (2019) |  | GALEX
---
SDSS DR14
WISE
| SPS modelling.
---
Observed SEDs are constructed based on UV, optical ($ugriz$-cModel magnitude),
and IR data.
SDSS cModel magnitude is calculated by fitting a surface brightness profile to
a linear combination of
the exponential and the de Vacoulers profile and integrating to infinite
radius.
SED templates are built using the SPS model by Bruzual & Charlot (2003)
assuming exponential star
formation history and Chabrier IMF.
Girelli et al. (2020) |  | SDSS DR4
---
(NYU-VAGC)
| Combination of photometric and SPS modelling.
---
Baldry et al. (2008) calculate the weighted-average of four independent
stellar mass measures using
NYU-VAGC Petrosian $ugriz$-magnitude, spectral features (e.g., D4000,
H$\rm\delta$), PEGASE
SPS models (Fioc & Rocca-Volmerange 1997), and Bruzual & Charlot (2003) SPS
models. Then the
results are rescaled to follow Planck 2015 cosmology and Chabrier IMF.
Table 4:  The stellar mass measuring methods of the observation references presented in Fig. 10 | Data |  | Method
---
Whitaker et al. (2012) | NMBS |  | SPS modelling.
---
SED templates are built using the SPS model by Bruzual & Charlot (2003)
assuming an exponential
star formation history, solar metallicity, and Chabrier IMF. Dust extinction
is taken into account
following Calzetti et al. (2000).
Mittal et al. (2015) |  | GALEX
---
2MASS
HST optical
SDSS DR10
| SPS modelling.
---
SEDs are constructed based on multi-wavelength data from UV to IR. The
aperture size is provided in
tables in the paper in arcsec unit. SED fitting is performed using GALAXEV
(Bruzual & Charlot 2003)
assuming Chabrier IMF.
Gozaliasl et al. (2016) |  | COSMOS
---
CFHTLS151515Canada–France–Hawaii Telescope Legacy Survey
| SPS modelling.
---
SEDs are constructed based on ugriz-magnitudes.
SED template fitting is performed using LE PHARE code (Arnouts et al. 1999;
Ilbert et al. 2006) and
FAST code (Kriek et al. 2009) with Bruzual & Charlot (2003) SSP and Chabrier
IMF.
Cooke et al. (2018) |  | GALEX
---
Subaru optical
Vista
Herschel
| SPS modelling
---
SEDs are constructed based on multi-wavelength data from FUV to FIR. SED
fitting is performed
using Bruzual & Charlot (2003) SSP and Chabrier IMF.

Converting observed photometric/spectroscopic properties to the stellar mass
is based on the stellar population synthesis modeling. Table 3 and Table 4
summarizes different stellar mass measurements in observation studies
presented in Fig. 1 and Fig. 10, respectively. Among the literatures, van
Uitert et al. (2018), Erfanianfar et al. (2019), Whitaker et al. (2012),
Mittal et al. (2015), Gozaliasl et al. (2016), and Cooke et al. (2018)
performed a direct SPS modeling; they constructed spectral energy
distributions (SEDs) of galaxies based on multi-wavelength data and searched
for a set of stellar populations that best fits the observed
spectra/magnitudes. Alternatively, one can adopt the stellar mass-to-light
ratio (M/L) estimated based on the properties that are well-known to correlate
with M/L. In Fig. 1, Yang et al. (2008); Yang et al. (2012), Kravtsov et al.
(2018), and Girelli et al. (2020) utilized a correlation between a photometric
colour and M/L from Bell et al. (2003). Similarly, Moster et al. (2013)
adopted M/L from Blanton & Roweis (2007) stellar population models. L18 used
M/L from Cappellari (2013); in contrast to the former mentioned M/L
references, Cappellari (2013) estimated the M/L from the 3D kinematic modeling
of early-type galaxies, therefore, the stellar mass corresponds to the
dynamical mass.

The stellar mass measure based on the photometry and the population synthesis
modeling depends on a number of factors, such as which initial mass function
(IMF) is assumed, how well the stellar mass-to-light ratio is constrained (see
Loubser et al., 2021, and references therein), and how far from the center the
stellar light is mapped (Gonzalez et al., 2013; Kravtsov et al., 2018; DeMaio
et al., 2020). The latter is a particularly vexing issue. Typically, the
stellar light associated with a BGG/BCG extends smoothly out from the
group/cluster center. Although this distribution is conventionally referred to
as BCG+ICL (or BGG+IGrL), there is no clear break or edge to indicate where
the galactic stellar component ends and the intragroup/intracluster light
(IGrL/ICL) begins (see, e.g., Contini 2021; Contini et al. 2022). For this
reason, most studies do not attempt to distinguish between the two. With
increasing distance from the center, the surface brightness profile drops and
eventually fade into the sky background (DeMaio et al., 2020). The over-
subtraction of the sky background light is one of the main causes of the
underestimated stellar mass (e.g., Von Der Linden et al. 2007; Bernardi et al.
2007; He et al. 2013). There have been several studies showing that the
estimated BGG & BCG stellar masses from SDSS photometry have been
systematically underestimated and that the bias becomes more severe with
increasing mass (Hill et al. 2011; Simard et al. 2011; Bernardi et al. 2013;
Gonzalez et al. 2013; Kravtsov et al. 2018). For example, Kravtsov et al.
(2018) reported an increased stellar mass at the high-mass end of the SMHM
relation when a careful treatment of background subtraction was applied.

###  A.2 Halo mass

Table 5:  The halo mass measuring methods of the observation references presented in Fig. 1 | Data |  | Method
---
Yang et al. (2008) |  | Halo mass function from
---
Warren et al. (2006) assuming
WMAP3 cosmology161616Spergel et al. (2007); $\Omega_{\rm m}=0.238$,
$\Omega_{\rm\Lambda}=0.762$, $h=0.73$, $\sigma{\rm 8}=0.75$.
| Abundance matching.
---
The ranking is based on (i) the total luminosity or (ii) the total stellar
mass of a
galaxy group.
Yang et al. (2012) |  | Halo mass function from
---
Warren et al. (2006) assuming
WMAP7 cosmology171717Komatsu et al. (2011); $\Omega_{\rm m}=0.275$,
$\Omega_{\rm\Lambda}=0.725$, $h=0.702$, $\sigma{\rm 8}=0.816$.
| Abundance matching.
---
Similar to Yang et al. (2008), with improved treatments on subhalos.
Moster et al. (2013) |  |  Halo catalogues from Millenium181818Springel et al. (2005)
---
& Millenium-II simulations191919Boylan-Kolchin et al. (2009)
converted to WMAP7 cosmology
| Abundance matching.
---
Kravtsov et al. (2018) | Chandra X-ray data |  | X-ray observations.
---
$Y_{X}\equiv M_{g}T_{X}$ is converted to the halo mass (see Kravtsov et al.
2006 for details).
L18 |  | Chandra X-ray data
---
XMM-Newton X-ray data
Weak lensing data in optical
|  For the CLoGS sample, X-ray system temperature $T_{\rm sys}$ is converted
to the
---
halo mass based on the scaling relations from Sun et al. (2009).
For the MENeaCS and the CCCP sample, weak lensing analysis is used
(Herbonnet et al. 2017; Hoekstra et al. 2015).
van Uitert et al. (2018) | KiDS survey |  | Weak lensing analysis.
---
Erfanianfar et al. (2019) |  | SPIDERS-CODEX
---
XMM-Newton
XMM-CFHTLSXMM-XXL
|  X-ray luminosity $L_{X}$ is converted to the halo mass based on the
redshift
---
dependent scaling relations from Leauthaud et al. (2010).
Girelli et al. (2020) |  |  Halo catalogues from the $\Lambda$CDM
---
DUSTGRAIN-pathfinder simulation
based on Planck 2015 cosmology.
| Abundance matching.
---

Table 5 summarizes how the galaxies’ host halo mass is measured in observation
studies presented in Fig. 1. One of the methods to derive the halo mass from
observables is to use X-ray observations assuming the X-ray emitting gas is in
hydrostatic equilibrium within the group/cluster potential. In Fig. 1,
Kravtsov et al. (2018), L18 BGGs, and Erfanianfar et al. (2019) used this
approach. Although the mass estimate resulting from this method has been shown
to be systematically biased (Mahdavi et al., 2013), the approach works well
for clusters and massive groups once the bias is accounted for. To correct the
bias, we scaled the halo mass of all the X-ray observation results by 1.25
when presenting them in Fig. 1. However, this method cannot, as of yet, be
straightforwardly applied to low-mass groups, because a significant fraction
of such systems tend to be X-ray under-luminous (Pearson et al., 2017;
O’Sullivan et al., 2017) and as a result do not have extended X-ray
observations on which to base the mass estimate. This brings up concerns that
systems which are X-ray bright might not be representative of the population
as a whole.

Alternatively, L18 BCGs and van Uitert et al. (2018) in Fig. 1 estimated the
halo masses using the weak gravitational lensing analysis. In terms of
reliably estimating the mass of an individual system, the approach works best
for massive clusters. Even then, the uncertainties in mass estimates for
individual systems is $\sim 15-25\%$ (Hoekstra et al. 2015) and these
uncertainties grow with decreasing halo masses.

For this reason, halo masses of lower mass systems are frequently estimated
using indirect methods that relate observed galaxies to their host halos. In
Fig. 1, Yang et al. (2008); Yang et al. (2012), Moster et al. (2013), and
Girelli et al. (2020) used this approach. One such method, the halo abundance
matching establishes a connection between observed galaxies and the halo mass
function predicted from analytic models or N-body simulations. Essentially,
more luminous galaxies are assigned to more massive halos. There are more
sophisticated approaches, such as halo occupation distribution (e.g., Berlind
& Weinberg 2002; Cooray & Sheth 2002; Tinker et al. 2005) and conditional
luminosity/stellar mass function (e.g., Yang et al. 2003; Cooray 2006), that
construct the connection between galaxies and halos using the number of
galaxies and the galaxy luminosity function in a given halo mass,
respectively. It is worth pointing out that the halo mass function is
dependent on cosmology. As a result, the halo mass assigned to galaxies via
abundance matching is subject to change depending on the choice of cosmology.
When comparing WMAP7 and Planck cosmology, Rodríguez-Puebla et al. (2016)
showed that their simulation based on Planck cosmology resulted in slightly
higher abundance of massive halos compared to the one based on WMAP7. However,
the variation in the SMHM relation is relatively small among cosmology models,
therefore, we omit the conversion between cosmologies when comparing results
presented in Fig. 1. Similarly in Yang et al. (2012), the authors compared the
SMHM relations derived from different cosmology models, namely, WMAP1, WMAP3,
WMAP5, and WMAP7. The differences in the resulting SMHM relations due to
cosmology are minor compared to the collective scatter in a compilation of
SMHM relations (see, for example, Fig. 10 of Coupon et al. 2015 or Fig. 9 of
Girelli et al. 2020.

The above references refer to several different definitions of the halo mass,
e.g., $M_{\rm 200}$, $M_{\rm 500}$. Conventionally, the subscript denotes the
overdensity ($\Delta$) and $M_{\rm\Delta}$ and $R_{\rm\Delta}$ are the mass
and radius of a sphere with the enclosed mass density equals to $\Delta$ times
the critical density ($\rho_{\rm crit}$):

$M_{\Delta}(r<R_{\Delta})=\frac{4}{3}\pi R_{\Delta}^{3}\Delta\rho_{\rm crit}.$
(2)

For a fair comparison between the results, we mapped the halo masses to
$M_{\rm 200}$, following the procedure described here. First, we calculated
the exact conversion factor that converts $M_{\rm\Delta_{1}}$ to
$M_{\rm\Delta_{2}}$, where $\Delta_{1}$ is the choice of overdensity in a
literature and $\Delta_{2}=200$ in our case. When assuming an NFW profile, the
radial density distribution of a system can be characterized by the scale
density $\rho_{0}$, the scale radius $r_{s}$, and the mass enclosed within
$R_{\rm\Delta}$. These are related as follows:

$M_{\rm\Delta}=4\pi\rho_{\rm 0}r_{\rm s}^{3}m(c_{\rm\Delta}),$ (3)

where $c_{\rm\Delta}\equiv R_{\rm\Delta}/r_{s}$ is a concentration parameter
and $m(x)=\ln(1+x)-x/(1+x)$. Combining equation (2) and equation (3), we get
the following relation between the two definitions of halo mass.

$\begin{cases}\frac{M_{\rm\Delta_{2}}}{M_{\rm\Delta_{1}}}=\left(\frac{c_{\rm\Delta_{2}}}{c_{\rm\Delta_{1}}}\right)^{3}\frac{\Delta_{2}}{\Delta_{1}}\\\
\frac{c_{\rm\Delta_{2}}}{c_{\rm\Delta_{1}}}=\left[\frac{\Delta_{1}}{\Delta_{2}}\frac{m(c_{\rm\Delta_{2}})}{m(c_{\rm\Delta_{1}})}\right]^{\frac{1}{3}}.\end{cases}$
(4)

Dutton & Macciò (2014) have shown that mass and concentration parameters are
tightly correlated. For the halo mass range of interest to us, we noticed that
the conversion factor ($M_{\Delta_{2}}/M_{200}$) correlates almost linearly
with mass in log scale. Therefore, we derived a linear conversion and used
this relation to scale a given halo mass to $M_{200}$.

###  A.3 Star formation rate

Table 6:  The SFR measuring methods of the observation references presented in Fig. 10 | Data |  | Method
---
Whitaker et al. (2012) |  NMBS202020The NEWFIRM Medium-Band Survey, Whitaker et al. (2011). |  |  UV and IR luminosities are converted to SFR using the relation from Kennicutt (1998)
---
(${SFR}_{\rm UV+IR}=0.98\times 10^{-10}(L_{\rm IR}+3.3L_{\rm 2800})$) assuming
Kroupa IMF.
The star-forming galaxies are selected based on $U-V$ and $V-J$ colours.
Mittal et al. (2015) |  | GALEX
---
2MASS
HST optical
SDSS DR10
| SFR is computed from SED constructed based on multi-wavelength data from UV
to IR.
---
SED fitting is performed using GALAXEV code (Bruzual & Charlot 2003) assuming
Chabrier IMF.
Gozaliasl et al. (2016) |  | COSMOS
---
CFHTLS
| SFR is computed from SED constructed based on ugriz-magnitudes.
---
SED template fitting is performed using LE PHARE code (Arnouts et al. 1999;
Ilbert et al. 2006)
and FAST code (Kriek et al. 2009) with Bruzual & Charlot (2003) SSP and
Chabrier IMF.
L18 |  | GALEX
---
2MASS/MIPS
IRAM 30m
| SFR is the sum of the UV- and IR-based SFRs measured separately.
---
For the BCGs, SFR is from Hoffer et al. (2012). For the BGGs, IR-based SFR is
from
O’Sullivan et al. (2015); O’Sullivan et al. (2018) and UV-based SFR is from
Kolokythas et al. (2021).
Cooke et al. (2018) |  | GALEX
---
Subaru optical
Vista
Herschel
| SFR is computed from SED constructed based on multi-wavelength data from UV
to IR.
---
SED fitting is performed using iSEDfit code (Moustakas et al., 2013;
Moustakas, 2017) and
Bruzual & Charlot (2003) SSP templates assuming Salpeter IMF

Table 6 summarizes how the star formation rate is measured in observations
presented in Fig. 10. The star formation rate estimations of Whitaker et al.
(2012), Mittal et al. (2015), Gozaliasl et al. (2016), and Cooke et al. (2018)
are based on SEDs constructed using infrared and/or optical magnitudes. The
star formation rate of L18 BCGs are from Hoffer et al. (2012). Similarly for
L18 BGGs, we take the total SFR to be the sum of the obscured and unobscured
SFR (i.e., IR-based star formation rates from O’Sullivan et al. 2015;
O’Sullivan et al. 2018 and UV-based star formation rates from Kolokythas et
al. 2021); many of these galaxies are lacking IR observations and
consequently, their SFRs are strictly speaking, lower limits.