How Does Topology Bias Distort Message Passing in Graph Recommender? A Dirichlet Energy Perspective
Authors: Yanbiao Ji
Yue Ding
Dan Luo
Chang Liu
Yuxiang Lu
Xin Xin
Hongtao Lu
Venue: NeurIPS 2025 poster ()
Keywords: Simplicial Complex
Recommender System
Dirichlet Energy
Primary area: applications

Abstract: Graph-based recommender systems have achieved remarkable effectiveness by modeling high-order interactions between users and items. However, such approaches are significantly undermined by popularity bias, which distorts the interaction graph’s structure—referred to as topology bias. This leads to overrepresentation of popular items, thereby reinforcing biases and fairness issues through the user-system feedback loop. Despite attempts to study this effect, most prior work focuses on the embedding or gradient level bias, overlooking how topology bias fundamentally distorts the message passing process itself. We bridge this gap by providing an empirical and theoretical analysis from a Dirichlet energy perspective, revealing that graph message passing inherently amplifies topology bias and consistently benefits highly connected nodes. To address these limitations, we propose Test-time Simplicial Propagation (TSP), which extends message passing to higher-order simplicial complexes. By incorporating richer structures beyond pairwise connections, TSP mitigates harmful topology bias and substantially improves the representation and recommendation of long-tail items during inference. Extensive experiments across five real-world datasets demonstrate the superiority of our approach in mitigating topology bias and enhancing recommendation quality. The implementation code is available at https://github.com/sotaagi/TSP.

How Does Topology Bias Distort Message Passing in Graph Recommender? A Dirichlet Energy Perspective
Yanbiao Ji
Yue Ding
Dan Luo2
Chang Liu
Yuxiang Lu
Xin Xin
Hongtao Lu²
Shanghai Jiao Tong University
Lehigh University
Shandong University
(jiyanbiao, dingyue, isonomialiu, 1uyuxiang_2011 2018, htlu]@sjtu. edu, danluo. ar@ggaall.com com
xinxin@sdu, cn
Abstract
Graph-based recommender systems have achieved remarkable effectiveness by modeling high-order interactions between users and items. However, such ap- proaches are significantly undermined by popularity bias, which distorts the in- teraction graph's s structure-referree to as topology bias. This leads to overrepre- sentation of popular items, thereby reinforcing biases and fairness issues through the user-system feedback loop. Despite attempts to study this effect, most prior work focuses the embedding gradient level bias, overlooking how topology bias fundamentally distorts the message passing process itself. We bridge this gap by providing empirical and theoretical analysis from Dirichlet energy per- spective, revealing that graph message passing inherently amplifies topology bias and consistently benefits highly connected nodes. To address these limitations, we propose Test-time Simplicial Propagation (TSP), which extends message passing higher-order simplicial complexes. By incorporating richer structures beyond pairwise connections, TSP mitigates harmful topology bias and substantially im- proves the representation and recommendation of long-tail items during inference. Extensive experiments across five real-world datasets demonstrate the superiority of our approach in mitigating topology bias and enhancing recommendation quality. The implementation code is available at httph:////// //github// com/sotaagi/T3p
1 Introduction
Recommender systems (RS) are fundamental components of modern online platforms, playing crucial role in connecting users with relevant items [16] 64, 8]. Recently, graph-based methods, such LighIGCN (2I) have gained prominence in RS, as they effectively capture collaborative
filtering signals by modeling high-order connectivity between users and items [34
66
63 722.
Despite their effectiveness, graph-based methods suffer from popularity bias, where popular items are recommended disproportionately more often than less popular items [ This notorious effect only undermines the accuracy and faimess of recommendation [74 35 55 but also exacerbates
the Matthew Effect and the filter bubble [38 45 through the user-system feedback loop.
In graph-based recommender systems, popularity bias mainly distorts the topology of the interaction graph, producing a highly biased degree distribution as shown in Figure [a), which we refer to topology bias. While research has shown that the topology bias gets further amplified [56] 76] in graph-basee methods, the root cause of this effect remains rather unexplored. Although some recent studies have attempted to understand this issue, they exhibit significant limitations. Their
"Corresponding Author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

analyses focus mainly on the embedding landscape [28 26 or gradient bias [O 71 31 after message
passing. The question of how topology bias distorts the message passing process remains unsolved.
To bridge this research gap, investigate the
Degree Distribution
(b]Degree Energy Distribution
message passing process in graph-based RS from the Dirichlet energy perspective. Our em- pirical and theoretical analysis reveals the fol- lowing results: (1) The graph message pass- ing mechanism is equivalent to minimizing the global graph Dirichlet energy. (2) Popular nodes with high degrees concentrate embedding norms. (3) The update of the norm of a node's embed- ding is upper bounded by local Dirichlet energy. Based on the above studies, we propose Test-time Simplicial Propagation (TSP), which performs message passing over simplicial complexes with intra- and inter-simplex smoothing to avoid the dominance of certain nodes or substructures, By incorporating simplices instead of mere edges, our proposed TSP mitigates harmful topology bias during inference, improving the performance of tail nodes without requiring RS retraining.
10a
Noce Eine1
Count
10.
0.0025
101
0.0000
1000
2000
1000
2000
Degree
Degree
Figure The degree and energy distribution of LightGCN embeddings on Gowalla dataset.
We summarize our contributions below:
We provide comprehensive theoretical analysis of how topology bias distorts message passing in graph-based recommender systems. We establish formal connections hetween graph message passing, Dirichlet energy optimization, and node embeddings, offering mathematical framework to understand why popular nodes are systematically favored.
We propose principled approach to overcome the limitations of pairwise edge message passing by introducing TSP, which leverages higher-order simplicial complexes for information propagation, enabling more balanced representation learning across nodes of varying popularity.
We develop TSP as plug-and and-alay solution that operates solely during inference, making it compatible with existing recommender systems without requiring costly retraining.
Through extensive experiments across five diverse weal-world datasets, we demonstrate that TSP can effectively mitigate topology bias without impairing the overall recommendation quality. Our results show consistent performance gains in tail nodes as well as competitive or better results in overall recommendation compared to ofate-oftthe-e baselines.
2 Preliminaries
2.1 Graph-based Recommendation
Let U = [u1, ... "bd) } denote the set of users and I = {i1, i2. } denote the set of items. User-item interactions are represented
implicit feedback matrix R {0, (M,4)I, where Rui 1 if user interacted with item i, and 0 otherwise. These interactions can be naturally modeled as a graph and the edge set contains edges (u, i) if and only if R., 1. Let A E {0, 1 be the adjacency matrix of the graph and D be the diagonal degree matrix where D = E, Asj- The
EV, specifically user-item bipartite graph where the node set V 2 UZ
normalized adjacency matrix is defined as À = D AD 3. In the widely adopted LightGCN 21 the message passing mechanism in layer I can be formally written as:
((++1) Ax
where (0) E xd is the node embedding matrix layer I, with d representing the embedding dimension. The recommendation score for user u and item is then computed as the inner product between their respective embeddings: Vus x x4.
2.2 Graph Laplacian and Dirichlet Energy
The graph Laplacian is a fundamental operator that captures the structural properties of a graph. The symmetrically normalized Laplacian 13
59
defined as:
i= D
AD 1
(2)

---

Given node embeddings X E RI the Dirichlet energy on the graph measures the smoothness of node embeddings with respect the graph structure: The Dirichlet energy of node embeddings X with re-
Definition (Graph Dirichlet Energy [3] spect to graph G is defined as:
Eg (X) TXX
1
Σ xx/Va-xx/VVa
2
(L,))E
where Tr(-) denotes the matrix trace, X; E Ed is the embedding of node i, and d, is its degree.
Building on this global measure, we define a localized node Dirichlet energy:
Definition (Local Dirichlet Energy) The local Dirichlet energy of node defined as:
(x,)
Σ
xolVad - -x 2
jEN(v)
where (v) denotes the set of neighbors of node v.
2.3 Simplicial Complexes
Simplicial complexes (SC) provide powerful framework to generalize graphs higher-order relationships [77]. These structures are combinations of simplices, which we define as follows:
Definition (k-simplex) k-simplex of is the convex hull of k + 1 affinely independent points S {vo, ,,,..., J. which constitute the vertices of the simplex. The dimension (or order) of the simplex is k. A j-simplex (j < formed by subset of S is called j j-face of o²
A k-simplex represents connections among k + 1 nodes: a O-simplex is node, 1-simplex is an edge, a 2-simplex is triangle, etc. These simplices are organized into simplicial complexes:
Definition 4 (Simplicial Complex) A simplicial complex X' is finite collection of simplices satis- fying the following properties: (1) Every face of a simplex in is also in X. (2) The non-empty intersection of any two simplices in X is a face of both.
The dimension (or order) of the simplicial complex X is the maximum dimension of any simplex X.
The structure of simplicial complex can be algebraically represented using boundary matrices:
Definition 5 (Boundary Matrix) The boundary matrix Bg maps k-simplices (k: 1 ))-sipllices by encoding the incidence relations between them. Specifically, (i, +1 if the (k: - 1)-simplex a is a face of the k-simplex o and 0 otherwise. The sign is determined by the orientation of simplices. Detailed discussion of orientations and examples can be found in Appendix
Now we can define the Hodge Laplacians, generalizing the graph Laplacian higher-order structures: Definition 6 (Hodge Laplacian 22 For K-dimensional simplicial complex, the k-th Hodge Laplacian L3 is defined as:
Lo = B Lg = B
(5)
L = B B&+ B (l<
(6)
3 Topology Bias from Dirichlet Energy Perspective
In this section, we provide a theoretical analysis of how topology bias shapes the message passing process in graph-based recommender systems. Our analysis uncovers the mechanism by which popular nodes are systematically favored, a phenomenon rooted in an energy minimization dynamic.
3.1 Message Passing as Dirichlet Energy Optimization
We begin by establishing a fundamental equivalence between graph message passing and Dirichlet energy minimization problem. This equivalence allows us to understand how the underlying graph topology directly governs information propagation.
3

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Lemma 3.1 The message passing mechanism is equivalent to optimizing the following energy function with Tikhonov regularization [46] and initial embedding matrix x
min T(X) - T= - -
1> O,
x
where L is the normalized graph Laplacian, and is regularization parameter.
D Proof in Appendix A./
Lemma 3.1 reveals that the message passing operation implicitly solves global smoothness objective. Topology bias directly alters this energy landscape, thus shaping the resulting embeddings a degree- dependent manner, as shown by the positive correlation in Figure [b).
3.2 Effect of Topology Bias on Embedding Norms
Next, we characterize how embeddings after message passing reflect the topology bias. Specifically, we show that nodes with higher degrees unavoidably accumulate larger embedding norms, a dynamic we refer to norm concentration.
(Norm Concentration) Let x, denote the embedding of node in the minimizer X* of Equation 7 Its squared norm is lower bounded by its degree d,:
Corollary 3.1
| 2 de C(L,x!)))
(8)
where C depends on the graph Laplacian L and initial embedding X
Proof in Appendix A.2
Corollary 3.1 formalizes that popular nodes (with larger de) necessarily acquire larger norms during message passing. often resulting in higher recommendation scores. This effect is also noted by Kim et al. 26 in their empirical observations.
3.3 Topology Bias in Embedding Update Dynamics
We further show how topology bias persists and is amplified every message passing step by bounding each node's ' embedding update in terms of its local graph energy.
Lemma 3.2 The squared norm of the embedding update for node v in a single message passing layer is upper bounded as follows:
| 122 S F((e),
(9)
where Ax, denotes the change in embedding for node after messuge passing, and Elv) the local Dirichlet energy of as defined in Desinition 2
Proof in Appendix A.3
Lemma 3.2 exposes self-reinforcing cycle: since the Dirichlet energy E(v) itself is greater for higher-degree (popular) nodes shown in Fi),re[[b,, these nodes receive larger embedding updates at each propagation step and potentially aggregate more information.
Methodology
4.1 Hodge Decomposition and Simplicial Dirichlet Energy
Theorem (Hodge Decomposition (2] The k-simplicial space N& admits an orthogonal direct sum decomposition
R iB @ kLr @ im(B
(10)
where im(-) and ker(-) are shorthand for the image and kernel spaces of the matrices, t represents the union of orthogonal subspaces, N₂ the cardinality of the space of k-simplex signals.
Theorem shows that any k-order simplicial signal x,* be decomposed
42
X,* X² G X,k, Xx*
(11)
where the gradient component X, BC - (corresponding to Equation 20 curl component
Xak.C B (corresponding to Equation 22 and the harmonic component X,*,u satisfies

---

LgX,+, . O. This decomposition provides theoretical guidance for the interaction and conversion between simplicial signals of different orders. Definition (Simplicial Dirichlet Energy be defined as:
61
The Dirichlet energy of simplices of order k can
E X = | 112
B R+ ot
l
(12)
Σ
1xo
x
Σ
x
xx
(13)
of
...0 Cak+ Cak+i
shared lower simplices
shared upper simplices
Different from the Dirichlet energy defined on a graph, the simplicial Dirichlet energy consists of two terms: energy between simplices sharing the same lower order component simplices (lower simplices) and energy between simplices consisting of the same higher order simplices (higher simplices).
Consider optimizing the simplicial Dirichlet energy similar to Equation 7
min X,t 2
1
Fo (X
1
gx=
X
(14)
The gradient with respect to xx is:
Vx,+ B, + B X5+ + 1X - xx L p(xx - xx (15)
One gradient descent step with learning rate can be written as:
x 0 V/xx,, x X (I n n X
(16)
Notably, Equation 16 the simplicial version of graph message passing. The orthonormal eigen- decomposition of Lg can be written as:
L Σ Nunuu
(17)
where 0 = Å1 =... A Ar+1 < .... S AN are the eigenvalues and are the corresponding eigenvectors. Similarly, we can define a resolvent this optimization problem as in Equation 31
R& LLL + +
(18)
For every initial embedding vector R&v yields the corresponding energy minimizer. Now show how simplicial message passing can alleviate the topology bias.
Lemma 4.1 For any (Br(1)+1) we have: |R < 11v12
Proof of Lemma 4.1 in Appendix A.4
Lemma reveals that simplicial message passing, for any embedding in the kernel of the upper adjacency operator (ker(B the norm is strictly reduced by the resolvent operator Rş- This means that, unlike traditional graph-based propagation which tends to inflate the norms of embeddings for popular (high-degree) nodes, simplicial message passing contract such dominant embeddings. By shrinking oversized embedding norms where bias accumulates, the simplicial approach helps regularize the embedding space, leading a more balanced embedding landscape.
4.2 The Proposed TSP
Guided by our theoretical analysis, we propose Test-time Simplicial Propagation (TSP) based on simplicial Dirichlet energy optimization, extending the message passing procedure to simplicial complexes. TSP comprises the following key modules: Semantic Graph Construction, Intra- Simplex Smoothing, Inter-Simplex Propagation, and Multi-Order Fusion. An overview of the We also provide algorithm description in pseudo code in Appendix B
whole pipeline is in Figure
Semantic Graph Construction The objective of this module is to furnish topology that is more reflective of semantic relationships between nodes, rather than relying solely on raw interaction patterns. Formally, given pretrained node embedding X and hyperparameter 0 controlling the
5

---

Interactions
Semantic
Simplices
massage passing
Recomiantde
Fretrained Gragh
Graph
Camplix
NimlleSimple
(noces)
Samples-
Samppie
Atawd node
B A1-2
(odges)
2-mpplex
nagreorng simp ex
U.
(1) Semantic Graph
[ringless]
(2) Intra- Simpl-Simpexx
(3) Inter-Simplex Propagation
Construction
Smoothing
(4) Mult-order Fusion
Figure 2: The pipeline of our proposed TSP, consisting of four main steps: (1) Semantic Graph
Construction, which builds a semantic graph based on node embeddings; (2) Intra-Simplex Smoothing, which aggregates information within each simplex; (3) Inter-Simplex Propagation, which performs message passing between simplices; and (4) Multi-Order Fusion, which combines embeddings from different orders of simplices for the final recommendation.
similarity threshold, we can construct the adjacency matrix of the semantic graph As:
1 (0.
x x; x= 0
As|i,jl =
(19)
otherwise
where is the embedding of node i, Assi. means the element the i-th row, j-th column in As. This process naturally densifies the tail nodes by giving low-degree nodes additional connections, thus partially alleviating the topology bias [4] 66]. Then we lift this semantic graph into simplicial complex, as described in Appendix
Intra-Simplex Smoothing Once have Ag, we lift it into simplicial complex by incorporating nodes, edges, and higher-order simplices (triangles, tetrahedrons, etc.) that capture multi-node connectivity. Each k-simplex 6² consists of k + nodes joined by semantic adjacency. Let Bg be the boundary matrix of k-simplices. These matrices enable us to convert node embeddings to high-order simplices signals as in Equation 11 component nodes as follows:
Formally, we can aggregate the information for ot from its
[ B B B
(20)
where X, is the embedding matrix of k-simplices. This process performs smoothing inside the simplex, and also provides initial embeddings for each simplex without the need for embedding learning. The factor II Bk converts the embeddings of component nodes into simplex embeddings.
Inter-Simplex Propagation Although intra aitra-simplex averaging helps to smooth embeddings within each simplex structure, it alone does not capture the relations between simplices. To address this, introduce inter-simplex propagation. More specifically, we utilize the Hodge Laplacians in Equation to define the message passing between simplices within each order l:
x (1+1) (I (LL) x gk
(21)
where L3 the Hodge Laplacian of order k., is the message passing layer, and controls the signal filtering on simplices [62]. This message passing process further refine the representation based on higher-order simplicial structure, incorporating simplex interactions.
Multi-Order Fusion from different orders of simplices. We can perform the reverse process of Equation 20 to fuse the embeddings of all the simplices into node embeddings:
To obtain the final recommendation scores, we need to fuse the embeddings
Xo- = B, B,) x(()
(22)
where Xo+ represents the node embeddings converted from k-simplices. We then produce the final representation for recommendation by performing mean pooling of multi-order representations:
X X, + Mean X ..., Xo.-k) k):
(23)
where Xo is the original pretrained embedding, and X a are the embeddings from k-order simplices after message passing. This multi-order fusion ensures that our final embeddings incorporate both interaction information and higher-order semantic relations.

---

Experiments
In this section, we empirically evaluate the effectiveness of our proposed method TSP. We aim to answer the following research questions:
RQ1: How does TSP perform compared to existing tfte-ooftthe-ar art debiasing methods?
RQ2: Can TSP improve recommendation fairness and diversity?
RQ3: Can TSP produce fairer item embeddings, suggested by the theoretical analysis?
RQ4: What the contribution of each component in TSP to model performance?
RQ5: How much computational overhead introduced by TSP?
5.1
Experimental Setup
Datasets We conduct experiments five real- world datasets representing diverse domains: Yelp2018 (business reviews), Gowalla (location check-ins), Adressa (news). Globo (news), and ML10M (movie ratings). User ratings in MLIOM are converted implicit feedback (1 if rated, otherwise). Details of these dataset statisties are in Appendix
Evaluation Protocols We split the datasets into 80%, 10%, 10% as train, validation, and test sets following the unbiased sampling method proposed by Wei et al. [56). We report Recall 20 (R @ 20) and NDCG@ 20 (N @ 20) averaged across all users. Specifically, also report performance of the 20% least popular items (tail items) beside overall performance. Evaluation details are in Appendix B
Backbone Models We demonstrate TSP's effectiveness by applying it to three strong ONNNbased backbone models: LightGCN (2] is simplified GCN model widely used for collaborative filtering. SimGCL 65] is graph contrastive learning method using noise injection for augmentation. Light- GCL. is the state-of-the--at contrastive learning model using SVD for graph view construction.
Debiasing Baselines We compare TSP against several representative debiasing methods: IPS [4
uses inverse propensity scoring reweight interactions based item popularity. CausE (6) relies causal inference to learn representations robust to exposure bias. MACR (56) employs counterfactual
reasoning based on a causal graph mitigate popularity bias. SDRO 58 uses distributionally robust optimization to improve worst-case performance.
Implementation GNNs use message passing layers and embedding dimension of 64. We use the Adam optimizer with learning rate 0.001, weight decay le-4, batch size 4096. Training runs for up to 500 epochs with early stopping patience 50. Further details can be found in Appendix G
5.2 Recommendation Performance (RQ1)
Table presents the recommendation performance specifically for tail items, while Figure 3 shows the overall performance across all items on Gowalla dataset. More empirical results of overall recommendation performance can be found in Appendix C
LeghGCL
LightCL
UehoCC SDRO
LigttcL SORD
MWC
LihHOCL MACK
GausE
LightGLL
Lightn
SmGGL
Simsa TSP
Smi3i MMC
simGEL MAZR
Simici Guust
SinGEl + Deadt
Simtic
Senda IPS
UghooCN TSP
UgMISch + TSP
UighsON SDR0
Lggisch + SD60
UgneSCN CausE
LightCy
UghiGON - IPS-
UghnSCh IPS
Lighesch-
LightOh
0.00
0.25
0.10
0.55
0.20
0.25
0.30
0.22
0.04
0.05
0.08
0.10
0.12
MRR820
ND06920
Figure 3: The overall performance of TSP and other baselines Gowalla dataset.
We have following key observations: (1) TSP consistently achieves substantial improvements in Recall and NDCG@ 20 for tail items across all datasets and backbone models (Table [ frequently outperforms the best baseline debiasing method by significant margin (% Improve). This validates the strong ability of TSP to improve tail performance. (2) While the primary goal mitigating topology bias, TSP often maintains
improves overall performance compared to
7

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Table I: Performance on tail items (20% least popular items). Best results are in bold, and second-best results are underlined. %Improve is TSP's relative gain over the second-best method. * indicates the statistically significant improvements (i.e, two-sided t-lest with p<0.05) over the best baseline.
Adressa
Gowalla
Yelp2018
ML.IM
Globo
Method
R@200
Nia20
R
NG:
R:820
N6
R.
N820 20
R@/20
N6:20
LightGICN
0.015
0.007
0,006
0.006
0.002
0.001
0.002
0.001
0.001
0.002
+ IPS
0.040
0.020
0,030
0.025
0.006
0.007
0.011
0.005
0.022
0.012
Caust!
0.023
0.011
0.020
0.013
0.007
0,004
0.007
0.002
0.014
0.008
MACR
0.007
0.014
0.088
0,0015
0.004
0,006
0.005
SDRO
0.014
0.0077
0.812
0.007
0.0015
0.003
0.007
8.002
0.003
0.004
+ TSP (Ours)
0.055
0.024
0.038
0.021
0.009
0.008
0.013
0.008
0.024
0.013
%/mpproe
37.50*
20,00
26.67
28.574
14.29*
18.
60.00
9.09
8.73
SimGCL
0.016
0.007
0.008
0.007
0.002
0,003
0.003
0.002
0.002
0.001
IPS
0.052
0.027
0.021
0.018
0.008
0,006
0.012
0.008
0,025
0.011
CausE
0.018
0.008
0.028
0.018
0.006
0,005
0.010
0.004
0,010
0.008
MACR
0.014
0.006
0,817
D0066
0,015
0,004
0.013
0.009
0.012
0.008
+ SDDO
0.016
0.007
0.014
0.007
0.006
0,003
0.008
0.003
0.007
0.006
TSP (Ours)
0.085
0,046
0.055
0.047
0.015
0.014
0
0.012
0.025
0.017
Ssippaos
70.37
96.47
161.17
87.50*
23.088
LightGCL
0.020
0.009
0.010
0.008
0.003
0,003
0.003
0.002
0.004
0.004
IPS
0.045
0.022
0,040
0.030
0.010
0.010
0.010
0.006
0.024
0.016
CausF
0.001
0,026
D0119
0.001
0,002
0.003
0.001
0.015
0.011
+ -MACR
0.008
0.005
0.015
0.012
0
0.012
0.011
0.007
0.013
0.007
-DDO
0.015
0.007
0,013
0.007
0.010
0.003
0.007
0.002
0.007
0.005
TSP (Ours)
0.047
0.050
0.032
0.020
0.019
0.014
0.010
0.032
0.018
Sdmmproee
66.67
113.64
25.00
6.67
81.82*
58
27.27
66.675
133.33
12.50*
the backbone (Figure 3 Appendix C). In many cases, it also outperforms other debiasing methods on
overall metrics, suggesting TSP can effectively balance representations without sacrificing overall recommendation quality. (3) The improvements offered by TSP are particularly pronounced when applied to contrastive learning backbones (SimGCL, LightGCL) compared to LightGCN. This might indicate that contrastive methods can benefit more from higher-order structures.
5.3 Recommendation Fairness and Diversity (RQ2)
To assess the effect of TSP on recommendation faimess and diversity, we evaluate two established metrics, Expected Free Discovery (EFD) [47 and Average Percentage of Tail Items (APT) (2), with detailed definitions provided in Appendix B As reported in Table 2 TSP consistently outperforms competitive baselines on both EFD@ 20 and APT@ 20 across all datasets when using the LightGCN backbone, indicating that our method improves the exposure of long-tail content while enhancing overall fairness.
Table 2: Performance on LightGCN backbone with respect to EFD @20 and APT@ @0. Best results are in bold, and second-ber results are underlined.
Adressa
Gowalla
Yelp2018
ML
Method
EFD4620
APT@22
EFD8
EFD
APT
EFD@:20
APT820
EFD@:20
APT6:20
LightGCN
0.823
0.124
0.791
0.131
0.752
0:085
0.715
0.072
0.724
0.064
IPS
0.831
0.133
0.801
0.156
0.761
0.101
0.732
0.080
0.714
0.073
CaudE
0.793
0.105
0.763
MACR
0.901
0.147
0.856
0.150
0,814
0.118
0.794
0.107
0.756
0.121
SDRO
0.832
0.136
0.781
0.138
0.738
0.110
0.705
D.197
0.721
0.112
TSP (Ours)
0.923
0.217
0,875
0,164
0813
0.133
0,802
0.116
0.773
0.149
5.4 Distribution of Item Embeddings (RQ3)
To explore the impact of our proposed method on the distribution of learned embeddings, we project the item embeddings learned by LightGCN and LightGCL on the Gowalla dataset into wwo-dimensional space using t-SNE visualization. We compare these results with competitive MACR baselines, as shown in Figure 4 balanced distribution of item embeddings compared to LightGCN, aligning with the expectation that contrastive learning methods can learn more uniform representations [51) (2) The incorporation of simplicial propagation in our method results a more uniform distribution of item embeddings, validating the effectiveness of simplicial message passing.
Our key observations are as follows: (1) LightGCL exhibits more
8

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Table 3: Ablation study Gowalla dataset. We report the overall and 20% tail item results of Top 20 performance.
Table 4: TSP preprocessing and inference time costs compared with LightGCN backbone train- ing/inference time.
Overall
Tail
Time Cost
Adressa
Gowalla
Yelp18
MI.IOM
Globo
Variant
R@20
N@-20
R@20
N@20
LGN Train
10mán
35min
1h40m
9h
3b
TSP Preproc
2.54s
15,97s
26.32s
91.34s
44.2%
TSP-Full
0.082
0.058
0.038
0.021
TSP-SE
0.056
0,041
0.017
0.009
LGN Infer
0.037s
0.080s
0.134s
0.575s
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Figure 4: I-SNE visualization of item embeddings learned on Gowalla dataset. The top figures plot the Gaussian KDE of embeddings projected to R The darker the color, the more items are in this region. The figures below show the KDE angles (i.e. arctan2(y, x) for each point (x, y)).
5.5 Ablation Study (RQ4)
To isolate the contribution of simplicial message passing and semantic graph construction to the final performance gains, we conduct an ablation study on Gowalla dataset. The results are in Table 3 with more results in Appendix H. Specifically, we compare following variants of TSP with same LightGCN backbone: (1) TSP-Full, the original model, (2) TSP-SE, which uses the same semantic graph as TSP, but only performs graph message passing, (3) LGN. the LightGCN baseline. The results show that while TSP-SE does provide some improvement, suggesting that denoising the graph structure can improve model performance. The full TSP framework, incorporating message passing on the simplicial complex structure, yields significantly larger gains. This confirms that explicitly modeling and propagating information through higher-order structures is crucial for effectively smoothing representations and achieving fairer recommendations.
5.6 Scalability (RQ5)
To assess practical viability (RQ5), measure the preprocessing time (building the complex and Laplacian) and the additional inference time introduced by TSP compared to the LightGCN backbone. Table 4 shows these times across datasets. The lightweight one-time preprocessing and efficient inference compared to the backbone LightGCN suggests that our approach introduces minimal additional computational overhead. Notably, the dataset size increases, the relative computational overhead introduced by TSP diminishes in comparison to the backbone model.
Related Works
Graph Neural Networks for Recommendation Graph Neural Networks (GNNs)
48 are
widely used for modeling high-order interactions in recommendation tasks. Early works like NGCF [52 utilize graph convolutions to iteratively aggregate collaborative signals. Recent advances focus on improving GNN architectures to enhance efficiency and accuracy, such as LightGCN 21].
UltraGCN (37. and SVD-GCN [40] The integration of contrastive learning [60, 65] 70 49 has
further improved representation robustness, with models like SimGCL [65] demonstrating the benefits
of self-supervised learning objectives and lightweight graph augmentations. Emerging research also
explores the utilization of large language models (LLMs) enhancing graph recommenders 24 57 54
expanding the applicability of graph-based methods.

---

Higher-Order and Simplicial Neural Networks While GNNs effectively capture pairwise rela- tionships, their expressive power limited by the L-Wais (1-WL) test (53), impeding their ability to model higher-order structures such as triangles, cliques, and communities [12]. To overcome this, recent studies employ simplicial complexes and higher-order message passing. Mod- els like Simplicial Neural Networks (SNN) [7]. BSCNet [I], and HSN [20] extend convolutions beyond simple edges, leveraging Hodge Laplacians for richer structural representations. Simplicial are also developed to model interactions among higher-order
attention mechanisms [19 18 29
simplices.
Popularity Bias Mitigation Popularity bias remains a critical challenge for fair recommendation. Earlier approaches utilize regularization 77 [50] adversarial frameworks [33] 68 27 reduce popularity-driven ranking bias. Causal modeling has emerged as a powerful tool to disentangle
popularity effects [56] 751, with counterfactual reasoning being applied to mitigate bias at the inference stage. Newer perspectives focus on employing topological interventions or stochastic graph augmentations
36 73.. Recent research also investigates debiasing via distributionally
robust optimization
58 embedding landscape modeling [32 28), and reinforcement learning
frameworks
44
30
to achieve a balance between recommendation quality and fairness.
Oversmoothing in Graph Neural Networks The performance of GNNs is strongly affected by the depth of the network (5
As the number of layers increases, the node embeddings contract and converge exponentially toward a constant vector, severely degrading the predictive accuracy (25). Several metries have been proposed quantify oversmoothing and characterize pessage-passing dynamics, including Dirichlet energy and Mean Average Distance (MAD). Although we also employ Dirichlet energy to analyze message passing, our focus is fundamentally different: we investigate the topology bias instead of network depth. Specifically, we provide the first analysis of how graph topology influences Dirichlet energy and extend this energy-based characterization to higher-order simplicial complexes.
14 39]
Conclusion
This paper addresses topology bias in graph-based recommender systems by analyzing message pass- ing through Dirichlet energy perspective. Based on theoretical analysis, propose Test-time Simplicial Propagation (TSP) to address this bias by using higher-order simplicial complex structures for more balanced message passing. Extensive experiments demonstrate that TSP effectively reduces the topology bias and improves recommendation quality.
Acknowlegments
This paper is supported by NSFC (No. 62176155), Shanghai Municipal Science and Technology Major Project, China, under grant No. 2021SHZDZX0102.
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Appendix
A Proofs
A.1 Proof of Lemma 3.1
Lemma 3.1 The message passing mechanism is equivalent to optimizing the following energy function with Tikhonov regularization [46] and initial embedding matrix X
min = = -I(TXX -
P 0,
(7)
where L is the normalized graph Laplacian, and y regularization parameter.
Proof The gradient of the objective function in Equation with respect to X is:
xx](X)= + m - x
(24)
With a learning rate of 17, one step of gradient descent yields:
( = ( xX (
(25)
= ( ,/LX(00 m(X - x
(26)
= ( nIx
(27)
(I -I- (1_)(()x())
(28)
When v Equation (28) becomes X (I L) (o) which exactly the message
passing equation described in Equation ① For 1j < 1, the operation corresponds to a low-pass
filtering on graph signals, consistent with prior work [67]]
Alternatvely, we can derive the closed-form solution by setting the gradient in Equation 24 to zero:
LX xXX' - -x:0))
(29)
[L (()) px (0)
(30)
= = (1 + μuI) -x(0)
(31)
Lerī= = UAUT be the eigendecompositton L. matrix of eigenvalues. Then:
L. where A diag(As A2 .... .N) is the diagonal
X = MUAU + -x
(32)
PUUAA -
(33)
-UAU x (0)
(34)
where
À μ(A + pd)-1 diag
f N
(35)
do + 11)
The embedding for node v can then be expressed as:
Σ
(u|ck,
(36)
AE 11
where C₁ (0) and ug [ is the v-th element of eigenvector ug The spectral filter Nete functions as a low-pass filter on graph signals, attenuating components corresponding to larger eigenvalues.
22

---

A.2 Proof of Corollary 3.1
Corollary (Norm Concentration) Let x; denote the embedding of node u the minimizer X* of Equation 7 Is squared norm is lower bounded by it degree d,g
|x 2 d, C((L.
(8)
where C depends on the graph Laplacian L and initial embedding X
Proof For the node embedding described in Equation 36
obtain lower bound of norm by only considering the smallest eigenvalue X1 = corresponding to eigenvector u1
(vd,, vdz. ......., VdN), where d, is the degree of node i.
1x
(A/(Ae + po)
(37)
k=1
pa + +)) [ |
(38)
= nn lulli c(i, x()) = d, C(C, x
(39)
A.3 Proof of Lemma 3.2
Lemma 3,2
The squared norm of the embedding update for node v in single message passing layer upper bounded as follows:
11x. S E(v),
where Ax. denotes the change in embedding for node u after message passing, and E(v) is the local Dirichlet energy as defined in Definition 2
Proof We can write Equation
as:
Ax, x!
x (1-1)
Σ
x,
(1-1)
-x (1-1)
(40)
V dd.j
jEN(v)
Σ
x
(1-1)
vdo x(-11)
(41)
dd jEN(1)
/dj
(42)
Then we have:
Ax, 12
Σ
(1-1)
(1-1)
(43)
dp
dly
Vd,
(E)
de
11-1)
(1 1)
CCuchhy-Schwazz Inequalityy
(44)
dy
ddj
Vd
(EN((v)
E(v)
(45)
A.4 Proof of Lemma 4.1
Lemma 4.1 For any v E ker(Br+1) we have:
| < ||v||
Proof
N
2 S ( + u
2 N
1 = Σ
# Ex4m
P
Σ
u 11 (u? v)2
(46)
Nrt1
And since u; span the kernel of B we have
Σ (uf v)² = ||v112
(47)
1=1
23

---

So we get
2 Art1+ + llv12 11v122
R
(48)
B Evaluation Protocols
B.1 Unbiased Sampling
To fairly evaluate debiasing performance, standard random train-test splits are inadequate they inherit the original data's popularity bias. We adopt the unbiased evaluation protocol from Wei al. [56]. Specifically, we uniformly sample 10% of interactions for the test set and 10% for validation, ensuring all items have an equal chance of being sampled, regardless of their popularity. The remaining 80% form the training set. This creates unbiased test and validation sets with balanced popularity distributions, allowing for accurate assessment of long-
maigtail tail performance.
B.2 Evaluation Metrics
Recall @ecall@k is the proportion of relevant items found in the top-k recommendations. A higher Recall value indicates that the system is better at identifying and ranking relevant items for users. Formally, Recall @k is defined as:
Number of Relevant Items in Top-k
Recall @k =
(49)
Total Number of Relevant Items
NDCG@K is widely used evaluation metric in recommender systems to measure the ranking quality of suggested items. NDCG@K k takes into account both the relevance of recommendations and their positions in the ranked list, assigning higher importance to relevant items appearing earlier. To compute NDCG@K k, first calculate the Discounted Cumulative Gain (DCG) up to position then normalize it by the ideal DCG (IDCG) obtained from the perfect ranking. Formally, for a list of recommendations T1, ....... Tk, the DCG@k defined as:
Normalized Discounted Cumulative Gain (NDCG) 23
k
2rel.
DCG@k
(50)
logg (i + 1)*
where rely is the relevance score of each item at position i. Then NDCG@K is the computed as:
DCG@K
NDCG@k IDCG@K'
(51)
where IDCG@k is the maximum possible DCG@k for the ideal ordering of items.
Here's polished version with the APT definition completed and implementation details clarified.
Expected Free Discovery (EFD) non-obvious items. We define
47
EFD quantifies how often a system surfaces relevant yet
EFD@k
disc(i) i u) loga (p(i seen)),
(52)
where disc(i) discounts lower-rank positions, p(rcl u)) is the user-specific relevance probability for the i-th recommended item, and p(i seen) denotes the likelihood of encountering item i given the user's previously seen items. In our implementation, p(rel i, u) using the softmax over cosine similarities between item embeddings and the user embedding; and approximate p(i seen) via smoothed conditional co-occurrence counts between item and the user's 's interaction history (Laplace smoothing, a 1) computed on the training set.
disc(i) 0.85 estimate
APT measures the share of long-tail content in users' recommendation lists. Let 24 be the evaluation user set, Tu,; the item at rank i for user u, and T the tail set (defined below). We report APT as:
Average Percentage of Tail Items (APT) 22]
1 a E ET))
APT
Σ
(53)
U
24

---

Tail items are determined by popularity on the training data: we compute item frequencies {f()} and define 7 {j f(i) S 7), where threshold chosen mark the long tail (e.g., bottom 20% by frequency). We use this percentile-based thresholding and average uniformly over users.
C More Experimental Results
Here present overall recommendation results datasets Adressa, Yelp2018, MLI0M and Globo extending results Figure
HR@20 Adressa
NDCG@20 Adressa
LightGCL + TSP-
LightGCL + TSP
LightGL + SDRO
LightGCL + SDRO
LightGCL + MACR
LightGCL MACR
LightGCL CausE
LightGCL CausE
LightGCL IPS
LightGCL IPS
SimGCL + TSP
SimGCL +
Method
SimGCL SDRO
Method
SimGCL + SDRO
SimGCL MACR-
SimGCL MACR
SimGCL CausE
SimGCL CausE
SimGCL IP5
SimGCL IPS
SimGCL
SimGCL
LighGCN + TSP
LighTCN + TSP
LightGCN + SDRO
LightGCN + SDRO
LightGCN MACR
LightGCN MACR
LightGCN CausE
LightGN CausE
LightGCN IPS
LightGCN IPS
LightGCN
LightGCN
0.000 0.025 0.050 0.075
0.100
0.125 0.150
0.00
0.02
0.04
0.06
HR@20
NDCG@20
Figure 5: The overall performance of TSP and other baselines Adressa dataset.
HR@20 Yelp2018
NDCG@20 Yelp2018
LightGCL + TSP
LightGCL + TSP
LighTGL + SDRO
LightGCL SDRO
LightGCL + MACR
LightGCL MACR
LightGCL CausE
LightGCL CausE
LightGCL IPS
LightGCL IPS
SimGCL + TSP-
SimGCL + TSP
Method
SimGCL SDRO
Method
SimGCL SDRO
SimGCL MACR
SimECL MACR
SimGCL CausE
Sim CausE
SimGCL IP5
SimGCL IPS
SimGCL
SimGCL
UghtGCN
LigiGCN TSP
UghtGCN + SDRO
UghtGCN SDRO
LightGCN MACR
LightGCN MACR
LightGCN CausE
LightGCN CausE
LightGCN IPS
LightGCN IPS
LightGCN
LightGCN
0.00
0.10
0.15
0.00
0.02
0.06
0.08
HR@20
NDCG@20
Figure 6: The overall performance of TSP and other baselines Yelp2018 8 dataset.
HR@22 ML10M
NDCG@2 ML10M
LightGCL + T5P
LightGCL TSP
LighGCL + SDRO-
LightCL SDRO
LightGCL + MACR
LightGCL MACR
LightGGL CausE
LightGCL CausE
LightGCL IPS
LightGCL IPS
LightGCL
SimGCL
SimGCL
Method
SimGCL SDRO
SimGCL SDRO
SimGCL MACR
SimGCL MACR
SimGCL CausE
SimGCL CausE
SimGCL IPS
SimGCL
SimGCL
SimGCL
LihtTCN
LighGCN
LightGCN + SDRO-
LightGCN SORO-
LightGCN MACR
LightGCN MACR
LightCN CausE
LghtGCN CausE
LightGEN IPS
LightGCN IPS
LightGON
LightGEN
0.00
0.05
0.10 HR@20
0.15
0.00
0.02
0.04
0.bs
NDCG@20
Figure 7: The overall performance of TSP and other baselines ML10M dataset.
25

---

HR@20 Globo
NDCG@20 Globo
LighGCL TSP-
LightCL TSP
LightGCL SDRO
LightGCL SDRO
LightGCL + MACR
LightGCL MACR
LightGCL CausE
LightecL + CausE
LightGCL IPS
LightGCL
LighGCL
LightGCL
TSP-
SimGCL
Methoo
SimGCL SDRO
Methoa
SimGCL SDRO
SimGCL + MACR
SimGEL MAGR
SimGCL CausE
Si Sim6CL + CausE
SimGCL IP5
SimGCL IPS
Sim6CL
SimGCL
LighTCN TSP-
LightGCN
LighGGN SDRO
LightG SDRO
LightGcN MACR
LightGNN MACR
LightGCN CausE
LightGEN + CausE
LightGCN IP5
LighTGN IPS
LightGCN
LightGCN
0.000
0.005
0.010
0.015
0.020
0.025
0,00000 00250.00500.00000
www.000750000000 www.:/.....1..02......r.
HR@20
NDCG@20
Figure 8: The overall performance of TSP and other baselines on Globo dataset.
D Examples of Simplicial Complexes
In simplicial complex, the boundary of k-simplex maps to its (k 1)-simplices (iee. the faces of the simplex). For a k-simplex a = (vo, UI,..,, Un the boundary Oo given by:
do Σ (-1) Ui...., 2=0
(54)
where úe indicates that vertex is omitted. This rule also determines the orientation of simplices in SC boundary matrix -1 or depending on the omitted of lower simplices.
a
2
1 2 3 4
A
0 -1 0 1 -1 0 0 0 -1 1 1
1 -1 0
0
a b c A f -1 1 0 0 0
d e
e
b
C
B
4
C-
B
0
1
d
e
B1
B2
Node((-simplex)) (1, 2, 4)
Edgee(1siiplex)xa, b, c, d, e) Triangle(2-sipplex)AAA. B)
Figure 9: An example of simplices and boundary matrices.
We present example simplicial complex consisting of vertices in Figure For triangle A = {1, 2,3), the edge (1,3) removes the ei Equation 54 so orientation (sign) in boundary matrix Bg is -1.
26

---

E TSP Algorithm
Algorithm 1: Test-time Simplicial Propagation
Input: Pretrained node embedding Xo
Boundary matrices B%
Hodge Laplacians L
Number of propagation layers T,
Simplicial filter coefficient, / Maximum SC order
Output: final node embeddings X
for k 1 to K do
/+ Intra-Simplex Smoothing
(11 BT) Xo
for k 0 to K do
/* Inter-Simplex Propagation
for to I. do
x ((++)) t (1 (11) X
for K
/* Multi-order Fusion +/
Xac- + Q1 Bi) x (L)
Xrec f- Xo + Mean Mean(Xo-1, .o..,.... Xo-k)
return X
Datasets
Five real-world recommendation datasets are used to evaluate the performance of TSP and other baselines with statistics in Table The details of these datasets are as follows:
Adressa is news dataset that includes news articles (in Norwegian) in connection with anonymized users.
Gowall: is location-based social networking website where users share their locations by checking-in. The friendship network is undirected and was collected using their public API, and consists of 196,591 nodes and 950,327 edges.
Yelp2018 is adopted from the 2018 edition of the yelp challenge, consisting of user interactions with local businesses like restaurants and bars. The 10-core setting (i.e., only users and items with at least 10 interaction records are included) is adopted in order to ensure data quality. Globaeconsine of users interactions logs (page views) from a news portal provided by Globo.come the most popular news portal in Brazil. This dataset also includes rich user contextual attributes such as click_ _os, click _couttyoo,,ryttt click_region and click referrer ptperrer_tyyr... ML10M is part of the movielens datasets with 10 million ratings by users. This dataset collects people's 's expressed preferences for movies as 0-5 star ratings. experiments we follow the common practice of converting ratings to binary interactions.
G Implementation Details
Here we list the hyperparameter search intervals we use on evaluation sets across different datasets in Table 6 All experiments conducted on single Nvidia RTX A6000 Ada GPU with 48GB memory, with 3 repeated runs averaged for final results.
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Table 5: Statistics of the datasets.
Dataset
Users
Items
Interactions
Sparsity
Adressa
13,485
744
1 116,321
0.011594
Globo
158,323
12,005
2,520,171
0.001326
ML10M
69,166
8,790
5,000,415
0.008225
Yelp2018
31,668
38,048
1,561,406
0.001300
Gowalla
29,858
40,981
1,027,370
0.000840
Table 6: Search intervals of hyperparameters for TSP and other baselines. Hyperparameters
Models
LightGCN
reg weight [le-4, 5e-4, 1e-3, Se-3, 1e-2]
LightGCL
CL loss weight X: [le-5_1e-6.1e-7.ee-8) rank SVD q: [2, 5, 8, 10]
SimGCL
CL loss weight A: [0.01, 0.05, 0.1, 0.2, 0.3, 1], noise magnitude c: [0, 0.01, 0.05, 0.1, 0.2, 0.5]
IPS
CausE
MACR
intermediate preference blocking c :[10, 20, 30, 40, 50, 60). item loss weight a :[ee-, le-4, le-3,le-2), user loss weight | :[1e-5, lc-4, Ie- 3,1c-2]
SDRO
TSP
maximum SC dimension K :3, 4, 5, 6], simplicial filter 3 :[]ee-, 5e-3, le-2, 5e-2, le-1, 5e-1, 1J, simplicial message passing layer L: [2, 3, 4]
H
Ablation Study
We provide full performance of the variants in Section 5.5 across datasets with LightGCN backbone in T
Table 7: Ablation study across datasets. We report the overall and 20% tail item results of Top 20 performance, Best results are in bold, and second-best results are underlined.
Adressa
Yclp2018
Gowalla
Globo
MLI0M
Scope
Variant
R@20
N@
R@:00
N@20
R@20
N@:00
R@-20
N@20
R@:20
N@20
TSP-Full
0.132
0.059
0.020
0.018
0.082
0.058
0.027
0.019
0.027
0.018
Overall
TSP-SE
0.108
0.046
0.010
0.006
0.056
0.041
0.016
0.012
0.015
0.01
LGN
0.096
0.042
0.007
0.002
0.047
0.038
0.010
0.007
0.009
0.008
TSP-Full
0.055
0.024
0.009
0.008
0.038
0.021
0.024
0.013
0.013
0.008
Tail
TSP-SE
0.024
0.013
0,006
0.003
0.017
0.009
0.007
0.006
0.006
0.003
LGN
0.015
0.007
0.004
0.002
0.006
0.006
0.001
0.002
0.002
0.001
I Lifting Graphs into Simplicial Complex
The implementation of lifting graphs into (clique) simplicial complexes centers finding every clique in the graph, where clique is a subset vertices with every pair connected by an edge. Efficient algorithms such as Bron- Kerbosce are often employed for this task. Each detected clique of k vertices then interpreted as a simplex of dimension k 1, so, for example, a triangle in the graph (a 3-clique) becomes 2-dimensional simplex in the complex. To ensure the resulting structure a simplicial complex, not only the maximal cliques but also all their subcliques should be included, since every subset of a clique is itself clique and thus forms fower-dimensions simplex.
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We simplify this complicated process with TopoNetX topological data.
41 suite specially designed to deal with
Broader Impact and Limitations
In this paper focus addressing the topology bias in recommender systems, ensuring fairer ommendation results. This can directly contribute to the development of more equitable information access and distribution, promoting diversity and inclusivity in user experiences across various plat- forms. However, our approach not without its limitations. Our current model solely concentrates on analyzing simplified graph message passing without incorporating weights non-linear activations. While this simplification is often adopted to simplify the analysis on graph message passing, this may lead to incomplete understanding of how topology bias influences this process. We leave this for future work.
29


Reviews:
[{"review_id": "CK8H1KsoLN", "reviewer": "Reviewer_HkJG", "summary": "This paper studied graph-based recommender methods. The authors started by analyzing the common popularity bias issue in the graph-based methods, from a Dirichlet energy perspective. Combined with the theoretical analysis, the authors proposed TSP, a training framework for existing graph-based recommender methods to alleviate the effect of popularity bias. The proposed TSP contains multiple steps, such as the semantic graph reconstruction, which leverages a pretrained model to add in predicted edges, similar to graph structure learning, and a few following steps as detailed in Sec. 4.2.", "questions": "1. The term \"semantic graph\" is quite confusing. IIUC, the graph-based collaborative filtering methods doesn't use any semantic features, but instead learn ID embeddings for each nodes. The way TSP reconstructs the semantic graph is very similar to existing literature in graph structure learning and graph data augmentation, which were not discussed in this work. Hence I wonder why the reconstructed graph is named \"semantic graph\" when it's more of a \"augmented graph\"?", "limitations": "see weakness", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. Popularity bias is a very important problem in collaborative filtering, which may even be enlarged in graph-based methods due to message passing. I'm glad to see papers trying to solve this critical issue.\n2. This paper is well supported by theoretical analysis.\n\nWeakness:\n1. Graph-based (or message passing-based) recommender methods are often criticized by their limitations on efficiency when it comes to larger graphs [1]. While the authors has shown that the proposed TSP added limited overhead to lightGCN, considering that TSP has steps such as calculating the graph laplacian, I'm concerned on the efficiency of the proposed method.\n2. Additional experiments on larger scale datasets would be appreciated.\n\n[1] UltraGCN: Ultra Simplification of Graph Convolutional Networks for Recommendation, CIKM 2021", "was_revised": true, "final_justification": "I appreciate the authors' rebuttal, which is very helpful with additional clarifications. So I raised my score.", "rebuttal": ""}, {"review_id": "mmMgYZabk1", "reviewer": "Reviewer_CeVg", "summary": "The paper provides, for the first time in the literature, a theoretical analysis of topology bias in message passing for graph-based recommender systems. Unlike similar previous analyses, mainly focusing on the node embeddings and gradients levels, the authors investigate this phenomenon under the lens of the Dirichlet energy. \n\nTo begin with, the paper theoretically demonstrates that performing the message passing is conceptually equivalent to setting a Dirichlet energy minimization problem. Secondly, they demonstrate how message passing leads to having a larger accumulation of embedding norms for high-degree nodes in the graph, and this phenomenon keeps amplifying the deeper we go in the message passing. \n\nOn such a basis, the authors propose a novel approach to tackle this issue, which interestingly runs in the inference step of the model's pipeline, thus it can be applied to any graph-based recommender systems during inference. The proposed pipeline creates a novel graph whose node connections are not necessarily based upon the topological structure of the original graph (which brings the bias information), but rather on the similarity of node embeddings conditioned on a hyperparameter $\\theta$. After that, the framework transitions to a simplicial complex (intra-simplex smoothing module) and a propagation function to propagate embeddings across simplicials (inter-simplex propagation module). Finally, a fusion strategy is adopted to combine all node representations from different orders of simplices. \n\nExtensive experimental results when applying the proposed solution during the inference process of three graph-based recommender models, against other debiasing baselines from the literature, largely demonstrate the efficacy of the proposed approach. This is true also across some recommendation datasets. Moreover, an ablation study on the main modules shows the goodness of the architectural choices made by the authors. Finally, an embedding distribution analysis empirically proves how the distribution is less concentrated in the case of the proposed solution, while a scalability study analyzes how the introduction of the methodology adds a very limited overhead on the simple inference of the original models.", "questions": "My main questions are related to the two weaknesses I outlined above.\n\nQ1) Could the authors provide additional results when running the model RP3beta on the tested datasets?\n\nQ2) How are the recommendation performance when considering additional standard bias and fairness metrics in the literature, such as the expected free discovery (EFD) and the average percentage of long-tail items (APLT)?\n\nQ3) Do the authors have any intuition on how their theoretical analysis and subsequent methodology could be differently applied to the user side and the item side? In principle, the two sides are different and show different properties, also in the topological bias (i.e., low-high activeness on the platform for the user vs. popular/niche products for the items).", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n$\\bullet$ The paper has the merit to not providing yet another graph-based recommendation model, but rather to analyze a critical issue in the current literature, topological bias, and disentangle it from a different theoretical perspective than other analyses provided in the recent literature.\n\n$\\bullet$ The paper is well-written and easy to follow.\n\n$\\bullet$ The methodology is quite clear and sound, as it is supported quite nicely by the background section. All theorems seem adequately sound and are rigorously proven in the Supplementary Material section.\n\n$\\bullet$ The proposed approach, which tackles the outlined issues, is nicely applied at inference time, thus not hurting the training of the models in any ways. The inference is also very limitedly hurt.\n\n$\\bullet$ Experimental settings are extensive with many evaluated dimensions. The code is released at review time, thus ensuring the complete reproducibility of the code and results presented in the paper.\n\n**Weaknesses**\n\n$\\bullet$ The authors might have considered to include a further baseline (RP3Beta [\\*]) which comes from a less-recent literature but works quite well on the user-item graph through random-walk and it is especially designed to not penalize less popular items. I think including this baseline as a standalone model to be compared against the other would provide a very nice complementary evaluation to the paper.\n\n$\\bullet$ In relation to the above, I think the authors might also consider to measure other recommendation measures which are usually adopted to assess the diversity in the recommendation lists [\\*\\*] and the presence of items from the long-tail [\\*\\*\\*].\n\n**References**\n\n[\\*] Christoffel et al., Blockbusters and Wallflowers: Accurate, Diverse, and Scalable Recommendations with Random Walks. RecSys 2015.\n\n[\\*\\*] S. Vargas, P. Castells, Rank and relevance in novelty and diversity metrics for recommender systems, in: RecSys, ACM, 2011, pp. 109–116. \n\n[\\*\\*\\*] Himan Abdollahpouri, Robin Burke, and Bamshad Mobasher. 2017. Controlling Popularity Bias in Learning-to-Rank Recommendation. In RecSys. ACM, 42–46.", "was_revised": true, "final_justification": "After the rebuttal phase, after reading the other reviews, and authors' responses to those, I decide to confirm my initial score I gave to the paper.\n\nSpecifically, all my concerns have been discussed and addressed by the authors.\n\nWhatever the final outcome for this paper, I think it will benefit quite a lot from the fruitful discussion we had along with the authors, the reviewers, and the Area Chair. I believe the paper will improve its quality a lot in any case.", "rebuttal": ""}, {"review_id": "ZAWgI4EsWc", "reviewer": "Reviewer_QJ5t", "summary": "Graph based recommender systems suffer from popularity bias. That is, the input graph (interaction graph between users and items) gets dominated by highly connected nodes which results in over-represenation of certain items. This results in reinforcing biases and fairness issues through the user-system feedback loop. This work analyses this topology bias through the lens of Dirichlet energy and show that message passing inherently minimises a global smoothing objective. Higher degree nodes accumulate larger embedding norms, a phenomenon called norm concentration. As a resolution, a test time simplicial propagation method is proposed which performs message passing over simplicial complexes with intra- and inter-simplex smoothing to avoid the dominance of certain nodes or substructures.", "questions": "See **weaknessess** above.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n1. The Dirichlet energy perspective to analyse graph based recommender systems is interesting.\n\n2. The formalization of norm concentration and how message passing favors highly connected nodes in recommender systems explaining over-representation of certain items is also fascinating.\n\n3. The paper is well-written and easy to follow.\n\n\n**Weaknessess**\n\n**Major weaknesses**\n\n1. I think the title of the paper should accurately reflect the contributions of the work. Currently, the title might prompt readers to think that the study is about topology bias in message passing GNNs covering more general form of GNNs such as GCNs, GATs, but the empirical analysis focuses specifically on **LightGCN-style architectures applied to graph based recommender systems**, which are simplified frameworks lacking feature transformations and non-linearities found in widely used GNNs. While I understand the creative liberty to have a catchy paper title, I think the title can be revised to better reflect the scope of the contributions.\n\n2. The idea that message passing (MP) is inherently minimizing Dirichlet energy is not a new insight. Extensive work studying over-smoothing in GNNs (see some of the references listed below [1], [2], [3]) have already established this fact. The way the work is currently presented doesn't adequately acknowledge this foundation or clearly articulate what's novel beyond existing theoretical understanding. It would make sense to properly position this work’s contributions in light of these related works.\n\n3. Although the norm concentration analysis is interesting, the idea that higher degree nodes receive larger message passing updates is obvious. For instance [4] also observes this phenomenon empirically (the current work also acknowledges this which is great). This leads again to my main issue which is properly discussing the contributions of the work in light of contemporary works. For instance works such as [9] and [10] treat this **degree bias/topology bias** more generally for GNNs and one can consider **norm concentration in LightGCN like architectures used specifically for recommender systems** as an application/special case of mitigating degree bias in GNNs ([4],[5],[6],[7],[8],[9],[10]).\n\n4. The resolution proposed to mitigate this topology bias is test-time simplicial complex propagation approach which although seems to work based on presented experimental validation, it is usually expensive to compute. Currently, the work compares with only few debiasing methods but what about works such as [5],[6],[7],[8],[9] and [10] which might provide a cheaper way to mitigate **degree bias in the general framework of GNNs** and not restricted to recommender systems? It would be interesting to check if such proposed methods already address the identified issue of topology bias. And why a more computationally expensive approach of lifting graphs to simplices is a better/preferred approach?\n\n\n**Minor weaknesses**\n\n1. L26-27 maybe explain in 1-2 lines what exactly is Matthew effect?\n\n2. The computational complexity for lifting the graphs into simplicial complex is missing.\n\n\n\n\n\n\n**References**\n\n1. A Note on Over-Smoothing for Graph Neural Networks. Cai et al, ICML 2020.\n\n2. PairNorm: Tackling Oversmoothing in GNNs. Zhao et al, ICLR 2020\n\n3. Not too little, not too much: a theoretical analysis of graph (over)smoothing. Keriven et al, 2022. \n\n4. Test-Time Embedding Normalization for Popularity Bias Mitigation. Kim et al, CIKM 2023.\n\n5. On Generalized Degree Fairness in Graph Neural Networks. Liu et al, AAAI 2023.\n\n6. GRAPHPATCHER: Mitigating Degree Bias for Graph Neural Networks via Test-time Augmentation. Ju et al, NeurIPS 2023.\n\n7. Tail-GNN: Tail-Node Graph Neural Networks. Liu et al, KDD, 2021.\n\n8. Investigating and Mitigating Degree-Related Biases in Graph Convolutional Networks. Tang et al. CIKM 2020.\n\n9. Theoretical and Empirical Insights into the Origins of Degree Bias in Graph Neural Networks. Subramonian et al, 2024.\n\n10. Degree-based stratification of nodes in Graph Neural Networks. Ali et al, ACML 2023.", "was_revised": true, "final_justification": "The authors have addressed my concerns and the experimental results strongly support their claims, I believe the current presentation does not adequately acknowledge existing work on over-smoothing and degree bias in the broader GNN literature. As I said in my review I believe the present work can be viewed as a special case of these established approaches applied to recommender systems. The theoretical contributions presented do not clearly articulate what is novel beyond the existing body of work on Dirichlet energy and over-smoothing. Given these concerns about theoretical positioning alongside the solid experimental validation, I have increased my score to 4.", "rebuttal": ""}]

Author rebuttal:

Meta-review: The paper addresses the problem of popularity bias—over-representation of popular items—in graph-based recommendation systems. It offers a fresh perspective by analyzing how graph topology influences message passing through the lens of Dirichlet energy. While the new ideas were appreciated, reviewers raised several concerns, particularly:
How the use of Dirichlet energy meaningfully differs from existing work on oversmoothing.
Evaluation of effectiveness on the long-tail of items.
Scalability of the proposed approach.
In response, the authors conducted additional experiments, which significantly strengthened the empirical section of the paper, leading to higher scores from some reviewers. However, during the discussion phase, the paper lacked a strong advocate. The ambivalence stemmed mainly from the perception that the theoretical contributions were modest. This impression was reinforced by the absence, in the main manuscript, of a clear summary of how the Dirichlet energy perspective connects to prior research on Graph Neural Networks. Although the rebuttal partially addressed this, questions about the extent of novelty remained.
Since a central motivation of the paper is to mitigate popularity bias, additional empirical evidence demonstrating improvements on long-tail recommendations would have been particularly impactful. Encouragingly, in the rebuttal, the authors presented results on one such dataset (as suggested by a reviewer), and the reviewer appreciated the results.

In summary, it will  be very difficult to make a strong recommendation for the paper. All the acceptance was contingent on the fact that substantive revision needs to be done. In case the paper is accepted the authors should mandatorily implement the changes discussed during rebuttal.
Decision: Accept (poster)