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[IRTalks@The University of Glasgow] A Topology-aware Analysis of Graph Collaborative Filtering

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Daniele Malitesta

Slides for the invited talk "A Topology-aware Analysis of Graph Collaborative Filtering" at the Glasgow Information Retrieval Group (University of Glasgow). Event link: https://samoa.dcs.gla.ac.uk/events/viewtalk.jsp?id=19320 Paper: https://arxiv.org/pdf/2308.10778.pdf

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A Topology-aware Analysis of Graph Collaborative Filtering Daniele Malitesta, PhD SisInfLab Research Group Politecnicodi Bari (Bari, Italy) The Universityof Glasgow (Glasgow, UK) 50 minutes February 05, 2024 (15.00 UK Time) IR Talks
TABLE OF CONTENTS 01 05 03 06 04 07 INTRODUCTION & BACKGROUND TOPOLOGICAL CHARACTERISTICS TOPOLOGY IN GRAPH CF PROPOSED ANALYSIS RESULTS TAKE-HOME MESSAGES 02 MOTIVATIONS
USEFUL RESOURCES The content of the following slides is taken from: β–  Daniele Malitesta, Claudio Pomo, Vito Walter Anelli, Alberto Carlo Maria Mancino, Eugenio Di Sciascio, Tommaso Di Noia: A Topology-aware Analysis of Graph Collaborative Filtering. CoRR abs/2308.10778 (2023) β–  Daniele Malitesta, Claudio Pomo, Tommaso Di Noia: Graph Neural Networks for Recommendation: Reproducibility, Graph Topology, and Node Representation. CoRR abs/2310.11270 (2023) Scan me!
INTRODUCTION & BACKGROUND 01
A RISING INTEREST 0 100 200 300 400 500 600 2020 2021 2022 2023 Papers As searched on DBLP through the keywords β€œgraph” and β€œrecommend”.
PREVIOUS TUTORIALS ON GNNs-RECSYS
A NON-EXHAUSTIVE TIMELINE Pioneer approaches proposing GCN-based aggregation methods [v an den Berg et al., Ying et al.] Explore inter-dependencies between nodes and their neighbours [Wang et al. (2019a)], use graph attention networks to recognize meaningful user-item interactions at higher-grained level [Wang et al. (2019b)] Lighten the graph convolutional layer [Chen e t al., He et al.], use graph attention networks to recognize meaningful user- item interactions at higher- grained level [Wang et al. (2020b), Tao et al.] Self-supervised and contrastive learning [Wu et al., Yu et al.] Simplify the message- passing formulation [Mao et al., Peng et al., Shen et al.] Explore other latent spaces [Shen et al., Sun e t al., Zhang et al.] Exploit hypergraphs [We i et al., Xia et al.] Use graph attention networks to recognize meaningful user- item interactions at finer- grained [Zhang et al.] level 2017-2018 2019 2020 2021-2022 2021-2022 2021-2022 2022 2022
USER-ITEM GRAPH 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 user-item interaction matrix 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 adjacency matrix 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6
USER-ITEM GRAPH 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 user-item interaction matrix 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 adjacency matrix 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 BIPARTITE
USER-ITEM GRAPH 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 user-item interaction matrix 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 adjacency matrix 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 UNDIRECTED
MESSAGE-PASSING & LATENT-FACTORS MESSAGE-PASSING(LAYER1) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ (οΏ½) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 οΏ½ ( ) ( ) , = οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) FACTORIZATION & MESSAGE-PASSING 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 < > , β‰ˆ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) Message-passing component Latent-factors component
MOTIVATIONS 02
Topologically WHY GRAPH CF WORKS WELL? β€’ The machine learning literature acknowledges that graph topology plays a crucial role in GNNs [Wei et al., Castellana and Errica]. β€’ For instance, revisit network topology to enhance the learning ability of GCNs [Shi et al.]. β€’ Established approaches such as LightGCN and DGCF adapt the GCN layer to suit the CF rationale. β€’ Recent graph-based RSs including UltraGCN and SVD-GCN go beyond the concept of message-passing. Algorithmically
[Adomavicius andZhang, Deldjoo et al.] Classical dataset characteristics such as dataset sparsity may impact the performance of recommendation models in diverse scenarios and tasks.
The topological nature of the user- item data and the new graph CF wave make it imperative to unveil the dependencies between topological properties and recommendation performance.
3) Build an explanatory model OUR CONTRIBUTIONS that studies the impact of classical and topological dataset characteristics on the performance of SOTA graph CF. which are recent and across-the-board in the literature. to calculate the linear dependences among characteristics and recommendation metrics. on their statistical significance, with varying settings of graph samplings. 2) Select 4 graph CF approaches 4) Validate the explanations 1) First analysis in the literature
TOPOLOGICAL CHARACTERISTICS 03
NODE DEGREE EXTENSION (1/3) DEFINITION Let 𝑁𝑒 (𝑙) and 𝑁𝑖 (𝑙) be the sets of neighborhood nodes for user 𝑒 and item 𝑖 at 𝑙 distance hops. The extended definition of node degrees for 𝑒 and 𝑖 is: πœŽπ‘’ = 𝑁𝑒 (1) , πœŽπ‘– = 𝑁𝑖 (1) . The average user and item node degrees on the whole user and item sets (π‘ˆ and 𝐼): πœŽπ‘ˆ = 1 π‘ˆ Οƒπ‘’βˆˆπ‘ˆ 𝑁𝑒 1 , 𝜎𝐼 = 1 𝐼 Οƒπ‘–βˆˆπΌ 𝑁𝑖 (1) .
NODE DEGREE EXTENSION (2/3) RECSYS RE-INTERPRETATION The node degree in the user-item graph stands for the number of items (users) interacted by a user (item). This is related to the cold-start issue in recommendation, where cold users denote low activity on the platform, while cold items are niche products. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 πœŽπ‘’2 = 5 (active user) πœŽπ‘’7 = 1 (non-active user)
NODE DEGREE EXTENSION (3/3) RECSYS RE-INTERPRETATION The node degree in the user-item graph stands for the number of items (users) interacted by a user (item). This is related to the cold-start issue in recommendation, where cold users denote low activity on the platform, while cold items are niche products. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 πœŽπ‘–7 = 4 (popular item) πœŽπ‘–4 = 1 (niche item)
CLUSTERING COEFFICIENT (1/5) DEFINITION Let 𝑣 and 𝑀 be two nodes from the same partition (e.g., user nodes). Their similarity is the intersection over union of their neighborhoods. By evaluating the metric node-wise: 𝛾𝑣 = Οƒ π‘€βˆˆπ‘π‘£ 2 𝛾𝑣,𝑀 𝑁𝑣 2 with 𝛾𝑣,𝑀 = 𝑁𝑣 1 βˆ©π‘π‘€ 1 𝑁𝑣 1 βˆͺ𝑁𝑀 1 , where 𝑁𝑣 (2) is the second-order neighborhood set of 𝑣. The average clustering coefficient [Latapy et al.] on π‘ˆ and 𝐼 is: π›Ύπ‘ˆ = 1 π‘ˆ Οƒπ‘’βˆˆπ‘ˆ 𝛾𝑒, 𝛾𝐼 = 1 𝐼 Οƒπ‘–βˆˆπΌ 𝛾𝑖.
CLUSTERING COEFFICIENT (2/5) RECSYS RE-INTERPRETATION High values of the clustering coefficient indicate that there exists a substantial number of co- occurrences among nodes of the same partition. For instance, user-wise, the value increases if several users share most of their interacted items. This intuition aligns with the rationale of collaborative filtering: two users are likely to share similar preferences if they interact with the same items. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝛾𝑒3,𝑒5 = 𝑖5,𝑖7 ∩ 𝑖5,𝑖6,𝑖7 𝑖5,𝑖6,𝑖7 = 2 3 = 0.667
CLUSTERING COEFFICIENT (3/5) RECSYS RE-INTERPRETATION High values of the clustering coefficient indicate that there exists a substantial number of co- occurrences among nodes of the same partition. For instance, user-wise, the value increases if several users share most of their interacted items. This intuition aligns with the rationale of collaborative filtering: two users are likely to share similar preferences if they interact with the same items. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝛾𝑒6,𝑒7 = 𝑖7,𝑖8 ∩ 𝑖8 𝑖7,𝑖8 = 1 2 = 0.500
CLUSTERING COEFFICIENT (4/5) RECSYS RE-INTERPRETATION High values of the clustering coefficient indicate that there exists a substantial number of co- occurrences among nodes of the same partition. For instance, user-wise, the value increases if several users share most of their interacted items. This intuition aligns with the rationale of collaborative filtering: two users are likely to share similar preferences if they interact with the same items. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝛾𝑒2,𝑒4 = 𝑖1,𝑖2,𝑖3,𝑖4,𝑖5 ∩ 𝑖3,𝑖7 𝑖1,𝑖2,𝑖3,𝑖4,𝑖5,𝑖7 = 1 6 = 0.167
CLUSTERING COEFFICIENT (5/5) RECSYS RE-INTERPRETATION High values of the clustering coefficient indicate that there exists a substantial number of co- occurrences among nodes of the same partition. For instance, user-wise, the value increases if several users share most of their interacted items. This intuition aligns with the rationale of collaborative filtering: two users are likely to share similar preferences if they interact with the same items. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝛾𝑒1,𝑒5 = 𝑖1,𝑖2 ∩ 𝑖5,𝑖6,𝑖7 𝑖1,𝑖2,𝑖5,𝑖6,𝑖7 = 0 6 = 0.000
DEGREE ASSORTATIVITY (1/3) DEFINITION Let 𝐷 = {𝑑1,𝑑2, …} be the set of unique node degrees in the graph, and let π‘’π‘‘β„Ž,π‘‘π‘˜ be the fraction of edges connecting nodes with degrees π‘‘β„Ž and π‘‘π‘˜. Then, let π‘žπ‘‘β„Ž be the probability distribution to choose a node with degree π‘‘β„Ž after having selected a node with the same degree (i.e., the excess degree distribution). The degree assortativity coefficient [M. E. J. Newman] is: 𝜌 = Οƒπ‘‘β„Žπ‘‘π‘˜ π‘‘β„Žπ‘‘π‘˜ π‘’π‘‘β„Ž,π‘‘π‘˜ βˆ’ π‘žπ‘‘β„Ž π‘žπ‘‘π‘˜ π‘ π‘‘π‘‘π‘ž 2 , where π‘ π‘‘π‘‘π‘ž is the standard deviation of the distribution π‘ž.
DEGREE ASSORTATIVITY (2/3) RECSYS RE-INTERPRETATION The degree assortativity calculated user- and item-wise is a proxy to represent the tendency of users with the same activity level on the platform and items with the same popularity to gather, respectively. To visualize the degree assortativity, we need the projected user-user and item-item graphs. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 user-item graph user-user projected graph
DEGREE ASSORTATIVITY (3/3) RECSYS RE-INTERPRETATION The degree assortativity calculated user- and item-wise is a proxy to represent the tendency of users with the same activity level on the platform and items with the same popularity to gather, respectively. To visualize the degree assortativity, we need the projected user-user and item-item graphs. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝜌 = βˆ’0.191 (high degree dis-assortativity)
Density CLASSICAL DATASET CHARACTERISTICS Estimates the number of all possible interactions: 𝜁 = π‘ˆπΌ. Defines the ratio between the number of users and items: πœ‹ = π‘ˆ 𝐼 . Measures the ratio of actual user-item interactions to all possible interactions: 𝛿 = 𝐸 π‘ˆπΌ . Calculates the interactions’ concentration for users (items): πœ…π‘ˆ = σ𝑒=1 π‘ˆβˆ’1 σ𝑣=𝑒+1 π‘ˆ π‘Žπ‘π‘  πœŽπ‘’βˆ’πœŽπ‘£ π‘ˆ σ𝑒=1 π‘ˆ πœŽπ‘’ . Shape Gini coefficient Space size
TOPOLOGY IN GRAPH CF 04
UltraGCN (CIKM 2021) SELECTED GRAPH CF APPROACHES Node degree used to normalize the adjacency matrix in the message-passing. Node degree used to normalize the adjacency matrix in the message- passing. Node degree used for normalization in the infinite layer message-passing. The model also learns from the item- projected graph. Node embeddings involve the largest singular values of the normalized user-item interaction matrix, whose maximum value is related to the maximum node degree of the user- item graph. The model learns from user- and item-projected graphs. DGCF (SIGIR 2020) SVD-GCN (CIKM 2022) LightGCN (SIGIR 2020)
The selected graph CF approaches explicitly utilize the node degree in the representation learning. However, clustering coefficient and degree assortativity do not have an evident representation in the formulations.
β€’ Which topological aspects graph-based models can (un)intentionally capture? β€’ Are (topological) dataset characteristics influencing the recommendation performance of graph CF models?
05 PROPOSED ANALYSIS
Our goal is to understand whether there exist dependencies among (topological) dataset characteristics and the performance of graph-based recommendation models. To this aim, we decide to build and fit an explanatory framework.
EXPLANATORY FRAMEWORK: STEPS
EXPLANATORY FRAMEWORK: STEPS STEP 1 Select models
EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 Select characteristics Select models
EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 Select datasets Select characteristics Select models
DATASETS # Users # Items # Interactions Yelp-2018 25,677 25,815 696,865 Amazon -Book 70,679 24,915 846,434 Gowalla 29,858 40,981 1,027,370
EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 Select characteristics Select models Select datasets
EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets Select characteristics Select models Select datasets
SUB-DATASETS GENERATION
NODE DROPOUT 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 users items
NODE DROPOUT 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 users items
NODE DROPOUT 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝑒1 𝑒2 𝑒3 𝑒7 𝑖2 𝑖5 𝑖6 𝑖7 𝑖8 users items users items
EDGE DROPOUT 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 users items
EDGE DROPOUT 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 users items 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 users items
EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets Select characteristics Select models Select datasets
EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 Get datasets’ statistics Select characteristics Select models Select datasets
STATISTICS CALCULATION (1/2)
STATISTICS CALCULATION (2/2)
EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 Get datasets’ statistics Select characteristics Select models Select datasets
EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 STEP 6 Train models Select characteristics Select models Select datasets Get datasets’ statistics
EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 STEP 6 Train models STEP 7 Evaluate models Select characteristics Select models Select datasets Get datasets’ statistics
EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 STEP 6 Train models STEP 7 Evaluate models STEP 8 Generate explanations Select characteristics Select models Select datasets Get datasets’ statistics
GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘.
GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. Predicted performance
GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. Predicted performance Prediction error
GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. Predicted performance Prediction error Global bias
GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. Predicted performance Prediction error Global bias Weight of characteristic c
GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. Predicted performance Prediction error Global bias Weight of characteristic c Characteristic c for all samples
GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. We use the ordinary least squares (OLS) optimization model: πœƒ0 βˆ—,πœ½π‘ βˆ— = min πœƒ0,πœƒπ‘ 1 2 π’š βˆ’ πœƒ0 βˆ’ πœ½π‘π‘Ώπ‘ 2 2 .
EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 STEP 6 Train models STEP 7 Evaluate models STEP 8 Generate explanations Select characteristics Select models Select datasets Get datasets’ statistics
RESULTS 06
IMPACT OF CHARACTERISTICS Characteristics’ weights [πœƒ0,πœƒ1,…, πœƒπ‘]
IMPACT OF CHARACTERISTICS How good is the linear regression at making predictions? 𝑅2 is, for many settings, above 95%. The regression model can provide good explanations.
IMPACT OF CHARACTERISTICS classical
IMPACT OF CHARACTERISTICS inverse Graph CF = neighborhood + latent factors. Graph CF behaves as factorization-based models. direct
IMPACT OF CHARACTERISTICS topological
IMPACT OF CHARACTERISTICS direct This confirms what observed in the models’ formulations. High users’ interactions correspond to better performance.
IMPACT OF CHARACTERISTICS strong direct Values for users- and items-side are more flattened for UltraGCN and SVD-GCN.
IMPACT OF CHARACTERISTICS LightGCN and DGCF have larger coefficients that SVD-GCN, because they explicitly propagate messages. direct inverse
IMPACT OF CHARACTERISTICS UltraGCN has bigger coefficients, probably because it uses the infinite-layer message passing. direct inverse
Probability distribution of node degrees on Gowalla
In the worst-case scenario, node- dropout drops many high-degree nodes; edge-dropout, drops all the edges connected to several nodes and thus disconnect them from the graph.
Is this undermining the goodness of the proposed explanatory framework?
INFLUENCE OF NODE- AND EDGE-DROPOUT NODE 100% [Setting A] 100% node-dropout
INFLUENCE OF NODE- AND EDGE-DROPOUT NODE EDGE 70% 30% [Setting B] 70% node-dropout 30% edge-dropout
INFLUENCE OF NODE- AND EDGE-DROPOUT NODE EDGE 30% 70% [Setting C] 30% node-dropout 70% edge-dropout
INFLUENCE OF NODE- AND EDGE-DROPOUT EDGE 100% [Setting D] 100% edge-dropout
INFLUENCE OF NODE- AND EDGE-DROPOUT Datasets’ statistics Node-dropout retains smaller portions of the dataset than edge-dropout.
INFLUENCE OF NODE- AND EDGE-DROPOUT Meaningful learned dependencies (this setting is like ours).
INFLUENCE OF NODE- AND EDGE-DROPOUT Not-statistically-significant at the extremes.
TAKE-HOME MESSAGES 07
WHAT WE HAVE LEARNED 01 LATENT-FACTORS vs. NEIGHBORHOOD
WHAT WE HAVE LEARNED 01 02 LATENT-FACTORS vs. NEIGHBORHOOD NODE DEGREE
WHAT WE HAVE LEARNED 01 02 03 LATENT-FACTORS vs. NEIGHBORHOOD NODE DEGREE CLUSTERING COEFFICIENT
WHAT WE HAVE LEARNED 01 04 02 03 LATENT-FACTORS vs. NEIGHBORHOOD NODE DEGREE CLUSTERING COEFFICIENT DEGREE ASSORTATIVITY
WHAT WE HAVE LEARNED 01 04 02 05 03 LATENT-FACTORS vs. NEIGHBORHOOD NODE DEGREE CLUSTERING COEFFICIENT DEGREE ASSORTATIVITY NODE- vs. EDGE- DROPOUT
REFERENCES (1/3) [Malitesta et al.] A Topology-aware Analysis of Graph Collaborative Filtering. CoRR abs/2308.10778 (2023) [Malitesta et al.] Graph Neural Networks for Recommendation: Reproducibility, Graph Topology, and Node Representation. CoRR abs/2310.11270 (2023) [Wang et al. (2020a)] Learning and Reasoning on Graph for Recommendation. WSDM 2020: 890-893 [El-Kishky et al.] Graph-based Representation Learning for Web-scale Recommender Systems. KDD 2022: 4784-4785 [Gao et al.] Graph Neural Networks for Recommender System. WSDM 2022: 1623-1625 [Purificato et al. (2023a)] Tutorial on User Profiling with Graph Neural Networks and Related Beyond- Accuracy Perspectives. UMAP 2023: 309-312 [Purificato et al. (2023b)] Leveraging Graph Neural Networks for User Profiling: Recent Advances and Open Challenges. CIKM 2023: 5216-5219 [van den Berg et al.] Graph Convolutional Matrix Completion. CoRR abs/1706.02263 (2017) [Ying et al.] Graph Convolutional Neural Networks for Web-Scale Recommender Systems. KDD 2018: 974-983 [Wang et al. (2019a)] Neural Graph Collaborative Filtering. SIGIR 2019: 165-174 [Wang et al. (2019b)] KGAT: Knowledge Graph Attention Network for Recommendation. KDD 2019: 950-958
REFERENCES (2/3) [Chen et al.] Revisiting Graph Based Collaborative Filtering: A Linear Residual Graph Convolutional Network Approach. AAAI 2020: 27-34 [He et al.] LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation. SIGIR 2020: 639-648 [Wang et al. (2020b)] Disentangled Graph Collaborative Filtering. SIGIR 2020: 1001-1010 [Tao et al.] MGAT: Multimodal Graph Attention Network for Recommendation. Inf. Process. Manag. 57(5): 102277 (2020) [Wu et al.] Self-supervised Graph Learning for Recommendation. SIGIR 2021: 726-735 [Yu et al.] Are Graph Augmentations Necessary?: Simple Graph Contrastive Learning for Recommendation. SIGIR 2022: 1294-1303 [Mao et al.] UltraGCN: Ultra Simplification of Graph Convolutional Networks for Recommendation. CIKM 2021: 1253-1262 [Peng et al.] SVD-GCN: A Simplified Graph Convolution Paradigm for Recommendation. CIKM 2022: 1625-1634 [Shen et al.] How Powerful is Graph Convolution for Recommendation? CIKM 2021: 1619-1629 [Sun et al.] HGCF: Hyperbolic Graph Convolution Networks for Collaborative Filtering. WWW 2021: 593-601
REFERENCES (3/3) [Zhang et al.] Geometric Disentangled Collaborative Filtering. SIGIR 2022: 80-90 [Wei et al.] Dynamic Hypergraph Learning for Collaborative Filtering. CIKM 2022: 2108-2117 [Xia et al.] Hypergraph Contrastive Collaborative Filtering. SIGIR 2022: 70-79 [Wei et al.] Designing the Topology of Graph Neural Networks: A Novel Feature Fusion Perspective. WWW 2022: 1381-1391 [Castellana and Errica] Investigating the Interplay between Features and Structures in Graph Learning. CoRR abs/2308.09570 (2023) [Shi et al.] Topology and Content Co-Alignment Graph Convolutional Learning. IEEE Trans. Neural Networks Learn. Syst. 33(12): 7899-7907 (2022) [Adomavicius and Zhang] Impactof data characteristics on recommender systems performance. ACM Trans. Manag. Inf. Syst. 3(1): 3:1-3:17 (2012) [Deldjoo et al.] How Dataset Characteristics Affect the Robustness of Collaborative Recommendation Models. SIGIR 2020: 951-960 [Latapy et al.] Basic notions for the analysis of large two-mode networks. Soc. Networks 30(1): 31-48 (2008) [M. E. J. Newman] Mixing patterns in networks. Phys. Rev. E, 67:026126, Feb 2003. doi: 10. 1103/PhysRevE.67.026126
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[IRTalks@The University of Glasgow] A Topology-aware Analysis of Graph Collaborative Filtering

  • 1. A Topology-aware Analysis of Graph Collaborative Filtering Daniele Malitesta, PhD SisInfLab Research Group Politecnicodi Bari (Bari, Italy) The Universityof Glasgow (Glasgow, UK) 50 minutes February 05, 2024 (15.00 UK Time) IR Talks
  • 2. TABLE OF CONTENTS 01 05 03 06 04 07 INTRODUCTION & BACKGROUND TOPOLOGICAL CHARACTERISTICS TOPOLOGY IN GRAPH CF PROPOSED ANALYSIS RESULTS TAKE-HOME MESSAGES 02 MOTIVATIONS
  • 3. USEFUL RESOURCES The content of the following slides is taken from: β–  Daniele Malitesta, Claudio Pomo, Vito Walter Anelli, Alberto Carlo Maria Mancino, Eugenio Di Sciascio, Tommaso Di Noia: A Topology-aware Analysis of Graph Collaborative Filtering. CoRR abs/2308.10778 (2023) β–  Daniele Malitesta, Claudio Pomo, Tommaso Di Noia: Graph Neural Networks for Recommendation: Reproducibility, Graph Topology, and Node Representation. CoRR abs/2310.11270 (2023) Scan me!
  • 5. A RISING INTEREST 0 100 200 300 400 500 600 2020 2021 2022 2023 Papers As searched on DBLP through the keywords β€œgraph” and β€œrecommend”.
  • 6. PREVIOUS TUTORIALS ON GNNs-RECSYS
  • 7. A NON-EXHAUSTIVE TIMELINE Pioneer approaches proposing GCN-based aggregation methods [v an den Berg et al., Ying et al.] Explore inter-dependencies between nodes and their neighbours [Wang et al. (2019a)], use graph attention networks to recognize meaningful user-item interactions at higher-grained level [Wang et al. (2019b)] Lighten the graph convolutional layer [Chen e t al., He et al.], use graph attention networks to recognize meaningful user- item interactions at higher- grained level [Wang et al. (2020b), Tao et al.] Self-supervised and contrastive learning [Wu et al., Yu et al.] Simplify the message- passing formulation [Mao et al., Peng et al., Shen et al.] Explore other latent spaces [Shen et al., Sun e t al., Zhang et al.] Exploit hypergraphs [We i et al., Xia et al.] Use graph attention networks to recognize meaningful user- item interactions at finer- grained [Zhang et al.] level 2017-2018 2019 2020 2021-2022 2021-2022 2021-2022 2022 2022
  • 8. USER-ITEM GRAPH 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 user-item interaction matrix 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 adjacency matrix 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6
  • 9. USER-ITEM GRAPH 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 user-item interaction matrix 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 adjacency matrix 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 BIPARTITE
  • 10. USER-ITEM GRAPH 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 user-item interaction matrix 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 adjacency matrix 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑒1 𝑒2 𝑒3 𝑒4 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 UNDIRECTED
  • 11. MESSAGE-PASSING & LATENT-FACTORS MESSAGE-PASSING(LAYER1) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ (οΏ½) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 οΏ½ ( ) ( ) , = οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) FACTORIZATION & MESSAGE-PASSING 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 < > , β‰ˆ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) οΏ½ ( ) Message-passing component Latent-factors component
  • 13. Topologically WHY GRAPH CF WORKS WELL? β€’ The machine learning literature acknowledges that graph topology plays a crucial role in GNNs [Wei et al., Castellana and Errica]. β€’ For instance, revisit network topology to enhance the learning ability of GCNs [Shi et al.]. β€’ Established approaches such as LightGCN and DGCF adapt the GCN layer to suit the CF rationale. β€’ Recent graph-based RSs including UltraGCN and SVD-GCN go beyond the concept of message-passing. Algorithmically
  • 14. [Adomavicius andZhang, Deldjoo et al.] Classical dataset characteristics such as dataset sparsity may impact the performance of recommendation models in diverse scenarios and tasks.
  • 15. The topological nature of the user- item data and the new graph CF wave make it imperative to unveil the dependencies between topological properties and recommendation performance.
  • 16. 3) Build an explanatory model OUR CONTRIBUTIONS that studies the impact of classical and topological dataset characteristics on the performance of SOTA graph CF. which are recent and across-the-board in the literature. to calculate the linear dependences among characteristics and recommendation metrics. on their statistical significance, with varying settings of graph samplings. 2) Select 4 graph CF approaches 4) Validate the explanations 1) First analysis in the literature
  • 18. NODE DEGREE EXTENSION (1/3) DEFINITION Let 𝑁𝑒 (𝑙) and 𝑁𝑖 (𝑙) be the sets of neighborhood nodes for user 𝑒 and item 𝑖 at 𝑙 distance hops. The extended definition of node degrees for 𝑒 and 𝑖 is: πœŽπ‘’ = 𝑁𝑒 (1) , πœŽπ‘– = 𝑁𝑖 (1) . The average user and item node degrees on the whole user and item sets (π‘ˆ and 𝐼): πœŽπ‘ˆ = 1 π‘ˆ Οƒπ‘’βˆˆπ‘ˆ 𝑁𝑒 1 , 𝜎𝐼 = 1 𝐼 Οƒπ‘–βˆˆπΌ 𝑁𝑖 (1) .
  • 19. NODE DEGREE EXTENSION (2/3) RECSYS RE-INTERPRETATION The node degree in the user-item graph stands for the number of items (users) interacted by a user (item). This is related to the cold-start issue in recommendation, where cold users denote low activity on the platform, while cold items are niche products. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 πœŽπ‘’2 = 5 (active user) πœŽπ‘’7 = 1 (non-active user)
  • 20. NODE DEGREE EXTENSION (3/3) RECSYS RE-INTERPRETATION The node degree in the user-item graph stands for the number of items (users) interacted by a user (item). This is related to the cold-start issue in recommendation, where cold users denote low activity on the platform, while cold items are niche products. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 πœŽπ‘–7 = 4 (popular item) πœŽπ‘–4 = 1 (niche item)
  • 21. CLUSTERING COEFFICIENT (1/5) DEFINITION Let 𝑣 and 𝑀 be two nodes from the same partition (e.g., user nodes). Their similarity is the intersection over union of their neighborhoods. By evaluating the metric node-wise: 𝛾𝑣 = Οƒ π‘€βˆˆπ‘π‘£ 2 𝛾𝑣,𝑀 𝑁𝑣 2 with 𝛾𝑣,𝑀 = 𝑁𝑣 1 βˆ©π‘π‘€ 1 𝑁𝑣 1 βˆͺ𝑁𝑀 1 , where 𝑁𝑣 (2) is the second-order neighborhood set of 𝑣. The average clustering coefficient [Latapy et al.] on π‘ˆ and 𝐼 is: π›Ύπ‘ˆ = 1 π‘ˆ Οƒπ‘’βˆˆπ‘ˆ 𝛾𝑒, 𝛾𝐼 = 1 𝐼 Οƒπ‘–βˆˆπΌ 𝛾𝑖.
  • 22. CLUSTERING COEFFICIENT (2/5) RECSYS RE-INTERPRETATION High values of the clustering coefficient indicate that there exists a substantial number of co- occurrences among nodes of the same partition. For instance, user-wise, the value increases if several users share most of their interacted items. This intuition aligns with the rationale of collaborative filtering: two users are likely to share similar preferences if they interact with the same items. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝛾𝑒3,𝑒5 = 𝑖5,𝑖7 ∩ 𝑖5,𝑖6,𝑖7 𝑖5,𝑖6,𝑖7 = 2 3 = 0.667
  • 23. CLUSTERING COEFFICIENT (3/5) RECSYS RE-INTERPRETATION High values of the clustering coefficient indicate that there exists a substantial number of co- occurrences among nodes of the same partition. For instance, user-wise, the value increases if several users share most of their interacted items. This intuition aligns with the rationale of collaborative filtering: two users are likely to share similar preferences if they interact with the same items. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝛾𝑒6,𝑒7 = 𝑖7,𝑖8 ∩ 𝑖8 𝑖7,𝑖8 = 1 2 = 0.500
  • 24. CLUSTERING COEFFICIENT (4/5) RECSYS RE-INTERPRETATION High values of the clustering coefficient indicate that there exists a substantial number of co- occurrences among nodes of the same partition. For instance, user-wise, the value increases if several users share most of their interacted items. This intuition aligns with the rationale of collaborative filtering: two users are likely to share similar preferences if they interact with the same items. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝛾𝑒2,𝑒4 = 𝑖1,𝑖2,𝑖3,𝑖4,𝑖5 ∩ 𝑖3,𝑖7 𝑖1,𝑖2,𝑖3,𝑖4,𝑖5,𝑖7 = 1 6 = 0.167
  • 25. CLUSTERING COEFFICIENT (5/5) RECSYS RE-INTERPRETATION High values of the clustering coefficient indicate that there exists a substantial number of co- occurrences among nodes of the same partition. For instance, user-wise, the value increases if several users share most of their interacted items. This intuition aligns with the rationale of collaborative filtering: two users are likely to share similar preferences if they interact with the same items. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝛾𝑒1,𝑒5 = 𝑖1,𝑖2 ∩ 𝑖5,𝑖6,𝑖7 𝑖1,𝑖2,𝑖5,𝑖6,𝑖7 = 0 6 = 0.000
  • 26. DEGREE ASSORTATIVITY (1/3) DEFINITION Let 𝐷 = {𝑑1,𝑑2, …} be the set of unique node degrees in the graph, and let π‘’π‘‘β„Ž,π‘‘π‘˜ be the fraction of edges connecting nodes with degrees π‘‘β„Ž and π‘‘π‘˜. Then, let π‘žπ‘‘β„Ž be the probability distribution to choose a node with degree π‘‘β„Ž after having selected a node with the same degree (i.e., the excess degree distribution). The degree assortativity coefficient [M. E. J. Newman] is: 𝜌 = Οƒπ‘‘β„Žπ‘‘π‘˜ π‘‘β„Žπ‘‘π‘˜ π‘’π‘‘β„Ž,π‘‘π‘˜ βˆ’ π‘žπ‘‘β„Ž π‘žπ‘‘π‘˜ π‘ π‘‘π‘‘π‘ž 2 , where π‘ π‘‘π‘‘π‘ž is the standard deviation of the distribution π‘ž.
  • 27. DEGREE ASSORTATIVITY (2/3) RECSYS RE-INTERPRETATION The degree assortativity calculated user- and item-wise is a proxy to represent the tendency of users with the same activity level on the platform and items with the same popularity to gather, respectively. To visualize the degree assortativity, we need the projected user-user and item-item graphs. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 user-item graph user-user projected graph
  • 28. DEGREE ASSORTATIVITY (3/3) RECSYS RE-INTERPRETATION The degree assortativity calculated user- and item-wise is a proxy to represent the tendency of users with the same activity level on the platform and items with the same popularity to gather, respectively. To visualize the degree assortativity, we need the projected user-user and item-item graphs. 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖8 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑒7 𝜌 = βˆ’0.191 (high degree dis-assortativity)
  • 29. Density CLASSICAL DATASET CHARACTERISTICS Estimates the number of all possible interactions: 𝜁 = π‘ˆπΌ. Defines the ratio between the number of users and items: πœ‹ = π‘ˆ 𝐼 . Measures the ratio of actual user-item interactions to all possible interactions: 𝛿 = 𝐸 π‘ˆπΌ . Calculates the interactions’ concentration for users (items): πœ…π‘ˆ = σ𝑒=1 π‘ˆβˆ’1 σ𝑣=𝑒+1 π‘ˆ π‘Žπ‘π‘  πœŽπ‘’βˆ’πœŽπ‘£ π‘ˆ σ𝑒=1 π‘ˆ πœŽπ‘’ . Shape Gini coefficient Space size
  • 31. UltraGCN (CIKM 2021) SELECTED GRAPH CF APPROACHES Node degree used to normalize the adjacency matrix in the message-passing. Node degree used to normalize the adjacency matrix in the message- passing. Node degree used for normalization in the infinite layer message-passing. The model also learns from the item- projected graph. Node embeddings involve the largest singular values of the normalized user-item interaction matrix, whose maximum value is related to the maximum node degree of the user- item graph. The model learns from user- and item-projected graphs. DGCF (SIGIR 2020) SVD-GCN (CIKM 2022) LightGCN (SIGIR 2020)
  • 32. The selected graph CF approaches explicitly utilize the node degree in the representation learning. However, clustering coefficient and degree assortativity do not have an evident representation in the formulations.
  • 33. β€’ Which topological aspects graph-based models can (un)intentionally capture? β€’ Are (topological) dataset characteristics influencing the recommendation performance of graph CF models?
  • 35. Our goal is to understand whether there exist dependencies among (topological) dataset characteristics and the performance of graph-based recommendation models. To this aim, we decide to build and fit an explanatory framework.
  • 38. EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 Select characteristics Select models
  • 39. EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 Select datasets Select characteristics Select models
  • 40. DATASETS # Users # Items # Interactions Yelp-2018 25,677 25,815 696,865 Amazon -Book 70,679 24,915 846,434 Gowalla 29,858 40,981 1,027,370
  • 41. EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 Select characteristics Select models Select datasets
  • 42. EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets Select characteristics Select models Select datasets
  • 49. EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets Select characteristics Select models Select datasets
  • 50. EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 Get datasets’ statistics Select characteristics Select models Select datasets
  • 53. EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 Get datasets’ statistics Select characteristics Select models Select datasets
  • 54. EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 STEP 6 Train models Select characteristics Select models Select datasets Get datasets’ statistics
  • 55. EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 STEP 6 Train models STEP 7 Evaluate models Select characteristics Select models Select datasets Get datasets’ statistics
  • 56. EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 STEP 6 Train models STEP 7 Evaluate models STEP 8 Generate explanations Select characteristics Select models Select datasets Get datasets’ statistics
  • 57. GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘.
  • 58. GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. Predicted performance
  • 59. GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. Predicted performance Prediction error
  • 60. GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. Predicted performance Prediction error Global bias
  • 61. GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. Predicted performance Prediction error Global bias Weight of characteristic c
  • 62. GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. Predicted performance Prediction error Global bias Weight of characteristic c Characteristic c for all samples
  • 63. GENERATE (LINEAR) EXPLANATIONS Characteristics Performance We aim to fit the following multivariate linear regression model: π’š = 𝝐 + πœƒ0 + πœƒπ‘π‘Ώπ‘. We use the ordinary least squares (OLS) optimization model: πœƒ0 βˆ—,πœ½π‘ βˆ— = min πœƒ0,πœƒπ‘ 1 2 π’š βˆ’ πœƒ0 βˆ’ πœ½π‘π‘Ώπ‘ 2 2 .
  • 64. EXPLANATORY FRAMEWORK: STEPS STEP 1 STEP 2 STEP 3 STEP 4 Generate sub-datasets STEP 5 STEP 6 Train models STEP 7 Evaluate models STEP 8 Generate explanations Select characteristics Select models Select datasets Get datasets’ statistics
  • 67. IMPACT OF CHARACTERISTICS How good is the linear regression at making predictions? 𝑅2 is, for many settings, above 95%. The regression model can provide good explanations.
  • 69. IMPACT OF CHARACTERISTICS inverse Graph CF = neighborhood + latent factors. Graph CF behaves as factorization-based models. direct
  • 71. IMPACT OF CHARACTERISTICS direct This confirms what observed in the models’ formulations. High users’ interactions correspond to better performance.
  • 72. IMPACT OF CHARACTERISTICS strong direct Values for users- and items-side are more flattened for UltraGCN and SVD-GCN.
  • 73. IMPACT OF CHARACTERISTICS LightGCN and DGCF have larger coefficients that SVD-GCN, because they explicitly propagate messages. direct inverse
  • 74. IMPACT OF CHARACTERISTICS UltraGCN has bigger coefficients, probably because it uses the infinite-layer message passing. direct inverse
  • 75. Probability distribution of node degrees on Gowalla
  • 76. In the worst-case scenario, node- dropout drops many high-degree nodes; edge-dropout, drops all the edges connected to several nodes and thus disconnect them from the graph.
  • 77. Is this undermining the goodness of the proposed explanatory framework?
  • 78. INFLUENCE OF NODE- AND EDGE-DROPOUT NODE 100% [Setting A] 100% node-dropout
  • 79. INFLUENCE OF NODE- AND EDGE-DROPOUT NODE EDGE 70% 30% [Setting B] 70% node-dropout 30% edge-dropout
  • 80. INFLUENCE OF NODE- AND EDGE-DROPOUT NODE EDGE 30% 70% [Setting C] 30% node-dropout 70% edge-dropout
  • 81. INFLUENCE OF NODE- AND EDGE-DROPOUT EDGE 100% [Setting D] 100% edge-dropout
  • 82. INFLUENCE OF NODE- AND EDGE-DROPOUT Datasets’ statistics Node-dropout retains smaller portions of the dataset than edge-dropout.
  • 83. INFLUENCE OF NODE- AND EDGE-DROPOUT Meaningful learned dependencies (this setting is like ours).
  • 84. INFLUENCE OF NODE- AND EDGE-DROPOUT Not-statistically-significant at the extremes.
  • 86. WHAT WE HAVE LEARNED 01 LATENT-FACTORS vs. NEIGHBORHOOD
  • 87. WHAT WE HAVE LEARNED 01 02 LATENT-FACTORS vs. NEIGHBORHOOD NODE DEGREE
  • 88. WHAT WE HAVE LEARNED 01 02 03 LATENT-FACTORS vs. NEIGHBORHOOD NODE DEGREE CLUSTERING COEFFICIENT
  • 89. WHAT WE HAVE LEARNED 01 04 02 03 LATENT-FACTORS vs. NEIGHBORHOOD NODE DEGREE CLUSTERING COEFFICIENT DEGREE ASSORTATIVITY
  • 90. WHAT WE HAVE LEARNED 01 04 02 05 03 LATENT-FACTORS vs. NEIGHBORHOOD NODE DEGREE CLUSTERING COEFFICIENT DEGREE ASSORTATIVITY NODE- vs. EDGE- DROPOUT
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