PROXI: Challenging the GNNs for Link Prediction
Authors: Astrit Tola
Jack Myrick
Baris Coskunuzer
Venue: Accepted by TMLR ()

Abstract: Over the past decade, Graph Neural Networks (GNNs) have transformed graph representation learning. In the widely adopted message-passing GNN framework, nodes refine their representations by aggregating information from neighboring nodes iteratively. While GNNs excel in various domains, recent theoretical studies have raised concerns about their capabilities. GNNs aim to address various graph-related tasks by utilizing such node representations, however, this one-size-fits-all approach proves suboptimal for diverse tasks.

Motivated by these observations, we conduct empirical tests to compare the performance of current GNN models with more conventional and direct methods in link prediction tasks. Introducing our model, PROXI, which leverages proximity information of node pairs in both graph and attribute spaces, we find that standard machine learning (ML) models perform competitively, even outperforming cutting-edge GNN models when applied to these proximity metrics derived from node neighborhoods and attributes. This holds true across both homophilic and heterophilic networks, as well as small and large benchmark datasets, including those from the Open Graph Benchmark (OGB). Moreover, we show that augmenting traditional GNNs with PROXI significantly boosts their link prediction performance. Our empirical findings corroborate the previously mentioned theoretical observations and imply that there exists ample room for enhancement in current GNN models to reach their potential.

Published in Transactions on Machine Learning Research (03/2025)
PROXI: Challenging the GNNs for Link Prediction
Astrit Tola
astrit.ool@@utdallss..cc eduu
Deportment of Mathemationl Sciences University of Tezas at Dallas
Jack Alec Myrick
juck..yrrcck@uuuullll... edu
Department of Computer Science University of Tezas at Dallas
Baris Coskunuzer
coskunusuutdallas..ee edu
Deportment of Mathematical Sciences
University of Tezas at Dallas
Reviewed OpenReview: https: opwprevieeeeveuueee.mmmee net/ forum? id= ugEHhdbb
Abstract
Over the past decade, Graph Neural Networks (GNNs) have transformed graph represenlation learning. In the widely adopted message- message-passing GNN framework, nodes refine their repre- sentations by aggregating information from neighboring nodes iteratively. While GNNs excol in various domains, recent theoretical studies have raised concerns about their capabilities. GNNs aim to address various graph-related tasks by utilizing such node representations, however, this fittssl approach proves suboptimal for diverse tasks.
Motivated by these observations, we conduct empirical tests to compare the performance of current GNN models with more conventional and direct methods in link prediction tasks. Introducing our model, PROXI, which leverages proximity information of node pairs in both graph and attribute spaces, we find that standard machine learning (ML) models perform competitively, even outperforming cutting-edge GNN models when applied to these proximity metrics derived from node neighborhoods and attributes. This holds true across both homophilic and heterophilic networks, as well as small and large benchmark datasots, including those from the Open Graph Benchmark (OGB). Moreover, we show that augmenting traditional GNNs with PROXI significantly boosts their link prediction performance. Our empirical findings corroborate the previously mentioned theoretical observations and imply that there exists ample room for enhancement in current GNN models to reach their potential. Our code available at https: //tg//gtttt//t/// com//vorkrrep22222/PXXXX
1 Introduction
In an era characterized by the complex web digital connections, understanding and predicting the formation of links between entities in complex networks has become a crucial challenge. Whether it's predicting social connections online social networks, anticipating collaborations between rescarchers, or forccasting potential interactions in recommendation systems, link prediction has emerged a fundamental task in graph representation learning. With the ongoing evolution of our societies and technologies, the significance of link prediction amplifies, considering its capability to refine broad spectrum of ioplications-rangeee from personalized content suggestions to strategically targeted marketing.
Over the past decade, Graph Neural Networks (GNNs) have achieved a significant breakthrough in graph representation learning (Wu et al., 2020). Despite their consistently superior performance compared state-of fhate-of-the-art (SOTA) results, seminal papers like Xu al. (2019); Morris al. (2019); Li & Leskovec (2022) have shown that the expressive capacity of message-passing GNN (MP-GNNs) models is not better

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than decades-old Weisfeiler-Lehman algorithm. Furthermore, Loukas (2020a;b); Sato (2020); Barceló et al. (2020) theoretically showed that MP-GNNs representational power is limited. These theoretical discoveries suggest that existing GNN models may not be fully leveraging their potential to integrate information from local neighborhoods and domain-specific knowledge when learning node embeddings, a fundamental process in the creation of node representations.
GNNs face another challenge with their one-size-fits-all approach. They employ range of methods and architectural designs to generate robust node embeddings, merging neighborhood information with node attributes. These embeddings are then tailored to different graph representation learning tasks such as node classification, graph classification, or link prediction by adjusting the prediction head and loss function. While this approach is straightfowwar for node classification, it is highly indirect for graph classification and link prediction tasks (Zhu et al., 2024; Zhang et al., 2021). In link prediction, for instance, understanding the "proximity" between given node pairs is crucial to determining whether they will form a link. Common GNN algorithms achiove this by first learning node representations, h and h in a latent space RT,, and then measuring 'the distance" between these representations, e-g., h h However, this method is not practically viable for link prediction. One reason is that distance is transitive operation, while link formation is not. For example, having links between node pairs u ~ v and v ~ does not guarantee a link between u ~ w. Yet, the proximity of h ~ he and h, ~ ha implies the proximity of h, ~ h due to the triangle inequality (Section 3.3). Hence, a more suitable approach for GNNs would be to directly learn the prediction heads for the tasks (e.g., node pair representation), bypassing the reliance on individual node embeddings.
In graph representation learning, homophily refers to the tendency of nodes with similar features or labels to be connected, reflecting the principle of like attracts like. In contrast, heterophily describes networks where nodes with dissimilar features or labels are more likely to connect, highlighting diversity in connections (See Section 3.1). Most GNNs, however, are designed based on the homophily assumption, where edges typically link nodes with similar attributes or labels, as commonly observed in citation networks. This homophilic bias poses a significant limitation in real-world applications, where hetcrophilic behavior is provalent, such as in protein interaction and web networks. In these settings, traditional GNN models (Hamilton el al., 2017; Klicpera et al., 2019) often suffer from substantial performance degradation as shown by recent studies (Zhou et al., 2022; Zheng et al., 2022; Pan et al., 2021; Zhu et al., 2020; Luan et al., 2022).
In this paper, motivated by these, we aimed to lest the performance of GNNs against conventional methods in link prediction tasks. First, we introduce simple yet very direct ML model: PROXI, which combines all relevant proximity information about node pairs. We approach the link prediction problem simply as a binary classification, whether a node pair will form a link or not. Our method is a fusion of two types of proximity information: structural prozimity, capturing the proximity of the node pair within the graph structure, and domain prorimity, measuring their similarity within the attribute space. These indices provide direct embeddings of node pairs encoding the proximity information. Through the integration of these spatial and domain proximities with conventional ML methods, models demonstrate highly competitive results compared to state-of-the-art GNNs across diverse range of datascts, spanning both homophilic and heterophilic settings. We further show that integration of our PROXI model significantly enhances the performance of conventional GNN models in link prediction tasks.
Our contributions:
o We propose a scalable ML model PROXI for link prediction task, by adeptly merging the local neighborhood information and domain-specific node attributes.
With only 20 indices, our model consistently achieves highly competitive results with SOTA GNN models in benchmark datasets for both homophilic and heterophilic settings. Hence, our simple model provides critical bascline for new GNNs in link prediction.
o Our PROXI indices can casily be integrated with existing GNN models, leading to significant performance improvements up to 11%
о Our results support recent theoretical studics, indicating that current GNNs may not be significantly better than traditional models. This underscores the need for novel approaches to unlock the full capabilities of GNNs.
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2 Related Work
2.1 GNNs for Link Prediction
Much of the recent work on link prediction has been using GNNs. Most GNNs follow the gressage-passing framework, in which node's representation is learned through aggregation operation, to pool local neighborhood attributes, and an update operation, which is a learned transformation (Guo et al., 2023). Another common framework is the encoder-decoder framework in which the encoder learns node representations and the decoder predicts the probability of a link between two nodes (Guo et al., 2023). There are four main groups of GNNs currently used (Wu et al., 2020); recurrent GNNs (RecGNN) (Dai et al., 2018), convolutional GNNs (ConvGNNss) (Chiang et al., 2019), graph autoencoders (GAEs) (Bojchevski et al., 2018), and spatial temporal GNNs (STGNNs) (Guo et al., 2019).
In the link prodiction task, GNNs have shown outstanding performance in the past decade (Zhang, 2022; Zhu et al., 2021; Wang et al., 2023; Srinivasan & Ribeiro, 2020; Rossi et al., 2021). Zhao et al. (2022) use counterfactual links as a data-augment augmentation method to obtain robust and high-performing GNN models. In Yun et al. (2021), the authors applicd novel approaches to improve learning structural information from graphs. In Zhu et al. (2021), the authors integrate the Bellman-Ford algorithm for path representations to their GNN model and obtain competitive results in both inductive and transductive settings. Wu et al. (2021) proposed an effective similarity computation method by employing a hashing technique boost the performance of GNNs in link prediction tasks. Liu et al. (2022) developed high-performing GNNs for dynamic interact ion graph setting. There is overwhelming erraure GNNs for link prediction in the past few years, and a comprehensive review of these developments can be found in Zhang (2022); Liu et al. (2023); Wu et al. (2022).
Note that the majority of the GNN models above rely on the homophily assumption and demonstrate suboptimal performance in heterophilic networks (Zhu et al., 2020). Therefore, over the recent years, several efforts have been made to create GNNs that perform well in heterophilic networks (Zhou al., 2022; Zheng et al., 2022; Pan et al., 2021; Luan et al., 2022).
Although this paper focuses on the link prediction task, we note that recent studics have demonstrated the competitive performance of traditional methods compared to GNNs in graph classification (Loiseaux el al., 2024; Chen et al., 2024) and node classification (Uddin et al., 2024: Luan et al., 2023a). These findings highlight significant opportunities for advancing GNNs through innovative approaches.
2.2 Brase Methods for Link Prediction
Before the GNNs, many of the common machine learning models often relied on feature engincering methods as primary approach (Kumar et al., 2020; Menon & Elkan, 2011). One of the simplest approaches to link prediction is through proximity-based methods (Lü & Zhou, 2011), including local proximity indices (Wu et al., 2016), global proximity indices (Jeh & Widom, 2002), quasi-local indices (Liu & Lü, 2010), the local path index (Lü et al., 2009) and its extension (Aziz et al., 2020), and the Katz index (Katz, 1953; Coşkun et al., 2021) In this case, the graphs are mostly assumed to be homophilic, and more similar nodes are deemed more likely to have link.
Most of the former proximity metrics for link prediction can be categorized as local proximity indices. Let S(u, v) denote a similarity score between two nodes u and v, let NN) denote the set of neighbors of and let kn denote the degree of u.
# Common Neighbors is the size of the intersection between two nodes' neighbors (Newman, 2001). This is equivalent to the number of paths of length 2 between two nodes, More common neighbors indicate a higher likelihood for a link. CA u) (1) (N)(
Juccard Coefficient is a normalized Common Neighbor score (Jaccard, 1901), i.e., the probability of selecting [N(w)W(t)) a common neighbor of two nodes from all neighbors of those nodes. J(u, [M())
o Salton Inder (cosine similarity) measures similarity using direction rather than magnitude (Singhal et al., ((( Vkakk
2001). Sa(u,v)
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Sørensen Inder was developed for ecological data samples (Sorensen, 1948), and it is more robust than W kuth.
Jaccard against outliers (McCune Grace, 2002). So(u, v))
Adamic Adar Inder measures the number of common neighbors between two nodes weighted by the inverse logarithm of their degrees (Adamic & Adar, 2003). It is defined as AA(u, -weNtwnwor log Aul
We use these similarity metries as our struclural prozimily indices for node pairs. We then effectively combine them with our domain prorimity indices, which is the relevant similarity measure between the node attributes, to complete our proximity indices for our ML model.
3 Methodology
3.1 Problem Statement
Link prediction problems in graph representation learning can be categorized into different types based on the nature of the problem and the availability of information during the prediction process. Three common types are transductive, inductive, and immi-inductive link prediction.
Let G = (V, X²) be a graph, where V= f= fui, , Un represents the set of nodes, & C V x V represents the set of edges and X' represents the attribute feature matrix (n m size) where X, E Rm the m-dimensional attribute feature vector of node v,- For the sake of simplicity, we focus on unweighted, undirected graphs, but our setup can easily be adapted to more general settings.
The main difference between these types comes from the availability of the information during the prediction process. We split the vertex and edge sets as observed (old) and unobserved (new) subsets, i.c. V = V. U V. and & = &, U E Hence, in the training process, we are provided G. = (V., Eo) information, and we are asked to predict the existence determined with respect to which subsets (i.c., Vo or Vu) these node pairs are chosen from:
link in E, for a given node pair in V. However, the type of the problem is
Transductive Setting: Predict whether eij E E. where Un Uy V..
Semi-induclive Selling: Predict whether eij E & where us Uj E Vo U V.-
Inductive Setting: Predict whether ei E & where U/ Vu- No local structure information is provided, only attribute vectors {X:} are provided for vi E Vu.
While, in the literature, the most common type is transductive setting, depending on the domain, the relevant question can come in any of these forms. To maintain focus in this paper, we align with the prevalent transductive setting, consistent with most contemporary GNN models. It important to note, however, that our proposed ML model exhibits a high degree of versatility and can casily adapt to any of these scttings.
Heterophily. Before presenting our model, we would like to recall the concepts of homophily and heterophily. Informally, homophily describes the tendency of edges in a graph to connect nodes that are similar, while heterophily describes the opposite tendency. Formally, homophily heapphilyyhheerooe hheteropheerryyyyeeeo is determined by the homophily ratio as follows: Let G = (V.E) be a graph with C: V - (1, 2,., node v let n(v) be the number of adjacent nodes with the same class, and let deg(v) denote the degree of node n(v)
......,N N} representing node classes. For each
v. Then, the node homophily ratio of G is defined as: Hg) Y&d EvEv deg(oy- By definition, Hg) E [o. 1] for any graph G. A common convention is that a graph & with (G) 2 0.5 is called homophilic, and otherwise heterophilic. Another important homophily metric especially for link prediction task is the edge homophily ratio, which is the proportion of edges connecting nodes in the same class to all edges in the graph.
[ where E represent the edges connecting same class nodes. Recently, various new metrics were introduced to study the homophily in graphs (Tuan et al., 2023b; Zhu et al., 2020; Jin et al., 2022).
i.e., He (G) = T
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3.2 PROXI for Link Prediction
In the following, we give the details of our simple proximity-based model, PROXT. Our primary aim is to furnish our ML classifier with comprehensive set of relevant and potentially valuable information about node pairs. In the domain of graph representation learning, most configurations offer two types of information about nodes. The first concerns the overall graph structure, providing spatial information about their local neighborhoods. The second involves node attributes derived from the specific problem domain, e.g. keywords of a paper in citation networks.
Essentially, the link prediction problem boils down to determining whether node pair (u, is poised to initiate interaction or not. Thercfore, their structural proximity and domain proximity (shared interests) play key roles in this decision-making process. In our model, intentionally opt to allow the classifier the discretion to determine which indices to leverage, depending on the dataset and the setting (e.g., homophilic or heterophilic) at hand. While existing literature typically employs these indices individually or in pairs, our intuition leads us to believe that by aggregating all of this information, the ML classificr can make finer determinations within the latent space. Furthermore, these indices can synergistically reinforce each other, resulting a more robust and accurate model.
In this context, our indices can be categorized into two distinct types: Structural Indices and Domain Indices. Structural indices draw upon the inherent graph structure and neighborhood information, while domain indices leverage proximity measures derived from the node attributes provided. We note that some of our indices have been integrated in earlier GNN modes like BUDDY and Labeling Trick, where we discuss them in Appendix B.1.
Novel Structural Proximity Indices for Node Pairs (u,e). Within our PROXT model, we incorporate range of structural indices. While subset of these indices corresponds to established similarity indices detailed in Section 2.2, along with their generalizations, we also introduce novel proximity indices. These newly defined indices are designed to capture finer insights from the local neighborhoods of node pairs.
The established indices are the Jacard, Salton, Sørensen, and Adamic Adar indices (Section 2.2). These are known to capture triadic closure information in networks, which is a key signature for link prediction problem (Huang al., 2015). Furthermore, for Jacard, Salton, and Sørensen indices, we use their natural generalizations for 3-ncighborhood versions as well new indices like length-k paths and distance index.
# Length-k paths: For a given u, E V, we define length-k paths inder Cu v) as the total number of length-k paths between the nodes u and v. Notice that length-2 paths index is the same with common neighbors index, i.c.,
Ca(u, v) = CM((,t)=
Graph Distance: For given u, U E V, we define distance indez D(u, v) as the length of the shortest path between and in G. Note that when computing D(u, v), we remove the edge between and from the graph if u and are adjacent nodes. Therefore, D(u, v) 2 2 for any u + V. The main motivation to define this index in this particular way is that in the test set, a priori, there won't be an edge between the nodes, Therefore, during the training ML classifier, this distance index provides valuable information to the ML classifier to distinguish positive and negative edges when combined with other indices.
JZ v) a natural generalization of the original Jaccard Index by using Ls(u, v) the length 3-paths instead of C2(u, v) length-2 paths.
3-Jaccard Inder:
La(u.) (u, = Wa
By using a similar idea, we implement a comparable adaptation to Salton and Sørensen indices:
3-Salton Index: By generalizing original Salton index to neighbooroos,, we introduce 3-Salton index: (u, Ls(u,
ki
3-Sprensen Inder: Similarly, by generalizing original Sorensen index to 3-neighborhoods, we introduce 2Lz(u, v)
3-Sorensen index: (3, he + ho
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Hence, for node pair u. E V. produce 10 structural PROXI indices as follows (u v), Sq AA(u, v), (2(u, u), u) and v(u,v).
[u,
Novel Domain Proximity Indices for Node Pairs (u, v). Next, we describe our domain indices. Contrary to our structural indices, our domain indices do not use graph structure, but only the node attribute vectors. For a given graph with node attributes 9 = (V, let X E R represent the attribute vector for the node u E V. In the following, for a given node pair u, E V, we extract our domain indices a(u, v) by using the similarity/ / /issimilarity of these node attribute vectors X. and X
While our overarching argument to produce the domain proximity indices is the same, we adapt our approach to accommodate various formats of node attribule vectors {Ka). These formats can generally be categorized as follows.
i. X, is binary vector: All entries X E {0,1}
In this case, we naturally interpret this as every binary digit in the vector X, represents the existence or nonexistence of property. For example, if G represents citation network, where nodes represent papers, X. can be binary vector representing the existence nonexistence of previously chosen keywords in the paper u. We define two similarity measures between X,, and X.
o Common Digils: X,, is a binary vector, we define our Common Digils domain index CD(u, v) as the number of matching '1's in the vectors X, and X i... CD(u, = #(i | = X = 13. similarly, one can define an analogous index as the number of common '0's to emphasize the common absent properties.
Normalized Common Digils: This domain index is a slight variation of the previous one with some normalization factor. In particular, if X, and X, have only few positive digits in their vectors, having equal nunber of common digits would result in them being considered more similar, in contrast to node pairs with numerous positive digits. We normalize this index by dividing it by the total number of positive digits #
in both vectors X, and X, (not counting the common positive digits twice). CD(u,)) = #U|(x.-x.> Notice that the vector X,, + X, would have only 0, 1, and 2 digits where 2s represent the common positive digits in X, and X.. i.e., CD(u, v) = # (X.. + xx))² = 2}
ii X takes finitely many values: X E (1.
In this case, make the assumption that a particular entry X of the vector X, can assume finite set of distinct values, such as X 21....... m. In such cases, we interpret this specific entry representing some sort of "node class information' of particular node attribute of u. Notably, if node classes are provided in the data, we take this information into account within this context. For instance, in citation networks, this data could correspond to the academic field of the paper (e.g,, Mathematics, Computer Science, History) as an attribute of the node. Within this category, we establish two distinct domain indices.
Class Identifier: CI(u, v) is an m-dimensional binary vector to identify the classes of u and v. i.c., if X and if X tt, then we define CI(u, v) as an m-dimensional binary vector with all entries are except tth and th entries, which are marked 1. Tf u and u belong to the same class (s t), then we CI(u, v) is all zeros except 9th entry.
10000. 0j. If u and For example, if there are 5 classes (m = 5), and X., = 2 and X. = 4, we have CI(u, v) = [0 1 1100100 v are in the same class, say X,, = X, 3, then we have CI(u, v) 000100.
Common Class: MC(u,v) u) is one-dimensional binary index that is one if and L are equal and zero otherwise.
ili. X, is a real-valued vector: X, E IR
When X., is represented as a reauevalued vector, it inherently serves as a node embedding within the attribute space R. Therefore, the similarity or dissimilarity between two nodes is intuitively associated with the distance between these embeddings. We employ two distinct types of distance measurements as domain indices.
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L'' Distancc: Simply, use L' -nom (Manhattan metric) in the attribute space R" Xf, [aq 02 ... am and X, bh b2 bml, we define
(1(,,)) = d(x,,,X.) >x a, bil
Cosine Distance: Another popular distance formula using some normalization is the cosine dis- tance/similarity. We define our cosine distance index as
D)(u,v) X. X X
Note that the node attributes may result from a combination these three calegories. such instances, we incorporate all of them by dissecting the node attribute vector based on their respective subtypes and acquiring the corresponding domain attributes for the node pairs.
3.3 Non-Transitivity of Link Prediction Problem
In this part, we will speculate about the challenges associated with using node representations for link prediction tasks. After generating node representations (hu}, the goal of the prediction head is to leverage these embeddings to determine whether given pair of nodes should form a positive negative edge. Typically, the prediction head employs similarity measure between node representations, implying that if two nodes have similar representations, they should form a link ((u, v) + + "); otherwise, they should not ((u, u) - " "). However, most similarity measures inherently adhere to some form of triangle incquality, such as cosine similarity.
cos(h,, hw) cos(hy, ho) cos(h.. bw) v(1 cos(hu; h2) (1 -cos(he. )w
In such cases, similarity measure becomes inherently transitive, i.e., if h, h, and h. h then h. h also holds (~ for similar). For instance, cos(h,, ho) is close to means h, ~ he, and cos(ha: h,) is close to -1 means h, and h," are highly dissimilar. Given these properties, by triangle inequality above, if both cos(h,, hy) and cos(h,, h are close to 1, then so is cos(h,, h.).
However, the link prediction task is generally not transitive. Specifically, u and v ~ w, does not necessarily imply that w. In Table we present the transitivity ratio for various benchmark datasets used in link prediction. We define the transitivity ratio as the proportion of node pairs (u, w) that have edge between them out of all node pairs that share at least one common neighbor v. The data shows that the link prediction task is highly non-transitive, as most node pairs with a common neighbor do not form a link. Consequently, using any distance or similarity-based operation as a prediction head can significantly hinder the performance of GNNs.
This why using individual node representations to represent node pairs not optimal as they can bring restrictions for GNNs. Instead, learning representations of node pairs (or edge representations) can be more effective for link prediction tasks. From this perspective, our approach, PROXI, can be seen direct node pair embedding where the coordinates are individual proximity indices.
For further discussion on the effect of similarity-based decoders, we refer to recent studies (Zhu et al., 2024; Cho, 2024). Furthermore, in Wang et al. (2022a), the authors evaluated the performance of various decoders for GNNs and demonstrated that models using similarity-basok prediction heads (c.g., dot product) exhibitod suboptimal performance. Their comparison table can be found in Table 12 in our appendix.
Table 1: Transitivily ratios of the datasets
TEXAS
WISC.
CHAM.
SQUIR. CROC.
CORA
CITESEER
PHOTO
0.046
0.049
0.009
0.002
0.002
0.092
0.145
0.010
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4 Experiments
4.1 Experimental Setup
Datasets. In our experiments, we used twelve benchmark datasets for link prediction tasks. All the datasets are used in the transductive setting like most other baselines in the domain. The dataset statistics are given in Table 2. The details of the datasets are given in Appendix A.
Experiment Settings. To compare our model's performance, we adopted the common method proposed in Lichtenwalter et al. (2010) with 85/5/10 split for all datasets except OGB datasets which come with their own predefined training and test sets. To expand the comparison baselines, we also report the performance of our model with different. split 70/10/20 in Table 9 in the Appendix.
Proximity Indices. In all datasets except OGBL-COLLAB, we used the same proximity indices described in Section 3.2. Since OGBL-COLLAN is dynamic and weighted, we needed to adjust our domain proximity indices to adapt our method to this context. The total number of indices patee///areeerrsss used for each dataset is given in Table 11. We gave the details of these proximity indices for each dataset in Appendix C.
Metrics. In all datasels, we used the common performance metric, AUC. In addition, in OGBL datasets and second split 70/10/20, to compare with the recent. baselines, we further used Hits@K metric which counts the ratio of positive links that are ranked at K-place or above, after ranking each true link among a set of 100,000 randomly sampled negative links (Hu al., 2021b).
Hyperparameter Settings. In our study, XGBoost acts the primary machine learning tool. The optimization objective is defined rank: pairwise, with logloss operating the evaluation metric. When assessing outcomes through the AUC metric, configure the maximum tree depth of 5, the learning rate varying in [0.01, 0.05], the number of estimators at 1000, and the regularization parameter lambda set to 10.0. For the more demanding metric, Hits&20, modifications are implemented. Notably, the maximum tree depth is upgraded to 5, the colsample bytree ratio is adjusted to 1, the learning rate is set to 0.1, and lambda is set to 1.0, while other parameters remain consistent. with those utilized for the AUC metric. Similarly, for the Hits@50 metric, the maximum tree depth is reset 11, the learning rate is incrcased to 0.5, and lambda is set to 1.0, with all other hyperparameters aligned with those used for the AUC metric. Lastly, in the context of the Hits@ 100 metric, adaptions are made within the AUC hyperparameter setting. These adjustments encompass changing the maximum tree depth to 5, setting the learning rate to 0.3, adjusting the subsample ratio to 0.5, and the colsample bytree ratio to 1.0, while lambda is maintained at 1.0.
Characteristics of our benchmark datasets for link prediction. FV Type represents the type of the Table 2:
node attribute voctor provided.
Datasets
Nodes
Edges
Classes
Attributes
FV Type
N. Hom.
E. Hom.
CORA
2,708
5.429
1,433
Binary
0.83
0.81
CITESEER
3,312
4,732
3,703
Biuary
0.71
0.74
PUBMED
19,717
44,338
500
Binary
0.79
0.80
PHOTO
7,650
119,081
745
Binary
0.77
0.83
COMPUTERS
13,752
245,861
767
Binary
0.82
0.78
OGBL-COLLAB
235,868
1,285,465
128
Real
OGBL-PPA
576,289
30,326,273
TEXAS
183
295
1,703
Binary
0.09
0.41
CORNELL
183
280
1,703
Biuary
0.39
0.57
WISCONSIN
251
466
1,703
Binary
0.15
0.45
CHAMELEO
2,277
31,421
2,325
Binary
0.25
0.28
SQUIRREL
5,201
198,493
2,089
Binary
0.22
0.24
CROCODILE
11,631
170,118
500
Biuary
0.25
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Table 3: Link Prediction Results. AUC results on benchmark datasets with baseline performances reported by Zhou al. (2022). Additional performance comparisons are given in Table 9 in the Appendix.
Models Hom.
TEXAS
WISC.
CHAM.
SQUIR.
CROC.
CORA
CITESEER
PHOTO
0.35
0.30
0.72
0.77
CN (Newman, 2001)
53.0 0..22
61.3=00.9
95.340.66
96.840.4
89.8
75.7=1.3
69.7
97.1+0.4
AA (Adamic Adar, 2003)
63.1
61.5#008
95.8+00.6
97.1+0.4
90.6
1..
69.8
97.4
VGAE (Alemi et al., 2016)
68.6
71.3++4.66
98.5++0.1
98.2+
98.8
96 0.2
97.3=002
94.9+0.8
SEAL (Zhang & Chen, 2018)
73.911.
72.3=2277
99.510.1
90. 101
97. 07.0=0.5
ARGVA (Pan et al., 2018)
67. 6.111
67. 3..00
96.5.10.2
93, 33.03:0.3
94.1=0.5
92.7=113
94.8
89.742.3
DisenGCN (Ma et al., 2019)
72.1
75.1=33.4
97.7+001
94.5+:02
96.4=004
96 0.33
96. 820.2
95.2
FactorGNN (Yaug ct al., 2020)
58.7=4411
68 18.80
98.3++033
96.92004
07.6200.4
92.3=14
87 3.36
97. 97.2+00.1
GPR-GNN (Chien et al., 2020)
76.3+225
80.1+4.55
98.7+0..1
96.0s:0.3
96.7#:0.1
94.8=00.3
96.0=00.3
97.0+:0.2
FAGCN (Bo et al., 2021)
68.7+7.3
73.7+44.9
93.8+2..6
94.8
95.3#+0.2
91.8=334
89.2#55.6
94.0+11.
LINKX (Lim et al., 2021)
75.8 8::.1
80.143.8
98.810011
98. 11:03
99.0.:0.1
93 4...3
93.5200.5
97.02:022
VGNAE (Ahn & Kim, 2021)
78 78.913.0
70.3
95.410.2
89. 89.41:.11
90.52:0.4
89.2=0.7
95.620.6
79.51:0.2
DisenLink (Zhou et al., 2022)
80.7+4.0
84.4#1.9
99.4+0.0
98.3+:0.1
99.1=001
97.1:0.4
98.3+1..8
97.9#0.1
VDGAE (Fu al., 2023)
81.3=28.5
85.0s-4.8
96.8+0.1
95. 8
93.9#0.2
95.9=0.4
97.8=003
94.7#:0.
PROXI
84.6++2.4
85.8+22.2
98.8+0.11
98.3:0.0
99.6+00.0
95.8=0.6
95.7=0066
99.1+0.1
Implementation and Runtime. We experiments single machine with 12th Generation Intel Core 17-1270P vPro Processor (E-cores up to 3.50 GHz, P-cores up to 4.80 GHz), and 32Gb of RAM (LPDDR5- 6400MHz). End-to-end runtime (computing proximity indices and ML classifier) for OGBL-COLLAB is 30 minutes. Similarly, the runtime is 15 minutes for COMPUTERS (250K nodes) for all indices. The computational complexitios of our similarity indices are O(1) (A9) where [V] is the total number nodes and & is the maximum degree in the network (Marlínez et al., 2016). By utilizing the powers of adjacency matrix, we would significantly reduced the computational time. We would like thank the reviewer for this insightful suggestion. We provide our code at attps://githhe comm//orrrepp2232/P000
4.2 Results
Baselines. We compare the link prediction performance of our model PROXI against the common embedding methods and GNNs. The baseline performances are reported from the references indicated in the captions of Tables 3, 4 and For OGB-COLLAB, we used the baseline performances from Li et al. (2023) and further reported the performance of the current leader (Wang et al., 2022b) at OGB Leaderboard (Hu et al., 2021b).
Results. We give our results the Tables 3, 4 and 9. We have five heterophilic and three homophilic datasets. In Table 3, we observe that our PROXI model outperforms SOTA GNN models in four out of five
Table 4: Performances for OGBL datasets with baselines by (Li et al., 2023).
OGBL-COLLAB
OGBL-PPA
Models
Hits@50
AUC
10t
AUC
MF (Menon & Elkan, 2011)
11.81=167
83.7511.77
28.40.2
99.46:20.10
N2Vec (Grover & Leskovee, 2016)
49.06=1.04
96.2410.15
26.2440.96
99.77=0.00
GCN (Kipf & Welling, 2016a)
54.96=3.18
97. 07.80+0.06
29.57+:2.00
99.8420.03
GAT (Veličković et al., 2018)
55.00=3.28
97.1110.09
OÓM
00M
SAGE (Hamilton et al., 2017)
1..44=1.37
98.08+0.03
41.02+1.94
99.82=0000
SEAT (Zhang & Chen, 2018)
63.37
95.65
48. 40.5.66
99.79=0.02
Neo-GNN (Yun et al., 2021)
66.13
98.23-10.05
48. 1.0
97.30
BUDDY (Chamberlain et al., 2022)
64.59
96.52+:0.0
47.33
99.56=0002
NCN (Wang et al., 2023)
63.86=10.11
97.83
6..63
09.95
NCNC (Wang et al., 2023)
1..
98.20
62.6110.76
99.9710.01
OGB Leader (Hu et al., 2021b)
71.29:10.18
1..2
PROXI
76.50=0.27
97.24+0.03
50.36+0.76
99.90-0.00
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Published in Transactions on Machine Learning Research (03/2025)
heterophilic datasets. Similarly, in homophilic setting, while outperforming SOTA in one dataset, it gives very competitive results in the other two.
Table 5: AUC results comparing GNN models w/ w/out
In OGBL datasets (Table 4) PROXI outperforms all SOTA GNNs and OGBL leader in the OGBL-COLLAB dataset for the OGB metric Hits550. In OGBI-PPA, the AUC performance of PROXI stands shoulder shoulder with the best-performing models.
PROXI integration on the CORA and TEXAS datasets.
Dataset
Backbone
GNN
PROXI-GNN
Up+
GON
87.71+22.6
02.90-0.92
5.19
CORA
GAT
86.86+1.78
94. 5.-666..87
7.70
SAGE
17.67
95. 95.11=1.32
7.14
Finally, in our second split for comparison with addi- tional baselines (Table 9), PROXI again outperforms SOTA GNNs in two out of three homophilic datasets, while giving highly competitive performance in the third one.
LINKX
93.40++0.30
93.57
0.17
GCN
63.34.16.63
75.35
11.01
TEXAS
GAT
65.65-+3.76
76.84-4.52
11.19
SAGE
09.58:47.14
77. 17.206
7.52
LINKX
75.80=44.70
79. 79.59=5.33
3.79
While our results show that GNNs are not significantly better than traditional models, a natural question is how use PROXI indices with current ML models. As discussed before, the link prediction task is a binary problem in the space of node pairs, and providing information on node pairs is very crucial to getting elfective prediction heads. We took this path, and concatenate our proximity indices {a(u, v)} through MLP to prediction heads h. @ h (Hadamard product) of classical GNN models (See Appendix B for details). Our results are outstanding as we see consistent major improvements for both homophilic and heterophilic settings (Table 5).
PROXI-GNNs.
1.
1.2
10
0.6
6.8
38-
196
29
100
epoch
epoch
epoch
(a) GON
(b) GAT
(c) SAGE
Figure 1: PROXI-GNN, Performance comparison of Vanilla-GNNs (orango) and PROXI-GNNs (blue) TEXAS dataset.
Ablation Study, We made two ablation studics on our method. In the first one, we evaluated the performances of our structural and domain indices alone (Table 6). In both homophilic and heterophilic settings, the outcomes are varied: we observe that one set of indices is not better than the other in general. Nonetheless, notable consistent finding across all datascts is the synergistic effect of these indices. Their combination consistently improves overall performance.
In the second ablation study, we evaluate the performance of various ML classifiers using PROXI indices (Table 8). The comparable results across all classifiers demonstrate that our PROXI indices are both highly effective and model-agnostic.
Index Importance Scores. In Table 7, we present the importance of individual proximity indices in our model across various datasets. We obtain index imporlance scores through XGBoost's 's built-in function: feature importance. Our index set proves to be highly versatile, effectively adapting to the unique characteristies of each dataset. Notably, certain indices exhibit substantial importance in specific datasets, while their
Table 6: Structural vs. Domain Indices. AUC results for our model for different proximity index subsets.
Indices
CORA
CITESEER
PUBMED
PHOTO
COMP.
TEXAS
WISC.
CHAM.
SQUIR.
CROC.
Structural only
88.60
81.33+1.21
92.60+0.31
93.48+0.15
98.60=0.08
68.51:3335
72.92+3.35
97.46=0.19
97.86-0006
09.00=0.0
Domain only
91.10:n.94
02.7740.80
87.40)40.34
92.98e0.16
88.97=0.14
81.09=22
81.K36222
40.44:0.7
82.97-0.18
92. 73:..000
All Indiees
95.8310..0
95.71100.58
96.08.10.16
99.15.0.00
98.9310.03
81.6112.37
85.8/12.14
98.7710.10
98.2710004
99.56...02
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Published in Transactions on Machine Learning Research (03/2025)
Table 7: Index Importance. For each dataset, the importance weights of our indices for XGBoost. The top three important indices in each dataset are given in blue, green, and red respectively. Newly introduced indices are marked in blue.
CORA
CITES.
PUBMED
PHOTO
COMP.
TEXAS
WISC.
CHAM.
SQUIR.
CROC.
graph distance
0.0385
0.0217
0.5976
0.0328
0.0275
0.0546
0.0709
0.1199
0.3835
0.0953
# 2-paths
0.0581
0.0607
0.0082
0.0066
0.0128
0.0165
0.0300
0.0159
0.0123
0.0089
Adamle Adar
0.0435
0.0426
0.0087
0.7436
0.7762
0.0798
0.1492
0.0344
0.4040
0.0282
Jaccard
0.0050
0.0049
0.0016
0.0063
0.0066
0.0531
0.0411
0.0069
0.0096
0.0055
Salton
0.0036
0.0035
0.0017
0.0028
0.0023
0.0445
0.0283
0.0090
0.0041
0.0024
Sorensen
0
0
0
0.0423
0.0332
0.0126
0.0182
0.0114
# 3-paths
0.1179
0.3056
0.1624
0.0117
0.0210
0.0479
0.1047
0.4093
0.0525
0.2424
3 3-Jaccard
0.0120
0.0406
0.0431
0.0039
0.0048
0.0355
0.0224
0.0123
0.0215
0.0076
3 Salton
0.0113
0.0654
0.0604
0.0455
0.0272
0.0467
0.0349
0.1402
0.0178
0.3649
3-Sorensen
0.0051
0.0078
0.0054
0.0037
0,0047
0.0507
0.0302
0.0092
0,0103
0.0083
common digits
0.0043
0.0052
0.0067
0.0040
0.0039
0.0496
0.0729
0.1378
0.0247
0.1058
common class
0.6254
0.1304
0.0386
0.0417
0.0266
0.1859
0.0652
0.0057
0.0018
0.0083
norm. digits
0.0264
0.2517
0.0356
0.0025
0.0027
0.0490
0.0497
0.0330
0.0112
0.0320
class identifier
0.0489
0.0600
0.0300
0.0949
0.0837
0.24400
0.2672
0.0539
0.0283
0.0791
impact minimal in others. Adamic Adar emerges as the most important index in four out ten datasets, Notably, in the remaining six datasets, our newly introduced proximity indices take precedence in five of them. Morcover, the table underscores the significant variation index importance, with no particular subsct of indices dominating others. Overall, the syneryy belween domain and structural prozimily indices emerges as pivotal factor contributing to improved performance.
Table 8 AUC results of PROXI for different ML Classifies..
ML Classifter
CORA
CITESEER
PUBMED
TEXAS
WISC.
CHAM.
Logistic Reg.
94.89
95.30=0882
95.68+0.14
81.8643.80
82.37+3.74
97.83=0.18
Naive Bayes
94.88
92.55=0.58
94.7740.16
76.8413.56
74.04=1995
06.6420.20
QDA
95.23+0.58
94.06=0.58
94.70+0.23
81.1643.07
79.29=1.45
96.71:0.20
XGBoost
95.83:10.59
95.71
96.0820.16
84.61-2.37
85.8412.18
98.77:00100
Discussion. Our experiments show that our simple ML model, which is a combination of proximity indices with a tree-based ML classifier, outperforms or gives an on-par performance with most of the current GNN models benchmark datasets. We would like to note that another traditional method (Adamic-Adar & Edge Proposal Sct) by Singh al. (2021) has a very high ranking the OGBL-COLLAB leaderboard. These findings may be seen unexpected since GNN models are generally perceived the frontrunners in graph representation learning. However, as recent theoretical studies suggest, our results underscore the pressing need for fresh and innovative approaches within the realm of GNNs to address unique challenges in graph representation learning and improve performance.
Limitations. The main limitation in our approach comes from the domain proximity indices depending on the format of the node altribute vectors and the context. Unfortunately, there is no general rule in this part, as the context of the dataset plays a crucial part in extracting useful domain indices. However, considering node attribute vectors as node embedding in the attribute space, we plan to use GNNs to obtain the most effective domain proximity indices by formulating the question in terms of learnable parameters. In our future projects, aim to further explore this direction.
5 Conclusion
In this paper, motivated by the theoretical studies, we have compared the performance of stato-of-tho-art GNN models with conventional methods in link prediction tasks. Our computationally efficient model, which utilizes collection of structural and domain proximity indices, outperforms or gives highly competitive results
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with state-of-the-art GNNs across both small/large benchmark datasets as well as homophilii/hhterophhile settings. Furthermore, when combined, our PROXT indices significantly enhances the performances of classical GNN methods. Our results not only empirically verify recent theoretical insights but also suggest significant untapped potential within GNNs. Looking ahead, we aim to delve into building GNN models tailored for link prediction, which utilizes edge embeddings. We intend to inlegrate our proximity indices to obtain stronger odge representations, thus enhancing the overall performance significantly.
Acknowledgements
This work was partially supported by National Science Foundation under grants DMS-2220613 and DMS- 2229417. The authors acknowledge the Texas Advancod Computing Center (TACC) at UT Austin for providing computational resources that have contributed to the research results reported within this paper. http://www. tacc. utexas.edu.
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Appendix
In this part, we provide extra experimental results and further details on our method. In Appendix A, we provide dataset details, the performance of our model with different ML classifiers (Table 8), and further performance results (Table 9). In Appendix B, we provide the details of how PROXI indices can boost the performance of GNN models. In Appendix C, we provide details of our proximity indices used for all datasets. Finally, in Appendix D.2, we discuss the relation between transitivity and heterophily.
A Datasets
In our experiments, we used twelve benchmark datasets for link prodiction tasks. All the datascts are used in the transductive setting like most other baselines in the domain. The dataset statistics are given in Table 2.
The citation network datasets, namely CORA, CITESEER, and PUBMED are introduced in (Yang et al., 2016), and they serve as valuable benchmark datasets for rescarch in the field of semi-supervised lcarning with graph representation learning. Within these datasets, individual nodes correspond to distinet documents, while the edges between them symbolize citation links, elucidating the interconnectedness of scholarly works within these domains.
In the context of co-purchasing networks, the benchmark datasets, PHOTO, and COMPUTERS, are introduced in (Shchur et al., 2018) representing the sales network at Amazon. In these networks, nodes correspond to various products, while edges signify the frequent co-purchasing of two products. The primary objective of this study is to leverage product reviews, represented as bag-of-worde node attributes, to establish a mapping between individual goods and their respective product categories, thus addressing a fundamental catogorization task within the context of these interconnected networks.
Next, the OGBI-COLLAB dataset is a part of the library of large benchmark datasets, namely Open Graph Benchmark (OGB) collection (Hu et al., 2020; 2021a). This is an undirected graph, representing a subset of the collaboration network between authors indexed by Microsoft Academic Graph (MAG) (Wang al., 2020). Each node represents author and edges indicate the collaboration between authors. All nodes come with 2o8ddimnsiional attributes, obtained by averaging the word embeddings of papers that are published by the authors. All edges are associated with two meta-information: the year and the edge weight, representing the number of co-authored papers published in that year. The graph can be viewed as a dynamic multi-graph since there can be multiple edges bet two nodes if they collaborate in more than one year.
Another OGB dataset is OGBL-PPA, which represents an unweighted, undirected graph structure. Nodes in this complex network carefully depict proteins from wide range of 58 different species, demonstrating the dataset's 's extensive coverage of the biological environment. Furthermore, this complex network's edges encode a wide range of biologically significant relationships between proteins. These relationships cover a wide range of interactions, such as physical contacts, co-expression palterns, homology determinations, and the defining of genomic regions (Szklarczyk et al., 2019).
Table 9: AUC and Hits@20 results on homophilic datasets with baselines by Zhao et al. (2022) for 70:10:20 split.
CORA
CITESEER
PUBMED
Models
AUC
Hits@20
AUC
Hits&20
AUC
Hits
Vode2Vie (Grover et al., 2016)
84.49.0.40
49.9642.51
80.00=0
4..78
80.3220.29
39.19:1.02
VGAE [Kipf& & Welling, 2016b)
88 48.68.1.40
43.9143.38
85.35.00
44.0411.86
95.8020.13
2..7321.61
GCN (Kipf & Welling, 2017)
90.25.10.58
49.0611..2
71.4711.0
55.56:1.32
96.3310.60
21.8413.87
GSACE (Hamilton et al., 2017)
90.24110.34
53.5411296
87.38
53.67.12.4
96. 00.7811..11
39.13.14.1
SEAL (Zhang & Chen, 2018)
92.55.10.50
51.3512.26
85.82
40.90113.6
96.3610.22
28.15133.81
JKNet (Xu al., 2018)
89.05=10.67
48.2113.86
88.58
55.6042.17
96.5810.22
25.611:..11
MVGRL (Hassani et al., 2020)
75.07+3.68
19.53+2264
G1. 00.20000
14.0744.79
10.78
14.1940.83
LGLP (Cai al., 2021)
91.30++0.05
62.98+0.55
89.41+0.13
57.43++3.71
CFLP (Zhao et al., 2022)
93.05=+0.24
65.57+1..05
92.12-0.47
68.09+1.44
97.58+0.17
44.90=2.00
PROXI
94.70.10.03
61.21112.90
94.97710.29
71.94::.82
94.66100113
12.10:22.1
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h
GNN
Graph
h
h
PROXI
Kuv
PROXI-GNN: The flowchart of the model used integrate PROXT indices with GNNss In the image 's" represents the Hadamard product hy & and '42 represents concatenation.
Figure 2:
The datasets WISCONSIN and TEXAS (Pei et al., 2020) are heterophilic website networks that are acquired via CMU. Web pages are represented as nodes in these networks, and the hyperlinks that connect them as odges. The lag-of-words representation of the corresponding web pages defines the node attributes.
Finally, the Wikipedia heterophilic webpage structures that are represented by the networks CROCODILE, CHAMELEON, and SQUIRREL (Pei et al., 2020) are each focused
particular topic, denoted by the name of the corrosponding datasct. Nodes in these networks are individual websites, while edges are hyperlinks that connect them. The attributes assigned to every node consist of variety of educational nouns taken from Wikipedia articles.
B Integrating PROXI indices with GNNs
In this section, we study the integration of PROXI indices to enhance the performance of GCN, GAT, SAGE and LINKX models, and give details of results presented in Table 5. The conventional approach without incorporating the indices follows the standard procedure. Initially, employing one of the GNN models mentioned, node embeddings denoted as h are generated over the training set using the node attributes available in the dataset. Next, within the predictor, for each edge v), the element-wise multiplication (Hadamard product) h, @ h, = (h), h.. h2 12 te,,...hh h h is computed, where h2 and h represent the tth element of node embeddings h2 and h, respectively. This resultant edge representation is then passed through a 3-layer MLP obtain the final edge prediction. The flowchart of our simple model integrating PROXI indices with GNNs is presented in Figure? 2.
To test the effect of PROXI indices in the performance GNN, introduce PROXI enhanced GNN model, PROXI-GNN. Our main motivation as discussed carlier, lcarning node representations is meaningful task for node classification lask, however, it is not very suitable for link prediction lask. Instead, the goal is to learn representations of node pairs, and consider this as binary classification problem (positive vs. negative edge). Our proximity indices directly provides embeddings of node pairs. We will use this direct information GNNs by enhancing their prediction heads for node pairs. Specifically, within the predictor, our model takes the indices of node pair (u, v) and processes them through l-yee MLP to extract meaningful representations.
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1.0
1.2
5.-
0.7
0.7
27-
244
100
100
epach
epoch
(a) GCN
(b) GAT
(c) SAGE
Figure 3: Performance comparison of vanilla-GNNs (orange) and PROXI-GNNs (blue) on CORA dataset.
These MLP layers help in capturing intricate patterns and characteristics present in the indicos. Following this, the resulting edge embedding from the MLP layers is concatenated with the element-wise product h, C he obtained earlier, and then it is passed through 3-layer MLP above. This concatenation operation combines the learned indices with the node embeddings, creating fused representation that captures both node-level and edge-level information. Our methodology seeks to utilize the extra information included in the relationships between nodes, as encoded in the indices, by integrating the indices in this way. With both node and edge information included, this enhanced representation offers a more complete picture of the network structure, which could enhance prediction performance. Our model's ability to accurately forocast. the existence attributes of edges is improved by incorporating indices directly into the prediction process. This helps the model to better capture the intricate interconnections and dependencies inside the graph.
In Figure 3, we present a comparative analysis between the standard algorithm, GNN, and our proposed modified algorithm, PROXI-GNN, on the Cora dataset, employing the before mentioned GNN models. The visualization illustrates the performance comparison across different GNN models, highlighting the impact of integrating our edge feature vectors into the models.
From the depicted results, it is evident that incorporating our edge feature vectors yields notable improve- ments in performance across all GNN models examined. Specifically, our modified algorithm PROXI-GNN consistently outperforms the standard algorithm by margin of at least 5% in terms of predictive accuracy. This substantial enhancement underscores the significance of integrating edge feature information into the modeling process. Same improvement can be noticed for heterophilic dataset Texas in Figure 1, where the AUC is considcrably improved by 8%.
The observed improvements in performance serve as compelling evidence of the efficacy of our proposed approach. By leveraging the additional contextual information embedded in the indices, our modified algorithm effectively enhances the models' ability to capture intricate relationships and dependencies within the graph structure. This enhancement ultimately leads to more accurate and robust predictions, as demonstrated by the substantial performance gains across the GNN models evaluated.
Overall, the outcomes shown in Figures 3 and 1 highlight the significance and effectiveness of the modifications we suggested. The noteworthy enhancements performance that may be obtained by using our proposed odge feature vectors underscore the possibility of our mothodology to augment the predictive powers of GNN models, especially in situations where edge-level data is pivotal to the underlying graph structure.
In Table 5, we present performance comparison of GNN models with PROXI-GNN on two distinct datasets, CORA (homophilic) and TEXAS (heterophilic). Notably, the inclusion of our indices demonstrates significant improvements in classification AUC, i.e., up to 8% in CORA, and 11% in TEXAS. This substantial increase in AUC suggests that the incorporation of our indices enhances the model's ability to learn from graph structures. Similar results are obscrved on the heterophilic dataset TEXAS. These results underscore the importance of considering proximity indices in graph-based learning tasks, showcasing the efficacy of our PROXT method in improving GNN performance.
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B.1 GNN Models with Proximity Indices
Several recent approaches, including BUDDY BUDDY/ELPH (Chamberlain et al., 2022), the Labeling Trick (Zhang et al., 2021) and HI-GNN (Zhang et al., 2024), have addressed the limitations of GNNs in link prediction tasks by focusing on generating meaningful representations for node sets or node pairs. BUDDY /ELPH enhances MPNNs by incorporating subgraph information into the message-passing process; implicitly capturing many structural indices akin to those used in our model. However, it does not utilize the domain-specific indices central to our methodology, which significantly improve performance. These indices provide direct and dataset-adaptive mechanism to inform the ML model about node-pair relationships, resulting in superior performance OGBL-COLLAB and OGBL-PPA datasets (Table 10). In contrast to BUDDY's node- centric approach, we are developing novel GNN model for link prediction that learns edge (or node-pair) representations, using our indices as initial messages, thereby offering a complementary perspective to the challenges of link prediction.
The Labeling Trick similarly focuses on addressing the challenges of generating meaningful node-set representations by leveraging node labeling schemes, such as the zero-one labeling. While this aligns with some of our domain-specific indices, including the "class identifier' and "common class" indices, it does not natively incorporate others, such as "common digits" and 'normalized digits," without additional adaptations. Notably, our PROXI model outperforms all GNN models enhanced with the Labeling Trick, reported in Zhang et al. (2021), both OGBL-PPA and OGBL-COLLAB dalasets (Table 10). This performance advantage highlights the effectiveness of directly integrating domain-specific indices into the prediction mechanism. The comparisons underscore the flcxibility and robustness of our approach, which addresses fundamental challenges in link prediction more comprehensively than existing methods.
Table 10: Comparison of model performance OGBL-PPA (Hits@50) and OGBL-COLLAB (Hits@100) datasets.
Model
OGBL-PPA
OGBL-COLLAB
BUDDY (Chamberlain et al., 2022)
17 13.93
61.5990.40
DCN (Zhang et al., 2021)
36.48
64 64.44gomm
GCN+DRNL (Zhang al., 2021)
45.24
64.401a.4b
SEAL (Zhang al., 2021)
48.5012.15
64.741o.1
HL-GNN (Zhang et al., 2024)
56.77
68.11
PROXI
70.361077
76.50
Similarly, HL-GNN (Zhang et al., 2024) effectively incorporates several structural indices that overlap with our methodology into its message aggregation scheme, outlined in (Zhang et al., 2024, Table 2). This approach significantly improves GNN performance. However, HL.GNN does not include our domain-specific indices, which limits its performance in certain datasets. Comparative evaluations reveal that PROXI outperforms HL-GNN on OGBL-COLLAB, achieves parity on the PHOTO dataset, and falls short on OGBL-PPA. These results underscore the importance integrating domain-speeific indices into GNN models to optimize performance across diverse datascts.
C Proximity Indices
In this part, we give the details of our structural and domain indices for each dataset.
Proximity Indices for All Datasets except OGB datasets. In all ten homophilic and heterophilic datasets, we used the following indices:
Structural Prozimàty Indices. (3, (i, Sa v), (), v), C2(u,v) v), L)(u,v), and D(u,v)
We have ten structural proximity indices: (),,
Domain Prozimity Indices.
We have four domain proximity indices: CI v) MC(u, (D(,,
Table 11: Total number of indices used for each dataset in our model.
CORA
CITESEER
PUBMED
PHOTO
COMP.
O-COLLAB
O-PPA TEXAS
WISC.
CHAM.
SQR CROC.
# Indicas
21
18
18
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In particular, in our PROXI, have 10 structural indices, while the domain indices vary according to the number of classes of each dataset. That is number of domain indices is equal to the number of classes that have fixed dimensions, which is three, added to the dimension of Cu, v), which varies according to the number of classes of the dataset. The details of these indices are given in Scction 2.2. The importance of each feature for each dataset given in Table 7.
The OGBL-COLLAB dataset a time-varying dataset, spanning between years 1963 to the year 2019, where the positive training set spans between years 1963 and 2017, the positive validation set is set the node pairs appearing in the year 2018, and year 2019 is the positive test set. Each link has weight which is the number of collaborations that occur wetw the authors pair for the given year. Since it is dynamic and weighted, we needed to adjust our indices in this dataset to adapt our method to this context.
Proximity Indices for OGBL-COLLAB.
Let (", v) be the number of collaborations that occur between nodes u and e in the year y. For simplicity let us define the total collaborations of pair (u, v) through all years
W(u, v) Σ (w, 1963 S S 2017).
We define the number of collaborations of the pair (u, v) between the years 2007 and 2017 as
Wio(u, = 1 w (u, 2007 S y S 2017),
and between years 2012 and 2017 as
Wblu, v) = 1 w0 (u, v) 2012 S S 2017].
Finally, considering g = [(u, v): : the collaboration between a and v occurs in years 2007 to 2017),
A(u) = Σ Wiolu, x E I(u)}.
Author's Oldest Paper Indez: For this feature track down the year of the earliest paper of each author. If this year is before year 1985, assign the value O, and otherwise.
o Author's Newest Paper Inder: For this feature we track down the year of the latest paper of each author. If this year is before year 1985, we assign the value O, and 1 otherwise.
All Time Collaborations: For each pair (u,v) we add up the weights of the pair through each year for which the link exists, which is W(u, v).
10- Year Collaborations: For each pair (u,v) we add up the weights of the pair through each year, from 2007 to 2017, for which the link exists, Wio(u, v).
5- Year Collaborations: For each pair (u,v) we add up the weights of the pair through each year, from 2012 to 2017, for which the link exists W)(u,v)
All Time Common Collaborators: For each pair (u.v) we find the neighborhood of node u and node over the graph created by combining all the years between 1963 and 2017, and we take the intersection of the neighborhoods.
10- Year Common Collaborators: For each pair (u,v) we find the neighborhood of node and node over the graph created by combining the years between 2007 and 2017, and we take the intersection of the neighborhoods.
5- Year Common Collaborators: For each pair (u.v) we find the neighborhood of node u and node v over the graph created by combining the years between 2012 and 2017, and we take the intersection of the neighborhoods.
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Preferential Attachment: We evaluate Preferential Attachment (Barabási et al., 2002) over the graph created by combining the years between 2007 and 2017, which formula is
PA(u, A(u) A(v)-
w- Adamic Adar: Considering the graph G,
Au,(u,))
Σ (wN(w)))()))
log AA(z)
w-Jaccard Index: Considering the graph g
(n 1 Wholu, 2) + Wio(z.u) :2 ((u) (N(
M2ouu, u, x) + Who(x, : (M)
u-Salton Inder: Considering the graph G,
(u, v) W2o(,, Who 2) Wholz N(N)) v AA(u)AAu)
Shortest Path Length: D(u, v) over the graph g.
For each node, a 128-dimensional feature vector of word embedings provided. We set up three proximity utilize them. These indices are:
Common Embedding: For given word embodding X, define Common Embedding domain fcature CE( v) as the number of matching 'I's in the vectors X., and X.. i.e,
CE(U, # x x
T Distance: Simply, we use 7.-morrm (Manhattan metric) the feature space TR. Tf X,, = [a1 02 am and X, Ibb d2 ... bm], we define D(u, v) d(X,,, Xx)) Em la--bi 6,]-
Cosine Distance: Another popular distance formula using some normalization is the cosine distanco/similarity We define our cosine distance feature
X, X X |X,||
D)
Year- wise label: For node pair (u, v), we define LAy, 1 if the pair exists in year y, and otherwise. In our experiments, we iterate between the years 2007 to 2016.
Proximity Indices for OGBL-PPA. OGBL-PPA was challenging dataset, because of its high number of odges, thus we were limited using a lot of indices due to memory reasons. For this datasct, we employod proximity indices:
Structural Indices.
Ju, V), Sa S., v), v), C2(u, v), and D(u, v)
Domain Indices.
CI(u, MC v)
Taking this into account, our PROXI for OGBI-PPA we have six struetural indices and three domain indices the vanillo Class Identifier CI(u, v) [class(u) (u). class(v)) 2-dimensional feature. The details of these indices given in Section 2.2.
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D More on Transitivity
D.1 Prediction Heads for Link Prediction
Building on our discussion in Section 3.3, we refer to a recent study that explores the impact of decoders on GNN performance. The authors of Wang et al. (2022a) evaluated varions prediction heads and, as previously mentioned, highlighted the poor performance of distance-based (dot-product) decoders due their inherent. limitations in learning efficacy.
Table 12:
Performance comparison of different decoders across benchmarks. The results are reported from Wang al. (2022a, Table 2). Each dataset has its own pre-defined comparison metric indicated below.
Decoder
Dot Product
Bilinear
ConcatMLP
Mad
OGBL-DDI (Hits@20)
GCN [5] GraphSAGE 12]
13.8
16.1
12.9=1.4
37.1
36.512.8
39.411.7
34.211.9
53.9
Node2Vec [22]
11.6+19
13.8+1.
10.8+1.7
23.3+2.1
OGBL-COLLAB (Hits@50)
GCN 55]
42.9=0.7
43.2+0.9
42.3#1.0
44.8=1.1
GraphSAGE 112]
37 37.9
41.510.8
37.0
48.1
Node2Vee [22]
27.7
31.5110
27.2a0.8
48.9
OGBL-PPA (Hits@100)
GCN [5]
5.1=0.4
5.820.5
6.2+0.6
18.7=1.3
GraphSAGE 12]
3.2=0.3
6.5
5.8+0.4
16.6=2.4
Node2Vee (22)
4.2
7.820.6
8.340.4
22.3
OGBL-CITATION CITATION2 (MRR)
GCN [5]
65.3+0.4
69.0+a.
62.7=03
84.7=002
GraphSAGE 12]
62.210.7
65.410.9
60.820.6
80.420.1
Node2Vee [22]
52.7#a.
54.1+0.6
51.4
61.4=0.1
D.2 Transitivity vs. Heterophily
In studying the transitivity behavior of networks, we identify a notable distinction between homophilic and heterophilic graph structures. Homophilic graphs generally exhibit significantly higher transitivity ratios, with the PHOTO dataset being notable exception. This can be attributed the strong clustering tendencies in homophilic graphs, where nodes of the same class are more likely to form densely interconnected local communities. These dense clusters positively influence the overall transitivity ratio. Conversely, heterophilic graphs are characterized by lower likelihood of edges connecting nodes of the same class, leading to reduced clustering and, consequently, lower transitivity. To further illustrate these patterns, we present 2D t-SNE visualizations of NODE2VEC embeddings for both graph types in Figure 4. For these embeddingg,, the dimensionality was set to 64, with a walk length of 50 and 10 walks per node. These initial findings highlight important structural patterns, but further analysis needed, which we plan to pursue in future work.
E Inductive Performance of PROXI
We present results for the inductive sctting on the CORA and TEXAS datascts in Table 13.
In this setting, test edges sampled exclusively between test nodes, which are removed from the original graph, and the model is trained on the training subgraph. While the performance drop in the homophilic CORA dataset is minimal, the helerophilic TEXAS dataset shows a significant decline, suggesting that struc- tural information alone may be less effective without node features in heterophilic graphs. For reference, Hao et al. (2021) reported AUC scores of 86.4 and 94.5 in inductive and transductive settings
Table 13: AUC results for inductive and transductive link prediction with PROXI.
Dataset
Inductive
Transductive
CORA TEXAS
86.07
10.33
48.53+6.22
84.61+2.37
24

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Published in Transactions on Machine Learning Research (03/2025)
63-
42
73-
Node
-00
-00
-5
-55
25
-2
(a) CROCODILE
(b) PHOTO
(c) CORA
Figure 4: 2D t-SNE visualizations of node2vec vectors of CROCODILE, PHOTO and CORA datasets.
on homophilic datasets, while Naddaf et al. (2023) highlighted that network density heavily influences generalization from transductive to inductive settings. In future work, we plan to explore the causes of this performance gap and investigate methods to mitigate information loss.
25


Reviews:
[{"review_id": "u0YGKBD9AZ", "reviewer": "Reviewer_izxa", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**\n1. The work is presented in a clear and easy to understand fashion\n2. The authors present experiments that are easy to understand and provide justification for many of their claims.\n3. The authors provide clear and intuitive arguments for the failing of learned structural node representations for link prediction. Insights like these are rarely shared in the literature, and I appreciate them.\n\n**Weaknesses:**\n1. The presented method seems like it might struggle with computational complexity. Could you detail the computational complexity for your PROXI scores? And is it possible to use techniques like those outlined in Chamberlin et al. to reduce this computational cost? Additionally, could you comment on whether this technique is suitable for situations where you have dynamic negative resampling?\n2. What is the operative difference between PROXI (or at least, the structural components of it) and BUDDY/ELPH, aside from the latter method using hashing based methods to approximate these neighborhood indices. It would seem to me that your $\\mathcal{L}_3$ index would be expressible through a mix of their approximate overlap operators.\n3. It seems as if PROXI seems to struggle in multiple settings -- could you comment as to why this might be the case?\n4. In what ways is PROXI similar to a labeling trick? Given that labeling tricks can also fed into a GNN in ways similar to PROXI, it is worth adding them as a baseline. Specifically double radius node labeling (DRNL) might be valuable and related.\n5. It seems as if you're missing other heuristic-learning baselines such as HLGNN [1] that might be relevant.\n6. The argument in 3.3 feels strange. Triangle closures are a common heuristic that performs quite well for link prediction, and has powered many methodological improvements [2]. Beyond this, the argument that link predictors that _only_ take in structural node representations can lead to degenerate learned representations is not new and has been proven rigorously [3] and described intuitively [4].\n7. The related work section is missing many relevant contributions. For structural indices it's missing for example the local path index [5], extensions of it [6], and the Katz index [7], among others. For structurally augmented GNN based link predictors the authors are missing references to SEAL, ELPH/BUDDY, NeoGNN, and HLGNN among others. Because of this, I would recommend that the authors give a more thorough review of the literature to help further situate the work.\n\n[1] https://arxiv.org/abs/2406.07979\n\n[2] https://dl.acm.org/doi/10.1145/3459637.3481920\n\n[3] https://arxiv.org/abs/1910.00452\n\n[4] https://proceedings.neurips.cc/paper_files/paper/2021/file/4be49c79f233b4f4070794825c323733-Paper.pdf\n\n[5] https://journals.aps.org/pre/abstract/10.1103/PhysRevE.80.046122\n\n[6] https://www.nature.com/articles/s41598-020-76860-2\n\n[7] https://link.springer.com/article/10.1007/bf02289026", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work explores the effect of augmenting modern machine learning methods with structural features (eg # of common neighbors). In doing so, the authors introduce PROXI, which is a structural feature augmentation for use in standard link-prediction settings that captures higher-order structure features. The authors carry out empirical experiments to justify their methodological claims, and show that PROXI provides some utility when used in conjunction with traditional backbones.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see the weaknesses section", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "gDu4U73uq2", "reviewer": "Reviewer_5U5K", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n1. Refreshing - the authors take inspiration from traditional approaches and show that applying deep neural networks is not always the best approach.\n2. Comprehensive - the authors present their method to a sufficient degree clearly and concisely.\n3. Well-motivated - the authors clearly present the motivation for their study and also cite recent work on existing issues with GNNs.\n### Weaknesses\n1. Adaptability - as the authors also point out, their method is difficult to adapt to new domains, which is a significant limitation compared to GNN-based approaches.\n2. Clarity - the paper's first two sections heavily rely on the difference between homophily and heterophily. An unfamiliar reader has to get to section 3.2 to get an understanding, but many paragraphs before it already assume familiarity with them.\n3. Section 3.3 - The speculative analysis in this section (and also in the introduction) lacks any references to existing models that use such distance metrics as prediction heads. It would be nice to see a performance comparison between \"similarity-based\" and learned (e.g. MLP) prediction heads in existing GNN models.\n4. Tables 3 and 4 - I noticed the reported performances in Table 4 do not appear in Table 3 (except for LINKX) and the baselines in Table 4 also have no citations. Is there any particular reason for this?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors challenge the current status quo of using Graph Neural Networks (GNNs) for link prediction. They present an alternative method, PROXI, which makes use of proximity information of node pairs, using both structural information and node attributes. They leverage several proximity-based methods to compute a set of features (or indices) and train a classifier model to predict links between node pairs. The authors provide a comprehensive summary of their method, clearly state their assumptions and compare the performance of their model to a multitude of baselines. Remarkably, they show that their method is not only competitive but outperforms the current state of the art on multiple benchmarks. Secondly, they show that PROXI can be combined with existing GNN architectures, resulting in significant performance improvements. Finally, the authors perform ablation studies to better understand the importance of different classes of proximity indices. Overall, the paper successfully challenges the over-usage of GNNs for link prediction tasks and shows that a traditional feature-engineering approach can indeed outperform the current state of the art.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. If possible, although I understand the space limitations, I would like to see an impact analysis on the adaptability of PROXI to other settings, where the current set of indices may not be applicable.\n2. Please try to improve the clarity of the paper by using less technical concepts or giving a short introduction to them the first time they appear.\n3. Section 3.3. needs some references, to make it more concrete and not overly speculative.\n4. Include a discussion/ justification for using different baselines in Tables 3 and 4.\n5. Please make sure to name the performance metric in Table 4 as well (I suppose it is AUC but I saw no mention of this).", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "QG6rFsxbb2", "reviewer": "Reviewer_h6dX", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n1. The paper is mostly well-written and the approach/details are presented clearly.\n2. The experiments consider graphs of varying sizes and the link-prediction results are promising when compared to GNN based approaches (especially for heterophilic graphs).\n\n**Weaknesses**\n\n1. Based on Table 8 in the paper, it seems like the 2 hop/3 hop neighborhood information along with the degrees of nodes seem to be the most influential handcrafted features for link-prediction. The paper can be strengthened if a discussion on \"why GNNs fail to approximate such 2/3 hop information\" can be added. For instance, consider the GCN architecture which relies on the normalized adjacency matrix (say $A$) for neighborhood feature aggregation. By using powers of such a matrix (say $A^2, A^3$), I think the multi-hop structural information can be captured (see [1]). \n2. Other than the \"graph distance\" metric, I believe that the rest of the metrics can be computed with 2 hop/3 hop neighborhood information. More importantly, I looked at the code and seems the authors are relying on the networkx package for graph operations. Based on Table 8, if one were to ignore the \"graph distance\" metric, then by computing the powers of the adjacency matrix, one can significantly improve the efficiency of the metric calculation functions. This can also be done on a GPU (if available). My concern is that if COMPUTERS dataset (with just 245K edges) takes 18 hours, then it is definitely not a scalable approach. In case, a GPU is not available, I would request the authors to discuss these computational aspects so that future works can address them.\n3. In section 3.3, the authors (informally) discuss why cosine similarity of node features might not work for link prediction problems. How does this argument differ for homophilic/heterophilic graphs? In my opinion, this homophily/heterophily nature of the graphs should be considered when presenting the Transitivity ratios in Table 1. Additionally, how do the class labels differ between nodes sharing a common neighbor? Such an analysis can possibly shed light on why PROXI beats the baselines on PHOTO dataset and not on CORA/CITESEER. Let me know if the authors feel otherwise.\n\n\n**References**\n\n1. Rossi, Ryan A., Nesreen K. Ahmed, and Eunyee Koh. \"Higher-order network representation learning.\" Companion Proceedings of the The Web Conference 2018. 2018.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents a feature engineering approach (called PROXI) to predict links/edges in a graph. Essentially, the results showcase the benefit of using traditional classification models (such as XGBoost) to leverage structural and domain specific feature information for link prediction tasks. Additionally, these hand-crafted features are incorporated into GNNs to improve their performance.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Throughout the paper, I couldn't find details about the GNN architectures used as baselines. I checked the code as well but couldn't find any information. More importantly, can the authors add details about the architectures of the GCN, GAT, or SAGE that are used for generating Figure 2?\n\n2. Is it possible to show some results for inductive tasks? In scenarios where node feature information is unavailable, then I think structural information alone on new graphs might not be as useful as expected (when compared to GNNs).\n\n3. **Optional:** Can the authors provide references of previous works that combined feature engineering approaches with GNNs or MLPs for other tasks such as node/graph classification? Did any such approach perform better than GNNs?\n\n4. nit: please adjust the white-space throughout the paper and avoid large empty spaces such as in Page 3.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: After the rebuttal period, all of the reviewers unanimously recommend acceptance of this work to TMLR. I concur with the reviewers, and believe this will be a valuable addition to the link prediction literature. Well done!
Decision: Accept (poster)