Disentangled and Self-Explainable Node Representation Learning
Authors: Simone Piaggesi
André Panisson
Megha Khosla
Venue: Accepted by TMLR ()

Abstract: Node embeddings are low-dimensional vectors that capture node properties, typically learned through unsupervised structural similarity objectives or supervised tasks. While recent efforts have focused on post-hoc explanations for graph models, intrinsic interpretability in unsupervised node embeddings remains largely underexplored. To bridge this gap, we introduce DiSeNE (Disentangled and Self-Explainable Node Embedding), a framework that learns self-explainable node representations in an unsupervised fashion. By leveraging disentangled representation learning, DiSeNE ensures that each embedding dimension corresponds to a distinct topological substructure of the graph, thus offering clear, dimension-wise interpretability. We introduce new objective functions grounded in principled desiderata, jointly optimizing for structural fidelity, disentanglement, and human interpretability. Additionally, we propose several new metrics to evaluate representation quality and human interpretability. Extensive experiments on multiple benchmark datasets demonstrate that DiSeNE not only preserves the underlying graph structure but also provides transparent, human-understandable explanations for each embedding dimension.

Published in Transactions on Machine Learning Research (07 (025)
Disentangled and Self- Self-Explainable Node Representation Learning
Simone Piaggesi
simone.piageextdd..mmpppp
University of Pisa, Pisa, Italy
André Panisson
andre.paanisoonncccttt...
CENTAI Institute, Turin, Italy
Megha Khosla
M. Khosla Studefffut
Delft University of Technology, Delft, Netherlands
Reviewed on OpenReview: https: // openreviewwweeeeooo net/ formm? id= s51708Eg1e
Abstract
Node embeddings are low-dimensional vectors that capture node properties, lypically learned through unsupervised structural similarity objectives or supervised tasks. While recent efforts have focused on post-hoc explanations for graph models, intrinsic interpretability in unsupervised node embeddings remains largely underexplored. To bridge this gap, introduce DISENF. (Disentangled and Self-Fxplainable Node Embedding), framework that learns self-explainable node representations in an unsupervised fashion. By leveraging disentangled representation learning. DISENE ensures that each embedding dimension corre- sponds to a distinct topological substructure of the graph, thus offering clear, dimension wise interpretability. We introduce new objective functions grounded principled desiderata, jointly optimizing for structural fidelity, disentanglement, and human interpretability. Ad- ditionally, propose several new metrics to evaluate representation quality and human interpretability. Extensive experiments on multiple benchmark datasets demonstrate that DISENE not only preserves the underlying graph structure but also provides transparent, human-understandlbbe explanations for each embedding dimension.
1 Introduction
Self-supervised and unsupervised node representation learning (Hamilton, 2020) provide powerful toolkit for extracting mcaningful insights from complex networks, making them essential in modern AI and machine learning applications for network analysis (Ding et al. 2024). These methods offer flexible and efficient ways to analyze high-dimensional networks by transforming them into low-dimensional vector spaces. This transformation enables dimensionality reduction, automatic feature extraction, and the deployment of standard machine learning algorithms for tasks such as node classification, clustering, and link prediction (Khosla et al. 2021). Moreover, unsupervised node representations, or embeddings, enable visualization complex networks and can be transferred across similar networks, enhancing understanding and predictive power in fields ranging from social networks to biological systems. Despite their widespread utility, these approaches often face substantial challenges in terms of interpretability, typically relying complex and post-hoc techniques to understand the latent information encoded within the embeddings (Piaggesi et al. 2024 Idahl et al. 2019 Gogoglou et al. 2019). This limitation raises a critical question: What information do these embeddings encode?
Despite a large body of research on explainable GNN models, embodding methods-the fundamental building blocks of graph-based systems-haws have received comparatively little attention. Most existing attempts explain embeddings predominantly post-hoc (Piaggesi et al. 2024 Gogoglou al.] 2019: Khoshraftar

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Figure 3: The node feature value indicates its proximity to the ex- planation substructure mapped to the corresponding dimension. The black node has a higher value for the dimension corresponding to the green subgraph (since it is hop away) than for the dimension cor- responding to the red subgraph (3 hops away).
Figure 2: explanations aligns with the cor- relation between the node feature values for those dimensions. The dimension referenced by the blue subgraph shows a stronger correla- tion with the red dimensions and a lower correlation with the green dimension.
The overlap in dimension
DISENE generales dimension- wise disentangled repre- sentations in which each embodding mesoscale
Figure 1:
dimension mapped substructure in the input graph. The vector represents the embed- ding for the node marked in blue and the bars depict feature values.
ct al.
and hcavily dependent on the specific embedding techniques used. Some extend methods from the post-processing of word embeddings (Subraaaiaan by minimizing reconstruction errors through over-complete auto-encoders to improve sparsity. Other works Goooglou et al. 2019 Khoshraftar et al., 2021 Dalmia et al. 2018) focus
2021 Dalmia et al.
2018
approaches ces
Piaggesi et al.
2024
et al. 2018 Chen & Zaki
2017]
solely on extracting meaningful explanations, without addressing the underlying embedding learning process.
We propose DISENE (Disentangled and Self- Exflaixxlle Node Embedding), framework that addresses the interpretability gap in unsupervised node embeddings by generating inherently self-explainable node representations. In our approach, 'self-explainable" means that each embedding dimension corresponds to a distinct subgraph that globally explains the structural information encoded within that dimension. These dimensional subgraphs highlight lumaat-comprehensible functional components of the input graph, providing clear and meaningful insights. DISENE leverages disentangled representation learning, an approach that encodes latent variables corresponding to distinct factors of variation in the data (Wang et al. 2024). produce node embeddings that are interpretable
per-dimension basis.
In graph data, node behaviour is strongly influenced by mesoscale structures such as communities, which shape the network's organization and drive dynamics (Barrat et al., (008). By leveraging disentangled representation learning, we capture these "topological" substructures more effectively, with cach embedding dimension reflecting an independent graph unit (see Figure D. We achieve this through novel objective function that ensures structural disentanglement. Specifically, we optimize the embeddings so that each dimension is predictive of a unique substructure in the input graph. To avoid degenerate solutions, we incorporate an entropy-basee basodd regularizer that ensures the resulting substructures are non-empty and informative.
Our paradigm represents a shift. in the language of explanations compared to the ones often considered when dealing with GNNs (Yuan et al. 2023). Explainability for GNNs often involves understanding which parts of the local computation graph (nodes, edges) and node attributes significantly influence the model's predictions Funke et al. 2023: Ying et al.] 2019 Schnake et al. 2022). On the other hand, the explanations that we aim discover are inherently non-local, since they could involve mesoscale structures such as node clusters PPaggeesi al. 2024), usually not included in the GNN computational graph.
We provide comprehensive evaluation of the embeddings and uncover novel insights by proposing several new metrics that capture the interplay between disentanglement and explainability (sce Scction 3.3 for details). For instance, our overlap consislency metric (illustrated in Figure 2 shows that the overlap of the topological substructures used as explanations matches the correlation among the corresponding embedding dimensions, providing insights into the interdependencics among node characteristics within the graph. Morcover, we attribute meaning to the embedding values by showing that the magnitude of node's entry a specific dimension correlates with its proximity (as depicted in Figure 3 to the topological subgraphs associated
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with that dimension. This relationship enhances our understanding of the relative positioning of nodes with respect to different graph components, thereby enhancing spatial awareness of the graph structure.
Contributions are summarized as follows: (i) we formalize new and essential criteria for achieving disen- tangled and explainable node representations, offering a fresh perspective on interpretability in unsupervised graph-based learning; (ti) we introduce novel evaluation metrics to help quantifying the goodness of node representation learning in disentangled and explainable settings (tit) we perform eztensive experimental analyses on synthetic and real-world data to establish state-of-the-art results in self-cxplainable node feature learning. We release code and data at httpp:///github. http://tttuu..oooommoo//eeepeppp commss/ioooiiiageeeeiassse
2 Preliminaries and Related Work
Given undirected graph G (V,E), node embeddings are obtained through encoding function h: KK that maps cach node to a point in a K -imennnsionaal vector space RK where typically D << (V]- We demole the K-dimensional embedding of a node e E V as h(v) = the value of the d-th feature of the embedding for node v. Alternatively, we can represent all node embeddings collectivcly as a matrix (g) E V where cach entry Hnd = ha(v) corresponds to the d-th feature for node v. We can also refer to columns of such matrix, H.,d as the dimensions of the embedding model space.
hx(v)], where ha(v) represents
Node embeddings interpretability. Node emboddings are shallow encoding techniques, often based matrix factorization or random walks (Qiu et al. 2018). Since the latent dimensions in these models are not aligned with high-level semantics (Senel et al.) 2018 Prouteau et al. 2022), interpreting embeddings
typically involves post-hoc explanations of their latent features (Gogoglou et al.
2019 Khoshraftar et al.)
2021). Other works propose alternative methods to modify existing node embeddings, making them casier to explain with human-understendllll) graph features (Piaggesi et. al.] 2024 Shafi et al. 2024). From a different. viewpoint, [Shakespeare & Roth (2024) explore how understandable are the embedded distances between nodes. Similarly, Dalmia et al. (2018) investigate whether specific topological features are predictable, and then encoded, in node representations.
Graph neural networks intorpretability. Graph Neural Networks (GNNs) Wu et al. 2021 are deep models that operate via complex feature transformation and message passing. In recent years, GNNs have gained significant research attention, also in addressing the opaque decision-making process. Several approaches have been proposed to explain GNN docision process YYuan et al. 2023), including perturbation approaches (Ying al. 2019 Yuan al.) 2021 Funke al., 2023), surrogate model-based methods (Vu & Thai 2020 Huang et al. 2023), and gradients-based methods (Pope et al., 2019 Sanchez-Lengelingg
al.j 2020). In parallel, alternative research directions focused concept-based explanations, i.e. high-level units of information that further facilitate human understandability (Magister et al. 2021 Xuanyuan et al. 2023
Disentangled learning on graphs. Disentangled representation learning seeks to uncover and isolate the fundamental explanatory factors within data (Wang et al. 2024). In recent years, these techniques have gained traction for graph-structured data [Liu et al., 2020 Li et al. 2022 Yang et al. 2020) Fan & Gao 2024).
For instance, FactorGCN (Yang et al. 2020) disentangles an input graph into multiple factorized graphs. resulting in distinct disentangled feature spaces that are aggregated afterwards. IPGDN (Liu et al.) 2020)
proposes a disentanglement using a neighborhood routing mechanism, enforcing independence between the latent representations as regularization term for GNN outputs. Mcanwhile, DGCL (Li et al.] 2021 focuses on learning disentangled graph-level representations through self-supervision, ensuring that the factorized components capture expressive information from distinct latent factors independently.
3 Our Proposed Framework: DiSeNE
In this section, we begin by outlining the key desiderata for achieving disentangled and self-explainable node representations. Next, we design a novel framework that meets these objectives by ensuring that the learned node representations are both disentangled and interpretable. Finally, we introduce new evaluation metries to effectively assess the quality of node representation learning in both disentangled and explainable settings.
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3.1 Core Objectives and Desiderata
In the context of unsupervised graph representation learning, we argue that learning self-explainable node embeddings amounts reconstructing the input graph in a numan-interpretable fashion. Traditionally, dot- product models based on NMF (Yang & Leskovee 2013) and LPCA (Chanpuriya al.) 2023) decompose the set of graph nodes into clusters, where each entry of the node embedding vector represents the strength of the participation of the node to a cluster. In this scenario, the dot-product of node embeddings becomes intuitively understandable, as it reflects the extent of shared community memberships between nodes, thereby providing a clear interpretation of edge likelihoods. This concept is also related to distance encoding methods (Li et al., 2020 Klemmer et al., 2023). where a node feature ha(u) is expressed as a function of the nodc's proximity ((u, Sa) = (GG(l v),y v E Sa)) to the anchor set Sa C V, using specific aggregation functions AGG, Typically, distance encodings are constructed by randomly sampling anchor sets (You et al.) 2019), and used augmented node features to enhance expressiveness and improve performance on downstream tasks.
Inspired by this idea, our goal is to oplimize unsupervised node embeddings encoded by GNN function h: RK trained on the graph G = EV, such that node features resemble non-random, structurally meaningful anchor sets, thus improving
iimman--nterpreabbiityyy To achieve this, we propose three key desiderata for learning general-purpose node representations: (i) connectivity preservation, (ti) dimensional interpretability, and (in) structural disentanglement. These desiderata serve as the foundational components of our approach, as detailed below.
Connectivity preservation. Ideally, node embeddings are constructed so that the geometric relationships in the low-dimensional space mirror the connectivity patterns of the original graph. Nodes with greater similarity in the network should be placed close to each other in the embedding space, and viceversa. To implement the approach, we train the node embedding function in recovering the graph structure. We employ random walk optimization framework based the skip-gram model with negative sampling (Huang et al.. 2021. The loss function for this framework is defined as:
Lrw
Σ (M
logo(h(u) h(w)) (h(u)Th(w)
Σ
log a ( h(u') h(v)),
(u', (u.,0)-Pr
where a-) is the sigmoid function, Prw the distribution of node pairs co-occurring on simulated random walks (positive samples), Pr, is a distribution over randomly sampled node pairs (negative samples), and Th(r)) represents the dot product between the embeddings of nodes and v. By optimizing this loss function, we encourage nodes that co-occur in random walks to have similar embeddings, effectively preserving the graph's 's structural information in the embedding space.
Dimensional interpretabilityy Embedding representations are dilti-ddiensional,, with specific latent factors contributing in representing each dimension (c.g., social similarity, functional proximity, shared interests). A given edge between u and may rely more heavily certain dimensions. This perspective shows how local relationships (edges) are directly informed by the global structure of the embeddings, with each dimension contributing uniquely to reconstructing particular relationships. Given structure-pesseringg preserving embeddings, meaning they effectively encode the input structure, we we should be able to interpret each embedding dimension in terms of the graph's topological structure. Specifically, our framework attributes 'responsibility' for reconstructing a local relationship (an edge) to specific dimensions, based on their contribution to the edge likelihood or the probability of reconstructing that edge through the embeddings. We achieve this by assigning local subgraphs to different latent dimensions. Consider the likelihood of an edge (u, v), defined as fu, h) = o Ek_ ha(u)ha(v)) The likelihood score is obtained by applying the sigmoid to a linear combination of per-dimension products ha(u)ha(w). Because the sigmoid is monotonic, each product's sign and magnitude directly determine its influence on ÿ. Thus, we can interpret the edge likelihood by inspecting the raw products This docomposition requires only lincar additivity and it does not depend on any statistical relationships among the terms. To understand how each dimension d contributes to this likelihood, we compute the edge-wise dimension importance Φa(u, v; h) as the deviation of the dimension-speciffc contribution
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from its average over all edges:
o, ha(u)ha(t)
Σ (a',0))E
halu')ha(o))
(1)
[8|
Since the dot-product is a linear function Ed= (ggha())ha(t)) + 3 with unitary coefficients ad = and zero intercept 3 = O, Eq. [ corresponds to the formulation of LincarSHAP attribution scores (Lundberg & Lee 2017), using the set of training edges as the background dataset. Essentially, the attribution function (a( v; h) indicates whether specific dimension contributes positively to an edge's likelihood. A positive attribution score means that the dimension increases the likelihood of predicting the edge. If two dimensions are correlated, they may often push an edge with similar attributions, but each still provides observable marginal deviation that we can measure. In other words, da quantifies marginal responsibility, not exclusive responsibility; therefore it remains valid in the presence of correlations. Each dimension contributes to reconstructing specific edges or substructures, and logether, these contributions create unified represent ation of the entire graph. Thus, the global nature of embeddings emerges as a direct consequence of their contributions local relationships. From this obscrvation, we generate dimension- wisse (global) explanations for the latent embedding model by collecting edge subsets with positive contributions:
E ((u, v) E & oa(u, u; > 0}-
These self-explanations take the form of global edge masks ( E 20 [ where each entry is defined as Mld (da; h) max(0, øa(u, e; h)}. By applying these masks to the adjacency matrix A through Hadamard product (O), we obtain A (d) = A @ M( Each masked adjacency matrix A (d) highlights the subgraph associated with dimension d. From these masked adjacency matrices, we construct edge-induced subgraphs Ga= = (Va, Ea), where Va is the set of nodes involved in edges E These subgraphs act as anchor sets for the model, providing interpretable representations of how each embedding dimension relates to specific structural patterns within the graph. We will refer to edge-induced subgraphs computed the aforementioned procedure (pseudo-code in Appendix D explanation subgraphs/ the embedding model.
sasrraphh/uubstrutturss /ubstrructures topological components of
Structural disentanglement. To enhance the effectiveness of dimensionally interpretable encodings, each dimension of the latent space should encode an independent structure of the imput graph, effectively acting as an anchor subgraph. Inspired by community-afiliationn models (Yang & Leskovec 2013: 2012), we introduce node affiliation matrix F V that captares the association between each node V and anchor subgraph Ga = (Va, Ex). Specifically, each entry Fud is proportional to the magnitude of predicted meaningful connections between node and other nodes in Ga, expressed using the per-dimension attribution scores from Eq. D (h) = Evey, Φa(u, v; h). This aggregates the contributions of dimension d to the likelihood of edges involving node u. To achieve structure-ase aware disentanglement, we enforce oot-oorthogonality among the columns of the affiliation m This ensures that different embedding dimensions capture independent structures, leading to nearly non-overlapping sets of predicted links for each dimension. We express the columns of the affiliation matrix as F..d and obtain the disentanglement loss function as:
K K
Cais
ícos (F F. F..) -5a,1
(3)
d=1==1
where cos (F.., F..) denotes the cosine similarity between the d-th and the I-th columns of F, and and da, is the Kronecker delta function (1 if d = 0 otherwise). This objective penalizes correlation between the edge importance attribution scores of different dimensions. By reducing redundancy, ensure that cach dimension contributes uniquely to the reconstruction of small set of edges, promoting interpretability. This approach enables us to obtain disentangled representations (Wang et al. 2024). where embedding dimensions correspond to orthogonal latent factors of the input graph. Although higher-order disentanglement is possible (e.g., with groups of dimensions), we focus on finle-feaatre disentanglement for achieving dimension wise interpretability.
NNoe that F Fuz(h)= = Evey ha(w)ha() 18T D' ba(u")ha(e)) and assuming [V] << [8], the second term becomes negligible, allowing us approximate F (h. and reduce computational costs.
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3.2 Our Approach: DiSeNE
Building upon the above components, introduced to satisty our desiderata for interpretable node embeddings, we present our approach, DISENE. Specifically, DISENE, takes as input identity matrix 1 [ node attributes and, depending on the encoder architecture, also the adjaceney matrix A IR | The input is encoded into an intermediate embedding layer Z R vixD. Next, DISENE processes the embedding matrix Z to compute the likelihood of link formation between node pairs, given by e0; h) = (b((u) (u) hh()) where h(u) = p(W z(v)) are the final node representations in H E M obtained by applying a linear transformation W E RDx followed by a non-linear activation function p. To encode 2, we employ architectures incorporating fully connected layers and graph convolutional layers (Wu et al. 2019). This process can be further enhanced by integrating more complex poossage-passingg mechanisms or MLP operations. For example, the message- passing could initiate from an M.P-trransfrmee node autribute matrix, ML,(X), incorporate more sophisticated architectures beyond simple graph convolutions for increased expressiveness (Xu et al. 2019 Velickovic et al. 2017
The embeddings are optimized by combining the previously described objective functions for preserving struc- tural faithfulness and achieving structural disentanglement, thereby improving dimensional interpretability. To avoid degenerate disentanglerent solutions we introduce a regularization strategy. Specifically, we aim avoiding the emergence of "empty" clusters characterized by near-zero columns in that, while orthogonal others, fail to convey meaningful information. This regularization ensures a minimal but significant level of connectivity within cach topological substructure. Specifically, contributions to predicted edges in each anchor subgraph GR: Σn.eyy Φu (u, v; h), to be non-zero. We found a more stable and precise approach by enforcing that the aggregated node features of each embedding dimension are non-zcro, achieved by maximizing the entropy? 2
enforce that the total amount of
H= = - E
Aa(u)
log
>,. Aa(a) h(w)lls)
Thus,
h(wi]l:
the model is optimized by minimizing the following comprehensive loss function:
L= Cyw + Lais + dent (1-
H.
log K
The hyperparameter Aen determines the strength of the regularization, controlling the stability for explanation
subgraph sizes across the various latent dimensions. We report the pseudo-code of DISENE in Appendix B along with a space-time complexity analysis, showing that our method has O(E)K + | + (]KTL)
runtime complexity and O(DV + L) space complexity (T refers to the window size, L to simulated walks length), which is in line with well-established techniques for node embeddings TTitsulin et al., 2021). Our approach deviates from typical GNNs by focusing solely on learning from graph topology, as node attributes may not always align with structural information (e.g., in cases of heterophily (Zhu et al., 2024 While semantic features can be integrated in various ways (Tan et al. 2024), chose more straightforward, broadly applicable method. Our node embeddings produce interpretable structural features h(x), which can be concatenated with node semantic attributes x(u), enabling transparent feature sets for effective explanations in rownstream (asks via post-hoe tabular techniques.
3.3 Proposed Evaluation Metrics
In the following, we present novel metrics to quantify interpretability and disentanglement in unsupervised node embeddings, which we use to compare models in our experiments. Unlike prior works, which primarily focus on explaining graph model decisions, our approach offers novel perspective by targeting the explanation of graph model encodings. Traditional graph- based explainers focus on interpreting GNN model decisions for specific tasks, highlighting subgraphs or motifs responsible for the predictions (Longa et al. 2025 Li et al. 2025). In contrast, our objective is to assess the interpretability of GNN models in a task-agnostic setting, by evaluating explanatory subgraphs derived from node embedding dimensions that are learned without supervision. In the next, we define our comprehensive vocabulary of evaluation metries. While alternative taxonomies exist in the literature, we propose a consistent framework highlighting similarities and differences
We assume non nen-tegatvve activation functions p (e.g., ReLU), thus every post-activation feature ha(u) is itself non negative. Therefore, their sum E. Na(u) non-negative as well, and an absolute-value operator in the entropy mass term unnecessary.
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between our indicators and other established metrics in the field of graph-based explainability Yuan et al. Kakkad et al. 2023)
2023
Topological A Aligmment. This metric measures how well the explanations of embedding dimensions align with human-interpretable graph structures. Given their importance in the organization of complex real-world systems (Girvan & Nowman 2002 Hric et al. 2014), community modules (or clusters) are often the most intuitive units for understanding a graph. We evaluate topological alignment by handling edges in explanation masks )
retrieved items from query, and measuring their overlap with the edges in the ground-truth communities using precision, recall, and Fi-score. While this metric captures one aspect of human-comprehenniiiii it does not exhaust the space of interpretable structures, such as motifs, roles, or domain-specific patterns. For example, in biological networks, structures like protein complexes or regulatory circuits are often considered as more interpretable. In social networks, roles like hubs, bridges, or peripheral nodes may give better explanations than just communities. Future work could incorporate alternative structural annotations to evaluate broader forms of topologiaal alignment.
Let C(E) {C()) ,...., C(()) denotes the set of truthful link communities of the input graph Associated to partition c(,), we define ground-truth edge masks CO) E {0, 10,1 with binary entries c [f v) E c(.). For a given mask M(4), topological alignment score is given by the maximum edge overlap (as Fi-score) of the explanation substructure computed across community structures in C(E), quantifying how closely the explanation match human--understanddable ground-bruth
(M) = max; { (M( c(O) =max; prec(M C(i) (()) + rec(M
2
(4)
CO
For precision, we weigh relevant item scores with normalized embedding masks values: (re CCO) E... Mi0 For recall, we weigh binarized embedding masks values with normalized ground-truth scores 4
M
( E... SIMMtsojcc! This approach is similar to the accuracy assessment used in GNNEx-
plainer and subsequent works (Ying et al. 2019 Luo et al. 2020), where explanations for model decisions are compared to planted graph substructures, perceived as hudaabreadable justifications for a node's prediction. However, instead of evaluating individual decisions' explanations by using synthetic ground-trui truth motifs, we perform global assossment of the unsupervised embedding dimensions by comparing their explanation subgraphs with reasonable ground-truth about the graph structure. An analogous evaluation of explanation correctness which specifically focuses on individual model decisions will be provided later by the plausibility metric.
Sparsity. We refer to sparsity as a measure of the localization of subgraph explanations, it is generally defined as the ratio of the number of bits nceded to encode an explanation compared to those required to encode the input (Pope et al.] 2019: Funke al., 2023 As we produce soft masks, we use entropy for this quantification. Given that compact explanations are more effective in delivering clear insights, evaluate sporsily by measuring the normalized Shannon entropy over the mask distribution:
M(d) M
M(d
Sp(M(4) log [
Σ (u,))E
log
En
Ene
M
A lower entropy in the mask distribution indicates higher pparsityccompaciin for explanation size quantification can be found in one of the existing works (Funke et al. 2023
Motivation for using entropy
Overlap Consistency. This metric assesses whether the correlation between two embedding dimensions is mirrored in their corresponding explanations. A well-structured, disentangled latent space should correspond
SSynthetic graphs be constructed with ground-truth relevant sub-structures (like BA-SHAPES (Ying et al. 2019) SBM graphs). In real-world graphs, it usually reasonable assume that the community structure [Fotunato 2010 can serve as ground-truth.
*For the precision, normalize with the sum of scores because they are continuous. For recall, we use the cardinality in place of the sum hecause the ground-truth has binary scores.
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to distinct, uncorrelated topological structures. For example, if two embedding dimensions are correlated, their explanation substructures should also overlap. In our context, this a proxy measure for explanation's is contrastivity (Wang et al., 2023; Pope et al. 2019), based on the intuition that explanations for unrelated dimensions should differ substantially particclllyyy in the specific substructures they highlight. We aim to quantily how different topological components affect pairwise feature correlations the latent space. To achieve this, we propose a metric that measures the strength of association between the physical overlap of the explanation substructures [Ga} and the correlation among corresponding latent dimensions {H.a). We compute the overlap between two subgraph components using the Jaccard similarity index of their edge sets from Eq. : ([ = between pairwise Jaccard values and squared Pearson correlation coefficients (p²) of the embedding features:
[Ln&, The overlap consistency (OvC) metric measures the linear correlation
[
OvC(h) = -:000=p((000 ((I(d,1;h))le. T Io (H H.
(6)
where [$]a<i denoles the condensed list of pair-wise similarities. By using p² we remain agnostic about the sign of the correlation among latent features, since high overlaps could originate from both cases.
Positional Coherence. This metric evaluates whether the feature value of a node representation in a specific embedding dimension corresponds to its spatial relationship with the explanation substructure for that dimension, In fact, an effective representation should preserve meaningful spatial properties that reflect node proximity and connectivity patterns. In our context, this is a proxy measure for explanation's foilhfulness (Zhao et al. 2023b), reflecting how well the dimension-based structures align with the embeddings they are intended to explain. To achieve this, we propose to measure the extent to which node entries reflect their relative positions with the subgraphs used as explanations. For example, if node has a high value in embedding dimension d, it should be positioned closer to the substructure that explains dimension d. Typically, positional encoding (Rampásek et al., 2022 Li et al. 2020 You et al.] 2019) involves the use of several sets of node anchors Sa C y that establish an intrinsic coordinate system. This system influences the node u's features based on the node's proximity ((u, = AGG((C(u, ()) v E Sa)), where AGG denotes a specific pooling operation. As node proximity, we used the inverse of the shortest path distance Sspa(u, (u, v) = (1+ dspa(u, uI) As the anchor sets, we chose the embedding substructures used for explanations, S. = Va- For a specified pair of dimensions (d, I), we assess the correlation between node features along dimension d and the corresponding distances to the topological component indexed by / via Teature-proximity correlation: [ (d, 1; h) = p( [Sspa(u, Va)] wev: H..). The posilional coherence metric (PoC) is defined to specifically evaluate the degree to which each feature d is uniquely correlated with its corresponding topological component Va, without being significantly influenced by correlations with other substructures. This metric is calculated the ratio of the average fPoor for the given dimensions to the average fPoor computed with pairs of permuted dimensions:
PoC(h) = Ea Peorr (d. d: h)
EEa (d, x h));
where (.), denotes an empirical average over multiple permutations. By comparing with random feature- subgraph pairs, the metric avoid promoting models with redundancies in the latent features, where high correlations with other topological components are possible.
Plausibility. Plausibility evaluates how closely subgraph explanations align with human reasoning, a concept also studied in existing works like BAGEL (Rathee et al. 2022). Unlike topological alignment, which measures the aligument of explanation substructures with community clusters in the graph (indepen- dent of task-specific information), plausibility focuses on explanations for decisions made in downstream tasks, using the embedding features predictor variables. To this end, relying on feature-based post- hoc explanation methods (Lundberg & Lee 2017 Ribeiro et al., 2016), we construct instance-specific explanations to determine feature importance scores related to the topological structures. Typical feature importance explainers (Bodria et al. 2023) are useless in this context because node embeddings have in- herently uninterpretable features, leading to uninformative explanations. Our approach overcomes this limitation by mapping explanations back to the graph's 's structural components, that are human-readable.
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We detail the procedure for link prediction here (see also Figure 4), but we also report the methodology for node classification in Appendix The process involves three key steps.
classifier prediction
ground-truth explanation
(1) Training a dounstream classifier. We train a binary classifier b BRR - [0,11] to perform the downstream link prediction task using node embeddings. This classification model can either leverage feature interpretability techniques, such as SHAP (Lundberg & Lee 2017 being inherently interpretable, such as logistic regression.
Plausibility
feature explanation
(2) Extracting post-hoc explanations, For an edge instance huu, v) (which could be derived from node-pair operations such as h(u) @ h()),, we employ post-hoc methods to determine the feature rele- vance for the classifier prediction on the node pair instance (u, v), (u, u; u;b)] b))))))..... .. for each of the embedding dimensions. Similarly to Eq. ① which defines task-agnostic masks, here calculate task- specific masks B() VR which aggregate the logic of the classifier
Figure 4: Sketch of Plausibility met- ric computation. High plausibility scores indicate that the dimensions deemed more comprehensible also re- ceived higher importance scores from the post-hoc attribution technique.
B (W;;b) max (,, vn 6)). Substructures corresponding to the most important features (identified by the post-hoc techniques), or the explanations directly returned by the interpretable model serve as the final explanations.
based individual feature importance:
(3) Evaluation. Explanations are then compared with the available human rationale behind the decisions. To do that, we resort the Fi-score in Eq. (4) with respect to the ground-truth structure of the edge under study, indexed by g(u, v), that it is known has driven the classifier decision. Specifically, we define plausibility for individual prediction b(u,e) v) the average Fi-accuracy relative to the ground-truth structure for the instance, weighted by the computed feature importance:
Pf(u, v; ΣØ= b; (B(), Cs(u.))
(8)
Σ/ I
where is a function guaranteeing the non-negativity of relevance weights. This ensures that only the features that both interpretable and significant to the local prediction contribute substantially to the score, penalizing instead those important features that are not human-comprehensiba.
It is worth to note that although our measure is similar to the accuracy index used to assess correctness of explanation subgraphs (Ying et al. 2019). we refer to it as "plausibility." This is because, despite their competitive classification performance, typical embedding representations may fail to capture the truthful substructures the most task- precctreeii features, as they are not constrained follow human reasoning. Thus, with plausibility, aim to understand when predictive embedding features are also congruent to human rationales. The choice of the weighting function f is case-specific and tailored to the objective of the analysis. For example, choosing f as indicator function such that f(ı, = Mij argmax 1 11,... T allows to study the plausibility of the most predictive substructure.
4 Experiments
We conduct extensive experiments to answer the following research questions:
(RQ1) Human understandability: How comprchensible and sparse are the explanation substructures generated by DISENE?
(RQ2) Structural disentanglement: Do the disentangled subgraphs reveal intrinsic properties of node embeddings, like feature correlations and latent positions?
(RQ3) Utility for downstream tasks: Are the identified substructures plausible and coherent enough to serve as explanations in downstream tasks?
To address our research questions, we extract topological components from multiple embodding methods, trained on different graph data, by computing edge subsets defined in Eq. (2), and analyzing embedding metrics defined in Section 3.3 In the following, we describe the data, models, experimental setup, and results.
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Moreover, in the Appendix we report the following supplementary experiments: in section D analyze accuragyinneeppreaabilty trade-offs for different embedding methods; in section G we perform ablation st studies on the interpretability performances of DISENE while varying the entropy regularization and the depth of the convolutional encoder; in section H we showcase the human-underssanndbbiity of embeddings in the presence of rich topic and text information graph data.
4.1 Datasets and Competitors
Datasets. We ran experiments on four real-world datasets (CORA, WIKI, FACEBoOK, PPI), and six synthetic datasets FRING-O--CLIQUES SBM, BA-CLIQUES, ER-CLIQUES, TREE-CLIQUES and (REE-GGIDS) with planted subgraphs, which serve as ground-truth human rationales for explaining specific node labels. Statistics for these datasets are provided in Table AA in the Appendix. Additionally we employ several biological datasets (see Appendix C for the evaluation on multi-label node classification). BA-CLIQUEE and ER-CLIQUES are variations of the BA-SHAPES (Ying et al. 2019) where we randomly attach cliques, instead of house molifs, to Barabási-A Aabbbis---Abbr. and Erdös-Rényí random graphs. TREE-CLIQUES and TREE-GRINE (Ying) et al. 2019) are composed of a 8-level balanced tree, with cliques and 3x3 grid motifs respectively. RING- CLIQUES and SBM (Abbe 2017] are implemented in N For synthetic data, we present only results for plausibility metrics, leaving the other findings in the Appendix
Methods. We compare with different node embedding methods in producing explainable feature dimensions. Competitors include shallow encoders DEEPWALK (Perozzi et al. 2014). WALK (Chanpuriya & Musco
2020), and deep graph models Graph Autoencoder (GRAPHAE) (Salha et al. 2020), GRAPHSAGE (Hamilton
ct al. 2017) and DGLFRM (Mehta et al. 2019). We also apply the DINE retrofitting approach (Piaggesi et al, 2024) to post-process embeddings from DEEPWALK and GRAPIAE. Moreover, we compare with graph-based
local explanation methods GNNEXPLAINER (Ying et al., 2019) and PGEXPLAINER (Luo et al.) 2020.. Post-hoc explainers are applied to the output of different GNN models for node classification: GCN (Wu)
ct a..
2019 GRAPHSAGE (Hamilton et al. 2017). and GATv2 (Brody et al., 2022 We evaluate method DISENE in two variants: a 1-layer fully- fully-comnected encoder (DiSE-FCAB) and 1-layer convolutional
encoder (DISE-GAE) GNN-based methods are trained using the identity matrix as node features. Specific
training settings are provided in Appendix A.
Setup. In experiments on real-world graphs, we investigate latent space interpretability and disentanglement metrics by keeping the output embedding dimension fixed 128. This dimensionality was chosen to ensure that all methods achieve optimal performance in terms of test accuracy, specifically for link prediction (see the Appendix sections C and D for extensive results on downstream tasks and embedding metrios). For synthetic data, since we investigated plausibility metric referred to downstrcam classifier, thus we did not focus on a specific dimension but we selected the best score metric varying the output dimensions in the list 2, 4, 8, 16, 32, 64, 128]. Each reported result is an average over 5 runs. For link prediction, we use a 90%/10% train/tccs test split, and for node classification, we use an 80% 90%/20% split. All results refer to the training set, except for downstream task experiments, where we present results for the test set.
4.2 Results and Discussion
(RQ1) Are the topological substructures both comprchensible and sparse to support human understandability? Here we explore how well the represented topological structures can serve as global explanations for node embeddings, quantifying the Topological alignment in the terms of associations between model parameters and human-nnderstandable units of the input graph, as well as the Sparsity of these associations. For Topological alignment, we apply modularity maximization to find meaningful clusters (Blondel et al. 2023). In Table 1 we show compact scores as the average values K Σ Align(M,) and 1 K Σd= Sp(M2) over all the embedding features. Since for sparsity we report the value subtracted from 1, all the scores present better results with higher values.
DEEPWALK, WNFWALK and GRAPISAGE show moderate performance in Topological Aligmment, excelling slightly on FB but underperforming on PPI, while GRAPHAE consistently lags behind, particularly on WIKI
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Table 1: Topological alignment and sparsity in real data, Best scores in bold, second best underlined.
Topological WIKI
Aligmment FB
Spansity WIKI
Method
CORA
PPI
CoRA
PPI
DEEPWALK
.363+.03
.356+.002
16022.044
.281=.002
.183=.001
.163:.002
.130+.004
.136:.003
CRAPHAE
2994:00 001
248+.002
.481
.263=.003
.1822.002
.164..00 004
.154+.003
.135:.003
INFWALK
.2811.002
.2881.001
.6581.006
.3121.002
.2111.003
.18511000
.3181.010
.1771-002
GRAPHSAGE
.3581.004
.3071.007
.5832.003
.3062.005
.189:.001
.1891.003
.145:1.02
.1724-001
DGLFRM
1511..01
.555+.014
.618+.00 005
.615+.008
.3144.014
3464.002
.373+.018
.383+.022
DW+DINE
.111+001
.496:.014
.813+.025
.6294.02
.3174.036
.2664.007
.226:.009
.1884.002
GAB+
.604
.591+.004
.843 005
.484 007
.200
.2522.001
.193:.002
.198 002
DISE-FCAE
.2211.011
.7551.003
.9711.001
.4841.001
15041.001
.4191.001
.2971.002
.28211001
DISE-CAE
.8344.033
.762#.001
.967 001
.155.001
4964.001
.184.001
.3044.003
.254+.002
Table 2: Overlap consistency and positional coherence in real data. Best scores in bold, second best underlined.
Overlap Consistency WIKI
Positional Coherence
Method
ConA
FB
PPI
CoRA
WIkI
FB
PPI
DERPWALK
.137:.009
.143:.006
.115:.007
.015=.003
1.07810.025
0.83510.025
1.119:100015
1.00910.015
GRAPHAE
.269:.002
.293:.001
.273:.017
.452=.008
1.023+0.006
1.040:0.00 0002
1.001+0.013
1.016:0.00
WALK
.008:.003
.023:.002
.0211.002
.134=.002
1.001:0.011
0.998:1.00
0.938:10.08
0.999:100002
GRAPHSAGE
.211+ 001+ .136:.017
.230+:007
.097=.040
1.09940.01
1.10340.00 0.00
1.005 007
1.018+0.002
DGLFRM
.061+.004 .047=.008
.0704.006
.102=.002
1.26340.00 052
1.19740.00 043
1.34940.000 065
1.291=0.049
DW+DINE
.9001.012
.8041.032
.8511.017
.8551.016
1.790.10.076
2.126
1.79210.058
1.24710.043
CAE DINE
.5601.010
.610:1006
.801
.646=.003
2.317:10.028
2.551:10.048
1.783110.077
1.098:10004
DIAE
.56+.0011
.911.0044
.9631.004
.934-0033
5.210++0080
3.540++0.022
3.348
1.28344.004
DISE-GAE
.911..001
.944:.005
.954+.002
.937:.001
5.300+t0.993
4.343:10.14 3.388 1.261
and PPI. DGLERM shows good topological alignment across all datasets. Incorporating DINE improves results, especially for GAE+DINE, which achieves improved scores on all datasets. The proposed models, DISE-FCAE and DASE-GAE, deliver the highest overall performance. DiSE-FCAE performs well on FB, while DISE-GAF. excels across CORA and WIKI. However, both models show sub-optimal results on PPI, suggesting potential for further improvement on this dataset.
DEEPWALK and GRAPHAE offer moderate Sparsity, peaking CORA, but underperform other datasets. WNLW excels on FB but shows moderate results elsewhere, while GRAPHSAGE performs poorly in terms of sparsity across all datascts. DEEPWALK and GAE significantly improve their sparsity with DINE, particularly on CORA. DGLFRM shows competitive sparsity, excelling in FB and PPI. For the proposed models, DIAE-FCAF, and DISE-GAF perform best across datasets CORA, WIKI.
(RQ2) Can the identified subgraphs explain the intrinsic characteristics of the node embeddings? Here we explore how well the defined topological units represent information in the node embedding space, providing insights into how the relative and absolute positioning of topological structures influences the feature encoding within a graph. By quantifying these relationships, we can better understand the underlying patterns and structural information encoded in graph embeddings. In Table 2 we report Positional Coherence and Overlap Consistency for the examined embedding methods. For the second metric, as node proximity we used the inverse of the shortest path distance with sum as pooling.
DEEPWALK, WALK and DGLFRM perform poorly for Overlap Consistency, while GRAPHAE shows moderate scores, particularly on PPI. GRAPHSAGE performs slightly worse, with the best overlap consistency FB and CORA. WW+DINE achieves strong scores across all datasets, while GAE+DINE performs solidly but slightly lower, with its best result on FB. The proposed models, DISE-FCAE and DISE-GAE, outperform all others, achieving the highest consistency across all datasets except on CORA. DISE-FCAE excels on FB and WIKI, while DISE-GAF achieves the best overall score on PPI. DEEPWALK, GRAPHAF, AE, and GRAPHSAGE demonstrate moderate Positional Coherence. INPWALK consistently scores around 1.0 on all datasets, indicating stable but unremarkable coherence. Incorporating DINE leads to substantial improvements for both DREPWALK and GAE, achieving notable gains CORA,
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Table 3: Plausibility for node embeddings in synthetic data. Best scores in bold, second best underlined.
Link Prediction SBM
Node Classification ER-Ci.
Method
RING-CL
BA-Ct
ER-Ct
BA-Ct.
Tn-Ct.
TR-GR
DEEPWALK
.234+.003
.205+.000
.173=.002
.160=.00
.1462.002
.141=.003
.103=.007
.011+.002
CRAPHAE
183+.008
.160+.002
.145+.00 001
.145=005
.130=.002
.135:.000 006
.083+.001
.022 001
INFWALK
.2241.005
.1811.005
.2181.007
.2121.008
.1291.002
.1411.004
.0971.002
.093110004
GRAPHSAGE
2524.005
.2174.003
.1862.006
.1782.005
.160:1.044
.1541.002
.0931.002
.0841-003
DGLFRM
3344+.006
2224+.002
1220+.010
.210=.007
.1494.003
.143:.002
.1662.001
.175:.009
DW+DINE
9434.012
.9044.002
.744+.00 .00
.724#.040
.3204.081
.3274.008
.549:.015
.6274.004
GAB+DING
5494.005
.547=.014
.418
.38=20002
.311=.011
.3772.003
.3662.013
.254
DISE-FCAE
.7881.001
1.241.006
.95011.066
.9381.014
.8201.011
.7911.012
.8601.004
.8101.008
DISE-CAE
969+ 002
.910 006
.936 003
.941+.00
8134.003
.797+.009
.791+0005
800+
Table 4: Plausibility for graph explainers in synthetic data. Best scores in bold, second best underlined.
Nade Classif.
GXNEXPLAINER GSAGE
PGEXPLAINRR GSAGE
DISENE
Dataset
GCN
GATV2
GCN
GATV2
FCAE
GAE
BA-CL
.7294.004
.703+.006
.7072.004
.855+.006
558120088
59642.066
.919=.001
.855.009
ER-CL
6384.005
.611.000
6334.002
.923+ 004
.7044.032
.724=.002
.881 000
.8724.008
TR-CL.
.8461.004
.8291.005
.8221.006
1.031.000
.374..000
.4221.074
.9261.00 0001
.8711.005
TR-GR
.8471.005
.8331.004
.8321.003
.5731.003
.6411.045
.765-1.039
.88911.006
.898-.001
WIKI and FB. DGLFRM shows moderate scores, nevertheless scoring the best in PPI. The proposed models, DISE-FCAE and DISE-GAE, far outperform other methods, with DISE-FCAE achieving top scores on PPI, like DGLFRM, while DISE-GAE dominates on CORA, WIKI and FB (though with higher variance): both models show consistent superiority.
(RQ3) Are the identified latent structures sufficiently meaningful to serve as explanations for downstream tasks? Node embeddings serve as versatile feature representations suitable for downstream tasks, though they typically function as 'tabular-like" feature vectors without semantic labels for each feature. This limitation restricts the use of established post-hoc analysis methods (Bodria et al., 2023) like LIME, SHAP, etc. Our method allows to link topological substructures with embedding features, thereby assigning semantic labels to node vectors. Consequently, we are able to explain downstream classifier trained with unsupervised embeddings using feature attribution. Our goal is to assess whether the lask-importanl features align with human understanding by measuring the Plausibility.
In these experiments we consider node classification and link prediction as binary downstream tasks, training a logistic regression classifier b(x; B) = a (E 3,h,(x) (x) + 3o), where x is a node/link instance. We use SHAP [Lundberg Lee 2017) to compute the instance- wise feature attribution values (vr b))==...... For node classification, we consider positive instances as the nodes inside a clique in the synthetic graph. Accordingly, the ground-truth explanation for node is the set of nodes within the clique it belongs to. For link prediction, we focus on test. edges that were inside a clique before removal, where the ground-truth explanation is again the set of edges inside the clique itself. We compute plausibility scores over test instances with correct predicted label, because local explanations extracted from wrong predictions are not reliable for analyzing model decisions. We report in Appendix D the corresponding accuracy scores downstream classifiers.
Table compares Plausibility scores where node features from different embeddings are used to train downstream predictors. Since typical embeddings exhibit semantic patterns distributed across many dimen- sions Elhage et al. 2022) here consider all the contributing dimensions employing a non-negative weighting function f(*) max(0, x). This choice prevents bias toward methods that inherently produce disentangled semantics (c.g., by analyzing only top-ranked dimensions). DEEPWALK, GRAPHAE, and INFWALK perform modestly, with DEEPWALK scoring the highest among these on RING-CL, and WNFW showing relative strength on BA-CL. Compared with earlier approaches, DGLFRM delivers a modest gain in Plausibility, most notably in link prediction, whereas GRAPHSAGE significantly underperforns across all tasks, especially in node classification. The addition of DINE improves both DEEPWALL and GAE. DW+DINB excels with strong performance on RING-CI, SBM, and TREE datasets, while GAE+DINE achieves slightly worst results,
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particularly on node classification tasks, such as in TR-GR. Within the proposed models, DISE-FCAE and DiSE-GAR consistently achieve the highest scores ranking as the best two methods overall.
In Table 1 we compare Plausibility for state-of-the-arr local post-hoe explainers for graphs in node clas- sification. We emphasize that our approach focuses on explaining model encodings, unlike methods such as GNNEXPLAINER and PGEXPLAINER, which explain model decisions. These methods present local explanation in the form of node and/or /or edge importance, whereas our method, combined with feature-basod explainer, the explanation format is a vector of feature importance, associated with a subgraph for each feature. To make a suitable comparison, subgraph associated to the most important embedding feature (according to the logistic classifier). Recalling Eq. (8) this approach is equivalent to choosing the function f(w,) = Ji = argmax ( wx to compute plausibility index. We observe GNNEXPLAINER has uniform results across different input GNN models, instead PGEXPLAINER performs best with GCN. DISE-FCAE and DISE- AAEE outperforms the competitors in most of the cases, except with G method is capable of producing subgraph-based local explanations with comparable, even better, plausibility scores than GNNEXPLAINER CNNEXPLANERR/PGEAPPAA PPEEE//EIEELLIIERR To enhance clarity, Appendix B includes qualitative examples that visualize the local explanations generated by the different methods.
consider as the explanation presented by method the
ER- ER-CLIQUES. Reported results show that our
5 Conclusions
We present DISENE, a novel framework for generating self-explainable unsupervised node embeddings. To build our framework, we design new objective functions that ensure connectivity preservation, dimensional explainability, and structural disentanglement. Unlike traditional GNN explanation methods that typically extract subgraph from node's local neighborhood, DISENE introduces paradigm shift by learning node embeddings where each dimension captures independent structural feature of the input graph. Additionally, we propose new metrics to evaluate the human interpretability of explanations, analyze the influence of spatial structures and node positions on latent features, and apply post-hoc feature attribution methods to derive task-specific instance-wizt explanations. Our results show that interpretable node representations for graphs can be obtained by disentangling topological substructures across embedding dimensions. Additionally, the most important node features identified by post-hoc techniques aligns with the true explanation subgraphs. These findings mark significant. step toward human-centric evaluations of node embeddings, pointing to critical directions for future work in advancing human-in-the-loon validations in graph feature learning.
Acknowledgments
S. Piaggesi acknowledges support from the European Community program under the funding schemes: ERC-2018-ADG G.A. 834756 "XAI: Science and technology for the eXplanation of AT decision making".
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Appendix
Table A1: Summary statistics of graph-structured data. In empirical data, we restrict our analysis to the
largest connected component of any graph.
CoRA
WIKI
FB
PPI
RING- Ct.
SBM
BA-Ci. ER- ER-CL
TR-Ct.
TR GR
# nodes
2,485
2,357
4,039
3,480
320
320
640
640
831
799
# edges
5,069
11,592
88,234
63,377
1,619
1,957
3,138
4,196
2,081
972
# clusters/ tassters/motiss motils
28
18
16
32
32
32
32
32
32
density
0.002
0.004
0.011
0.009
0.032
0.038
0.015
0.021
0.006
0.003
clust. cocff.
0,238
0.383
0.606
0.173
0.807
0.561
0.486
0.456
0.360
0.002
A Training Settings
For DEEPWALK Perozzi et al. 2014), we train NODE2VEd algorithm for 5 epochs with the following parameters: p=1, 1= walk length 20, num sam_walks-10, window windox_size- size=5.
For WALW Chanpuriya & Musco 2020), a matrix factorization based method linked to DEEPWALK and spectral graph embeddings, we set the same value window_sizen size=5 used for WEEPWALK.
In GRAPHAE (Salha et al. 2020), we optimize l-layer GCN encoder with a random-walk loss setting analogous to DEEPWALK. The model is trained for 50 iterations using Adam optimizer and learning rate of 0.01.
In GRAPHSAGE Hamilton et al. 2017 we optimize a 2-layer SAGE encoder with mean aggregation and with a random- walk loss setting analogous to WALK. The model is trained for 50 iterations using Adam optimizer, learning rate of 0.01.
In DGLFRM MMehta et al. 2019), optimize a 2-layer GCN encoder with the standard hyperparameters for variational inference: ag ap=10, 10, Aprior and >post==. The hidden embedding size is tuned in the list [8, 16, 32, 64, 128, 512). The model is trained for 300 iterations using Adam optimizer, learning rate of 0.01.
DINE 10
Piaggesi et al. 2024h, autocncodor-based post-processing process trained for 2000 iterations, and learning rate of 0.1. Inpul embeddings are from DEEPWALK and GAE methods, tuning the input embedding size in the list [8, 16, 32, 64, 128, 256, 512].
DISE-FCAE and DISE-GAE trained for 50 iterations using Adam optimizer and lcarning rate of 0.01. The hidden embedding size is luned in the list [8, 16, 32, 64, 128, 256, 512]. Random walk sampling follows the same setting as DEEPWALK GRAPHAE and GRAPHSAGE.
GNNEXPLAINER is trainod for 30 epochs for cach test node, while PGEXPLAINER is trained for 5 epochs on trained nodes before being applied on test nodes. Moreover, since PGEXPLAINER is based on edge masks, we derive node masks for that model with the average mask value from incident edges.
Graph explainers applied top of the following GNN models trained node classification (two-layer for clique-based data and three-layer for grid-based data): GCN (Wu et al. 2019), GRAPHSAGE (Hamilton ct al. 2017], and GATv2 (Brody et al. 2022 All the graph models (not the explainers) are tuned by searching the best output embedding size from the list [2, 4, 8, 16, 32, 64, 128], as the input to the classification layer.
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httpp::/ppyorrce http::/::........e..e.ottteettee algorithh.n....n.ommm PCExplainer wtw.biiner.ttttttrrrrroo htmldtorch.
gaonetric. axplcc...ppppa.. aww.arricc.oplaanaalgottt PiExplainer
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B Algorithm Complexity
Space and time complexity of DISENE can be analyzed by looking at the pseudo-code in Algorithm A1 Part of the complexity depends on the complexity of the encoder. Here, assume GCN encoding functions, with its own set of learnable parameters θ But, in the experiments, we have also tested fully-connected
encoders.
Algorithm A1: DI
Xent)
Table A2: Time and space complexity for various embed- ding methods in terms of number of nodes [V], number of edges |&] and latent dimensions K. For models trained with random walk sequences, T denotes the context window size and L the random walks length. For models trained with neighborhood sampling, r denotes the number of sampled neighbors per node. For simplicity, single-layer architectures and we omitl the O(K2) memory usage of model weights since they are negligible compared to storing the embeddings.
Input
Graph 9 = (V.8)
Adjaceny matrix A E [0, MI
Embedding size K, Context window T,
Walks length L, Regularization Aent
Output: Embedding matrix H e |
Init. encoder network Ence (+); Init. identity matrix features X;
GNNs we consider
while not converged do p T Ence (A, X)); Sample batch of nodes: B + Somple(V);
Encoding slep:
Tnit. random walks W <
foreach E B do
Method
Time Complexity
Space Complexity
8
Sample random walk:
DEEPWALK
O(V]KTL)
O(IVK + (VI))
W WU Random W alk(A, v, L); Random-wall walk loss: Low (H, T);
INFWALK
O(IV(K)
O(123)
10
foreach d (1....) do
GRAPHAE
O(EEK + (VIK3)
O(VVK)
Aggregate rows of H: fa Σ Hod;
12
Compute 1-norm: [0/ < Σ Lv,d Hodi
GRAPHSAGE
O(r[V[K²)
O(rV]K)
13
Node affiliation matrix: F + H ff;
O(D)
O(K?)
14
Disentanglement loss Ldis(F)
DINE
+base model
+base model
15
Regularization loss (F)
16
Total loss: Liw + Ciia Ann Cont
O(E + +
17
Backpropagate and update θ W:
DISENE
+IVIKT))
O(VIK + MIL)
18
return H;
Our method consists of four main steps:
1. Encoding step generates the node embeddings and has the same per-layer time/space complexity of standard GCNs (Duan et al. 2022 i.c. O(IAa|oK + ()]3) and O(V K) respectively.
Random walk sampling and loss calculation has time/space complexity O(VIKTL) and O(V)L) respoctively (Rozemberczki et al. 2019), where T the context window size and L is the random- walk length (we sample 1 random walk per node, fixing well the number of negative samples to 1 for each positive sample). RandomW walk function sample first-order random walk starting from source node v of length L.
3. Node affiliation matrix involves computing the entries Fud = Eveys őalu, v; h) as Fud = Σ, Hug = Hudfa, ie. by multiplying node embedding entries Hud with quantities f, = =,, Hod- This step involves ((V)) K) operations for computing and storing matrix F.
4. Disentanglement and regularization losses involve respectively O(V) and O(K) operations for cosine similarity (matrix products) and entropy (vector sum).
Overall, given that ||Aloo is 2/E), DISENE results in O(EE) + 2 + KTL) time complexity and O(V]K + L)) space complexity. In Table A2 we compare these findings with computational complexities of competitor methods as they are reported from previous works (Tsitsulin al. 2021 Chiang et al.) 2019 Piaggesi al. 22024,, showing that our approach is in line with established node embeddings (runtime scales linearly with the number of nodes and edges).
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C Downstream Tasks Results for Real Datasets
We tested link prediction for the datascts reported in the main paper. For node classification, we tested PPI and other benchmark biological datasets in multi-label setting Zhao et al. 2023ah: the PCG dataset for the protein phenotype prediction, the HUMLoC, and FuKLoc datasets for the human and eukaryote protein subcellular location prediction tasks, respectively. Characteristics of additional biological datasets are reported in Table A3 We concatenated node attributes to node embeddings to get an enriched set of prodictors that, given our method extract interpretable features, can be used in combination with leature-based explainers (e.g., SHAP) for building fully transparent prediction pipelines. In Figure A1 report AUC-PR scores for link prediction and node classification in real-world graph data. Generally, scores increase with the number of latent embedding dimensions. Tables A4 and A5 show the maximum scores for link prediction and node classification, demonstrating that our approach can consistently achieve reasonable performances within the expected range of the pefforuance-iteerrrtaablllii trade-off.
Explanations for Real Datasets
Figures A2 and A3 display results for Topological Alignment, Sparsity, Overlap Consistency and Positional Coherence, for real-world datasets. These curves complement the results presented in Tables | and 2 by covering a broader range of output embedding sizes K 1K= in the main paper) and tuning for the hidden size D, as detailed in Section A for DINE (i.e., D [8, 16, 32, 64, 128, 256, 512)). Generally speaking, DISE-GAL yields the strongest interpretability scores when the embedding dimension is large, whereas performance differences are less systematic in the low-dimensional regime. Our subsequent analysis therefore focuses on the trade-off between these interpretability metrics and downstream task accuracy (e.g., link prediction).
Figure A4 plots link pue-ppreiittoon AUC-PR against each interpretability metric for all the considered embedding mothods. Every cluster of points represents single method with its range hidden and output dimensions considered hyper-parameters. To enable principled assessment. of which method offers the most favorable balance between predictive power and transparency, the left panel of Figure A5 highlights the Pareto frontier in each scatter plot. Models on this frontier optimal in the sense that alternative model attains both higher accuracy and greater interpretability, therefore making any other point lying below the frontier strictly dominated with respect to the two criteria, and thus sub-optimal (dominance was assessed while incorporating the statistical uncertainty each metric, estimated with the standard error from five independent runs per model).
Once the Pareto frontier is identified, we aim to quantify which embedding technique most consistently reaches that frontier. For every method E collect all of its hyperparameter instantiations E = (F1, E2,... (e.g,, 4R) Let P be the set of all configurations that lie on the frontier. We rank methods with two complementary scores:
cou E) = Pn E
eff(E) = PNE
P
E
Coverage the proportion of methods in the Pareto-frontier that belongs E. Efficiency the proportion of E's configurations that are Pareto-optimal. Scores reported in Table A6 combine Coverage and Elficiency via their harmonic mean to have compact measure quantifying the 'average' Pareto-optimality of every embedding method. Using this composite ranking, DISE-GAE consistently finishes within the top two methods across all datasets.
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Table A3: Summary statistics of graph biological data used for multi-label node classification.
PCG
HUMLoc
EukLoe
# nodes
3,480
3,177
2,552
2,969
# edges
53,377
37,314
15,971
11,130
labels
15
14
22
density
0.009
0.007
0.005
0.003
clust. coeff.
0.173
0.346
0.132
0.150
DeepWalk GraphAE InfWalk
GraphSAGE DGLFRM
GAE+
DiSe-FCAE DiSe-GAE
DW+DINE
cora
wiki
facebook
ppi
1.0
AUC-PR
0.9
0.9
0.8
0.7
0.8
0.7
2
16
64
128
32
128
16
32
128
16
32
64 128
output dimensions
output dimensions
output dimensions
output dimensions
DeepWalk GraphAE
InfWalk GraphSAGE
DW+DINE
DiSe-FCAE DiSe-GAE
GAE+ DINE
ppi
pcg
humloc
eukloc
0.25
AUC-PR
0.30
0.45
0.20
200
0.25
0.40
0.20
0.15
128
64 output dimensions
output dimensions
output dimensions
output dimensions
Figure A1: Downstream tasks results on real-world datasets (link prediction on the top panel, multi-label node classification on the bottom pancl) with varying feature dimensions size.
Table A4: Link prediction results (AUC-PR) on real- world datasets. Best scores are in bold, while scores with a relative performance loss of no more than 2% respect the best score are underlined.
Table A5: Node classification results (AUC-PR) on real-world datasets, Best scores are in bold, while scores with relative performance loss of more than 5% respect to the best score are underlined.
CORA
WIkI
FB
PPI
PPI
PCG
HuMLoc
EugLoc
DEEPWALK
.8924.005
.974.002 002
990++000 001
.794+.002
WALK
.4764.003
.2104.001
.3144.012
.241+.010
GRAPHAR
.911+.003
.50+..0 001
.994+.001
.16+.001
GRAPHAF
.517:4003
.2411.001
.336=.004
.2494.005
TNFW
.923:.003
.9364..00 002
.941+.006
.854+.003
INFWALK
.4424.001
.207:.002
.287..004
212
GRAPUSAGE
.13:1.005
.9411.00
.9914.001
.9244.003
GRAPESAGE
.506:1.001
.231:.002
.316:1.004
.237.1001
DGLFRM
.9721.001
.967
.9791.001
.89011.01
DW DINE
.88.1.002
.217.10001
.308.1.04
.231
DW DINE
8964.004
.9314
.987.0001
.84
CAE DINE
.5262.001
.24++000 001
333+ 006
.344 008
GAE+DINE
9264.001
.957
.9222 002
.919-.002
DISE-PCAE
.98.1.001
.233.1.003
.291 29911.006
.230-1.006
DISE-FCAE
8564.007
.9111.00
004
.9774 001
8844
DISE-GAE
518:t.001
.242+.00/
3315:.0003
.2384.006 006
GAF
8854.0002
.9774
.933
.933..01
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DeepWalk
GraphSAGE
GAE+ DINE
GraphAE
DGLFRM
DiSe-FCAE DiSe-GAE
InfWalk
DW+DINE
Topological Alignment
cora
wiki
facebook
ppi
1.0
0.8
0.4
0.6
0.4
2
8
16
32
64
128
4
8
16
32
64
128
2
4
16
32
64
128
2
8
16
32
64
128
output dimensions
output dimensions
output dimensions
output dimensions
cora
wiki
facebook
ppi
0.4
0.3
0.4
Sparsity
0.3
0.3
0.2
0.2
0.1
0.1
2
8
16
32
64
128
8
16
32
64
128
2
4
16
32
64
128
2
8
16
32
64
128
output dimensions
output dimenslons
output dimensions
output dimensions
Figure A2: Topological alignment and sparsity results real datasets with varying feature dimensions size.
DeepWalk
GraphSAGE
GAE+ DINE
GraphAE
DGLFRM
DiSe-FCAE DiSe-GAE
InfWalk
DW+DINE
cora
wiki
facebook
ppi
Overlap Consistency
1.0
1.0
1.0
0.5
0.5
0.0
0.0
0.0
0.0
2
8
16
32
64
128
4
8
16
32
64
128
2
4
16
32
64
128
2
8
16
32
64
128
output dimensions
output dimensions
output dimensions
output dimensions
cora
wiki
facebook
ppi
Positional chherence
10a
100
10a
10a
4
8
16
32
64
128
2
8
16
32
64
128
4
16
32
64
128
2
8
16
32
64
128
output dimensions
output dimensions
output dimensions
output dimensions
Figure A3: Overlap consistency and positional coherence results real datasets with varying feature dimensions size.
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DeepWalk GraphAE
GraphSAGE DGLFRM DW+ DINE
GAE+ DiSe-FCAE DíSe-GAE
InfWalk
Topologica Aligriment
cora
wiki
facebook
ppi
0.8
0.8
0.6
0.5
0.2
0.2
0.6
0.8
1.5
0..
0.8
1.0
0.5
c.R
1.0
0.5
0.8
D.7
0.
6.9
AUC-PR
AUC-PR
AUC-PR
AUC-PR
cora
wiki
facebook
ppi
0.4
0.4
Sparsity
0.6
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.0
1.9
AUC-PR
AUC-PR
AUC-PR
AUC-PR
cora
wiki
facebook
ppi
Overlap Consistency
1.00
1.00
1.00
0.75
0.75
0.75
0.75
0.50
0.50
0.25
0.25
0.8
0.6
0.8
1.0
0.6
0.8
0.5
0.6
0.7
0.8
0.9
AUC-FR
AUC-PR
AUC-PR
AUC-PR
Posiioona Coherence
wiki
facebook
ppi
10.
15F
10"
10f
Aaah
0.6
1.0
0.8
0.6
0.9
AUC-PR
AUC-PR
AUC-PR
AUC.PR
Figure A4: Scatter plots with link prediction performance (x-axis) and interpretability metrics (y-axis) on real datasets with varying feature dimensions size.
cora
wiki
facebook
ppi
Table A6: Pareto-optimal methods for real
Top. Align.
datascts.
Datasct
Metric
#1
#2
Alpg.l
DIŠE-GAE/
GAE+ (.130)
Sparsity
CORA
Over Cons.
Sparsity
DiSt-GAE 667
DiSe-GAE (541
DISE, 16665)
DGLFRML, 190)
Pos.Cober:
DiSe-GAE (68)
GAE+DINB 213)
Allgg
DiSi-GAEE
GAE+DINE (ICHDIIEE(1144
Sparsity
DiSe-GAE(.622
FIAE.FCAE, (31)) 135)
Pos. Coher. Over. Cons.
WIkI
Over. Over.Cons.
DiSR-GAE(
DGLFRML
Pos. DiSe-GAE(5119
GAE+DINE( 198)
TopgAlign.
DiSR-GAFF,G80
DiSs-FCA (X.CAF(.200. 23225)
Sparaity
DiSe-GAE 515
DGLFRMi, 156)
FB
Over. Cous.
DiSE-GAE
GAE+DINE (.06))
Pos.Co
DiSe-GAE.6990
FCAE, (616)))()))
Tol.Align.
DiSg-GAE
GAE+DINEY ((1)
Spansity
DGLPHM(,324
DiSE- GAE, 206)
PPI
Over Cous.
DiSg-GAE DGLFRMi (66)
Pus. Cober.
DiSe-GAE(4469)
DGLFHM,.77 173)
Figure A5: Pareto frontiers for the optimal models considering the interpretabilly-accrraayy (rade-olls defined
by the four proposed metrics and the link prediction accuracy.
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E
Explanations Visualization for Synthetic Datasets
Global Subgraph Explanations In Figure A6
we show
subgraph-level global explanations on synthetic dataset BA- CLIQUES. Subgraphs are generated for each feature dimension using the procedure described in Section 3.1 (summarized on the left in Algorithm A2) and are basod on various unsuper- vised embedding methods. The explanatory subgraphs demon- strate that our method effectively aligns embedding dimensions with meaningful, non-random functional components of the graph. In contrast, standard methods such as DEEPWALK 2 Compute background average attribution: and GRAPHAE E struggle to isolate individual structural units within dimensions. Instead, their embeddings often associate dimensions with groups cliques or subgraphs that include ele- ments from the random Barabási-Albert scaffold. Additionally, the visualization the right shows the correlation between latent features, further underscoring that the alignment be- tween embedding dimensions and graph structure is closely tied to the ability to disentangle feature correlation through non- collincarity. Specifcially, we highlight that for DEEPWALK and GRAPHAE the subgraphs exhibit significant overlap, which can be attributed to non-zero correlations within their latent. features. Tn contrast, the uncorrelated features of DISENE produce distinct, non-overlapping explanations.
A Algorithm A2:
UNSUPEDGESUDGRAPII(( Z,d)
Input
Graph 9 (V,E)
Embedding function V BK
Dimension explain d [1 ....) .K}
Output: Craph mask M R RM
Init. graph mask: M < 01
Ga [ट E(w.)ee =a(u)=a(v): for (u, E do
Compute edge attribution:
(a(u, =u(u)=a(v) Cas
Add explanation:
MiS max oz(u,
return M(d)
DEEP WALK
8.0
GRAPHAE
1.0
De Co
-34
222
8.0
DISENE
0.3// COTT
38
36
34
velue
8.
-22
dimension
dimension 1
dimension dimension
dimension 4
dimension
Figure A6: Subgraph-level global explanations for a representative subset of embedding dimensions, along with corresponding pairwise feature correlation plots, on synthetic dataset -LQUUI
24

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Published in Transactions on Machine Learning Research (07 (07/2025)
Local Subgraph Explanations Local explanations for node embeddings are extracted by using post-hoc feature importance method SHAP. For a given em- bedding model : V + RK we train a downstream classifier, e.g., in node classification task or link predic- tion. For simplicity, here we write the case when the classifier is a (binary) linear model, but it be any arbitrary complex model. It is anyway reasonable to assume that node embeddings come from deep graph model and downstream classifier is a simple l-layer neural network on top of the embedding layers.
Algorithm A3:
NoDECLASSSUUGRAPIIG K,
Input : Graph 9 (V,E)
Feature-base explamation matrix RR
Dimension to explain j C [1...K]
Output: Node mask B C R(
Tnit. node mask: B + 00
for E V do
Add explanation:
BQ) max (0, V, (e;b)) return B
K b(v) = 0 Σ
(node classification)
Byhg(r) + 30)
j=1
(link classification)
b(u,v) o > B,h Bu)
Given a vector representation of a graph instance (e.g., a node embedding h(v) or an edge embedding h(u, u)); and the corresponding prediction from classifier b, we compute feature importance with SHAP (1) ( (1; b)}j=1... or (w) (u, u; b=).)...... and the corresponding task-based graph masks (we illustrate the pesudo-code for node classification masks in Algorithm A3
B6) (0(0)))) E RR
BBD) max v (1) (w: b))
B (T)(())) E RIIII
BG) max (0, / (u, v; b)) ue
It valuable to remark that, training with logistic regression and applying SHAP, the resulting importance
scores are simply the coefficients of the regression (Lundberg & Lee 2017) (x 8y(h(z)) Elhy]). Thus, combining this methodology to interpretable graph features of DISENE, obtain fully transparent node/edge classification pipeline for graph data.
Figure A7 present examples of local explanations for node classification tasks on the small-sized synthetic datasets BA- MA-CLIQUES and TREE-GRIDS, using different methods. The experimental settings are consistent with those described in the main paper. On the left, we highlight the local ground-truth structures for the instance nodes depicted in the illustrations. On the right, we display the explanation subgraphs generated by each method, with nodes color-coded according to the respective explanation masks. For GRAPHAE and DISENE, the visualized subgraphs represent the most relevant structures determined by fcature importance attribution from the logistic regression classifier. For GNNEXPLAINER and PGEXPLAINER, the node masks correspond to the algorithm's 's output in explaining 2-layer GCN (Wu et al. 2019). Notably, DISENE demonstrates a strong ability to produce meaningful and interpretable node masks, effectively competing with state-of thtt--t GNN explanation methods.
25

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Published in Transactions on Machine Learning Research (07/2025)
Ground-truth
GRAPHAE
GNNEXP
PGExP
DISENE
10
0.8
0.6
0.4
anl
02
0.0
1.0
anlen mas4
0.0
1.0
0.8
amie mseu
Ground-truth
GRAPHAE
GNNEXP
PGExP
DISENE
1.0
0.8
0.6
1.0
0.8
Useu BA anjen
1.0
0.8
0.6
Figure A7: Subgraph local explanations for node classification in BA-CLIQUES (top) and TREE-GRID (bottom). On the leftmost column, we highlight the local ground-truth structures for the considered instance nodes. On the other columns, color-coded according to the respective explanation masks. For GRAPHAF, and DISENE, the visualized subgraphs represent the most relevant structures extracted with Algorithm importance attribution from the logistic regression classifier. For GNNEXPLAINER and PGEXPLAINER, the node masks correspond to the algorithm's output in explaining a GCN [Wu et al. 2019 in node classification.
display the explanation subgraphs generated by each method, with nodes
A3 and determined by feature
26

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Published in Transactions on Machine Learning Research (07/2025)
Topalogical Alignment
wiki
wiki
Overiap Consistancy
wiki
Positional Coherence
4.5
wiki
0.5
0.76
0.9
Sparsity
4.0
Aon
0.74
0.4
3.5
Aant =
0.7
Aent 0.01
3.0
Aurr 0.001
16
32
64
128
256
512
8
16
32
64
128
256
512
8
16
32
64
128
256
512
16
32
64
128
256
512
Topological Alignment
wiki
wiki
Overiap Consistency
wiki
Positional Coherence
4.5
wiki
0.95
Sparsity
0.40
0.90
4.0
0.75
0.85
3.5
depth =1
depth 3
0.35
depth
depth
0.80
3.0
0.70
16
32 128
16
128 256 512
32
64
128
256
512
hidden dimensions (output dimension=118))
hidden dimensions
hidden dimensions (output dimensions=22222
hidden dimensions
(outout dimenssins==22
(output dimensions-128
Figure A8: Interpretability metrics computed in WIKI with varying entropy regularization parameters, number of convolutional layers and hidden dimension sizes in DISE-GAE.
F Explanations for Synthetic Datasets
Global Explanations In Figure A9 we plot results for Topological Alignment and Sparsity, on the top and the bottom respectively, on synthetic datasets. Generally, DIŠE-FCAE outperforms aISE-GAB and the other competitors in all the datasets. In Figure A10 we plot results for Overlap Consistency and Positional Coherence, the top and the bottom respectively, on synthetic datasets. For the overlap metrie, UISE-FCAB and DISE-GAE consistently outperform the competitors, especially with more than 8 dimensions where they achieve almost perfect overlap. For the positional metric, the competitors GAE+DINE and DW+DINE slightly outperform DISE methods, especially in large dimensions, while DEEPWALK also show good results.
Local Explanations In Figure A1l we plot results for the plausibility metric on link prediction and node classification, on the top and the bottom respectively, while comparing different unsupervised methods that output node embeddings. Plausibility seems to benefit larger dimension values for DISE methods and DW-DINE for link prediction. Figure A12 shows the corresponding downstream task accuracy results.
G Influence of Entropy Regularizer and Model Depth for Wiki Dataset
Figure A8 explores how both the reguaiz-rggnllrizzttiin coefficient, Aents and the GCN depth in DISE-GAE influence the interpretability metrics on the WIKI datasct. Increasing Xent and using deeper networks boost Topological Aligument and Overlap Consistency, but also reduce Sparsity, revealing trade-off between compactness and explanatory power. Positional Coherence is weakly sensitive to Aenty and tends to improve with shallower architectures.
H Qualitative Examples for Cora Dataset
Tables A7 and A8 presents qualitative examples regarding the embedding interpretability the CORA citation graph. The dataset comprises seven classes, each corresponding distinct computer science topic. We employ different embeddiigg predictor variables for the node classification task with one-vs-rest multiclass strategy using Logistic Regression. Comparing DISE-GAE with the vanilla GAE and GAE+DINE, we selected for each class the most task-relevant embedding dimension corresponding to the highest coefficient. of the linear classifier. For every selected dimension, we analyze different properties of the corresponding anchor subgraph, computed with Eq. (2). First, without inspecting node class labels, the Topological Aligument scores and the Density scores (i.e., the ratio of the number of edges with respect the maximum possible edges in the subgraph) tell us that DiSE-GAF produces denser subgraphs compared to the overall
27

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Published in Transactions on Machine Learning Research (07 (0/5))2))
DeepWalk
GraphSAGE
GAE+ DINE DiSe-FCAE DiSe-GAE
GraphAE
DGLFRM
InfWalk
DW+DINE
ring_op of cliques
stochastic bloccc_lloooko model
ba cliques
cliques
Topological Alignment
1.00
1.00
0.8
0.75
0.75
0.75
0.6
0.50
0.50
0.50
0.4
0.25
0.25
0.25
16
64
128
32
128
32
128
16
64
128
output dimensions
output dimensions
output dimensions
output dimensions
ring_ of cliques
stochastic wlccsicc_loooo..c bloec_modelllllllo
ba cliques
er cliques
0.4
Sparsity
0.4
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
2
16
64
128
32
64
128
2
4
16
32
128
16
64
128
output dimensions
output dimensions
output dimensions
output dimensions
Figure A9: Topological alignment and sparsity results on synthetic datasets with varying feature dimensions size.
DeepWalk
GraphSAGE
GAE+
GraphAE
DGLFRM
DiSe-FCAE
InfWalk
DW+DINE
DiSe-GAE
ring_ot of cliques
stochastic_block model
_ldee
ba ba_ cliques
er cliques
Overlap Consistency
1.0
1.0
0.5
0.0
2
16
64
128
8
32
128
16
32
128
16 64
128
output
output dimensions
output dimensions
output dimensions
ring_of of cliques
stochastic block model
ba_ bl_quiuues
er clicuiq
Positional Coherence
101
10a
10a
101
10a
10a
10a
10a
128
64
32
128
64
output dimensions
output dimensions
output dimensions
output dimensions
Figure A10: Overlap consistency and positional coherence results on synthetic datasets with varying feature dimensions size.
graph, indicating stronger structural cohesion in the most task-relevant subgraph. Morcover, considering the bar plots that display distributions of class labels inside the subgraphs, nodes grouped within a single dimension of OSSE-GAB share research themes more consistently. This is quantified the lower entropy of labels' distributions within each subgraph. Furthermore, by looking at the top-20 most frequent keywords extracted from each cluster (after lemmatization and stop-words removal), we represent the word-cloud for each dimension, indicating clusters that more thematically coherent and easier to interpret. For example,
28

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Published in Transactions on Machine Learning Research (07 (025)
DeepWalk
GraphSAGE
GAE+ DINE
GraphAE
DGLFRM
DiSe-FCAE
InfWalk
DW+DINE
DiSe-GAE
ring of cliques
stochastic block model
ba al
er cliques
1.0
Plausibliity
0.75
0.5
0.50
0.50
0.25
0.25
0.00
2
16
64
128
8
32
128
16
32
128
32 Output dimensions
16
64
128
Output dimensions
Output dimensions
Output dimensions
ba_ cliques
er er_ciiuues
tree_cliquee
gre_grids
0.75
0.75
0.75
0.75
Plausibility
0.50
0.50
0.50
0.50
0.25
0.25
0.25
0.25
0.00
0.00
16
64
128
32
128
32
16
64
Output dimensions
Output dimensions
Output dimenslons
Output dimensions
Figure A11: Plausibility results on synthetic datasets (link prediction on the top panel, binary node classification on the bottom panel) with varying feature dimensions size.
DeepWalk
GraphSAGE
GAE+ DiSe-FCAE DiSe-GAE
GraphAE
DGLFRM
InfWalk
DW+DINE
ring_op of cliques
stochastic_bloce block model
ba_ cliques
er cliques
0.9
0.9
0.9
AUC-PR
0.8
0.8
0.7
0.7
2
16
64
128
8
32
128
2 output dimensions
16
32
128
16
64
128
output dimensions
output
output dimensions
ba cliques
cliques
Cree_cliquee
tree tree_grids
Accuracy
0.9
0.95
0.9
0.90
0.8
0.8
0.7
0.7
0.7
2
64 128
32 Output dimensions
8
16
128
64 Output dimensions
Output dimensions
Output dimensions
Figure A12: Downstream tasks results synthetic datasets (link prediction the top panel, binary node classification on the bottom panel) with varying feature dimensions size.
looking at the most predictive dimensions for the label Probabilistic methods, we observe with DISE-GAB we identify papers mentioning topic-relevant keywords such as "bayesian netwoks", "inference' and 'probabilistic': while papers identified with GAF+ frequently refer to "reinforcement learning' and "control'. These qualitative examples therefore show that our model learns latent dimensions whose structural and semantic meanings are markedly more human-intelligibbe than those of the competing methods.
29

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Published in Transactions on Machine Learning Research (07/2025)
Table A7: Most task-relevant dimensions in different methods for node labels Newral Networks, Reinforcement Learning, and Probabilistic Methods and their interpretations based on topic and text information in CORA.
Node Label
DISE-GAE (Macro-Fl m.864
GraphAE (Macro-FI 866)
GAE+DINE Macro-F1m.883
Topolog- Al84
Topolog. Align.=.277
Topolog. Align. .69
Density=.035 (16x)
Density=.001 (1.6x)
Deusity=.005 (2.2x)
Labels Entropy=.216
Labels Entropy=.88
Labels Entropy=.355
Relative Freq.
Relative Frea.
031
Relative Freq.
9.75
02
0,50
9.25
025
Cave Besedt
9.00
Genetic Alg
Neural Hete:
Probath Beet
Paant Leam
Lean. Theiry
Case Banad
Neural Neter
Probab. Hath
Baint Lawn
Rule Lewrn
Theary
cass Baved
oog
Ganetic Alg
Neural Nate Probati Matt:
Reint Leam.
Rule Lewn. Theary
Gerettic
Neural Networks
Tearning
parallel
ppattern
model
approach
using
paper
problem meblenmethhd equence
time
mode
system
dynanic
Lnmpurt
a
classifier
gorithm
appr each
training
task
systa
S
tem
rule
used
paper
new
learning
system
data
mode
ses
pape aaagrr
data
ne
network k
domati
use behavior
me
thod
learning
result
domatn
HMM
probl em em
neural network
structure dapptive
result Bayesian
show
algorithm
Topolog- Align==.955
Topolog. Align...779
Topolog. Align.
Density= (27x)
Density .00 0005 x2.1x)
Density=.01 (2.2x)
Labels Entropy=.151
Labels Entropy=.88 =.82
Labels Entropy=.499
Relative Freq.
Relative Freq.
0.
Relative Freq.
06-
0.25
Cave Bexeed
0.00
Genetti Alg.
Neural Nette
Protate bleth.
Palnt Learn.
Rute Learn
Case Banad
Genetic All
Neural Netsr.
Probus. Nath Reint Lawn. Lean.
Theary
Case Bissed
Gereste Alg.
Neural Netw.
Proist Metth
Reint Learn.
Rude Learn. Theery
Reinforcement Learning
reinforcement
learning
using
lyyten
systen
geent aagen
model meo use
paper
task
k
use
funch
solution
tion
based
pr
network
used
result
em
learn
network
problem
k
paper
I a tearning a g aargotthin
spstem
me
thod
use
task
result
lear
Learning
present
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a
algorithm
probl problem paper
endromeent
a algorim rthm
informat met
Tear marnin ngg ming used
teo
new system
new
search
show
state
Topolog- Align.=.897
Topolog. Aliga.=.258
Topolog. Align.=.720 Density=001 (4.2x)
Density=.0ff .073 (32x)
Density maasstty0003 .003 (1.4x)
Labels Entropy=.212
Labela Entropy=.699
Labels Entropy=
Relabive Freq.
Relative Freq.
Relative Freq.
2.35
02
0.2
Caye Based
2.20
Gereetc Alg.
Protuth dleth
Pelnt Learn.
Rute Leam. Thety
Case Banad
Geretic Ag.
Heural Naite.
Probat. Neth.
Reint Learn. Leamn.
Theary
Case Bosse
Gemettc Alg.
Neural Note.
Proisb: Mett.
Rein Leem.
Rde Leem. Theery
Heural
Probabilistic Methods
pe probabilEstic
teo
representation
approach system
theory
me
plan
method
nixture
task
task
paper
grovide
pr
problem
paper
Bayesian
ne
twork
data
ana
lysis
method
task
use
network k
mode
learning
D
al goortthm
model
inference,
proble
show
state
em
present
work
algoritie
learning
result
algorit
show
model
lear
based
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learning
work
problem
using
distrabuttin
results
30

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Published in Transactions on Machine Learning Research (07/2025)
Table A8: Most task-relevant dimensions in different methods for node labels Theory, Genetic Algorithms, and Rule Learning and their interpretations based on topic and text information in CORA.
Node Label
DISE-GAE (Macro-Fim.86k
GraphAF (Macro-F1m.88)
GAE+DINE (Macro-FI m838)
Topolog- Align.=,900 .37
Topolog. Aliga.=.402 Denaity=.001 (1.7x)
Topalog, Align.=.677
Deusity=.00e (27x)
Deusity=.009 (3.5x)
Labels Entropy -417
Labels Entropy=.887 .878
Labels Entropy=.Jee
Relative Freq.
5.5
Rolative free.
1034
Relative Freq.
0.75
0.25
0.2
Care Basso
6.00
Neural Nete
frobab Melh Baint Lear. Rute Learn Theory
Case Based
Gonettc Alg
Neural Hetse Frotab. Meth Reint Learn. Rule Lawrn. Theary
Casee Based
0.03
Cameetic Alg
Neural Nete: Probab. Reint Leam.
Rute Leam Theary
Genetic
Theory
distribution
data
system decision tree new
distribution
number
ahgorithm
learning
Learn
exomple
result
state
model
result classifiaatonn
used case
Learnin
one
one
Da
1 learning
algorit
show
mworithr
bound
F function
trethed
prediction
pr ob Lebleman
file
result
concept
show
paper
set model
problem
present
efficient
present
using
algor i it Teature
give model
online,
learning algorithn
Topolog. Align. =,992 .992
Topolog. Aliga.=.300
Topalog, Align.=.600 6008
Density .07 (32x)
Density=.0on -003 (1.4x)
Density 0000 (3.6x)
Labels Entropy=.055
Labels Entropy=.755
Labels Entropy=.43%
Relative Freg.
1.0
Relative free.
03
Relalive Freq.
02.
02
Case Based
0.0
Neural Hotse
Protub Netth Renn Loum.
Rote Lewrn Theory
Cass Based
0.
Gurotic Alp.
Neanal Hitte
Probub Neth Reint Lear.
Rute Leurn. Thatry
Case Boned
e.
Gunetic Aig
Neural Pratse
Probab Mers
Rant Leurn. Rale Leurn Theay
Gevetic
Genetic Algorithms
representation show based us ing population result
learning
model
set
nethod
axthod
program
Detaor
kep
am
paper
program am
GP
net
networ
tree
worr
used
neural
ne
network
usang
k
problem
genetic programee amming system
function ne
thod
pr
show program
mroblem
paper
mobble
emm
data
besed
knowsedge
lapproac
leaaning
ob em
task
al
Forithm
search used solution
sys eentemuss usingg
system eystem search per fornnce
Picisonn tree
Topolog. Align.
Topolog- Align.=.394
Topolog Aligu.=.366
Density
106s 046 (20x)
Density=.008 (2.1x)
Density=000 0005 (2.3x)
Labela Entropy=
Labela Entropy=.808
Labels Entropy= 641
Relative Freq.
Relative Freq.
Relative Freq.
82
0.2
0.0
Gunertcc.a Alg. Reural Notse
Based
Protab Neth.
Renn Loum.
Rute Lewnn Theory
Case Basd
ea
Neural Hitsee
Frohab. Neth
Reint Leam.
Rute Lear. Thery
Case Basad
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Genetec Ag
Neural hetse
Frobal, Mets
Rein, Leurn.
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Ganotic
Rule Learning
result
paper
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Lgoor
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rule
progran
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31


Reviews:
[{"review_id": "SzowmjEz1w", "reviewer": "Reviewer_EpKW", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1.) Learning explainable node embeddings in an unsupervised setting is well-motivated, as it provides task-agnostic explainability.\n\n2.) Interpretability is evaluated along several axes.\n\n3.) Several real and synthetic datasets are used in the empirical evaluation.\n\nWeaknesses:\n\n1.) It is not clear that the evaluation metric called \"comprehensibility\" deserves the name.  Specifically, while community structure is an important structural characteristic of graphs, other structures may be important and human-understandable in some contexts.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a method, DiSeNE, for unsupervised node representation learning, that yields node embeddings that are both _disentangled_ and _self-explainable_.  The former implies that distinct embedding dimensions correspond to subgraphs that are in a precise sense minimally correlated.  The latter means that distinct embedding dimensions are associated with distinct subgraphs whose structures are encoded by those dimensions.\n\nThe paper additionally proposes several novel metrics for evaluating interpretable node representations and uses them to compare DiSeNE empirically with state of the art post-hoc explanation generators (e.g., GNNExplainer).", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1.) I recommend that \"comprehensibility\" be renamed in light of weakness #1.  Additionally, I wonder if the authors can discuss other notions of comprehensibility that measure the extent to which mask matrices capture other human-understandable structures.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "jlM1NTnAZX", "reviewer": "Reviewer_j9WJ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The paper is well-motivated. The paper persuasively argues that post-hoc explainers are insufficient for mission-critical applications, making intrinsic interpretability in node embeddings an important contribution.\n2. The proposed loss for optimizing the interpretable embedding that blends random-walk preservation, soft orthogonality (for disentanglement) and an entropy term (to avoid empty factors) is conceptually simple yet effective.\n3. The five proposed metrics fill the evaluation gap and move beyond purely accuracy-centred comparisons.\n\nWeaknesses:\n1. There's no ablation on $\\lambda_{ent}$ or on GCN depth.\n2. There lacks theoretical guarantees that the entropy + cosine objective yields disentangled factors; empirical evidence suffices but theory would strengthen claims.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes DiSeNE (Disentangled & Self-Explainable Node Embedding), a framework that learns embeddings where each latent dimension aligns with a specific topological sub-structure (a global “anchor” subgraph) of the input graph. To obtain such embeddings, the authors propose to jointly optimize a random-walk reconstruction loss $\\mathcal{L}\\_{rw}$ for structural fidelity, a disentanglement loss $\\mathcal{L}\\_{dis}$ that soft-orthogonalises a node–subgraph affiliation matrix, and an entropy regulariser that avoids empty dimensions.\nThey also propose five new metrics—Comprehensibility, Sparsity, Overlap Consistency, Positional Coherence, and Plausibility—to quantify how human-understandable and disentangled the embeddings are.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Provide ablation on $\\lambda_{ent}$ and the number of encoder layers to show robustness.", "broader_impact_concerns": "No ethical implications.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "JP3RRyVSQI", "reviewer": "Reviewer_dLkq", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The writing is clear and easy to understand, the notations and figures are presented properly.\n2. The proposed evaluation metric looks interesting to me, and can be considered as useful information to this community in the future.\n\nWeaknesses:\n1. From the statement made above Eq(1), how could the authors assume that each dimension $d$ of the node representation is independent or weakly dependent such that each dimension can be evaluated separately, and contribute to the likelihood? \n2. It is acceptable if the authors treat $\\phi_d(\\cdot)$ as a score metric, but in the training, a non-linear activation is applied on, so I am concerned about whether it can be simply used as a valid metric for evaluating dimension importance (Eq(2)) like the other linear model.\n3. The dimension of node embedding, $d$, which also determines the number of global edge mask $M^{d}$ is determined by the graph structure such that it is data-dependent, how to choose it is empirical.\n4. The 'node affiliation matrix' $F$ is essentially a block model to me that captures the community correlation underlying the whole graph. This is a well-studied problem and approaches of explicitly learning the block model can be found in [1,2]. The authors are expected to discuss and compare with the closely related works, which use a probabilistic model and explicitly learns the block model which was shown to capture the community structure. \n5. I am confused why in entropy model $H$ the numerator is not applied absolute value. Also, the intuition of using entropy model here is not clear to me, since maximizing the entropy will force each dimension to be equally learned (assume the embeddings are non-negative at this point), but the goal of this model is to disentangle the dimension, which contradicts with each other.\n6. The human understandability can be improved, especially when the dataset (CORA, WIKI) used has rich topic and text information, the authors are expected to show more on how the learned node representation is interpretable. \n\n[1] Mehta, Nikhil, Lawrence Carin Duke, and Piyush Rai. \"Stochastic blockmodels meet graph neural networks.\" International Conference on Machine Learning. PMLR, 2019.\n\n[2] Chen, Xiaohui, Xi Chen, and Liping Liu. \"Interpretable Node Representation with Attribute Decoding.\" Transactions on Machine Learning Research, 2022.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents a stochastic block model (SBM) motivated graph model that attempts to disentangle the node embedding, unveil the underlying community and improve interpretability. The node representation is encoded by GNN, and loss terms are proposed to better learn the block model.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please refer to the weakness part and answer the questions. The authors are supposed to compare the mentioned interpretable probabilistic graph generation model that captures the community structure and learns interpretable node representations.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "bPQa6gM7Gr", "reviewer": "Reviewer_nkUJ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths：\n1. By decoupling representation learning and interpretability objectives, DISENE enables each embedding dimension to correspond to a unique topological sub-structure, which addresses the lack of interpretability in existing unsupervised node embedding methods.\n2. DISENE shows excellent performance on multiple real-world and synthetic datasets in terms of human understandability, structural decoupling, and downstream task utility. The high scores in understandability, overlap consistency, and credibility in the experiments demonstrate its effectiveness.\n3. The introduction of new metrics helps to evaluate the understandability of explanations and analyze the impact of spatial structure and node position on latent features, providing a more comprehensive understanding of the method.\n\nWeaknesses:\n1. The results are based on specific experimental settings, such as the fixed output embedding dimension in real-world data and the range of dimensions in synthetic data. It is unclear how the method performs under different or more complex experimental settings, and there may be limitations in generalization.\n2. Although new metrics are introduced, the understanding and evaluation of interpretability are still subjective to some extent. There may be other aspects of interpretability that are not fully captured by the current metrics, and further research is needed to improve and expand the interpretability evaluation system.\n3. The framework involves multiple objectives and constraints, such as the random walk optimization framework, calculation of edge importance, and soft orthogonality constraints. This may lead to relatively high computational complexity, especially for large-scale graphs, which could limit its scalability in real-world applications.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents DISENE, a novel framework for learning self - explainable node embeddings in an unsupervised way. It decouples representation learning and interpretability objectives, enabling each embedding dimension to correspond to a unique topological sub - structure. With three key objectives (connectivity preservation, dimension interpretability, and structural decoupling) and a comprehensive loss function, it ensures high - quality embeddings. Experiments on real - world and synthetic datasets show that DISENE excels in human understandability, structural decoupling, and downstream task utility, while also introducing new metrics for evaluation.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please refer to the weaknesses section.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)