Paper: MPXGAT: An Attention based Deep Learning Model for Multiplex Graphs Embedding (ICLR-2024)

Reviewer comment (Presentation):
Unclear representation, figure with toy examples could help

Author response:
We thank the Referee for the valuable comments. We tried to improve the readability of our manuscript by clarifying the definitions in accordance with the suggestions provided. In particular, we have now added a graphical illustration of how our algorithm would represent a toy example of a multiplex network in terms of horizontal and vertical layers.

Reviewer comment (Writing):
Need clarity on whether horizontal network refers to multiple edge types for a set of nodes and vertical network is a set of cross edges among different layers. Reviewer notes they have research experience on graph construction but are almost lost by the description. It's recommended to illustrate the abstract definition with a toy graph or from a real-world example, e.g., two people can build relationships via multiple social media (Facebook, Twitter, YouTube, TikTok), and then explain where the horizontal and vertical subnetworks are in the example.

Author response:
We thank the Referee for the valuable comments. We tried to improve the readability of our manuscript by clarifying the definitions in accordance with the suggestions provided. In particular, we have now added a graphical illustration of how our algorithm would represent a toy example of a multiplex network in terms of horizontal and vertical layers.

Reviewer comment (Theory):
Clarification on "Equations 1 use the convolutional layer as described in the GAT model" (Page 3) — does the convolutional layer refer to the unit block in GCN? Since one main difference between GCN and GAT is the aggregation mechanism (auto-weighted vs unweighted), can a GAT layer be considered as a convolutional layer? Any reference?

Author response:
We agree with the reviewer that a GAT layer can be considered convolutional since it also uses a message-passing mechanism between the nodes of the graph. The main difference between a standard GCN and a GAT layer is the aggregation method used, which in GAT is based on attention mechanisms, while this is not the case for GCN, as GCN uses a simple weighted average of linear applications of the messages.

Reviewer comment (Reproducibility):
Clarification on how many MPXGAT-H layers are needed before forwarding node embedding to MPXGAT-V, and vice versa. The authors seem not to mention the detail. Is the default 1 MPXGAT-H layer → 1 MPXGAT-V layer → 1 MPXGAT-H layer...? How many rounds of H-V block are needed?

Author response:
The MPXGAT model consists of two submodels: MPXGAT-H and MPXGAT-V. MPXGAT-H generates K embeddings for each node, one for each of the K horizontal networks (the layers of the multiplex network). While, MPXGAT-V uses these embeddings to compute another set of embeddings for each node, considering the vertical network. The final output consists of K+1 embeddings per node: the first K is used for intra-layer link prediction, and the last one is used for inter-layer link prediction. The MPXGAT model applies MPXGAT-H and MPXGAT-V only once, without any loops or iterations.

Reviewer comment (Experiments):
A simple baseline is applying a single-layer graph embedding method (e.g., DeepWalk, Node2vec, LINE, GCN, GAT, etc.) and concatenating them via multiple graph layers to do link prediction.

Author response:
The Referee is correct in stating that this can be another interesting baseline to consider. In our case, we have considered three baselines: one is a single-layer graph embedding method, i.e., GraphSAGE, as suggested by the Referee. The other two, i.e., GATNE and MultiplexSAGE, are instead algorithms developed to embed heterogeneous graphs and multiplex networks, respectively.

Reviewer comment (Novelty):
The motivation of the paper is unclear in the current version. In what applications do we need to predict inter-layer connections? Does predicting inter-layer connections in multiplex networks mean alignment of entities and so equivalent to the graph alignment problem? Why can link prediction in heterogeneous graphs not be used for predicting the inter-layer connections? That is, one can consider a multiplex network as a special case of heterogeneous graphs and then apply link prediction methods in heterogeneous graphs, which can both predict inter-layer and intra-layer links.

Author response:
We thank the reviewer for raising these relevant aspects. We tried to clarify them through the text.  The inter-layer link prediction problem finds many crucial applications such as: i) in online social network (OSN) analysis, the linkage of user identities across different OSNs, an emerging task in social media that has attracted increasing attention in recent years (Shu et al., 2017). User identity linkage finds potential impact in different domains, from recommendation systems to cybersecurity (Tang et al., 2020); ii) identifying the same genes or proteins across different biological networks, such as gene expression, protein interaction, metabolic pathways, etc. This can help to discover the molecular mechanisms of diseases and to find potential drug targets (Jain et al., 2023); iii) matching the same entities across different knowledge graphs (Azmy et al., 2019). This can help to enrich the semantic information and to improve the query answering and reasoning capabilities. The range of possible applications of inter-layer link prediction ultimately motivates the development of embedding algorithms for multiplex networks that are able to distinguish intra-layer and inter-layer links and to reconstruct both connectivity patterns.
We remark that predicting inter-layer connections in multiplex networks is related but not equivalent to either the graph matching problem or the graph alignment problem. Indeed, the goal of unlabeled graph matching is to identify an isomorphism between two graphs (Mathon, 1979). Therefore, the topology of two isomorphic graphs has to be exactly the same. Instead, the aim of inter-layer prediction is to find missing links between the same nodes in different layers of a multiplex network, that have in general different connectivity patterns. Concerning the global graph alignment (Ma & Liao, 2020), the goal is to find the best correspondences between two or more graphs by taking into account also the label of the nodes. The alignment task may lead to multiple solutions, including the one desired for the inter-layer link prediction, as various alignments can be equally good. Therefore, the inter-layer link prediction problem differs, as its objective is to find a given bijection between two graphs, i.e., the layers of the multiplex network. Moreover, graph alignment is a more general problem and can be applied to any graphs, while inter-layer prediction is a problem specific to multiplex networks.
The Referee is also right in pointing out that one can use heterogeneous graph embedding algorithms to perform inter-layer link predictions by distinguishing two types of edges, i.e., intra-layer and inter-layer. This is actually how we run GATNE in our experiments. However, one can clearly see that MPXGAT outperforms such a method. However, we want to clarify that link prediction in heterogeneous graphs should not be directly used for predicting the inter-layer connections, because heterogeneous graphs have different types of nodes and edges, while multiplex networks have the same type of nodes but different types of edges. Link prediction in heterogeneous graphs requires to consider the node and edge attributes and semantics, while inter-layer prediction requires to consider the layer and network properties.

Reviewer comment (Evaluation):
The experiments show that GATNE, which is not state-of-the-art for link prediction in heterogeneous graphs, can outperform the proposed method in intra-layer link prediction. Based on this result, it seems that the only application of the proposed method is inter-layer link prediction, which has unclear motivation.

Author response:
The performance of GATNE is clearly better in the case of intra-layer prediction. However, as discussed in the previous comment, inter-layer link prediction has several well-motivated applications ranging from recommendation systems to biology and knowledge graph analysis. For such applications, our model clearly outperforms competitors, yielding more reliable results.
Furthermore, if we consider the general behavior of the methods without distinguishing intra- and inter-layer connections, one can see that our algorithm is clearly the best solution. Such an analysis is reported in Table 4 of our revised paper.

Reviewer comment (Experiments):
Multi-relational data can be modeled as heterogeneous and multiplex models even when different instances of the same nodes have different features. In this paper, the authors only compare multiplex methods and do not compare with heterogeneous methods, which are more popular. In the heterogeneous case, it will be about predicting different types of self-loop. Also, evaluate your model on the Heterogeneous benchmark. References [1] Are we really making much progress? Revisiting, benchmarking, and refining heterogeneous graph neural networks [2] Revisiting Link Prediction on Heterogeneous Graphs with a Multi-view Perspective.

Author response:
We appreciate the reviewer's suggestion of using self-loops, but they may not reflect the information in the multiplex model. Self-loops could indicate information that is easy to obtain in multi-relational graphs. For example if you have a multi-relational graph with three different types of edges (red, green, yellow) you can easily put three self-loops in each node (red-green, red-yellow, green-yellow). This comes from the fact that using a multi-graph representation for multi-relational data is based on the assumption of knowing which nodes are linked, namely the identity of each node in every layer of relationships.
Therefore, link prediction in heterogeneous graphs cannot be directly used for predicting the inter-layer connections, because heterogeneous graphs have different types of nodes and edges, while multiplex networks have the same type of nodes but different types of edges. Link prediction in heterogeneous graphs requires to consider the node and edge attributes and semantics, while inter-layer prediction requires to consider the layer and network properties. Therefore, different methods and features are needed for these two tasks.