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<base href="/html/2406.11504v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S1" title="In On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S2" title="In On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Related Work</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S2.SS0.SSS0.Px1" title="In 2 Related Work ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title">Fidelity.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S2.SS0.SSS0.Px2" title="In 2 Related Work ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title">Graph pruning.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S3" title="In On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Methodology</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S3.SS1" title="In 3 Methodology ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Basic Notations and Problem Settings</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S3.SS2" title="In 3 Methodology ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Fidelity<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">-</span></sup>-inspired Pruning Framework</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4" title="In On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Empirical Observations</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS1" title="In 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Basic Settings</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS2" title="In 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Explanation Methods</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS2.SSS0.Px1" title="In 4.2 Explanation Methods ‣ 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title">Attention (Att).</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS2.SSS0.Px2" title="In 4.2 Explanation Methods ‣ 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title">Saliency (SA).</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS2.SSS0.Px3" title="In 4.2 Explanation Methods ‣ 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title">Integrated Gradient (IG).</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS2.SSS0.Px4" title="In 4.2 Explanation Methods ‣ 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title">Guided Backpropagation (GB).</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS2.SSS0.Px5" title="In 4.2 Explanation Methods ‣ 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title">GNNExplainer (GNNEx).</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS2.SSS0.Px6" title="In 4.2 Explanation Methods ‣ 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title">PGExplainer (PGEx).</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS2.SSS0.Px7" title="In 4.2 Explanation Methods ‣ 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title">FastDnX (FDnX).</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS3" title="In 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Experimental Results on Graph Pruning</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS4" title="In 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.4 </span>Visualizations of Graph Pruning</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.SS5" title="In 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.5 </span>Relationship with Fidelity Scores</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S5" title="In On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Discussion and Conclusion</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">On the Feasibility of Fidelity<sup class="ltx_sup" id="id6.id1"><span class="ltx_text ltx_font_italic" id="id6.id1.1">-</span></sup> for Graph Pruning</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Yong-Min Shin<sup class="ltx_sup" id="id7.2.id1">1</sup> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Won-Yong Shin<sup class="ltx_sup" id="id8.3.id1"><span class="ltx_text ltx_font_italic" id="id8.3.id1.1">1,</span></sup><span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>Corresponding author.</span></span></span> <sup class="ltx_sup" id="id9.4.id2">1</sup>Yonsei University, Seoul, South Korea <br class="ltx_break"/>{jordan3414, wy.shin}@yonsei.ac.kr </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id5.1">As one of popular quantitative metrics to assess the quality of explanation of graph neural networks (GNNs), <span class="ltx_text ltx_font_italic" id="id5.1.1">fidelity</span> measures the output difference after removing unimportant parts of the input graph. Fidelity has been widely used due to its straightforward interpretation that the underlying model should produce similar predictions when features deemed unimportant from the explanation are removed. This raises a natural question: “Does fidelity induce a global (soft) mask for graph pruning?” To solve this, we aim to explore the potential of the fidelity measure to be used for graph pruning, eventually enhancing the GNN models for better efficiency. To this end, we propose <span class="ltx_text ltx_font_bold" id="id5.1.2">F</span>idelity<sup class="ltx_sup" id="id5.1.3">-</sup>-<span class="ltx_text ltx_font_bold" id="id5.1.4">i</span>nspired <span class="ltx_text ltx_font_bold" id="id5.1.5">P</span>runing (<span class="ltx_text ltx_font_bold" id="id5.1.6">FiP</span>), an effective framework to construct global edge masks from local explanations. Our empirical observations using 7 edge attribution methods demonstrate that, surprisingly, general eXplainable AI methods outperform methods tailored to GNNs in terms of graph pruning performance.</p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Alongside the recent popularity of graph neural networks (GNNs) for graph-related tasks spanning across domains from social network recommendations <cite class="ltx_cite ltx_citemacro_cite">Wu <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib24" title="">2023</a>)</cite> to molecular property predictions <cite class="ltx_cite ltx_citemacro_cite">Reiser <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib14" title="">2022</a>)</cite>, such developments have resulted in an increasing demand in developing eXplainable AI (XAI) methods for GNN models. While early work focused on extending various edge attribution methods into GNNs <cite class="ltx_cite ltx_citemacro_cite">Baldassarre and Azizpour (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib4" title="">2019</a>); Pope <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib12" title="">2019a</a>)</cite> for explaining the underlying model’s behavior, many XAI methods specifically designed with GNN models in mind have been since proposed <cite class="ltx_cite ltx_citemacro_cite">Yuan <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib28" title="">2023</a>)</cite>, e.g., GNNExplainer <cite class="ltx_cite ltx_citemacro_cite">Ying <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib27" title="">2019</a>)</cite>. More recent studies include FastDnX <cite class="ltx_cite ltx_citemacro_cite">Pereira <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib11" title="">2023</a>)</cite>, relying on training SGC <cite class="ltx_cite ltx_citemacro_cite">Wu <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib22" title="">2019</a>)</cite> as a simpler surrogate model to the original GNN to extract relevant subgraphs. Although there are alternative forms of explanations for GNN models, the most prevalent one lies in the form of locally identifying the most relevant subgraph structure to the GNN’s output for a given node (i.e., the predicted node class).</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">One of the broader objectives of XAI is to ultimately enhance the performance based on the knowledge gained from the explanation <cite class="ltx_cite ltx_citemacro_cite">Samek and Müller (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib15" title="">2019</a>); Ali <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib2" title="">2023</a>)</cite>. In this regard, even though the majority of XAI methods of GNNs have successfully developed effective explanations, studies on utilizing such explanations to <span class="ltx_text ltx_font_italic" id="S1.p2.1.1">improve</span> the underlying GNN model have been vastly underexplored. In the context of to graph datasets and GNN models, we focus on the problem of <span class="ltx_text ltx_font_italic" id="S1.p2.1.2">graph pruning</span>, which is related to increasing the GNN model’s efficiency by removing unimportant edges from the underlying graph. In other words, we are interested in removing edges from the input graph altogether, guided by edge attributions from XAI methods. If such utilization of XAI is shown to be successful, then we are capable of naturally boosting the efficiency of the underlying GNN model, since the time complexity of most GNN models is directly determined by the number of input edges <cite class="ltx_cite ltx_citemacro_cite">Wu <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib23" title="">2021</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.4">Our work aims at making a connection between graph pruning and <span class="ltx_text ltx_font_italic" id="S1.p3.4.1">fidelity</span>, a quantitative metric that is often used to assess the quality of (graph) explanations <cite class="ltx_cite ltx_citemacro_cite">Ancona <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib3" title="">2017</a>); Yeh <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib26" title="">2019</a>); Yuan <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib28" title="">2023</a>)</cite>. In the context of GNN explanations, two variants of fidelity are commonly used. First, fidelity<sup class="ltx_sup" id="S1.p3.4.2"><span class="ltx_text ltx_font_italic" id="S1.p3.4.2.1">-</span></sup> measures the output difference between two instances when the original input graph is used and when the ‘unimportant’ parts (i.e., edges) are removed from the input graph. The intuition behind this metric is quite straightforward: if the explanation is valid, then structures deemed less important (i.e., assigned low edge attribution scores) should have less impact to the model’s output after removal from the input. Second, for fidelity<sup class="ltx_sup" id="S1.p3.4.3"><span class="ltx_text ltx_font_italic" id="S1.p3.4.3.1">+</span></sup>, the definition and interpretations are vice versa (i.e., removing ‘important’ parts). Revisiting on the intuition of the fidelity<sup class="ltx_sup" id="S1.p3.4.4"><span class="ltx_text ltx_font_italic" id="S1.p3.4.4.1">-</span></sup> measure, we hypothesize that frequently removed edges, when measuring fidelity<sup class="ltx_sup" id="S1.p3.4.5"><span class="ltx_text ltx_font_italic" id="S1.p3.4.5.1">-</span></sup>, may potentially be simply removed from the original graph, provided that the quality of the given explanation is good enough.</p> </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="212" id="S1.F1.g1" src="x1.png" width="747"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S1.F1.2.1.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S1.F1.3.2" style="font-size:90%;">The overview of the FiP framework.</span></figcaption> </figure> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">In this work, we investigate the feasibility of this hypothesis, i.e., we are interested in using the intuition of fidelity itself in the context of pruning the edges from the input graph. In other words, we attempt to use the aggregate of given local explanations for each node for pruning edges from the input graph. Towards this end, we first provide a simple yet effective framework, built upon any edge attribution methods, to prune the input graph based on local explanations. Our empirical results from 7 different edge attribution methods comprehensively demonstrate the feasibility of using XAI methods for graph pruning. Surprisingly, we find that explanation methods that are specifically designed for GNN methods does not perform well in graph pruning, although they have superb performance in fidelity<sup class="ltx_sup" id="S1.p4.1.1"><span class="ltx_text ltx_font_italic" id="S1.p4.1.1.1">-</span></sup>. Our analysis further validates this finding by explicitly visualizing pruned graphs, while emphasizing the necessity of developing a more sophisticated aggregation method.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Related Work</h2> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Fidelity.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p1.2">The fidelity metric, i.e., the measurement on the subset of input features highlighted by the explanation for the actual relevance to the model, has been acknowledged as one of the core properties for explanation <cite class="ltx_cite ltx_citemacro_cite">Yeh <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib26" title="">2019</a>)</cite>. In GNN explanations, two variants, i.e., fidelity<sup class="ltx_sup" id="S2.SS0.SSS0.Px1.p1.2.1"><span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px1.p1.2.1.1">+</span></sup> and fidelity<sup class="ltx_sup" id="S2.SS0.SSS0.Px1.p1.2.2"><span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px1.p1.2.2.1">-</span></sup>, are widely used depending on the unimportant/important part to be removed from the input <cite class="ltx_cite ltx_citemacro_cite">Yuan <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib28" title="">2023</a>)</cite>. Although the fidelity measures are empirical measurements, theoretical analysis on its robustness <cite class="ltx_cite ltx_citemacro_cite">Agarwal <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib1" title="">2022</a>); Zheng <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib30" title="">2023</a>)</cite> has also been performed.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Graph pruning.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px2.p1.1">Removing irrelevant edges from the underlying graph is a common strategy for reducing the computation complexity of the GNN models, as the complexity is known to be proportional to the number of edges <cite class="ltx_cite ltx_citemacro_cite">Chiang <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib5" title="">2019</a>)</cite>. Different ways to identify such irrelevant edges, including training a separate neural network <cite class="ltx_cite ltx_citemacro_cite">Zheng <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib29" title="">2020</a>)</cite>, using effective resistance measures <cite class="ltx_cite ltx_citemacro_cite">Srinivasa <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib19" title="">2020</a>); Liu <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib8" title="">2023b</a>)</cite> and using graph lottery ticket hypothesis <cite class="ltx_cite ltx_citemacro_cite">Liu <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib7" title="">2023a</a>)</cite>, were presented.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px2.p2"> <p class="ltx_p" id="S2.SS0.SSS0.Px2.p2.1">Our work focuses on the unique task of utilizing the intuition of the fidelity measurement on graph explanations to be used to prune edges from the input graph. To the best of our knowledge, <cite class="ltx_cite ltx_citemacro_cite">Naik <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib10" title="">2024</a>)</cite> lies in a similar objective to our study; however, it focuses on providing additional node features as a result.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Methodology</h2> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Basic Notations and Problem Settings</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.15">We denote an undirected and unweighted graph as <math alttext="G=(\mathcal{V},\mathcal{E},X,\mathcal{A})" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.4"><semantics id="S3.SS1.p1.1.m1.4a"><mrow id="S3.SS1.p1.1.m1.4.5" 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xref="S3.SS1.p1.1.m1.4.5.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.4b"><apply id="S3.SS1.p1.1.m1.4.5.cmml" xref="S3.SS1.p1.1.m1.4.5"><eq id="S3.SS1.p1.1.m1.4.5.1.cmml" xref="S3.SS1.p1.1.m1.4.5.1"></eq><ci id="S3.SS1.p1.1.m1.4.5.2.cmml" xref="S3.SS1.p1.1.m1.4.5.2">𝐺</ci><vector id="S3.SS1.p1.1.m1.4.5.3.1.cmml" xref="S3.SS1.p1.1.m1.4.5.3.2"><ci id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1">𝒱</ci><ci id="S3.SS1.p1.1.m1.2.2.cmml" xref="S3.SS1.p1.1.m1.2.2">ℰ</ci><ci id="S3.SS1.p1.1.m1.3.3.cmml" xref="S3.SS1.p1.1.m1.3.3">𝑋</ci><ci id="S3.SS1.p1.1.m1.4.4.cmml" xref="S3.SS1.p1.1.m1.4.4">𝒜</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.4c">G=(\mathcal{V},\mathcal{E},X,\mathcal{A})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.4d">italic_G = ( caligraphic_V , caligraphic_E , italic_X , caligraphic_A )</annotation></semantics></math>, where <math alttext="\mathcal{V}" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml">𝒱</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><ci id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">𝒱</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">\mathcal{V}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">caligraphic_V</annotation></semantics></math> is the set of nodes, <math alttext="\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mrow id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.3.m3.1.1.2" xref="S3.SS1.p1.3.m3.1.1.2.cmml">ℰ</mi><mo id="S3.SS1.p1.3.m3.1.1.1" xref="S3.SS1.p1.3.m3.1.1.1.cmml">⊆</mo><mrow id="S3.SS1.p1.3.m3.1.1.3" xref="S3.SS1.p1.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.3.m3.1.1.3.2" xref="S3.SS1.p1.3.m3.1.1.3.2.cmml">𝒱</mi><mo id="S3.SS1.p1.3.m3.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.3.m3.1.1.3.1.cmml">×</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.3.m3.1.1.3.3" xref="S3.SS1.p1.3.m3.1.1.3.3.cmml">𝒱</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><apply id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1"><subset id="S3.SS1.p1.3.m3.1.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1.1"></subset><ci id="S3.SS1.p1.3.m3.1.1.2.cmml" xref="S3.SS1.p1.3.m3.1.1.2">ℰ</ci><apply id="S3.SS1.p1.3.m3.1.1.3.cmml" xref="S3.SS1.p1.3.m3.1.1.3"><times id="S3.SS1.p1.3.m3.1.1.3.1.cmml" xref="S3.SS1.p1.3.m3.1.1.3.1"></times><ci id="S3.SS1.p1.3.m3.1.1.3.2.cmml" xref="S3.SS1.p1.3.m3.1.1.3.2">𝒱</ci><ci id="S3.SS1.p1.3.m3.1.1.3.3.cmml" xref="S3.SS1.p1.3.m3.1.1.3.3">𝒱</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">caligraphic_E ⊆ caligraphic_V × caligraphic_V</annotation></semantics></math> is the set of edges, <math alttext="X\in\mathbb{R}^{|\mathcal{V}|\times d}" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.1"><semantics id="S3.SS1.p1.4.m4.1a"><mrow id="S3.SS1.p1.4.m4.1.2" xref="S3.SS1.p1.4.m4.1.2.cmml"><mi id="S3.SS1.p1.4.m4.1.2.2" xref="S3.SS1.p1.4.m4.1.2.2.cmml">X</mi><mo id="S3.SS1.p1.4.m4.1.2.1" xref="S3.SS1.p1.4.m4.1.2.1.cmml">∈</mo><msup id="S3.SS1.p1.4.m4.1.2.3" xref="S3.SS1.p1.4.m4.1.2.3.cmml"><mi id="S3.SS1.p1.4.m4.1.2.3.2" xref="S3.SS1.p1.4.m4.1.2.3.2.cmml">ℝ</mi><mrow id="S3.SS1.p1.4.m4.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.cmml"><mrow id="S3.SS1.p1.4.m4.1.1.1.3.2" xref="S3.SS1.p1.4.m4.1.1.1.3.1.cmml"><mo id="S3.SS1.p1.4.m4.1.1.1.3.2.1" stretchy="false" xref="S3.SS1.p1.4.m4.1.1.1.3.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.4.m4.1.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.1.cmml">𝒱</mi><mo id="S3.SS1.p1.4.m4.1.1.1.3.2.2" rspace="0.055em" stretchy="false" xref="S3.SS1.p1.4.m4.1.1.1.3.1.1.cmml">|</mo></mrow><mo id="S3.SS1.p1.4.m4.1.1.1.2" rspace="0.222em" xref="S3.SS1.p1.4.m4.1.1.1.2.cmml">×</mo><mi id="S3.SS1.p1.4.m4.1.1.1.4" xref="S3.SS1.p1.4.m4.1.1.1.4.cmml">d</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><apply id="S3.SS1.p1.4.m4.1.2.cmml" xref="S3.SS1.p1.4.m4.1.2"><in id="S3.SS1.p1.4.m4.1.2.1.cmml" xref="S3.SS1.p1.4.m4.1.2.1"></in><ci id="S3.SS1.p1.4.m4.1.2.2.cmml" xref="S3.SS1.p1.4.m4.1.2.2">𝑋</ci><apply id="S3.SS1.p1.4.m4.1.2.3.cmml" xref="S3.SS1.p1.4.m4.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.1.2.3.1.cmml" xref="S3.SS1.p1.4.m4.1.2.3">superscript</csymbol><ci id="S3.SS1.p1.4.m4.1.2.3.2.cmml" xref="S3.SS1.p1.4.m4.1.2.3.2">ℝ</ci><apply id="S3.SS1.p1.4.m4.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1"><times id="S3.SS1.p1.4.m4.1.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1.2"></times><apply id="S3.SS1.p1.4.m4.1.1.1.3.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.3.2"><abs id="S3.SS1.p1.4.m4.1.1.1.3.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.3.2.1"></abs><ci id="S3.SS1.p1.4.m4.1.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1">𝒱</ci></apply><ci id="S3.SS1.p1.4.m4.1.1.1.4.cmml" xref="S3.SS1.p1.4.m4.1.1.1.4">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">X\in\mathbb{R}^{|\mathcal{V}|\times d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.1d">italic_X ∈ blackboard_R start_POSTSUPERSCRIPT | caligraphic_V | × italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is the node feature matrix, and <math alttext="\mathcal{A}:\mathcal{E}\rightarrow\mathbb{R}" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.1"><semantics id="S3.SS1.p1.5.m5.1a"><mrow id="S3.SS1.p1.5.m5.1.1" xref="S3.SS1.p1.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.5.m5.1.1.2" xref="S3.SS1.p1.5.m5.1.1.2.cmml">𝒜</mi><mo id="S3.SS1.p1.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p1.5.m5.1.1.1.cmml">:</mo><mrow id="S3.SS1.p1.5.m5.1.1.3" xref="S3.SS1.p1.5.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.5.m5.1.1.3.2" xref="S3.SS1.p1.5.m5.1.1.3.2.cmml">ℰ</mi><mo id="S3.SS1.p1.5.m5.1.1.3.1" stretchy="false" xref="S3.SS1.p1.5.m5.1.1.3.1.cmml">→</mo><mi id="S3.SS1.p1.5.m5.1.1.3.3" xref="S3.SS1.p1.5.m5.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.1b"><apply id="S3.SS1.p1.5.m5.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1"><ci id="S3.SS1.p1.5.m5.1.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1.1">:</ci><ci id="S3.SS1.p1.5.m5.1.1.2.cmml" xref="S3.SS1.p1.5.m5.1.1.2">𝒜</ci><apply id="S3.SS1.p1.5.m5.1.1.3.cmml" xref="S3.SS1.p1.5.m5.1.1.3"><ci id="S3.SS1.p1.5.m5.1.1.3.1.cmml" xref="S3.SS1.p1.5.m5.1.1.3.1">→</ci><ci id="S3.SS1.p1.5.m5.1.1.3.2.cmml" xref="S3.SS1.p1.5.m5.1.1.3.2">ℰ</ci><ci id="S3.SS1.p1.5.m5.1.1.3.3.cmml" xref="S3.SS1.p1.5.m5.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.1c">\mathcal{A}:\mathcal{E}\rightarrow\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.1d">caligraphic_A : caligraphic_E → blackboard_R</annotation></semantics></math> maps each edge in <math alttext="\mathcal{E}" class="ltx_Math" display="inline" id="S3.SS1.p1.6.m6.1"><semantics id="S3.SS1.p1.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.6.m6.1.1" xref="S3.SS1.p1.6.m6.1.1.cmml">ℰ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.6.m6.1b"><ci id="S3.SS1.p1.6.m6.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1">ℰ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.6.m6.1c">\mathcal{E}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.6.m6.1d">caligraphic_E</annotation></semantics></math> to a real number (representing the set of edge weights or attributions). Also, we denote the set of neighbors for node <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.p1.7.m7.1"><semantics id="S3.SS1.p1.7.m7.1a"><mi id="S3.SS1.p1.7.m7.1.1" xref="S3.SS1.p1.7.m7.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.7.m7.1b"><ci id="S3.SS1.p1.7.m7.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.7.m7.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.7.m7.1d">italic_v</annotation></semantics></math> as <math alttext="\mathcal{N}_{v}" class="ltx_Math" display="inline" id="S3.SS1.p1.8.m8.1"><semantics id="S3.SS1.p1.8.m8.1a"><msub id="S3.SS1.p1.8.m8.1.1" xref="S3.SS1.p1.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.8.m8.1.1.2" xref="S3.SS1.p1.8.m8.1.1.2.cmml">𝒩</mi><mi id="S3.SS1.p1.8.m8.1.1.3" xref="S3.SS1.p1.8.m8.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.8.m8.1b"><apply id="S3.SS1.p1.8.m8.1.1.cmml" xref="S3.SS1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.8.m8.1.1.1.cmml" xref="S3.SS1.p1.8.m8.1.1">subscript</csymbol><ci id="S3.SS1.p1.8.m8.1.1.2.cmml" xref="S3.SS1.p1.8.m8.1.1.2">𝒩</ci><ci id="S3.SS1.p1.8.m8.1.1.3.cmml" xref="S3.SS1.p1.8.m8.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.8.m8.1c">\mathcal{N}_{v}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.8.m8.1d">caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. We focus on node classification, where a set of classes <math alttext="C=\{1,...,c\}" class="ltx_Math" display="inline" id="S3.SS1.p1.9.m9.3"><semantics id="S3.SS1.p1.9.m9.3a"><mrow id="S3.SS1.p1.9.m9.3.4" xref="S3.SS1.p1.9.m9.3.4.cmml"><mi id="S3.SS1.p1.9.m9.3.4.2" xref="S3.SS1.p1.9.m9.3.4.2.cmml">C</mi><mo id="S3.SS1.p1.9.m9.3.4.1" xref="S3.SS1.p1.9.m9.3.4.1.cmml">=</mo><mrow id="S3.SS1.p1.9.m9.3.4.3.2" xref="S3.SS1.p1.9.m9.3.4.3.1.cmml"><mo id="S3.SS1.p1.9.m9.3.4.3.2.1" stretchy="false" xref="S3.SS1.p1.9.m9.3.4.3.1.cmml">{</mo><mn id="S3.SS1.p1.9.m9.1.1" xref="S3.SS1.p1.9.m9.1.1.cmml">1</mn><mo id="S3.SS1.p1.9.m9.3.4.3.2.2" xref="S3.SS1.p1.9.m9.3.4.3.1.cmml">,</mo><mi id="S3.SS1.p1.9.m9.2.2" mathvariant="normal" xref="S3.SS1.p1.9.m9.2.2.cmml">…</mi><mo id="S3.SS1.p1.9.m9.3.4.3.2.3" xref="S3.SS1.p1.9.m9.3.4.3.1.cmml">,</mo><mi id="S3.SS1.p1.9.m9.3.3" xref="S3.SS1.p1.9.m9.3.3.cmml">c</mi><mo id="S3.SS1.p1.9.m9.3.4.3.2.4" stretchy="false" xref="S3.SS1.p1.9.m9.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.9.m9.3b"><apply id="S3.SS1.p1.9.m9.3.4.cmml" xref="S3.SS1.p1.9.m9.3.4"><eq id="S3.SS1.p1.9.m9.3.4.1.cmml" xref="S3.SS1.p1.9.m9.3.4.1"></eq><ci id="S3.SS1.p1.9.m9.3.4.2.cmml" xref="S3.SS1.p1.9.m9.3.4.2">𝐶</ci><set id="S3.SS1.p1.9.m9.3.4.3.1.cmml" xref="S3.SS1.p1.9.m9.3.4.3.2"><cn id="S3.SS1.p1.9.m9.1.1.cmml" type="integer" xref="S3.SS1.p1.9.m9.1.1">1</cn><ci id="S3.SS1.p1.9.m9.2.2.cmml" xref="S3.SS1.p1.9.m9.2.2">…</ci><ci id="S3.SS1.p1.9.m9.3.3.cmml" xref="S3.SS1.p1.9.m9.3.3">𝑐</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.9.m9.3c">C=\{1,...,c\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.9.m9.3d">italic_C = { 1 , … , italic_c }</annotation></semantics></math> are given. Then, we denote the one-hot label matrix <math alttext="Y\in\{0,1\}^{|\mathcal{V}|\times|\mathcal{C}|}" class="ltx_Math" display="inline" id="S3.SS1.p1.10.m10.4"><semantics id="S3.SS1.p1.10.m10.4a"><mrow id="S3.SS1.p1.10.m10.4.5" xref="S3.SS1.p1.10.m10.4.5.cmml"><mi id="S3.SS1.p1.10.m10.4.5.2" xref="S3.SS1.p1.10.m10.4.5.2.cmml">Y</mi><mo id="S3.SS1.p1.10.m10.4.5.1" xref="S3.SS1.p1.10.m10.4.5.1.cmml">∈</mo><msup id="S3.SS1.p1.10.m10.4.5.3" xref="S3.SS1.p1.10.m10.4.5.3.cmml"><mrow id="S3.SS1.p1.10.m10.4.5.3.2.2" xref="S3.SS1.p1.10.m10.4.5.3.2.1.cmml"><mo id="S3.SS1.p1.10.m10.4.5.3.2.2.1" stretchy="false" xref="S3.SS1.p1.10.m10.4.5.3.2.1.cmml">{</mo><mn id="S3.SS1.p1.10.m10.3.3" xref="S3.SS1.p1.10.m10.3.3.cmml">0</mn><mo id="S3.SS1.p1.10.m10.4.5.3.2.2.2" xref="S3.SS1.p1.10.m10.4.5.3.2.1.cmml">,</mo><mn id="S3.SS1.p1.10.m10.4.4" xref="S3.SS1.p1.10.m10.4.4.cmml">1</mn><mo id="S3.SS1.p1.10.m10.4.5.3.2.2.3" stretchy="false" xref="S3.SS1.p1.10.m10.4.5.3.2.1.cmml">}</mo></mrow><mrow id="S3.SS1.p1.10.m10.2.2.2" xref="S3.SS1.p1.10.m10.2.2.2.cmml"><mrow id="S3.SS1.p1.10.m10.2.2.2.4.2" xref="S3.SS1.p1.10.m10.2.2.2.4.1.cmml"><mo id="S3.SS1.p1.10.m10.2.2.2.4.2.1" stretchy="false" xref="S3.SS1.p1.10.m10.2.2.2.4.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.10.m10.1.1.1.1" xref="S3.SS1.p1.10.m10.1.1.1.1.cmml">𝒱</mi><mo id="S3.SS1.p1.10.m10.2.2.2.4.2.2" rspace="0.055em" stretchy="false" xref="S3.SS1.p1.10.m10.2.2.2.4.1.1.cmml">|</mo></mrow><mo id="S3.SS1.p1.10.m10.2.2.2.3" rspace="0.222em" xref="S3.SS1.p1.10.m10.2.2.2.3.cmml">×</mo><mrow id="S3.SS1.p1.10.m10.2.2.2.5.2" xref="S3.SS1.p1.10.m10.2.2.2.5.1.cmml"><mo id="S3.SS1.p1.10.m10.2.2.2.5.2.1" stretchy="false" xref="S3.SS1.p1.10.m10.2.2.2.5.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.10.m10.2.2.2.2" xref="S3.SS1.p1.10.m10.2.2.2.2.cmml">𝒞</mi><mo id="S3.SS1.p1.10.m10.2.2.2.5.2.2" stretchy="false" xref="S3.SS1.p1.10.m10.2.2.2.5.1.1.cmml">|</mo></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.10.m10.4b"><apply id="S3.SS1.p1.10.m10.4.5.cmml" xref="S3.SS1.p1.10.m10.4.5"><in id="S3.SS1.p1.10.m10.4.5.1.cmml" xref="S3.SS1.p1.10.m10.4.5.1"></in><ci id="S3.SS1.p1.10.m10.4.5.2.cmml" xref="S3.SS1.p1.10.m10.4.5.2">𝑌</ci><apply id="S3.SS1.p1.10.m10.4.5.3.cmml" xref="S3.SS1.p1.10.m10.4.5.3"><csymbol cd="ambiguous" id="S3.SS1.p1.10.m10.4.5.3.1.cmml" xref="S3.SS1.p1.10.m10.4.5.3">superscript</csymbol><set id="S3.SS1.p1.10.m10.4.5.3.2.1.cmml" xref="S3.SS1.p1.10.m10.4.5.3.2.2"><cn id="S3.SS1.p1.10.m10.3.3.cmml" type="integer" xref="S3.SS1.p1.10.m10.3.3">0</cn><cn id="S3.SS1.p1.10.m10.4.4.cmml" type="integer" xref="S3.SS1.p1.10.m10.4.4">1</cn></set><apply id="S3.SS1.p1.10.m10.2.2.2.cmml" xref="S3.SS1.p1.10.m10.2.2.2"><times id="S3.SS1.p1.10.m10.2.2.2.3.cmml" xref="S3.SS1.p1.10.m10.2.2.2.3"></times><apply id="S3.SS1.p1.10.m10.2.2.2.4.1.cmml" xref="S3.SS1.p1.10.m10.2.2.2.4.2"><abs id="S3.SS1.p1.10.m10.2.2.2.4.1.1.cmml" xref="S3.SS1.p1.10.m10.2.2.2.4.2.1"></abs><ci id="S3.SS1.p1.10.m10.1.1.1.1.cmml" xref="S3.SS1.p1.10.m10.1.1.1.1">𝒱</ci></apply><apply id="S3.SS1.p1.10.m10.2.2.2.5.1.cmml" xref="S3.SS1.p1.10.m10.2.2.2.5.2"><abs id="S3.SS1.p1.10.m10.2.2.2.5.1.1.cmml" xref="S3.SS1.p1.10.m10.2.2.2.5.2.1"></abs><ci id="S3.SS1.p1.10.m10.2.2.2.2.cmml" xref="S3.SS1.p1.10.m10.2.2.2.2">𝒞</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.10.m10.4c">Y\in\{0,1\}^{|\mathcal{V}|\times|\mathcal{C}|}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.10.m10.4d">italic_Y ∈ { 0 , 1 } start_POSTSUPERSCRIPT | caligraphic_V | × | caligraphic_C | end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="Y_{v,:}={\bf y}_{v}" class="ltx_Math" display="inline" id="S3.SS1.p1.11.m11.2"><semantics id="S3.SS1.p1.11.m11.2a"><mrow id="S3.SS1.p1.11.m11.2.3" xref="S3.SS1.p1.11.m11.2.3.cmml"><msub id="S3.SS1.p1.11.m11.2.3.2" xref="S3.SS1.p1.11.m11.2.3.2.cmml"><mi id="S3.SS1.p1.11.m11.2.3.2.2" xref="S3.SS1.p1.11.m11.2.3.2.2.cmml">Y</mi><mrow id="S3.SS1.p1.11.m11.2.2.2.4" xref="S3.SS1.p1.11.m11.2.2.2.3.cmml"><mi id="S3.SS1.p1.11.m11.1.1.1.1" xref="S3.SS1.p1.11.m11.1.1.1.1.cmml">v</mi><mo id="S3.SS1.p1.11.m11.2.2.2.4.1" xref="S3.SS1.p1.11.m11.2.2.2.3.cmml">,</mo><mo id="S3.SS1.p1.11.m11.2.2.2.2" xref="S3.SS1.p1.11.m11.2.2.2.2.cmml">:</mo></mrow></msub><mo id="S3.SS1.p1.11.m11.2.3.1" xref="S3.SS1.p1.11.m11.2.3.1.cmml">=</mo><msub id="S3.SS1.p1.11.m11.2.3.3" xref="S3.SS1.p1.11.m11.2.3.3.cmml"><mi id="S3.SS1.p1.11.m11.2.3.3.2" xref="S3.SS1.p1.11.m11.2.3.3.2.cmml">𝐲</mi><mi id="S3.SS1.p1.11.m11.2.3.3.3" xref="S3.SS1.p1.11.m11.2.3.3.3.cmml">v</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.11.m11.2b"><apply id="S3.SS1.p1.11.m11.2.3.cmml" xref="S3.SS1.p1.11.m11.2.3"><eq id="S3.SS1.p1.11.m11.2.3.1.cmml" xref="S3.SS1.p1.11.m11.2.3.1"></eq><apply id="S3.SS1.p1.11.m11.2.3.2.cmml" xref="S3.SS1.p1.11.m11.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.p1.11.m11.2.3.2.1.cmml" xref="S3.SS1.p1.11.m11.2.3.2">subscript</csymbol><ci id="S3.SS1.p1.11.m11.2.3.2.2.cmml" xref="S3.SS1.p1.11.m11.2.3.2.2">𝑌</ci><list id="S3.SS1.p1.11.m11.2.2.2.3.cmml" xref="S3.SS1.p1.11.m11.2.2.2.4"><ci id="S3.SS1.p1.11.m11.1.1.1.1.cmml" xref="S3.SS1.p1.11.m11.1.1.1.1">𝑣</ci><ci id="S3.SS1.p1.11.m11.2.2.2.2.cmml" xref="S3.SS1.p1.11.m11.2.2.2.2">:</ci></list></apply><apply id="S3.SS1.p1.11.m11.2.3.3.cmml" xref="S3.SS1.p1.11.m11.2.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.11.m11.2.3.3.1.cmml" xref="S3.SS1.p1.11.m11.2.3.3">subscript</csymbol><ci id="S3.SS1.p1.11.m11.2.3.3.2.cmml" xref="S3.SS1.p1.11.m11.2.3.3.2">𝐲</ci><ci id="S3.SS1.p1.11.m11.2.3.3.3.cmml" xref="S3.SS1.p1.11.m11.2.3.3.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.11.m11.2c">Y_{v,:}={\bf y}_{v}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.11.m11.2d">italic_Y start_POSTSUBSCRIPT italic_v , : end_POSTSUBSCRIPT = bold_y start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> is the ground-truth label for node <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.p1.12.m12.1"><semantics id="S3.SS1.p1.12.m12.1a"><mi id="S3.SS1.p1.12.m12.1.1" xref="S3.SS1.p1.12.m12.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.12.m12.1b"><ci id="S3.SS1.p1.12.m12.1.1.cmml" xref="S3.SS1.p1.12.m12.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.12.m12.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.12.m12.1d">italic_v</annotation></semantics></math>. We assume that we are given a pre-trained <math alttext="L" class="ltx_Math" display="inline" id="S3.SS1.p1.13.m13.1"><semantics id="S3.SS1.p1.13.m13.1a"><mi id="S3.SS1.p1.13.m13.1.1" xref="S3.SS1.p1.13.m13.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.13.m13.1b"><ci id="S3.SS1.p1.13.m13.1.1.cmml" xref="S3.SS1.p1.13.m13.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.13.m13.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.13.m13.1d">italic_L</annotation></semantics></math>-layered GNN model <math alttext="f" class="ltx_Math" display="inline" id="S3.SS1.p1.14.m14.1"><semantics id="S3.SS1.p1.14.m14.1a"><mi id="S3.SS1.p1.14.m14.1.1" xref="S3.SS1.p1.14.m14.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.14.m14.1b"><ci id="S3.SS1.p1.14.m14.1.1.cmml" xref="S3.SS1.p1.14.m14.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.14.m14.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.14.m14.1d">italic_f</annotation></semantics></math>, which produces a prediction <math alttext="\hat{Y}\in\mathbb{R}^{|\mathcal{V}|\times|\mathcal{C}|}" class="ltx_Math" display="inline" id="S3.SS1.p1.15.m15.2"><semantics id="S3.SS1.p1.15.m15.2a"><mrow id="S3.SS1.p1.15.m15.2.3" xref="S3.SS1.p1.15.m15.2.3.cmml"><mover accent="true" id="S3.SS1.p1.15.m15.2.3.2" xref="S3.SS1.p1.15.m15.2.3.2.cmml"><mi id="S3.SS1.p1.15.m15.2.3.2.2" xref="S3.SS1.p1.15.m15.2.3.2.2.cmml">Y</mi><mo id="S3.SS1.p1.15.m15.2.3.2.1" xref="S3.SS1.p1.15.m15.2.3.2.1.cmml">^</mo></mover><mo id="S3.SS1.p1.15.m15.2.3.1" xref="S3.SS1.p1.15.m15.2.3.1.cmml">∈</mo><msup id="S3.SS1.p1.15.m15.2.3.3" xref="S3.SS1.p1.15.m15.2.3.3.cmml"><mi id="S3.SS1.p1.15.m15.2.3.3.2" xref="S3.SS1.p1.15.m15.2.3.3.2.cmml">ℝ</mi><mrow id="S3.SS1.p1.15.m15.2.2.2" xref="S3.SS1.p1.15.m15.2.2.2.cmml"><mrow id="S3.SS1.p1.15.m15.2.2.2.4.2" xref="S3.SS1.p1.15.m15.2.2.2.4.1.cmml"><mo id="S3.SS1.p1.15.m15.2.2.2.4.2.1" stretchy="false" xref="S3.SS1.p1.15.m15.2.2.2.4.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.15.m15.1.1.1.1" xref="S3.SS1.p1.15.m15.1.1.1.1.cmml">𝒱</mi><mo id="S3.SS1.p1.15.m15.2.2.2.4.2.2" rspace="0.055em" stretchy="false" xref="S3.SS1.p1.15.m15.2.2.2.4.1.1.cmml">|</mo></mrow><mo id="S3.SS1.p1.15.m15.2.2.2.3" rspace="0.222em" xref="S3.SS1.p1.15.m15.2.2.2.3.cmml">×</mo><mrow id="S3.SS1.p1.15.m15.2.2.2.5.2" xref="S3.SS1.p1.15.m15.2.2.2.5.1.cmml"><mo id="S3.SS1.p1.15.m15.2.2.2.5.2.1" stretchy="false" xref="S3.SS1.p1.15.m15.2.2.2.5.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.15.m15.2.2.2.2" xref="S3.SS1.p1.15.m15.2.2.2.2.cmml">𝒞</mi><mo id="S3.SS1.p1.15.m15.2.2.2.5.2.2" stretchy="false" xref="S3.SS1.p1.15.m15.2.2.2.5.1.1.cmml">|</mo></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.15.m15.2b"><apply id="S3.SS1.p1.15.m15.2.3.cmml" xref="S3.SS1.p1.15.m15.2.3"><in id="S3.SS1.p1.15.m15.2.3.1.cmml" xref="S3.SS1.p1.15.m15.2.3.1"></in><apply id="S3.SS1.p1.15.m15.2.3.2.cmml" xref="S3.SS1.p1.15.m15.2.3.2"><ci id="S3.SS1.p1.15.m15.2.3.2.1.cmml" xref="S3.SS1.p1.15.m15.2.3.2.1">^</ci><ci id="S3.SS1.p1.15.m15.2.3.2.2.cmml" xref="S3.SS1.p1.15.m15.2.3.2.2">𝑌</ci></apply><apply id="S3.SS1.p1.15.m15.2.3.3.cmml" xref="S3.SS1.p1.15.m15.2.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.15.m15.2.3.3.1.cmml" xref="S3.SS1.p1.15.m15.2.3.3">superscript</csymbol><ci id="S3.SS1.p1.15.m15.2.3.3.2.cmml" xref="S3.SS1.p1.15.m15.2.3.3.2">ℝ</ci><apply id="S3.SS1.p1.15.m15.2.2.2.cmml" xref="S3.SS1.p1.15.m15.2.2.2"><times id="S3.SS1.p1.15.m15.2.2.2.3.cmml" xref="S3.SS1.p1.15.m15.2.2.2.3"></times><apply id="S3.SS1.p1.15.m15.2.2.2.4.1.cmml" xref="S3.SS1.p1.15.m15.2.2.2.4.2"><abs id="S3.SS1.p1.15.m15.2.2.2.4.1.1.cmml" xref="S3.SS1.p1.15.m15.2.2.2.4.2.1"></abs><ci id="S3.SS1.p1.15.m15.1.1.1.1.cmml" xref="S3.SS1.p1.15.m15.1.1.1.1">𝒱</ci></apply><apply id="S3.SS1.p1.15.m15.2.2.2.5.1.cmml" xref="S3.SS1.p1.15.m15.2.2.2.5.2"><abs id="S3.SS1.p1.15.m15.2.2.2.5.1.1.cmml" xref="S3.SS1.p1.15.m15.2.2.2.5.2.1"></abs><ci id="S3.SS1.p1.15.m15.2.2.2.2.cmml" xref="S3.SS1.p1.15.m15.2.2.2.2">𝒞</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.15.m15.2c">\hat{Y}\in\mathbb{R}^{|\mathcal{V}|\times|\mathcal{C}|}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.15.m15.2d">over^ start_ARG italic_Y end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT | caligraphic_V | × | caligraphic_C | end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.10">We consider a GNN explanation method <math alttext="\Phi" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mi id="S3.SS1.p2.1.m1.1.1" mathvariant="normal" xref="S3.SS1.p2.1.m1.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">roman_Φ</annotation></semantics></math> that takes a target node <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mi id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><ci id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">italic_v</annotation></semantics></math>, a target output <math alttext="t\in\mathcal{C}" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><mrow id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml"><mi id="S3.SS1.p2.3.m3.1.1.2" xref="S3.SS1.p2.3.m3.1.1.2.cmml">t</mi><mo id="S3.SS1.p2.3.m3.1.1.1" xref="S3.SS1.p2.3.m3.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.3.m3.1.1.3" xref="S3.SS1.p2.3.m3.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><apply id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1"><in id="S3.SS1.p2.3.m3.1.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1.1"></in><ci id="S3.SS1.p2.3.m3.1.1.2.cmml" xref="S3.SS1.p2.3.m3.1.1.2">𝑡</ci><ci id="S3.SS1.p2.3.m3.1.1.3.cmml" xref="S3.SS1.p2.3.m3.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">t\in\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">italic_t ∈ caligraphic_C</annotation></semantics></math> as input and assigns a non-negative edge attribution value to a given edge <math alttext="e_{i,j}" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m4.2"><semantics id="S3.SS1.p2.4.m4.2a"><msub id="S3.SS1.p2.4.m4.2.3" xref="S3.SS1.p2.4.m4.2.3.cmml"><mi id="S3.SS1.p2.4.m4.2.3.2" xref="S3.SS1.p2.4.m4.2.3.2.cmml">e</mi><mrow id="S3.SS1.p2.4.m4.2.2.2.4" xref="S3.SS1.p2.4.m4.2.2.2.3.cmml"><mi id="S3.SS1.p2.4.m4.1.1.1.1" xref="S3.SS1.p2.4.m4.1.1.1.1.cmml">i</mi><mo id="S3.SS1.p2.4.m4.2.2.2.4.1" xref="S3.SS1.p2.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p2.4.m4.2.2.2.2" xref="S3.SS1.p2.4.m4.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m4.2b"><apply id="S3.SS1.p2.4.m4.2.3.cmml" xref="S3.SS1.p2.4.m4.2.3"><csymbol cd="ambiguous" id="S3.SS1.p2.4.m4.2.3.1.cmml" xref="S3.SS1.p2.4.m4.2.3">subscript</csymbol><ci id="S3.SS1.p2.4.m4.2.3.2.cmml" xref="S3.SS1.p2.4.m4.2.3.2">𝑒</ci><list id="S3.SS1.p2.4.m4.2.2.2.3.cmml" xref="S3.SS1.p2.4.m4.2.2.2.4"><ci id="S3.SS1.p2.4.m4.1.1.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1.1.1">𝑖</ci><ci id="S3.SS1.p2.4.m4.2.2.2.2.cmml" xref="S3.SS1.p2.4.m4.2.2.2.2">𝑗</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m4.2c">e_{i,j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.2d">italic_e start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math> on the GNN model, i.e., <math alttext="\Phi(e_{i,j};v,t)\coloneqq\phi_{v}^{t}(i,j)\in\mathbb{R}" class="ltx_Math" display="inline" id="S3.SS1.p2.5.m5.7"><semantics id="S3.SS1.p2.5.m5.7a"><mrow id="S3.SS1.p2.5.m5.7.7" xref="S3.SS1.p2.5.m5.7.7.cmml"><mrow id="S3.SS1.p2.5.m5.7.7.1" xref="S3.SS1.p2.5.m5.7.7.1.cmml"><mi id="S3.SS1.p2.5.m5.7.7.1.3" mathvariant="normal" xref="S3.SS1.p2.5.m5.7.7.1.3.cmml">Φ</mi><mo id="S3.SS1.p2.5.m5.7.7.1.2" xref="S3.SS1.p2.5.m5.7.7.1.2.cmml">⁢</mo><mrow id="S3.SS1.p2.5.m5.7.7.1.1.1" xref="S3.SS1.p2.5.m5.7.7.1.1.2.cmml"><mo id="S3.SS1.p2.5.m5.7.7.1.1.1.2" stretchy="false" xref="S3.SS1.p2.5.m5.7.7.1.1.2.cmml">(</mo><msub id="S3.SS1.p2.5.m5.7.7.1.1.1.1" xref="S3.SS1.p2.5.m5.7.7.1.1.1.1.cmml"><mi id="S3.SS1.p2.5.m5.7.7.1.1.1.1.2" xref="S3.SS1.p2.5.m5.7.7.1.1.1.1.2.cmml">e</mi><mrow id="S3.SS1.p2.5.m5.2.2.2.4" xref="S3.SS1.p2.5.m5.2.2.2.3.cmml"><mi id="S3.SS1.p2.5.m5.1.1.1.1" xref="S3.SS1.p2.5.m5.1.1.1.1.cmml">i</mi><mo id="S3.SS1.p2.5.m5.2.2.2.4.1" xref="S3.SS1.p2.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p2.5.m5.2.2.2.2" xref="S3.SS1.p2.5.m5.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S3.SS1.p2.5.m5.7.7.1.1.1.3" xref="S3.SS1.p2.5.m5.7.7.1.1.2.cmml">;</mo><mi id="S3.SS1.p2.5.m5.3.3" xref="S3.SS1.p2.5.m5.3.3.cmml">v</mi><mo id="S3.SS1.p2.5.m5.7.7.1.1.1.4" xref="S3.SS1.p2.5.m5.7.7.1.1.2.cmml">,</mo><mi id="S3.SS1.p2.5.m5.4.4" xref="S3.SS1.p2.5.m5.4.4.cmml">t</mi><mo id="S3.SS1.p2.5.m5.7.7.1.1.1.5" stretchy="false" xref="S3.SS1.p2.5.m5.7.7.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p2.5.m5.7.7.3" xref="S3.SS1.p2.5.m5.7.7.3.cmml">≔</mo><mrow id="S3.SS1.p2.5.m5.7.7.4" xref="S3.SS1.p2.5.m5.7.7.4.cmml"><msubsup id="S3.SS1.p2.5.m5.7.7.4.2" xref="S3.SS1.p2.5.m5.7.7.4.2.cmml"><mi id="S3.SS1.p2.5.m5.7.7.4.2.2.2" xref="S3.SS1.p2.5.m5.7.7.4.2.2.2.cmml">ϕ</mi><mi id="S3.SS1.p2.5.m5.7.7.4.2.2.3" xref="S3.SS1.p2.5.m5.7.7.4.2.2.3.cmml">v</mi><mi id="S3.SS1.p2.5.m5.7.7.4.2.3" xref="S3.SS1.p2.5.m5.7.7.4.2.3.cmml">t</mi></msubsup><mo id="S3.SS1.p2.5.m5.7.7.4.1" xref="S3.SS1.p2.5.m5.7.7.4.1.cmml">⁢</mo><mrow id="S3.SS1.p2.5.m5.7.7.4.3.2" xref="S3.SS1.p2.5.m5.7.7.4.3.1.cmml"><mo id="S3.SS1.p2.5.m5.7.7.4.3.2.1" stretchy="false" xref="S3.SS1.p2.5.m5.7.7.4.3.1.cmml">(</mo><mi id="S3.SS1.p2.5.m5.5.5" xref="S3.SS1.p2.5.m5.5.5.cmml">i</mi><mo id="S3.SS1.p2.5.m5.7.7.4.3.2.2" xref="S3.SS1.p2.5.m5.7.7.4.3.1.cmml">,</mo><mi id="S3.SS1.p2.5.m5.6.6" xref="S3.SS1.p2.5.m5.6.6.cmml">j</mi><mo id="S3.SS1.p2.5.m5.7.7.4.3.2.3" stretchy="false" xref="S3.SS1.p2.5.m5.7.7.4.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p2.5.m5.7.7.5" xref="S3.SS1.p2.5.m5.7.7.5.cmml">∈</mo><mi id="S3.SS1.p2.5.m5.7.7.6" xref="S3.SS1.p2.5.m5.7.7.6.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.5.m5.7b"><apply id="S3.SS1.p2.5.m5.7.7.cmml" xref="S3.SS1.p2.5.m5.7.7"><and id="S3.SS1.p2.5.m5.7.7a.cmml" xref="S3.SS1.p2.5.m5.7.7"></and><apply id="S3.SS1.p2.5.m5.7.7b.cmml" xref="S3.SS1.p2.5.m5.7.7"><ci id="S3.SS1.p2.5.m5.7.7.3.cmml" xref="S3.SS1.p2.5.m5.7.7.3">≔</ci><apply id="S3.SS1.p2.5.m5.7.7.1.cmml" xref="S3.SS1.p2.5.m5.7.7.1"><times id="S3.SS1.p2.5.m5.7.7.1.2.cmml" xref="S3.SS1.p2.5.m5.7.7.1.2"></times><ci id="S3.SS1.p2.5.m5.7.7.1.3.cmml" xref="S3.SS1.p2.5.m5.7.7.1.3">Φ</ci><list id="S3.SS1.p2.5.m5.7.7.1.1.2.cmml" xref="S3.SS1.p2.5.m5.7.7.1.1.1"><apply id="S3.SS1.p2.5.m5.7.7.1.1.1.1.cmml" xref="S3.SS1.p2.5.m5.7.7.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.5.m5.7.7.1.1.1.1.1.cmml" xref="S3.SS1.p2.5.m5.7.7.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p2.5.m5.7.7.1.1.1.1.2.cmml" xref="S3.SS1.p2.5.m5.7.7.1.1.1.1.2">𝑒</ci><list id="S3.SS1.p2.5.m5.2.2.2.3.cmml" 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xref="S3.SS1.p2.5.m5.7.7.4.2.2.3">𝑣</ci></apply><ci id="S3.SS1.p2.5.m5.7.7.4.2.3.cmml" xref="S3.SS1.p2.5.m5.7.7.4.2.3">𝑡</ci></apply><interval closure="open" id="S3.SS1.p2.5.m5.7.7.4.3.1.cmml" xref="S3.SS1.p2.5.m5.7.7.4.3.2"><ci id="S3.SS1.p2.5.m5.5.5.cmml" xref="S3.SS1.p2.5.m5.5.5">𝑖</ci><ci id="S3.SS1.p2.5.m5.6.6.cmml" xref="S3.SS1.p2.5.m5.6.6">𝑗</ci></interval></apply></apply><apply id="S3.SS1.p2.5.m5.7.7c.cmml" xref="S3.SS1.p2.5.m5.7.7"><in id="S3.SS1.p2.5.m5.7.7.5.cmml" xref="S3.SS1.p2.5.m5.7.7.5"></in><share href="https://arxiv.org/html/2406.11504v1#S3.SS1.p2.5.m5.7.7.4.cmml" id="S3.SS1.p2.5.m5.7.7d.cmml" xref="S3.SS1.p2.5.m5.7.7"></share><ci id="S3.SS1.p2.5.m5.7.7.6.cmml" xref="S3.SS1.p2.5.m5.7.7.6">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.5.m5.7c">\Phi(e_{i,j};v,t)\coloneqq\phi_{v}^{t}(i,j)\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.5.m5.7d">roman_Φ ( italic_e start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ; italic_v , italic_t ) ≔ italic_ϕ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_i , italic_j ) ∈ blackboard_R</annotation></semantics></math>. We set <math alttext="t" class="ltx_Math" display="inline" id="S3.SS1.p2.6.m6.1"><semantics id="S3.SS1.p2.6.m6.1a"><mi id="S3.SS1.p2.6.m6.1.1" xref="S3.SS1.p2.6.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.6.m6.1b"><ci id="S3.SS1.p2.6.m6.1.1.cmml" xref="S3.SS1.p2.6.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.6.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.6.m6.1d">italic_t</annotation></semantics></math> as the predicted class of <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.p2.7.m7.1"><semantics id="S3.SS1.p2.7.m7.1a"><mi id="S3.SS1.p2.7.m7.1.1" xref="S3.SS1.p2.7.m7.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.7.m7.1b"><ci id="S3.SS1.p2.7.m7.1.1.cmml" xref="S3.SS1.p2.7.m7.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.7.m7.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.7.m7.1d">italic_v</annotation></semantics></math> unless otherwise stated.<span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span>Although explanation methods often provide feature masks, we focus on edge-wise explanations in this work.</span></span></span> By collecting the edge attribution values <math alttext="\phi_{v}^{t}(i,j)" class="ltx_Math" display="inline" id="S3.SS1.p2.8.m8.2"><semantics id="S3.SS1.p2.8.m8.2a"><mrow id="S3.SS1.p2.8.m8.2.3" xref="S3.SS1.p2.8.m8.2.3.cmml"><msubsup id="S3.SS1.p2.8.m8.2.3.2" xref="S3.SS1.p2.8.m8.2.3.2.cmml"><mi id="S3.SS1.p2.8.m8.2.3.2.2.2" xref="S3.SS1.p2.8.m8.2.3.2.2.2.cmml">ϕ</mi><mi id="S3.SS1.p2.8.m8.2.3.2.2.3" xref="S3.SS1.p2.8.m8.2.3.2.2.3.cmml">v</mi><mi id="S3.SS1.p2.8.m8.2.3.2.3" xref="S3.SS1.p2.8.m8.2.3.2.3.cmml">t</mi></msubsup><mo id="S3.SS1.p2.8.m8.2.3.1" xref="S3.SS1.p2.8.m8.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.p2.8.m8.2.3.3.2" xref="S3.SS1.p2.8.m8.2.3.3.1.cmml"><mo id="S3.SS1.p2.8.m8.2.3.3.2.1" stretchy="false" xref="S3.SS1.p2.8.m8.2.3.3.1.cmml">(</mo><mi id="S3.SS1.p2.8.m8.1.1" xref="S3.SS1.p2.8.m8.1.1.cmml">i</mi><mo id="S3.SS1.p2.8.m8.2.3.3.2.2" xref="S3.SS1.p2.8.m8.2.3.3.1.cmml">,</mo><mi id="S3.SS1.p2.8.m8.2.2" xref="S3.SS1.p2.8.m8.2.2.cmml">j</mi><mo id="S3.SS1.p2.8.m8.2.3.3.2.3" stretchy="false" xref="S3.SS1.p2.8.m8.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.8.m8.2b"><apply id="S3.SS1.p2.8.m8.2.3.cmml" xref="S3.SS1.p2.8.m8.2.3"><times id="S3.SS1.p2.8.m8.2.3.1.cmml" xref="S3.SS1.p2.8.m8.2.3.1"></times><apply id="S3.SS1.p2.8.m8.2.3.2.cmml" xref="S3.SS1.p2.8.m8.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.p2.8.m8.2.3.2.1.cmml" xref="S3.SS1.p2.8.m8.2.3.2">superscript</csymbol><apply id="S3.SS1.p2.8.m8.2.3.2.2.cmml" xref="S3.SS1.p2.8.m8.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.p2.8.m8.2.3.2.2.1.cmml" xref="S3.SS1.p2.8.m8.2.3.2">subscript</csymbol><ci id="S3.SS1.p2.8.m8.2.3.2.2.2.cmml" xref="S3.SS1.p2.8.m8.2.3.2.2.2">italic-ϕ</ci><ci id="S3.SS1.p2.8.m8.2.3.2.2.3.cmml" xref="S3.SS1.p2.8.m8.2.3.2.2.3">𝑣</ci></apply><ci id="S3.SS1.p2.8.m8.2.3.2.3.cmml" xref="S3.SS1.p2.8.m8.2.3.2.3">𝑡</ci></apply><interval closure="open" id="S3.SS1.p2.8.m8.2.3.3.1.cmml" xref="S3.SS1.p2.8.m8.2.3.3.2"><ci id="S3.SS1.p2.8.m8.1.1.cmml" xref="S3.SS1.p2.8.m8.1.1">𝑖</ci><ci id="S3.SS1.p2.8.m8.2.2.cmml" xref="S3.SS1.p2.8.m8.2.2">𝑗</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.8.m8.2c">\phi_{v}^{t}(i,j)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.8.m8.2d">italic_ϕ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_i , italic_j )</annotation></semantics></math> for node <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.p2.9.m9.1"><semantics id="S3.SS1.p2.9.m9.1a"><mi id="S3.SS1.p2.9.m9.1.1" xref="S3.SS1.p2.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.9.m9.1b"><ci id="S3.SS1.p2.9.m9.1.1.cmml" xref="S3.SS1.p2.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.9.m9.1d">italic_v</annotation></semantics></math>, we can construct a soft mask over the input graph, denoted as <math alttext="G^{\phi_{v}^{t}}" class="ltx_Math" display="inline" id="S3.SS1.p2.10.m10.1"><semantics id="S3.SS1.p2.10.m10.1a"><msup id="S3.SS1.p2.10.m10.1.1" xref="S3.SS1.p2.10.m10.1.1.cmml"><mi id="S3.SS1.p2.10.m10.1.1.2" xref="S3.SS1.p2.10.m10.1.1.2.cmml">G</mi><msubsup id="S3.SS1.p2.10.m10.1.1.3" xref="S3.SS1.p2.10.m10.1.1.3.cmml"><mi id="S3.SS1.p2.10.m10.1.1.3.2.2" xref="S3.SS1.p2.10.m10.1.1.3.2.2.cmml">ϕ</mi><mi id="S3.SS1.p2.10.m10.1.1.3.2.3" 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id="S3.SS1.p2.10.m10.1.1.3.3.cmml" xref="S3.SS1.p2.10.m10.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.10.m10.1c">G^{\phi_{v}^{t}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.10.m10.1d">italic_G start_POSTSUPERSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="373" id="S3.F2.g1" src="x2.png" width="747"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F2.2.1.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S3.F2.3.2" style="font-size:90%;">Graph pruning performance of FiP for 7 edge attribution methods as well as a random baseline on 4 benchmark datasets. The grey area indicates the performance of random attributions, and the dashed line indicate the test performance without any pruning.</span></figcaption> </figure> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>Fidelity<sup class="ltx_sup" id="S3.SS2.2.1"><span class="ltx_text ltx_font_italic" id="S3.SS2.2.1.1">-</span></sup>-inspired Pruning Framework</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">To utilize the local explanations for graph pruning, we propose <span class="ltx_text ltx_font_bold" id="S3.SS2.p1.1.1">F</span>idelity<sup class="ltx_sup" id="S3.SS2.p1.1.2">-</sup>-<span class="ltx_text ltx_font_bold" id="S3.SS2.p1.1.3">i</span>nspired <span class="ltx_text ltx_font_bold" id="S3.SS2.p1.1.4">P</span>runing, <span class="ltx_text ltx_font_bold" id="S3.SS2.p1.1.5">FiP</span>, a simple yet effective framework that aggregates the edge attribution scores and creates a global edge mask (see Figure <a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_tag">1</span></a>). The step-by-step description of FiP is as follows:</p> <ol class="ltx_enumerate" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.1">Explanations (i.e., local soft edge masks <math alttext="G^{\phi_{v}^{t}}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics id="S3.I1.i1.p1.1.m1.1a"><msup id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.2.cmml">G</mi><msubsup id="S3.I1.i1.p1.1.m1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.3.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.3.2.2" xref="S3.I1.i1.p1.1.m1.1.1.3.2.2.cmml">ϕ</mi><mi id="S3.I1.i1.p1.1.m1.1.1.3.2.3" xref="S3.I1.i1.p1.1.m1.1.1.3.2.3.cmml">v</mi><mi id="S3.I1.i1.p1.1.m1.1.1.3.3" xref="S3.I1.i1.p1.1.m1.1.1.3.3.cmml">t</mi></msubsup></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><apply 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encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">italic_G start_POSTSUPERSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>) for each target node (yellow nodes in the figure) obtained by a specific edge attribution method are taken as input.</p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.5">The local soft edge masks <math alttext="G^{\phi_{v}^{t}}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.1"><semantics id="S3.I1.i2.p1.1.m1.1a"><msup id="S3.I1.i2.p1.1.m1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.2.cmml">G</mi><msubsup id="S3.I1.i2.p1.1.m1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.3.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.3.2.2" xref="S3.I1.i2.p1.1.m1.1.1.3.2.2.cmml">ϕ</mi><mi id="S3.I1.i2.p1.1.m1.1.1.3.2.3" xref="S3.I1.i2.p1.1.m1.1.1.3.2.3.cmml">v</mi><mi id="S3.I1.i2.p1.1.m1.1.1.3.3" xref="S3.I1.i2.p1.1.m1.1.1.3.3.cmml">t</mi></msubsup></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.1b"><apply id="S3.I1.i2.p1.1.m1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1">superscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2">𝐺</ci><apply id="S3.I1.i2.p1.1.m1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.3.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3">superscript</csymbol><apply id="S3.I1.i2.p1.1.m1.1.1.3.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.3.2.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.3.2.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.2.2">italic-ϕ</ci><ci id="S3.I1.i2.p1.1.m1.1.1.3.2.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.2.3">𝑣</ci></apply><ci id="S3.I1.i2.p1.1.m1.1.1.3.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.1c">G^{\phi_{v}^{t}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.1d">italic_G start_POSTSUPERSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> are aggregated over all <math alttext="v\in\mathcal{V}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.2.m2.1"><semantics id="S3.I1.i2.p1.2.m2.1a"><mrow id="S3.I1.i2.p1.2.m2.1.1" xref="S3.I1.i2.p1.2.m2.1.1.cmml"><mi id="S3.I1.i2.p1.2.m2.1.1.2" xref="S3.I1.i2.p1.2.m2.1.1.2.cmml">v</mi><mo id="S3.I1.i2.p1.2.m2.1.1.1" xref="S3.I1.i2.p1.2.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.I1.i2.p1.2.m2.1.1.3" xref="S3.I1.i2.p1.2.m2.1.1.3.cmml">𝒱</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.2.m2.1b"><apply id="S3.I1.i2.p1.2.m2.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1"><in id="S3.I1.i2.p1.2.m2.1.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.1"></in><ci id="S3.I1.i2.p1.2.m2.1.1.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2">𝑣</ci><ci id="S3.I1.i2.p1.2.m2.1.1.3.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3">𝒱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.2.m2.1c">v\in\mathcal{V}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.2.m2.1d">italic_v ∈ caligraphic_V</annotation></semantics></math> via summing or averaging the edge attributions to generate <math alttext="G^{\phi}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.3.m3.1"><semantics id="S3.I1.i2.p1.3.m3.1a"><msup id="S3.I1.i2.p1.3.m3.1.1" xref="S3.I1.i2.p1.3.m3.1.1.cmml"><mi id="S3.I1.i2.p1.3.m3.1.1.2" xref="S3.I1.i2.p1.3.m3.1.1.2.cmml">G</mi><mi id="S3.I1.i2.p1.3.m3.1.1.3" xref="S3.I1.i2.p1.3.m3.1.1.3.cmml">ϕ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.3.m3.1b"><apply id="S3.I1.i2.p1.3.m3.1.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.3.m3.1.1.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1">superscript</csymbol><ci id="S3.I1.i2.p1.3.m3.1.1.2.cmml" xref="S3.I1.i2.p1.3.m3.1.1.2">𝐺</ci><ci id="S3.I1.i2.p1.3.m3.1.1.3.cmml" xref="S3.I1.i2.p1.3.m3.1.1.3">italic-ϕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.3.m3.1c">G^{\phi}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.3.m3.1d">italic_G start_POSTSUPERSCRIPT italic_ϕ end_POSTSUPERSCRIPT</annotation></semantics></math> (i.e., turning <math alttext="\phi_{v}^{t}(i,j)" class="ltx_Math" display="inline" id="S3.I1.i2.p1.4.m4.2"><semantics id="S3.I1.i2.p1.4.m4.2a"><mrow id="S3.I1.i2.p1.4.m4.2.3" xref="S3.I1.i2.p1.4.m4.2.3.cmml"><msubsup id="S3.I1.i2.p1.4.m4.2.3.2" xref="S3.I1.i2.p1.4.m4.2.3.2.cmml"><mi id="S3.I1.i2.p1.4.m4.2.3.2.2.2" xref="S3.I1.i2.p1.4.m4.2.3.2.2.2.cmml">ϕ</mi><mi id="S3.I1.i2.p1.4.m4.2.3.2.2.3" xref="S3.I1.i2.p1.4.m4.2.3.2.2.3.cmml">v</mi><mi id="S3.I1.i2.p1.4.m4.2.3.2.3" xref="S3.I1.i2.p1.4.m4.2.3.2.3.cmml">t</mi></msubsup><mo id="S3.I1.i2.p1.4.m4.2.3.1" xref="S3.I1.i2.p1.4.m4.2.3.1.cmml">⁢</mo><mrow id="S3.I1.i2.p1.4.m4.2.3.3.2" xref="S3.I1.i2.p1.4.m4.2.3.3.1.cmml"><mo id="S3.I1.i2.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S3.I1.i2.p1.4.m4.2.3.3.1.cmml">(</mo><mi id="S3.I1.i2.p1.4.m4.1.1" xref="S3.I1.i2.p1.4.m4.1.1.cmml">i</mi><mo id="S3.I1.i2.p1.4.m4.2.3.3.2.2" xref="S3.I1.i2.p1.4.m4.2.3.3.1.cmml">,</mo><mi id="S3.I1.i2.p1.4.m4.2.2" xref="S3.I1.i2.p1.4.m4.2.2.cmml">j</mi><mo id="S3.I1.i2.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S3.I1.i2.p1.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.4.m4.2b"><apply id="S3.I1.i2.p1.4.m4.2.3.cmml" xref="S3.I1.i2.p1.4.m4.2.3"><times id="S3.I1.i2.p1.4.m4.2.3.1.cmml" xref="S3.I1.i2.p1.4.m4.2.3.1"></times><apply id="S3.I1.i2.p1.4.m4.2.3.2.cmml" xref="S3.I1.i2.p1.4.m4.2.3.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.4.m4.2.3.2.1.cmml" xref="S3.I1.i2.p1.4.m4.2.3.2">superscript</csymbol><apply id="S3.I1.i2.p1.4.m4.2.3.2.2.cmml" xref="S3.I1.i2.p1.4.m4.2.3.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.4.m4.2.3.2.2.1.cmml" xref="S3.I1.i2.p1.4.m4.2.3.2">subscript</csymbol><ci id="S3.I1.i2.p1.4.m4.2.3.2.2.2.cmml" xref="S3.I1.i2.p1.4.m4.2.3.2.2.2">italic-ϕ</ci><ci id="S3.I1.i2.p1.4.m4.2.3.2.2.3.cmml" xref="S3.I1.i2.p1.4.m4.2.3.2.2.3">𝑣</ci></apply><ci id="S3.I1.i2.p1.4.m4.2.3.2.3.cmml" xref="S3.I1.i2.p1.4.m4.2.3.2.3">𝑡</ci></apply><interval closure="open" id="S3.I1.i2.p1.4.m4.2.3.3.1.cmml" xref="S3.I1.i2.p1.4.m4.2.3.3.2"><ci id="S3.I1.i2.p1.4.m4.1.1.cmml" xref="S3.I1.i2.p1.4.m4.1.1">𝑖</ci><ci id="S3.I1.i2.p1.4.m4.2.2.cmml" xref="S3.I1.i2.p1.4.m4.2.2">𝑗</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.4.m4.2c">\phi_{v}^{t}(i,j)</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.4.m4.2d">italic_ϕ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_i , italic_j )</annotation></semantics></math> into a global soft edge mask <math alttext="\phi(i,j)" class="ltx_Math" display="inline" id="S3.I1.i2.p1.5.m5.2"><semantics id="S3.I1.i2.p1.5.m5.2a"><mrow id="S3.I1.i2.p1.5.m5.2.3" xref="S3.I1.i2.p1.5.m5.2.3.cmml"><mi id="S3.I1.i2.p1.5.m5.2.3.2" xref="S3.I1.i2.p1.5.m5.2.3.2.cmml">ϕ</mi><mo id="S3.I1.i2.p1.5.m5.2.3.1" xref="S3.I1.i2.p1.5.m5.2.3.1.cmml">⁢</mo><mrow id="S3.I1.i2.p1.5.m5.2.3.3.2" xref="S3.I1.i2.p1.5.m5.2.3.3.1.cmml"><mo id="S3.I1.i2.p1.5.m5.2.3.3.2.1" stretchy="false" xref="S3.I1.i2.p1.5.m5.2.3.3.1.cmml">(</mo><mi id="S3.I1.i2.p1.5.m5.1.1" xref="S3.I1.i2.p1.5.m5.1.1.cmml">i</mi><mo id="S3.I1.i2.p1.5.m5.2.3.3.2.2" xref="S3.I1.i2.p1.5.m5.2.3.3.1.cmml">,</mo><mi id="S3.I1.i2.p1.5.m5.2.2" 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style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S3.I1.i3.p1"> <p class="ltx_p" id="S3.I1.i3.p1.1">Hard edge masks are generated via discarding edges with the lowest aggregated edge attribution scores <math alttext="\phi(i,j)" class="ltx_Math" display="inline" id="S3.I1.i3.p1.1.m1.2"><semantics id="S3.I1.i3.p1.1.m1.2a"><mrow id="S3.I1.i3.p1.1.m1.2.3" xref="S3.I1.i3.p1.1.m1.2.3.cmml"><mi id="S3.I1.i3.p1.1.m1.2.3.2" xref="S3.I1.i3.p1.1.m1.2.3.2.cmml">ϕ</mi><mo id="S3.I1.i3.p1.1.m1.2.3.1" xref="S3.I1.i3.p1.1.m1.2.3.1.cmml">⁢</mo><mrow id="S3.I1.i3.p1.1.m1.2.3.3.2" xref="S3.I1.i3.p1.1.m1.2.3.3.1.cmml"><mo id="S3.I1.i3.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S3.I1.i3.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S3.I1.i3.p1.1.m1.1.1" xref="S3.I1.i3.p1.1.m1.1.1.cmml">i</mi><mo id="S3.I1.i3.p1.1.m1.2.3.3.2.2" xref="S3.I1.i3.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.I1.i3.p1.1.m1.2.2" xref="S3.I1.i3.p1.1.m1.2.2.cmml">j</mi><mo 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expect that the performance gradually decreases when we prune more edges (eventually resulting in information loss of the input), but a <span class="ltx_text ltx_font_italic" id="S3.SS2.p2.5.1">good</span> global soft edge mask <math alttext="G^{\phi}" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><msup id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml"><mi id="S3.SS2.p2.1.m1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.2.cmml">G</mi><mi id="S3.SS2.p2.1.m1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.3.cmml">ϕ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><apply id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1">superscript</csymbol><ci id="S3.SS2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.2">𝐺</ci><ci id="S3.SS2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3">italic-ϕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">G^{\phi}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_G start_POSTSUPERSCRIPT italic_ϕ end_POSTSUPERSCRIPT</annotation></semantics></math> will assign a low score to noisy edges and a higher score to edges that severely hurt the performance when removed. Note that the process of FiP can be interpreted as a global version of fidelity<sup class="ltx_sup" id="S3.SS2.p2.5.2">-</sup>, since both FiP and fidelity<sup class="ltx_sup" id="S3.SS2.p2.5.3">-</sup> discard unimportant edges in <math alttext="G^{\phi_{v}^{t}}" class="ltx_Math" display="inline" id="S3.SS2.p2.4.m4.1"><semantics id="S3.SS2.p2.4.m4.1a"><msup id="S3.SS2.p2.4.m4.1.1" xref="S3.SS2.p2.4.m4.1.1.cmml"><mi id="S3.SS2.p2.4.m4.1.1.2" xref="S3.SS2.p2.4.m4.1.1.2.cmml">G</mi><msubsup id="S3.SS2.p2.4.m4.1.1.3" xref="S3.SS2.p2.4.m4.1.1.3.cmml"><mi id="S3.SS2.p2.4.m4.1.1.3.2.2" xref="S3.SS2.p2.4.m4.1.1.3.2.2.cmml">ϕ</mi><mi id="S3.SS2.p2.4.m4.1.1.3.2.3" xref="S3.SS2.p2.4.m4.1.1.3.2.3.cmml">v</mi><mi id="S3.SS2.p2.4.m4.1.1.3.3" xref="S3.SS2.p2.4.m4.1.1.3.3.cmml">t</mi></msubsup></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.4.m4.1b"><apply id="S3.SS2.p2.4.m4.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.4.m4.1.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1">superscript</csymbol><ci id="S3.SS2.p2.4.m4.1.1.2.cmml" xref="S3.SS2.p2.4.m4.1.1.2">𝐺</ci><apply id="S3.SS2.p2.4.m4.1.1.3.cmml" xref="S3.SS2.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p2.4.m4.1.1.3.1.cmml" xref="S3.SS2.p2.4.m4.1.1.3">superscript</csymbol><apply id="S3.SS2.p2.4.m4.1.1.3.2.cmml" xref="S3.SS2.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p2.4.m4.1.1.3.2.1.cmml" xref="S3.SS2.p2.4.m4.1.1.3">subscript</csymbol><ci id="S3.SS2.p2.4.m4.1.1.3.2.2.cmml" xref="S3.SS2.p2.4.m4.1.1.3.2.2">italic-ϕ</ci><ci id="S3.SS2.p2.4.m4.1.1.3.2.3.cmml" xref="S3.SS2.p2.4.m4.1.1.3.2.3">𝑣</ci></apply><ci id="S3.SS2.p2.4.m4.1.1.3.3.cmml" xref="S3.SS2.p2.4.m4.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.4.m4.1c">G^{\phi_{v}^{t}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.4.m4.1d">italic_G start_POSTSUPERSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> or <math alttext="G^{\phi}" class="ltx_Math" display="inline" id="S3.SS2.p2.5.m5.1"><semantics id="S3.SS2.p2.5.m5.1a"><msup id="S3.SS2.p2.5.m5.1.1" xref="S3.SS2.p2.5.m5.1.1.cmml"><mi id="S3.SS2.p2.5.m5.1.1.2" xref="S3.SS2.p2.5.m5.1.1.2.cmml">G</mi><mi id="S3.SS2.p2.5.m5.1.1.3" xref="S3.SS2.p2.5.m5.1.1.3.cmml">ϕ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.5.m5.1b"><apply id="S3.SS2.p2.5.m5.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.5.m5.1.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1">superscript</csymbol><ci id="S3.SS2.p2.5.m5.1.1.2.cmml" xref="S3.SS2.p2.5.m5.1.1.2">𝐺</ci><ci id="S3.SS2.p2.5.m5.1.1.3.cmml" xref="S3.SS2.p2.5.m5.1.1.3">italic-ϕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.5.m5.1c">G^{\phi}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.5.m5.1d">italic_G start_POSTSUPERSCRIPT italic_ϕ end_POSTSUPERSCRIPT</annotation></semantics></math>. Although there may be more sophisticated methods to aggregate local edge attributions aside summation and averaging them, we leave the design of such methods as future work.</p> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Empirical Observations</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this section, we evaluate the graph pruning performance of FiP using various GNN explanation methods.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Basic Settings</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">In our experiments, we train a 2-layer GAT model <cite class="ltx_cite ltx_citemacro_cite">Velickovic <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib21" title="">2018</a>)</cite> on 4 benchmark datasets, BA-Shapes <cite class="ltx_cite ltx_citemacro_cite">Ying <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib27" title="">2019</a>)</cite>, Cora, Citeseer, and Pubmed <cite class="ltx_cite ltx_citemacro_cite">Yang <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib25" title="">2016</a>)</cite>, where the model achieves test performance of 0.9857, 0.8531, 0.7389, and 0.8056, respectively. As mentioned, we only consider average or summation when aggregating local edge attributions in FiP.</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Explanation Methods</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">We adopt the following seven edge attribution methods commonly used in the literature.</p> </div> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Attention (Att).</h4> <div class="ltx_para" id="S4.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p1.1">The edge attention weights are treated as a proxy of edge attribution. We average the attention weights over all layers, similarly as in <cite class="ltx_cite ltx_citemacro_cite">Ying <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib27" title="">2019</a>); Sánchez-Lengeling <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib16" title="">2020</a>)</cite>.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Saliency (SA).</h4> <div class="ltx_para" id="S4.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="S4.SS2.SSS0.Px2.p1.1"><cite class="ltx_cite ltx_citemacro_cite">Simonyan <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib17" title="">2014</a>)</cite> is the absolute value of the gradient with respect to the input.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Integrated Gradient (IG).</h4> <div class="ltx_para" id="S4.SS2.SSS0.Px3.p1"> <p class="ltx_p" id="S4.SS2.SSS0.Px3.p1.1"><cite class="ltx_cite ltx_citemacro_cite">Sundararajan <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib20" title="">2017</a>)</cite> calculates an edge attribution score via approximating the integral of gradients of the model’s output with respect to the input along the path from a baseline to the input.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">Guided Backpropagation (GB).</h4> <div class="ltx_para" id="S4.SS2.SSS0.Px4.p1"> <p class="ltx_p" id="S4.SS2.SSS0.Px4.p1.1"><cite class="ltx_cite ltx_citemacro_cite">Springenberg <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib18" title="">2015</a>)</cite> is similar to SA, except that the negative gradients are clipped during backpropagation, basically focusing on features with an excitation effect.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px5"> <h4 class="ltx_title ltx_title_paragraph">GNNExplainer (GNNEx).</h4> <div class="ltx_para" id="S4.SS2.SSS0.Px5.p1"> <p class="ltx_p" id="S4.SS2.SSS0.Px5.p1.1"><cite class="ltx_cite ltx_citemacro_cite">Ying <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib27" title="">2019</a>)</cite> is the most widely used explanation method tailored to GNNs, where it identifies a local subgraph most relevant to the model’s predictions by maximizing the mutual information.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px6"> <h4 class="ltx_title ltx_title_paragraph">PGExplainer (PGEx).</h4> <div class="ltx_para" id="S4.SS2.SSS0.Px6.p1"> <p class="ltx_p" id="S4.SS2.SSS0.Px6.p1.1"><cite class="ltx_cite ltx_citemacro_cite">Luo <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib9" title="">2020</a>)</cite> trains a separate parameterized mask predictor to generate edge masks that identify edges important to the prediction.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px7"> <h4 class="ltx_title ltx_title_paragraph">FastDnX (FDnX).</h4> <div class="ltx_para" id="S4.SS2.SSS0.Px7.p1"> <p class="ltx_p" id="S4.SS2.SSS0.Px7.p1.1"><cite class="ltx_cite ltx_citemacro_cite">Pereira <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib11" title="">2023</a>)</cite> is a recently proposed method for explaining GNNs, where it basically relies on a surrogate SGC model <cite class="ltx_cite ltx_citemacro_cite">Wu <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib22" title="">2019</a>)</cite> to explain the model’s behavior.</p> </div> <figure class="ltx_figure" id="S4.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="125" id="S4.F3.g1" src="x3.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.2.1.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text" id="S4.F3.3.2" style="font-size:90%;">Visualizations of graph pruning using different edge attribution methods by removing 50% of the edges from the original graph.</span></figcaption> </figure> </section> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Experimental Results on Graph Pruning</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.3">We show the experimental results of using various explanation methods in FiP with our findings. As our main result, Figure <a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S3.F2" title="Figure 2 ‣ 3.1 Basic Notations and Problem Settings ‣ 3 Methodology ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_tag">2</span></a> shows the test performance when we remove <math alttext="p" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.1"><semantics id="S4.SS3.p1.1.m1.1a"><mi id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.1b"><ci id="S4.SS3.p1.1.m1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.1d">italic_p</annotation></semantics></math>% of the edges with the lowest global soft edge mask <math alttext="G^{\phi}" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m2.1"><semantics id="S4.SS3.p1.2.m2.1a"><msup id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml"><mi id="S4.SS3.p1.2.m2.1.1.2" xref="S4.SS3.p1.2.m2.1.1.2.cmml">G</mi><mi id="S4.SS3.p1.2.m2.1.1.3" xref="S4.SS3.p1.2.m2.1.1.3.cmml">ϕ</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><apply id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.2.m2.1.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1">superscript</csymbol><ci id="S4.SS3.p1.2.m2.1.1.2.cmml" xref="S4.SS3.p1.2.m2.1.1.2">𝐺</ci><ci id="S4.SS3.p1.2.m2.1.1.3.cmml" xref="S4.SS3.p1.2.m2.1.1.3">italic-ϕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">G^{\phi}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m2.1d">italic_G start_POSTSUPERSCRIPT italic_ϕ end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="p\in\{5,10,\cdots,100\}" class="ltx_Math" display="inline" id="S4.SS3.p1.3.m3.4"><semantics id="S4.SS3.p1.3.m3.4a"><mrow id="S4.SS3.p1.3.m3.4.5" xref="S4.SS3.p1.3.m3.4.5.cmml"><mi id="S4.SS3.p1.3.m3.4.5.2" xref="S4.SS3.p1.3.m3.4.5.2.cmml">p</mi><mo id="S4.SS3.p1.3.m3.4.5.1" xref="S4.SS3.p1.3.m3.4.5.1.cmml">∈</mo><mrow id="S4.SS3.p1.3.m3.4.5.3.2" xref="S4.SS3.p1.3.m3.4.5.3.1.cmml"><mo id="S4.SS3.p1.3.m3.4.5.3.2.1" stretchy="false" xref="S4.SS3.p1.3.m3.4.5.3.1.cmml">{</mo><mn id="S4.SS3.p1.3.m3.1.1" xref="S4.SS3.p1.3.m3.1.1.cmml">5</mn><mo id="S4.SS3.p1.3.m3.4.5.3.2.2" xref="S4.SS3.p1.3.m3.4.5.3.1.cmml">,</mo><mn id="S4.SS3.p1.3.m3.2.2" xref="S4.SS3.p1.3.m3.2.2.cmml">10</mn><mo id="S4.SS3.p1.3.m3.4.5.3.2.3" xref="S4.SS3.p1.3.m3.4.5.3.1.cmml">,</mo><mi id="S4.SS3.p1.3.m3.3.3" mathvariant="normal" xref="S4.SS3.p1.3.m3.3.3.cmml">⋯</mi><mo id="S4.SS3.p1.3.m3.4.5.3.2.4" xref="S4.SS3.p1.3.m3.4.5.3.1.cmml">,</mo><mn id="S4.SS3.p1.3.m3.4.4" xref="S4.SS3.p1.3.m3.4.4.cmml">100</mn><mo id="S4.SS3.p1.3.m3.4.5.3.2.5" stretchy="false" xref="S4.SS3.p1.3.m3.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.3.m3.4b"><apply id="S4.SS3.p1.3.m3.4.5.cmml" xref="S4.SS3.p1.3.m3.4.5"><in id="S4.SS3.p1.3.m3.4.5.1.cmml" xref="S4.SS3.p1.3.m3.4.5.1"></in><ci id="S4.SS3.p1.3.m3.4.5.2.cmml" xref="S4.SS3.p1.3.m3.4.5.2">𝑝</ci><set id="S4.SS3.p1.3.m3.4.5.3.1.cmml" xref="S4.SS3.p1.3.m3.4.5.3.2"><cn id="S4.SS3.p1.3.m3.1.1.cmml" type="integer" xref="S4.SS3.p1.3.m3.1.1">5</cn><cn id="S4.SS3.p1.3.m3.2.2.cmml" type="integer" xref="S4.SS3.p1.3.m3.2.2">10</cn><ci id="S4.SS3.p1.3.m3.3.3.cmml" xref="S4.SS3.p1.3.m3.3.3">⋯</ci><cn id="S4.SS3.p1.3.m3.4.4.cmml" type="integer" xref="S4.SS3.p1.3.m3.4.4">100</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.3.m3.4c">p\in\{5,10,\cdots,100\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.3.m3.4d">italic_p ∈ { 5 , 10 , ⋯ , 100 }</annotation></semantics></math>. For comparison, we also show the performance when we use random attributions (averaged over 10 independent trials), depicted as the grey area. From Figure <a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S3.F2" title="Figure 2 ‣ 3.1 Basic Notations and Problem Settings ‣ 3 Methodology ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_tag">2</span></a>, we make the following observations:</p> <ul class="ltx_itemize" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.1">Overall, edge attribution methods reveal their potential in edge pruning. For example, on BAShapes, we can delete half of the edges using IG only with the performance degradation of less than <math alttext="4\%" class="ltx_Math" display="inline" id="S4.I1.i1.p1.1.m1.1"><semantics id="S4.I1.i1.p1.1.m1.1a"><mrow id="S4.I1.i1.p1.1.m1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.cmml"><mn id="S4.I1.i1.p1.1.m1.1.1.2" xref="S4.I1.i1.p1.1.m1.1.1.2.cmml">4</mn><mo id="S4.I1.i1.p1.1.m1.1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.1.m1.1b"><apply id="S4.I1.i1.p1.1.m1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1"><csymbol cd="latexml" id="S4.I1.i1.p1.1.m1.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1.1">percent</csymbol><cn id="S4.I1.i1.p1.1.m1.1.1.2.cmml" type="integer" xref="S4.I1.i1.p1.1.m1.1.1.2">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.1c">4\%</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.1d">4 %</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.1">Despite being rare, there are cases where the accuracy after pruning outperforms the original test accuracy (e.g., on Citeseer, 5% pruning with FastDnX by summation).</p> </div> </li> </ul> <p class="ltx_p" id="S4.SS3.p1.4">Table <a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.T1" title="Table 1 ‣ 4.3 Experimental Results on Graph Pruning ‣ 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_tag">1</span></a> summarizes the rank in performance averaged over 20 different pruning cases for 7 edge attribution methods (as well as a random baseline) on 4 benchmark datasets. Our findings are as follows.</p> <ul class="ltx_itemize" id="S4.I2"> <li class="ltx_item" id="S4.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I2.i1.p1"> <p class="ltx_p" id="S4.I2.i1.p1.1">The best edge attribution method for graph pruning tends be one of Att, SA, and IG. This is quite unexpected, as SA and IG are ‘general’ XAI methods (i.e., the ones not tailored to GNNs). In a similar regard, GNNExplainer tends to exhibit the worst performance for most of the datasets.</p> </div> </li> <li class="ltx_item" id="S4.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I2.i2.p1"> <p class="ltx_p" id="S4.I2.i2.p1.1">There are no significant differences between using summation and average for aggregating local edge attributions in FiP.</p> </div> </li> </ul> </div> <figure class="ltx_table" id="S4.T1"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T1.2" style="width:433.6pt;height:282.1pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(92.3pt,-60.0pt) scale(1.74109260427206,1.74109260427206) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S4.T1.2.1"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T1.2.1.1.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="S4.T1.2.1.1.1.1"><span class="ltx_text" id="S4.T1.2.1.1.1.1.1" style="font-size:90%;">Method</span></th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T1.2.1.1.1.2"><span class="ltx_text" id="S4.T1.2.1.1.1.2.1" style="font-size:90%;">BAShapes</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T1.2.1.1.1.3"><span class="ltx_text" id="S4.T1.2.1.1.1.3.1" style="font-size:90%;">Cora</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T1.2.1.1.1.4"><span class="ltx_text" id="S4.T1.2.1.1.1.4.1" style="font-size:90%;">Citeseer</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T1.2.1.1.1.5"><span class="ltx_text" id="S4.T1.2.1.1.1.5.1" style="font-size:90%;">Pubmed</span></td> </tr> <tr class="ltx_tr" id="S4.T1.2.1.2.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S4.T1.2.1.2.2.1"><span class="ltx_text" id="S4.T1.2.1.2.2.1.1" style="font-size:90%;">Att</span></th> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.2.1.2.2.2"><span class="ltx_text" id="S4.T1.2.1.2.2.2.1" style="font-size:90%;">3.63/2.26</span></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.2.1.2.2.3"><span class="ltx_text" id="S4.T1.2.1.2.2.3.1" style="font-size:90%;">4.89/4.11</span></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.2.1.2.2.4"> <span class="ltx_text ltx_font_bold" id="S4.T1.2.1.2.2.4.1" style="font-size:90%;">1.74</span><span class="ltx_text" id="S4.T1.2.1.2.2.4.2" style="font-size:90%;">/</span><span class="ltx_text ltx_font_bold" id="S4.T1.2.1.2.2.4.3" style="font-size:90%;">1.89</span> </td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.2.1.2.2.5"><span class="ltx_text" id="S4.T1.2.1.2.2.5.1" style="font-size:90%;">6.32/6.58</span></td> </tr> <tr class="ltx_tr" id="S4.T1.2.1.3.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T1.2.1.3.3.1"><span class="ltx_text" id="S4.T1.2.1.3.3.1.1" style="font-size:90%;">SA</span></th> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.3.3.2"> <span class="ltx_text" id="S4.T1.2.1.3.3.2.1" style="font-size:90%;">2.42/</span><span class="ltx_text ltx_font_bold" id="S4.T1.2.1.3.3.2.2" style="font-size:90%;">1.58</span> </td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.3.3.3"><span class="ltx_text" id="S4.T1.2.1.3.3.3.1" style="font-size:90%;">2.58/2.47</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.3.3.4"><span class="ltx_text" id="S4.T1.2.1.3.3.4.1" style="font-size:90%;">1.89/2.11</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.3.3.5"> <span class="ltx_text ltx_font_bold" id="S4.T1.2.1.3.3.5.1" style="font-size:90%;">2.21</span><span class="ltx_text" id="S4.T1.2.1.3.3.5.2" style="font-size:90%;">/</span><span class="ltx_text ltx_font_bold" id="S4.T1.2.1.3.3.5.3" style="font-size:90%;">2.11</span> </td> </tr> <tr class="ltx_tr" id="S4.T1.2.1.4.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T1.2.1.4.4.1"><span class="ltx_text" id="S4.T1.2.1.4.4.1.1" style="font-size:90%;">IG</span></th> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.4.4.2"> <span class="ltx_text ltx_font_bold" id="S4.T1.2.1.4.4.2.1" style="font-size:90%;">1.53</span><span class="ltx_text" id="S4.T1.2.1.4.4.2.2" style="font-size:90%;">/2.58</span> </td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.4.4.3"> <span class="ltx_text ltx_font_bold" id="S4.T1.2.1.4.4.3.1" style="font-size:90%;">1.95</span><span class="ltx_text" id="S4.T1.2.1.4.4.3.2" style="font-size:90%;">/</span><span class="ltx_text ltx_font_bold" id="S4.T1.2.1.4.4.3.3" style="font-size:90%;">1.84</span> </td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.4.4.4"><span class="ltx_text" id="S4.T1.2.1.4.4.4.1" style="font-size:90%;">5.32/4.26</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.4.4.5"><span class="ltx_text" id="S4.T1.2.1.4.4.5.1" style="font-size:90%;">6.58/6.58</span></td> </tr> <tr class="ltx_tr" id="S4.T1.2.1.5.5"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T1.2.1.5.5.1"><span class="ltx_text" id="S4.T1.2.1.5.5.1.1" style="font-size:90%;">GB</span></th> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.5.5.2"><span class="ltx_text" id="S4.T1.2.1.5.5.2.1" style="font-size:90%;">3.84/3.68</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.5.5.3"><span class="ltx_text" id="S4.T1.2.1.5.5.3.1" style="font-size:90%;">2.16/2.26</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.5.5.4"><span class="ltx_text" id="S4.T1.2.1.5.5.4.1" style="font-size:90%;">4.42/5.05</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.5.5.5"><span class="ltx_text" id="S4.T1.2.1.5.5.5.1" style="font-size:90%;">5.16/5.42</span></td> </tr> <tr class="ltx_tr" id="S4.T1.2.1.6.6"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T1.2.1.6.6.1"><span class="ltx_text" id="S4.T1.2.1.6.6.1.1" style="font-size:90%;">GNNEx</span></th> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.6.6.2"><span class="ltx_text" id="S4.T1.2.1.6.6.2.1" style="font-size:90%;">5.11/7.42</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.6.6.3"><span class="ltx_text" id="S4.T1.2.1.6.6.3.1" style="font-size:90%;">7.42/7.95</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.6.6.4"><span class="ltx_text" id="S4.T1.2.1.6.6.4.1" style="font-size:90%;">7.68/7.79</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.6.6.5"><span class="ltx_text" id="S4.T1.2.1.6.6.5.1" style="font-size:90%;">7.26/6.74</span></td> </tr> <tr class="ltx_tr" id="S4.T1.2.1.7.7"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T1.2.1.7.7.1"><span class="ltx_text" id="S4.T1.2.1.7.7.1.1" style="font-size:90%;">PGEx</span></th> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.7.7.2"><span class="ltx_text" id="S4.T1.2.1.7.7.2.1" style="font-size:90%;">5.58/4.84</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.7.7.3"><span class="ltx_text" id="S4.T1.2.1.7.7.3.1" style="font-size:90%;">5.84/5.58</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.7.7.4"><span class="ltx_text" id="S4.T1.2.1.7.7.4.1" style="font-size:90%;">4.00/3.84</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.7.7.5"> <span class="ltx_text ltx_font_bold" id="S4.T1.2.1.7.7.5.1" style="font-size:90%;">2.21</span><span class="ltx_text" id="S4.T1.2.1.7.7.5.2" style="font-size:90%;">/2.42</span> </td> </tr> <tr class="ltx_tr" id="S4.T1.2.1.8.8"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T1.2.1.8.8.1"><span class="ltx_text" id="S4.T1.2.1.8.8.1.1" style="font-size:90%;">FDnX</span></th> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.8.8.2"><span class="ltx_text" id="S4.T1.2.1.8.8.2.1" style="font-size:90%;">5.32/6.16</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.8.8.3"><span class="ltx_text" id="S4.T1.2.1.8.8.3.1" style="font-size:90%;">4.53/4.42</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.8.8.4"><span class="ltx_text" id="S4.T1.2.1.8.8.4.1" style="font-size:90%;">4.68/4.53</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.2.1.8.8.5"><span class="ltx_text" id="S4.T1.2.1.8.8.5.1" style="font-size:90%;">3.16/3.05</span></td> </tr> <tr class="ltx_tr" id="S4.T1.2.1.9.9"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb" id="S4.T1.2.1.9.9.1"><span class="ltx_text" id="S4.T1.2.1.9.9.1.1" style="font-size:90%;">Random</span></th> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.2.1.9.9.2"><span class="ltx_text" id="S4.T1.2.1.9.9.2.1" style="font-size:90%;">7.53/6.47</span></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.2.1.9.9.3"><span class="ltx_text" id="S4.T1.2.1.9.9.3.1" style="font-size:90%;">6.42/6.68</span></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.2.1.9.9.4"><span class="ltx_text" id="S4.T1.2.1.9.9.4.1" style="font-size:90%;">5.47/5.68</span></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.2.1.9.9.5"><span class="ltx_text" id="S4.T1.2.1.9.9.5.1" style="font-size:90%;">3.05/3.00</span></td> </tr> </tbody> </table> </span></div> <figcaption class="ltx_caption ltx_centering" style="font-size:90%;"><span class="ltx_tag ltx_tag_table">Table 1: </span>Rank in performance averaged over 20 different pruning percentages (average/summation).</figcaption> </figure> </section> <section class="ltx_subsection" id="S4.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.4 </span>Visualizations of Graph Pruning</h3> <div class="ltx_para" id="S4.SS4.p1"> <p class="ltx_p" id="S4.SS4.p1.1">We observe the effect of using different edge attributions for graph pruning by visualizing the resulting graphs. Figure <a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.F3" title="Figure 3 ‣ FastDnX (FDnX). ‣ 4.2 Explanation Methods ‣ 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_tag">3</span></a> visualizes results on BA-Shapes for a specific target node (yellow nodes) using FiP with summation. We choose BA-Shapes since it is the only dataset that contains ground-truth explanation edges (see blue arrows). Note that only ground-truth explanation edges are meaningful in the sense that they construct house-shapes (i.e., motifs), influencing the ground-truth node labels included in the motif (dark blue and yellow nodes). The remaining edges do not have any semantic meanings and merely serve as a backbone structure of the graph. By setting <math alttext="p=50" class="ltx_Math" display="inline" id="S4.SS4.p1.1.m1.1"><semantics id="S4.SS4.p1.1.m1.1a"><mrow id="S4.SS4.p1.1.m1.1.1" xref="S4.SS4.p1.1.m1.1.1.cmml"><mi id="S4.SS4.p1.1.m1.1.1.2" xref="S4.SS4.p1.1.m1.1.1.2.cmml">p</mi><mo id="S4.SS4.p1.1.m1.1.1.1" xref="S4.SS4.p1.1.m1.1.1.1.cmml">=</mo><mn id="S4.SS4.p1.1.m1.1.1.3" xref="S4.SS4.p1.1.m1.1.1.3.cmml">50</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.1.m1.1b"><apply id="S4.SS4.p1.1.m1.1.1.cmml" xref="S4.SS4.p1.1.m1.1.1"><eq id="S4.SS4.p1.1.m1.1.1.1.cmml" xref="S4.SS4.p1.1.m1.1.1.1"></eq><ci id="S4.SS4.p1.1.m1.1.1.2.cmml" xref="S4.SS4.p1.1.m1.1.1.2">𝑝</ci><cn id="S4.SS4.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS4.p1.1.m1.1.1.3">50</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.1.m1.1c">p=50</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.1.m1.1d">italic_p = 50</annotation></semantics></math> (i.e., removing half of the edges), we observe the following:</p> <ul class="ltx_itemize" id="S4.I3"> <li class="ltx_item" id="S4.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I3.i1.p1"> <p class="ltx_p" id="S4.I3.i1.p1.1">Edge attribution methods showing superior performance on BA-Shapes (i.e., Att, SA, IG, and GB) tend to prune edges that are not included in the ground-truth explanation edges compared to GNNex, PGEx, and FDnX.</p> </div> </li> <li class="ltx_item" id="S4.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I3.i2.p1"> <p class="ltx_p" id="S4.I3.i2.p1.1">Especially for Att and SA, the resulting graphs after pruning tend to be less noisy and explainable.</p> </div> </li> </ul> </div> <figure class="ltx_table" id="S4.T2"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T2.28" style="width:433.6pt;height:237.8pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(85.5pt,-46.9pt) scale(1.6515211805988,1.6515211805988) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S4.T2.28.28"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T2.28.28.29.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="S4.T2.28.28.29.1.1"><span class="ltx_text" id="S4.T2.28.28.29.1.1.1" style="font-size:90%;">Method</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T2.28.28.29.1.2"><span class="ltx_text" id="S4.T2.28.28.29.1.2.1" style="font-size:90%;">BAShapes</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T2.28.28.29.1.3"><span class="ltx_text" id="S4.T2.28.28.29.1.3.1" style="font-size:90%;">Cora</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T2.28.28.29.1.4"><span class="ltx_text" id="S4.T2.28.28.29.1.4.1" style="font-size:90%;">Citeseer</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T2.28.28.29.1.5"><span class="ltx_text" id="S4.T2.28.28.29.1.5.1" style="font-size:90%;">Pubmed</span></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T2.4.4.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S4.T2.4.4.4.5"><span class="ltx_text" id="S4.T2.4.4.4.5.1" style="font-size:90%;">Att</span></th> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.1.1.1.1"><math alttext="4.06\times 10^{-2}" class="ltx_Math" display="inline" id="S4.T2.1.1.1.1.m1.1"><semantics id="S4.T2.1.1.1.1.m1.1a"><mrow id="S4.T2.1.1.1.1.m1.1.1" xref="S4.T2.1.1.1.1.m1.1.1.cmml"><mn id="S4.T2.1.1.1.1.m1.1.1.2" mathsize="90%" xref="S4.T2.1.1.1.1.m1.1.1.2.cmml">4.06</mn><mo id="S4.T2.1.1.1.1.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="S4.T2.1.1.1.1.m1.1.1.1.cmml">×</mo><msup id="S4.T2.1.1.1.1.m1.1.1.3" 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xref="S4.T2.5.5.5.1.m1.1.1.2.cmml">3.54</mn><mo id="S4.T2.5.5.5.1.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="S4.T2.5.5.5.1.m1.1.1.1.cmml">×</mo><msup id="S4.T2.5.5.5.1.m1.1.1.3" xref="S4.T2.5.5.5.1.m1.1.1.3.cmml"><mn id="S4.T2.5.5.5.1.m1.1.1.3.2" mathsize="90%" xref="S4.T2.5.5.5.1.m1.1.1.3.2.cmml">10</mn><mrow id="S4.T2.5.5.5.1.m1.1.1.3.3" xref="S4.T2.5.5.5.1.m1.1.1.3.3.cmml"><mo id="S4.T2.5.5.5.1.m1.1.1.3.3a" mathsize="90%" xref="S4.T2.5.5.5.1.m1.1.1.3.3.cmml">−</mo><mn id="S4.T2.5.5.5.1.m1.1.1.3.3.2" mathsize="90%" xref="S4.T2.5.5.5.1.m1.1.1.3.3.2.cmml">7</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.5.5.5.1.m1.1b"><apply id="S4.T2.5.5.5.1.m1.1.1.cmml" xref="S4.T2.5.5.5.1.m1.1.1"><times id="S4.T2.5.5.5.1.m1.1.1.1.cmml" xref="S4.T2.5.5.5.1.m1.1.1.1"></times><cn id="S4.T2.5.5.5.1.m1.1.1.2.cmml" type="float" xref="S4.T2.5.5.5.1.m1.1.1.2">3.54</cn><apply id="S4.T2.5.5.5.1.m1.1.1.3.cmml" xref="S4.T2.5.5.5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.5.5.5.1.m1.1.1.3.1.cmml" xref="S4.T2.5.5.5.1.m1.1.1.3">superscript</csymbol><cn id="S4.T2.5.5.5.1.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.5.5.5.1.m1.1.1.3.2">10</cn><apply id="S4.T2.5.5.5.1.m1.1.1.3.3.cmml" xref="S4.T2.5.5.5.1.m1.1.1.3.3"><minus id="S4.T2.5.5.5.1.m1.1.1.3.3.1.cmml" xref="S4.T2.5.5.5.1.m1.1.1.3.3"></minus><cn id="S4.T2.5.5.5.1.m1.1.1.3.3.2.cmml" type="integer" xref="S4.T2.5.5.5.1.m1.1.1.3.3.2">7</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.5.5.5.1.m1.1c">3.54\times 10^{-7}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.5.5.5.1.m1.1d">3.54 × 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T2.6.6.6.2"><math alttext="2.21\times 10^{-7}" class="ltx_Math" display="inline" id="S4.T2.6.6.6.2.m1.1"><semantics id="S4.T2.6.6.6.2.m1.1a"><mrow id="S4.T2.6.6.6.2.m1.1.1" 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ltx_align_center" id="S4.T2.8.8.8.4"><math alttext="2.46\times 10^{0}" class="ltx_Math" display="inline" id="S4.T2.8.8.8.4.m1.1"><semantics id="S4.T2.8.8.8.4.m1.1a"><mrow id="S4.T2.8.8.8.4.m1.1.1" xref="S4.T2.8.8.8.4.m1.1.1.cmml"><mn id="S4.T2.8.8.8.4.m1.1.1.2" mathsize="90%" xref="S4.T2.8.8.8.4.m1.1.1.2.cmml">2.46</mn><mo id="S4.T2.8.8.8.4.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="S4.T2.8.8.8.4.m1.1.1.1.cmml">×</mo><msup id="S4.T2.8.8.8.4.m1.1.1.3" xref="S4.T2.8.8.8.4.m1.1.1.3.cmml"><mn id="S4.T2.8.8.8.4.m1.1.1.3.2" mathsize="90%" xref="S4.T2.8.8.8.4.m1.1.1.3.2.cmml">10</mn><mn id="S4.T2.8.8.8.4.m1.1.1.3.3" mathsize="90%" xref="S4.T2.8.8.8.4.m1.1.1.3.3.cmml">0</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.8.8.8.4.m1.1b"><apply id="S4.T2.8.8.8.4.m1.1.1.cmml" xref="S4.T2.8.8.8.4.m1.1.1"><times id="S4.T2.8.8.8.4.m1.1.1.1.cmml" xref="S4.T2.8.8.8.4.m1.1.1.1"></times><cn id="S4.T2.8.8.8.4.m1.1.1.2.cmml" type="float" 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class="ltx_Math" display="inline" id="S4.T2.9.9.9.1.m1.1"><semantics id="S4.T2.9.9.9.1.m1.1a"><mrow id="S4.T2.9.9.9.1.m1.1.1" xref="S4.T2.9.9.9.1.m1.1.1.cmml"><mn id="S4.T2.9.9.9.1.m1.1.1.2" mathsize="90%" xref="S4.T2.9.9.9.1.m1.1.1.2.cmml">6.25</mn><mo id="S4.T2.9.9.9.1.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="S4.T2.9.9.9.1.m1.1.1.1.cmml">×</mo><msup id="S4.T2.9.9.9.1.m1.1.1.3" xref="S4.T2.9.9.9.1.m1.1.1.3.cmml"><mn id="S4.T2.9.9.9.1.m1.1.1.3.2" mathsize="90%" xref="S4.T2.9.9.9.1.m1.1.1.3.2.cmml">10</mn><mn id="S4.T2.9.9.9.1.m1.1.1.3.3" mathsize="90%" xref="S4.T2.9.9.9.1.m1.1.1.3.3.cmml">0</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.9.9.9.1.m1.1b"><apply id="S4.T2.9.9.9.1.m1.1.1.cmml" xref="S4.T2.9.9.9.1.m1.1.1"><times id="S4.T2.9.9.9.1.m1.1.1.1.cmml" xref="S4.T2.9.9.9.1.m1.1.1.1"></times><cn id="S4.T2.9.9.9.1.m1.1.1.2.cmml" type="float" xref="S4.T2.9.9.9.1.m1.1.1.2">6.25</cn><apply id="S4.T2.9.9.9.1.m1.1.1.3.cmml" xref="S4.T2.9.9.9.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.9.9.9.1.m1.1.1.3.1.cmml" xref="S4.T2.9.9.9.1.m1.1.1.3">superscript</csymbol><cn id="S4.T2.9.9.9.1.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.9.9.9.1.m1.1.1.3.2">10</cn><cn id="S4.T2.9.9.9.1.m1.1.1.3.3.cmml" type="integer" xref="S4.T2.9.9.9.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.9.9.9.1.m1.1c">6.25\times 10^{0}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.9.9.9.1.m1.1d">6.25 × 10 start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T2.10.10.10.2"><math alttext="1.26\times 10^{0}" class="ltx_Math" display="inline" id="S4.T2.10.10.10.2.m1.1"><semantics id="S4.T2.10.10.10.2.m1.1a"><mrow id="S4.T2.10.10.10.2.m1.1.1" xref="S4.T2.10.10.10.2.m1.1.1.cmml"><mn id="S4.T2.10.10.10.2.m1.1.1.2" mathsize="90%" xref="S4.T2.10.10.10.2.m1.1.1.2.cmml">1.26</mn><mo id="S4.T2.10.10.10.2.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="S4.T2.10.10.10.2.m1.1.1.1.cmml">×</mo><msup id="S4.T2.10.10.10.2.m1.1.1.3" xref="S4.T2.10.10.10.2.m1.1.1.3.cmml"><mn id="S4.T2.10.10.10.2.m1.1.1.3.2" mathsize="90%" xref="S4.T2.10.10.10.2.m1.1.1.3.2.cmml">10</mn><mn id="S4.T2.10.10.10.2.m1.1.1.3.3" mathsize="90%" xref="S4.T2.10.10.10.2.m1.1.1.3.3.cmml">0</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.10.10.10.2.m1.1b"><apply id="S4.T2.10.10.10.2.m1.1.1.cmml" xref="S4.T2.10.10.10.2.m1.1.1"><times id="S4.T2.10.10.10.2.m1.1.1.1.cmml" xref="S4.T2.10.10.10.2.m1.1.1.1"></times><cn id="S4.T2.10.10.10.2.m1.1.1.2.cmml" type="float" xref="S4.T2.10.10.10.2.m1.1.1.2">1.26</cn><apply id="S4.T2.10.10.10.2.m1.1.1.3.cmml" xref="S4.T2.10.10.10.2.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.10.10.10.2.m1.1.1.3.1.cmml" xref="S4.T2.10.10.10.2.m1.1.1.3">superscript</csymbol><cn id="S4.T2.10.10.10.2.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.10.10.10.2.m1.1.1.3.2">10</cn><cn id="S4.T2.10.10.10.2.m1.1.1.3.3.cmml" type="integer" xref="S4.T2.10.10.10.2.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.10.10.10.2.m1.1c">1.26\times 10^{0}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.10.10.10.2.m1.1d">1.26 × 10 start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T2.11.11.11.3"><math alttext="5.68\times 10^{-1}" class="ltx_Math" display="inline" id="S4.T2.11.11.11.3.m1.1"><semantics id="S4.T2.11.11.11.3.m1.1a"><mrow id="S4.T2.11.11.11.3.m1.1.1" xref="S4.T2.11.11.11.3.m1.1.1.cmml"><mn id="S4.T2.11.11.11.3.m1.1.1.2" mathsize="90%" xref="S4.T2.11.11.11.3.m1.1.1.2.cmml">5.68</mn><mo id="S4.T2.11.11.11.3.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="S4.T2.11.11.11.3.m1.1.1.1.cmml">×</mo><msup id="S4.T2.11.11.11.3.m1.1.1.3" xref="S4.T2.11.11.11.3.m1.1.1.3.cmml"><mn id="S4.T2.11.11.11.3.m1.1.1.3.2" mathsize="90%" xref="S4.T2.11.11.11.3.m1.1.1.3.2.cmml">10</mn><mrow id="S4.T2.11.11.11.3.m1.1.1.3.3" xref="S4.T2.11.11.11.3.m1.1.1.3.3.cmml"><mo id="S4.T2.11.11.11.3.m1.1.1.3.3a" mathsize="90%" xref="S4.T2.11.11.11.3.m1.1.1.3.3.cmml">−</mo><mn id="S4.T2.11.11.11.3.m1.1.1.3.3.2" mathsize="90%" xref="S4.T2.11.11.11.3.m1.1.1.3.3.2.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.11.11.11.3.m1.1b"><apply id="S4.T2.11.11.11.3.m1.1.1.cmml" xref="S4.T2.11.11.11.3.m1.1.1"><times id="S4.T2.11.11.11.3.m1.1.1.1.cmml" xref="S4.T2.11.11.11.3.m1.1.1.1"></times><cn id="S4.T2.11.11.11.3.m1.1.1.2.cmml" type="float" xref="S4.T2.11.11.11.3.m1.1.1.2">5.68</cn><apply id="S4.T2.11.11.11.3.m1.1.1.3.cmml" xref="S4.T2.11.11.11.3.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.11.11.11.3.m1.1.1.3.1.cmml" xref="S4.T2.11.11.11.3.m1.1.1.3">superscript</csymbol><cn id="S4.T2.11.11.11.3.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.11.11.11.3.m1.1.1.3.2">10</cn><apply id="S4.T2.11.11.11.3.m1.1.1.3.3.cmml" xref="S4.T2.11.11.11.3.m1.1.1.3.3"><minus id="S4.T2.11.11.11.3.m1.1.1.3.3.1.cmml" xref="S4.T2.11.11.11.3.m1.1.1.3.3"></minus><cn id="S4.T2.11.11.11.3.m1.1.1.3.3.2.cmml" type="integer" xref="S4.T2.11.11.11.3.m1.1.1.3.3.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.11.11.11.3.m1.1c">5.68\times 10^{-1}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.11.11.11.3.m1.1d">5.68 × 10 start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T2.12.12.12.4"><math alttext="\mathbf{2.25\times 10^{0}}" class="ltx_Math" display="inline" id="S4.T2.12.12.12.4.m1.1"><semantics id="S4.T2.12.12.12.4.m1.1a"><mrow id="S4.T2.12.12.12.4.m1.1.1" xref="S4.T2.12.12.12.4.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S4.T2.12.12.12.4.m1.1.1.2" mathsize="90%" mathvariant="bold" xref="S4.T2.12.12.12.4.m1.1.1.2.cmml">2.25</mn><mo id="S4.T2.12.12.12.4.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="S4.T2.12.12.12.4.m1.1.1.1.cmml">×</mo><msup id="S4.T2.12.12.12.4.m1.1.1.3" xref="S4.T2.12.12.12.4.m1.1.1.3.cmml"><mn id="S4.T2.12.12.12.4.m1.1.1.3.2" mathsize="90%" xref="S4.T2.12.12.12.4.m1.1.1.3.2.cmml">𝟏𝟎</mn><mn id="S4.T2.12.12.12.4.m1.1.1.3.3" mathsize="90%" xref="S4.T2.12.12.12.4.m1.1.1.3.3.cmml">𝟎</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.12.12.12.4.m1.1b"><apply id="S4.T2.12.12.12.4.m1.1.1.cmml" xref="S4.T2.12.12.12.4.m1.1.1"><times id="S4.T2.12.12.12.4.m1.1.1.1.cmml" xref="S4.T2.12.12.12.4.m1.1.1.1"></times><cn id="S4.T2.12.12.12.4.m1.1.1.2.cmml" type="float" xref="S4.T2.12.12.12.4.m1.1.1.2">2.25</cn><apply id="S4.T2.12.12.12.4.m1.1.1.3.cmml" xref="S4.T2.12.12.12.4.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.12.12.12.4.m1.1.1.3.1.cmml" xref="S4.T2.12.12.12.4.m1.1.1.3">superscript</csymbol><cn id="S4.T2.12.12.12.4.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.12.12.12.4.m1.1.1.3.2">10</cn><cn id="S4.T2.12.12.12.4.m1.1.1.3.3.cmml" type="integer" xref="S4.T2.12.12.12.4.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.12.12.12.4.m1.1c">\mathbf{2.25\times 10^{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.12.12.12.4.m1.1d">bold_2.25 × bold_10 start_POSTSUPERSCRIPT bold_0 end_POSTSUPERSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T2.16.16.16"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T2.16.16.16.5"><span class="ltx_text" id="S4.T2.16.16.16.5.1" style="font-size:90%;">GB</span></th> <td class="ltx_td ltx_align_center" id="S4.T2.13.13.13.1"><math alttext="3.77\times 10^{0}" class="ltx_Math" display="inline" id="S4.T2.13.13.13.1.m1.1"><semantics id="S4.T2.13.13.13.1.m1.1a"><mrow id="S4.T2.13.13.13.1.m1.1.1" xref="S4.T2.13.13.13.1.m1.1.1.cmml"><mn 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xref="S4.T2.14.14.14.2.m1.1.1.1.cmml">×</mo><msup id="S4.T2.14.14.14.2.m1.1.1.3" xref="S4.T2.14.14.14.2.m1.1.1.3.cmml"><mn id="S4.T2.14.14.14.2.m1.1.1.3.2" mathsize="90%" xref="S4.T2.14.14.14.2.m1.1.1.3.2.cmml">10</mn><mn id="S4.T2.14.14.14.2.m1.1.1.3.3" mathsize="90%" xref="S4.T2.14.14.14.2.m1.1.1.3.3.cmml">0</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.14.14.14.2.m1.1b"><apply id="S4.T2.14.14.14.2.m1.1.1.cmml" xref="S4.T2.14.14.14.2.m1.1.1"><times id="S4.T2.14.14.14.2.m1.1.1.1.cmml" xref="S4.T2.14.14.14.2.m1.1.1.1"></times><cn id="S4.T2.14.14.14.2.m1.1.1.2.cmml" type="float" xref="S4.T2.14.14.14.2.m1.1.1.2">1.42</cn><apply id="S4.T2.14.14.14.2.m1.1.1.3.cmml" xref="S4.T2.14.14.14.2.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.14.14.14.2.m1.1.1.3.1.cmml" xref="S4.T2.14.14.14.2.m1.1.1.3">superscript</csymbol><cn id="S4.T2.14.14.14.2.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.14.14.14.2.m1.1.1.3.2">10</cn><cn id="S4.T2.14.14.14.2.m1.1.1.3.3.cmml" type="integer" 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rspace="0.222em" xref="S4.T2.16.16.16.4.m1.1.1.1.cmml">×</mo><msup id="S4.T2.16.16.16.4.m1.1.1.3" xref="S4.T2.16.16.16.4.m1.1.1.3.cmml"><mn id="S4.T2.16.16.16.4.m1.1.1.3.2" mathsize="90%" xref="S4.T2.16.16.16.4.m1.1.1.3.2.cmml">10</mn><mn id="S4.T2.16.16.16.4.m1.1.1.3.3" mathsize="90%" xref="S4.T2.16.16.16.4.m1.1.1.3.3.cmml">0</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.16.16.16.4.m1.1b"><apply id="S4.T2.16.16.16.4.m1.1.1.cmml" xref="S4.T2.16.16.16.4.m1.1.1"><times id="S4.T2.16.16.16.4.m1.1.1.1.cmml" xref="S4.T2.16.16.16.4.m1.1.1.1"></times><cn id="S4.T2.16.16.16.4.m1.1.1.2.cmml" type="float" xref="S4.T2.16.16.16.4.m1.1.1.2">2.40</cn><apply id="S4.T2.16.16.16.4.m1.1.1.3.cmml" xref="S4.T2.16.16.16.4.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.16.16.16.4.m1.1.1.3.1.cmml" xref="S4.T2.16.16.16.4.m1.1.1.3">superscript</csymbol><cn id="S4.T2.16.16.16.4.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.16.16.16.4.m1.1.1.3.2">10</cn><cn id="S4.T2.16.16.16.4.m1.1.1.3.3.cmml" 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xref="S4.T2.19.19.19.3.m1.1.1.3.3.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.19.19.19.3.m1.1c">3.52\times 10^{-1}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.19.19.19.3.m1.1d">3.52 × 10 start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T2.20.20.20.4"><math alttext="2.46\times 10^{0}" class="ltx_Math" display="inline" id="S4.T2.20.20.20.4.m1.1"><semantics id="S4.T2.20.20.20.4.m1.1a"><mrow id="S4.T2.20.20.20.4.m1.1.1" xref="S4.T2.20.20.20.4.m1.1.1.cmml"><mn id="S4.T2.20.20.20.4.m1.1.1.2" mathsize="90%" xref="S4.T2.20.20.20.4.m1.1.1.2.cmml">2.46</mn><mo id="S4.T2.20.20.20.4.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="S4.T2.20.20.20.4.m1.1.1.1.cmml">×</mo><msup id="S4.T2.20.20.20.4.m1.1.1.3" xref="S4.T2.20.20.20.4.m1.1.1.3.cmml"><mn id="S4.T2.20.20.20.4.m1.1.1.3.2" mathsize="90%" xref="S4.T2.20.20.20.4.m1.1.1.3.2.cmml">10</mn><mn id="S4.T2.20.20.20.4.m1.1.1.3.3" mathsize="90%" xref="S4.T2.20.20.20.4.m1.1.1.3.3.cmml">0</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.20.20.20.4.m1.1b"><apply id="S4.T2.20.20.20.4.m1.1.1.cmml" xref="S4.T2.20.20.20.4.m1.1.1"><times id="S4.T2.20.20.20.4.m1.1.1.1.cmml" xref="S4.T2.20.20.20.4.m1.1.1.1"></times><cn id="S4.T2.20.20.20.4.m1.1.1.2.cmml" type="float" xref="S4.T2.20.20.20.4.m1.1.1.2">2.46</cn><apply id="S4.T2.20.20.20.4.m1.1.1.3.cmml" xref="S4.T2.20.20.20.4.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.20.20.20.4.m1.1.1.3.1.cmml" xref="S4.T2.20.20.20.4.m1.1.1.3">superscript</csymbol><cn id="S4.T2.20.20.20.4.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.20.20.20.4.m1.1.1.3.2">10</cn><cn id="S4.T2.20.20.20.4.m1.1.1.3.3.cmml" type="integer" xref="S4.T2.20.20.20.4.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.20.20.20.4.m1.1c">2.46\times 10^{0}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.20.20.20.4.m1.1d">2.46 × 10 start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T2.24.24.24"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T2.24.24.24.5"><span class="ltx_text" id="S4.T2.24.24.24.5.1" style="font-size:90%;">PGEx</span></th> <td class="ltx_td ltx_align_center" id="S4.T2.21.21.21.1"><math alttext="3.83\times 10^{-7}" class="ltx_Math" display="inline" id="S4.T2.21.21.21.1.m1.1"><semantics id="S4.T2.21.21.21.1.m1.1a"><mrow id="S4.T2.21.21.21.1.m1.1.1" xref="S4.T2.21.21.21.1.m1.1.1.cmml"><mn id="S4.T2.21.21.21.1.m1.1.1.2" mathsize="90%" xref="S4.T2.21.21.21.1.m1.1.1.2.cmml">3.83</mn><mo id="S4.T2.21.21.21.1.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="S4.T2.21.21.21.1.m1.1.1.1.cmml">×</mo><msup id="S4.T2.21.21.21.1.m1.1.1.3" xref="S4.T2.21.21.21.1.m1.1.1.3.cmml"><mn id="S4.T2.21.21.21.1.m1.1.1.3.2" mathsize="90%" xref="S4.T2.21.21.21.1.m1.1.1.3.2.cmml">10</mn><mrow id="S4.T2.21.21.21.1.m1.1.1.3.3" xref="S4.T2.21.21.21.1.m1.1.1.3.3.cmml"><mo id="S4.T2.21.21.21.1.m1.1.1.3.3a" mathsize="90%" xref="S4.T2.21.21.21.1.m1.1.1.3.3.cmml">−</mo><mn id="S4.T2.21.21.21.1.m1.1.1.3.3.2" mathsize="90%" xref="S4.T2.21.21.21.1.m1.1.1.3.3.2.cmml">7</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.21.21.21.1.m1.1b"><apply id="S4.T2.21.21.21.1.m1.1.1.cmml" xref="S4.T2.21.21.21.1.m1.1.1"><times id="S4.T2.21.21.21.1.m1.1.1.1.cmml" xref="S4.T2.21.21.21.1.m1.1.1.1"></times><cn id="S4.T2.21.21.21.1.m1.1.1.2.cmml" type="float" xref="S4.T2.21.21.21.1.m1.1.1.2">3.83</cn><apply id="S4.T2.21.21.21.1.m1.1.1.3.cmml" xref="S4.T2.21.21.21.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.21.21.21.1.m1.1.1.3.1.cmml" xref="S4.T2.21.21.21.1.m1.1.1.3">superscript</csymbol><cn id="S4.T2.21.21.21.1.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.21.21.21.1.m1.1.1.3.2">10</cn><apply 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rspace="0.222em" xref="S4.T2.22.22.22.2.m1.1.1.1.cmml">×</mo><msup id="S4.T2.22.22.22.2.m1.1.1.3" xref="S4.T2.22.22.22.2.m1.1.1.3.cmml"><mn id="S4.T2.22.22.22.2.m1.1.1.3.2" mathsize="90%" xref="S4.T2.22.22.22.2.m1.1.1.3.2.cmml">10</mn><mrow id="S4.T2.22.22.22.2.m1.1.1.3.3" xref="S4.T2.22.22.22.2.m1.1.1.3.3.cmml"><mo id="S4.T2.22.22.22.2.m1.1.1.3.3a" mathsize="90%" xref="S4.T2.22.22.22.2.m1.1.1.3.3.cmml">−</mo><mn id="S4.T2.22.22.22.2.m1.1.1.3.3.2" mathsize="90%" xref="S4.T2.22.22.22.2.m1.1.1.3.3.2.cmml">2</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.22.22.22.2.m1.1b"><apply id="S4.T2.22.22.22.2.m1.1.1.cmml" xref="S4.T2.22.22.22.2.m1.1.1"><times id="S4.T2.22.22.22.2.m1.1.1.1.cmml" xref="S4.T2.22.22.22.2.m1.1.1.1"></times><cn id="S4.T2.22.22.22.2.m1.1.1.2.cmml" type="float" xref="S4.T2.22.22.22.2.m1.1.1.2">2.04</cn><apply id="S4.T2.22.22.22.2.m1.1.1.3.cmml" xref="S4.T2.22.22.22.2.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.22.22.22.2.m1.1.1.3.1.cmml" xref="S4.T2.22.22.22.2.m1.1.1.3">superscript</csymbol><cn id="S4.T2.22.22.22.2.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.22.22.22.2.m1.1.1.3.2">10</cn><apply id="S4.T2.22.22.22.2.m1.1.1.3.3.cmml" xref="S4.T2.22.22.22.2.m1.1.1.3.3"><minus id="S4.T2.22.22.22.2.m1.1.1.3.3.1.cmml" xref="S4.T2.22.22.22.2.m1.1.1.3.3"></minus><cn id="S4.T2.22.22.22.2.m1.1.1.3.3.2.cmml" type="integer" xref="S4.T2.22.22.22.2.m1.1.1.3.3.2">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.22.22.22.2.m1.1c">2.04\times 10^{-2}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.22.22.22.2.m1.1d">2.04 × 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T2.23.23.23.3"><math alttext="7.11\times 10^{-3}" class="ltx_Math" display="inline" id="S4.T2.23.23.23.3.m1.1"><semantics id="S4.T2.23.23.23.3.m1.1a"><mrow id="S4.T2.23.23.23.3.m1.1.1" xref="S4.T2.23.23.23.3.m1.1.1.cmml"><mn id="S4.T2.23.23.23.3.m1.1.1.2" mathsize="90%" xref="S4.T2.23.23.23.3.m1.1.1.2.cmml">7.11</mn><mo id="S4.T2.23.23.23.3.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="S4.T2.23.23.23.3.m1.1.1.1.cmml">×</mo><msup id="S4.T2.23.23.23.3.m1.1.1.3" xref="S4.T2.23.23.23.3.m1.1.1.3.cmml"><mn id="S4.T2.23.23.23.3.m1.1.1.3.2" mathsize="90%" xref="S4.T2.23.23.23.3.m1.1.1.3.2.cmml">10</mn><mrow id="S4.T2.23.23.23.3.m1.1.1.3.3" xref="S4.T2.23.23.23.3.m1.1.1.3.3.cmml"><mo id="S4.T2.23.23.23.3.m1.1.1.3.3a" mathsize="90%" xref="S4.T2.23.23.23.3.m1.1.1.3.3.cmml">−</mo><mn id="S4.T2.23.23.23.3.m1.1.1.3.3.2" mathsize="90%" xref="S4.T2.23.23.23.3.m1.1.1.3.3.2.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.23.23.23.3.m1.1b"><apply id="S4.T2.23.23.23.3.m1.1.1.cmml" xref="S4.T2.23.23.23.3.m1.1.1"><times id="S4.T2.23.23.23.3.m1.1.1.1.cmml" xref="S4.T2.23.23.23.3.m1.1.1.1"></times><cn id="S4.T2.23.23.23.3.m1.1.1.2.cmml" type="float" xref="S4.T2.23.23.23.3.m1.1.1.2">7.11</cn><apply id="S4.T2.23.23.23.3.m1.1.1.3.cmml" xref="S4.T2.23.23.23.3.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.23.23.23.3.m1.1.1.3.1.cmml" xref="S4.T2.23.23.23.3.m1.1.1.3">superscript</csymbol><cn id="S4.T2.23.23.23.3.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.23.23.23.3.m1.1.1.3.2">10</cn><apply id="S4.T2.23.23.23.3.m1.1.1.3.3.cmml" xref="S4.T2.23.23.23.3.m1.1.1.3.3"><minus id="S4.T2.23.23.23.3.m1.1.1.3.3.1.cmml" xref="S4.T2.23.23.23.3.m1.1.1.3.3"></minus><cn id="S4.T2.23.23.23.3.m1.1.1.3.3.2.cmml" type="integer" xref="S4.T2.23.23.23.3.m1.1.1.3.3.2">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.23.23.23.3.m1.1c">7.11\times 10^{-3}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.23.23.23.3.m1.1d">7.11 × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T2.24.24.24.4"><math alttext="2.46\times 10^{0}" class="ltx_Math" display="inline" id="S4.T2.24.24.24.4.m1.1"><semantics id="S4.T2.24.24.24.4.m1.1a"><mrow id="S4.T2.24.24.24.4.m1.1.1" xref="S4.T2.24.24.24.4.m1.1.1.cmml"><mn id="S4.T2.24.24.24.4.m1.1.1.2" mathsize="90%" xref="S4.T2.24.24.24.4.m1.1.1.2.cmml">2.46</mn><mo id="S4.T2.24.24.24.4.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="S4.T2.24.24.24.4.m1.1.1.1.cmml">×</mo><msup id="S4.T2.24.24.24.4.m1.1.1.3" xref="S4.T2.24.24.24.4.m1.1.1.3.cmml"><mn id="S4.T2.24.24.24.4.m1.1.1.3.2" mathsize="90%" xref="S4.T2.24.24.24.4.m1.1.1.3.2.cmml">10</mn><mn id="S4.T2.24.24.24.4.m1.1.1.3.3" mathsize="90%" xref="S4.T2.24.24.24.4.m1.1.1.3.3.cmml">0</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.24.24.24.4.m1.1b"><apply id="S4.T2.24.24.24.4.m1.1.1.cmml" xref="S4.T2.24.24.24.4.m1.1.1"><times id="S4.T2.24.24.24.4.m1.1.1.1.cmml" xref="S4.T2.24.24.24.4.m1.1.1.1"></times><cn id="S4.T2.24.24.24.4.m1.1.1.2.cmml" type="float" 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xref="S4.T2.27.27.27.3.m1.1.1.3.2.cmml">10</mn><mrow id="S4.T2.27.27.27.3.m1.1.1.3.3" xref="S4.T2.27.27.27.3.m1.1.1.3.3.cmml"><mo id="S4.T2.27.27.27.3.m1.1.1.3.3a" mathsize="90%" xref="S4.T2.27.27.27.3.m1.1.1.3.3.cmml">−</mo><mn id="S4.T2.27.27.27.3.m1.1.1.3.3.2" mathsize="90%" xref="S4.T2.27.27.27.3.m1.1.1.3.3.2.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.27.27.27.3.m1.1b"><apply id="S4.T2.27.27.27.3.m1.1.1.cmml" xref="S4.T2.27.27.27.3.m1.1.1"><times id="S4.T2.27.27.27.3.m1.1.1.1.cmml" xref="S4.T2.27.27.27.3.m1.1.1.1"></times><cn id="S4.T2.27.27.27.3.m1.1.1.2.cmml" type="float" xref="S4.T2.27.27.27.3.m1.1.1.2">7.05</cn><apply id="S4.T2.27.27.27.3.m1.1.1.3.cmml" xref="S4.T2.27.27.27.3.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.27.27.27.3.m1.1.1.3.1.cmml" xref="S4.T2.27.27.27.3.m1.1.1.3">superscript</csymbol><cn id="S4.T2.27.27.27.3.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.27.27.27.3.m1.1.1.3.2">10</cn><apply 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mathsize="90%" rspace="0.222em" xref="S4.T2.28.28.28.4.m1.1.1.1.cmml">×</mo><msup id="S4.T2.28.28.28.4.m1.1.1.3" xref="S4.T2.28.28.28.4.m1.1.1.3.cmml"><mn id="S4.T2.28.28.28.4.m1.1.1.3.2" mathsize="90%" xref="S4.T2.28.28.28.4.m1.1.1.3.2.cmml">10</mn><mn id="S4.T2.28.28.28.4.m1.1.1.3.3" mathsize="90%" xref="S4.T2.28.28.28.4.m1.1.1.3.3.cmml">0</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.28.28.28.4.m1.1b"><apply id="S4.T2.28.28.28.4.m1.1.1.cmml" xref="S4.T2.28.28.28.4.m1.1.1"><times id="S4.T2.28.28.28.4.m1.1.1.1.cmml" xref="S4.T2.28.28.28.4.m1.1.1.1"></times><cn id="S4.T2.28.28.28.4.m1.1.1.2.cmml" type="float" xref="S4.T2.28.28.28.4.m1.1.1.2">2.46</cn><apply id="S4.T2.28.28.28.4.m1.1.1.3.cmml" xref="S4.T2.28.28.28.4.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.28.28.28.4.m1.1.1.3.1.cmml" xref="S4.T2.28.28.28.4.m1.1.1.3">superscript</csymbol><cn id="S4.T2.28.28.28.4.m1.1.1.3.2.cmml" type="integer" xref="S4.T2.28.28.28.4.m1.1.1.3.2">10</cn><cn 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measuring the average fidelity<sup class="ltx_sup" id="S4.SS5.p1.7.1"><span class="ltx_text ltx_font_italic" id="S4.SS5.p1.7.1.1">-</span></sup> scores over all nodes in the graph for each edge attribution method. In our setting, we measure fidelity<sup class="ltx_sup" id="S4.SS5.p1.7.2"><span class="ltx_text ltx_font_italic" id="S4.SS5.p1.7.2.1">-</span></sup> as the average output (logit) difference between using the original graph and a sparser graph as input, where the sparse graph is generated by removing 50% of the edges with the lowest edge attribution scores in <math alttext="G^{\phi}" class="ltx_Math" display="inline" id="S4.SS5.p1.3.m3.1"><semantics id="S4.SS5.p1.3.m3.1a"><msup id="S4.SS5.p1.3.m3.1.1" xref="S4.SS5.p1.3.m3.1.1.cmml"><mi id="S4.SS5.p1.3.m3.1.1.2" xref="S4.SS5.p1.3.m3.1.1.2.cmml">G</mi><mi id="S4.SS5.p1.3.m3.1.1.3" xref="S4.SS5.p1.3.m3.1.1.3.cmml">ϕ</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS5.p1.3.m3.1b"><apply id="S4.SS5.p1.3.m3.1.1.cmml" xref="S4.SS5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS5.p1.3.m3.1.1.1.cmml" xref="S4.SS5.p1.3.m3.1.1">superscript</csymbol><ci id="S4.SS5.p1.3.m3.1.1.2.cmml" xref="S4.SS5.p1.3.m3.1.1.2">𝐺</ci><ci id="S4.SS5.p1.3.m3.1.1.3.cmml" xref="S4.SS5.p1.3.m3.1.1.3">italic-ϕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p1.3.m3.1c">G^{\phi}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p1.3.m3.1d">italic_G start_POSTSUPERSCRIPT italic_ϕ end_POSTSUPERSCRIPT</annotation></semantics></math>. Lower fidelity<sup class="ltx_sup" id="S4.SS5.p1.7.3"><span class="ltx_text ltx_font_italic" id="S4.SS5.p1.7.3.1">-</span></sup> indicates a higher quality explanation for each node. Table <a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#S4.T2" title="Table 2 ‣ 4.4 Visualizations of Graph Pruning ‣ 4 Empirical Observations ‣ On the Feasibility of Fidelity- for Graph Pruning"><span class="ltx_text ltx_ref_tag">2</span></a> summarizes the measurement of fidelity<sup class="ltx_sup" id="S4.SS5.p1.7.4"><span class="ltx_text ltx_font_italic" id="S4.SS5.p1.7.4.1">-</span></sup> scores for all edge attribution methods and datasets. Here, we find that the fidelity<sup class="ltx_sup" id="S4.SS5.p1.7.5"><span class="ltx_text ltx_font_italic" id="S4.SS5.p1.7.5.1">-</span></sup> does not necessarily translate to graph pruning performance. As an example, GNNEx shows the best performance in fidelity<sup class="ltx_sup" id="S4.SS5.p1.7.6"><span class="ltx_text ltx_font_italic" id="S4.SS5.p1.7.6.1">-</span></sup> for the Cora dataset; however, the average rank when using GNNEx in FiP is 5.84 and 5.58 for average and summation aggregation cases, respectively.</p> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Discussion and Conclusion</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">In this work, we have empirically validated the feasibility of using local edge attribution methods for edge pruning. Our empirical analysis demonstrated that 1) local edge attributions can be effectively used for graph pruning with our FiP framework and 2) general XAI methods outperform XAI methods tailored to GNN models. We believe that FiP will not only improve efficiency but also eventually result in sparser explanations, which makes manual inspection feasible <cite class="ltx_cite ltx_citemacro_cite">Pope <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib13" title="">2019b</a>)</cite>, as sparse explanations are generally considered to be more human-comprehensible <cite class="ltx_cite ltx_citemacro_cite">Yuan <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib28" title="">2023</a>); Funke <span class="ltx_text ltx_font_italic">et al.</span> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11504v1#bib.bib6" title="">2023</a>)</cite>. Potential avenues of future work include development of more sophisticated aggregation methods as well as investigation of the relationship between sparsity levels and human comprehensiveness on the explanations.</p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgements</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1A2C3004345, No. RS-2023- 00220762).</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Agarwal <span class="ltx_text ltx_font_italic" id="bib.bib1.2.2.1">et al.</span> [2022]</span> <span class="ltx_bibblock"> Chirag Agarwal, Marinka Zitnik, and Himabindu Lakkaraju. </span> <span class="ltx_bibblock">Probing GNN explainers: A rigorous theoretical and empirical analysis of GNN explanation methods. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib1.3.1">AISTATS</span>, Virtual event, Mar. 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Ali <span class="ltx_text ltx_font_italic" id="bib.bib2.2.2.1">et al.</span> [2023]</span> <span class="ltx_bibblock"> Sajid Ali, Tamer Abuhmed, Shaker H. 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