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SoRec: social recommendation using probabilistic matrix factorization

Hao MaHaixuan YangMichael R. LyuIrwin King

article2008CIKM1,572 citations

Proposes a scalable probabilistic matrix factorization framework that fuses user social network relations with rating matrices to improve recommendation accuracy for users with sparse or missing feedback.

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Commercial recommender systems are essential for helping users discover relevant products, media, and services. However, conventional collaborative filtering methods struggle with severe data sparsity—where available rating density is often below one percent—and perform poorly when users have rated very few or no items. Additionally, traditional systems assume individual users act independently, ignoring how real-world social connections and trusted relationships shape personal tastes and consumer decisions.

The article demonstrates a novel framework called Social Recommendation (SoRec), which uses probabilistic matrix factorization to fuse user social network structures with user-item rating histories through a shared latent feature representation. The primary objective is to evaluate whether integrating social trust networks with rating matrices can significantly improve recommendation accuracy and resolve cold-start challenges for inactive users while maintaining computational scalability.

To evaluate this approach, the researchers conducted extensive empirical testing on a real-world dataset from Epinions, comprising 40,163 users, 139,529 items, 664,824 ratings, and 487,183 directed trust statements. The authors benchmarked the proposed method against three state-of-the-art matrix factorization approaches across varying training set sizes ranging from 20% to 99% of total ratings, measuring accuracy through Mean Absolute Error (MAE) and analyzing computational runtime complexity.

The analysis produced several critical findings. First, the proposed social recommendation framework consistently outperformed existing state-of-the-art models, improving average prediction accuracy by approximately 7.8% to 11.0% across all evaluated training splits. Second, the model demonstrated dramatic improvements for inactive users with zero prior ratings, outperforming competing methods by more than 36% to 41%. Third, the framework achieves linear computational scalability relative to the number of observed interactions, converging within 5 to 18 minutes on standard computing hardware. Finally, tuning the balance between social ties and user ratings proved critical; incorporating moderate social influence prevented model overfitting and optimized prediction quality.

These findings indicate that organizations operating digital platforms can substantially reduce cold-start friction for new or low-engagement users by incorporating social connection data. Deploying this approach can enhance user retention, boost personalization quality, and increase conversion rates without requiring expensive, specialized computing infrastructure. Unlike older heuristic trust algorithms that suffered from poor scalability, this unified probabilistic approach scales efficiently to large enterprise datasets.

Organizations seeking to enhance their recommendation platforms should consider piloting matrix factorization architectures that integrate user relationship graphs with transaction or rating histories. When implementing these systems, engineering teams must tune the balancing parameter between social and rating data to prevent overfitting. Future development should explore nonlinear kernel representations to capture complex feature relationships, model information diffusion dynamics across social graphs, and incorporate user distrust signals, which were excluded from this evaluation due to privacy constraints and modeling complexity.

The Chinese University of Hong Kongdata sparsitymatrix factorizationsnetwork analysisprobabilistic matrix factorizationsocial recommendationsocial network matrix factorization
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Abstract

Data sparsity, scalability and prediction quality have been recognized as the three most crucial challenges that every collaborative filtering algorithm or recommender system confronts. Many existing approaches to recommender systems can neither handle very large datasets nor easily deal with users who have made very few ratings or even none at all. Moreover, traditional recommender systems assume that all the users are independent and identically distributed; this assumption ignores the social interactions or connections among users. In view of the exponential growth of information generated by online social networks, social network analysis is becoming important for many Web applications. Following the intuition that a person's social network will affect personal behaviors on the Web, this paper proposes a factor analysis approach based on probabilistic matrix factorization to solve the data sparsity and poor prediction accuracy problems by employing both users' social network information and rating records. The complexity analysis indicates that our approach can be applied to very large datasets since it scales linearly with the number of observations, while the experimental results shows that our method performs much better than the state-of-the-art approaches, especially in the circumstance that users have made few or no ratings.

Topics

data sparsitymatrix factorizationsnetwork analysisonline social networksrecommendationsocial network analysisuser ratingweb applicationcollaborative filteringsocial relationships

Keywords

probabilistic matrix factorizationsocial recommendationsocial network matrix factorizationcollaborative filtering methodsuser-item rating matrixMaximum Margin Matrix Factorizationonline social networkslatent feature vectors

Table of Contents

  • Categories and Subject Descriptors
  • General Terms
  • Keywords
  • 1. INTRODUCTION
  • 2. RELATED WORK
  • 3. SOCIAL RECOMMENDATION FRAMEWORK
  • 3.1 Toy Example
  • 3.2 Social Network Matrix Factorization
  • 3.3 User-Item Matrix Factorization
  • 3.4 Matrix Factorization for Social Recommendation
  • 3.5 Complexity Analysis
  • 4. EXPERIMENTAL ANALYSIS
  • 4.1 Description of the Epinions Dataset
  • 4.2 Metrics
  • 4.3 Comparison
  • 4.4 Impact of Parameter λ C
  • 4.5 Performance on Different Users
  • 4.6 Efficiency Analysis
  • 5. CONCLUSIONS AND FUTURE WORK
  • 6. ACKNOWLEDGMENTS
  • 7. REFERENCES

Knowls

  1. Knowl 1 — SoRec Probabilistic Matrix Factorization Objective Function

    model/method

    Social Recommendation (SoRec) is a factor analysis framework that jointly models a user-item rating matrix and a directed user-user social trust graph by coupling them through a shared user latent feature space.

    Let mm be the number of users and nn be the number of items. The observed user-item ratings are denoted by R∈Rm×nR \in \mathbb{R}^{m \times n} where integer ratings from 11 to Rmax⁡R_{\max} are mapped to [0,1][0, 1] via f(x)=(x−1)/(Rmax⁡−1)f(x) = (x - 1) / (R_{\max} - 1). The social network is represented as an asymmetric adjacency matrix C∈Rm×mC \in \mathbb{R}^{m \times m}, where cik∈(0,1]c_{ik} \in (0, 1] denotes the trust weight from user ii to user kk, and cik∗c_{ik}^* is its degree-adjusted trust value.

    Let U=[U1,…,Um]∈Rl×mU = [U_1, \dots, U_m] \in \mathbb{R}^{l \times m} be the user latent feature matrix, V=[V1,…,Vn]∈Rl×nV = [V_1, \dots, V_n] \in \mathbb{R}^{l \times n} be the item latent feature matrix, and Z=[Z1,…,Zm]∈Rl×mZ = [Z_1, \dots, Z_m] \in \mathbb{R}^{l \times m} be the social factor latent feature matrix in an ll-dimensional latent space. Assuming independent Gaussian observations with logistic mean mapping g(x)=1/(1+exp⁡(−x))g(x) = 1 / (1 + \exp(-x)) and zero-mean spherical Gaussian priors on UU, VV, and ZZ, maximizing the posterior distribution is equivalent to minimizing the following regularized sum-of-squared-errors loss function:

    L(R,C,U,V,Z)=12∑i=1m∑j=1nIijR(rij−g(UiTVj))2+λC2∑i=1m∑k=1mIikC(cik∗−g(UiTZk))2+λU2∥U∥F2+λV2∥V∥F2+λZ2∥Z∥F2\mathcal{L}(R, C, U, V, Z) = \frac{1}{2} \sum_{i=1}^m \sum_{j=1}^n I_{ij}^R \left( r_{ij} - g(U_i^T V_j) \right)^2 + \frac{\lambda_C}{2} \sum_{i=1}^m \sum_{k=1}^m I_{ik}^C \left( c_{ik}^* - g(U_i^T Z_k) \right)^2 + \frac{\lambda_U}{2} \|U\|_F^2 + \frac{\lambda_V}{2} \|V\|_F^2 + \frac{\lambda_Z}{2} \|Z\|_F^2

    where IijRI_{ij}^R is an indicator variable equal to 11 if user ii rated item jj and 00 otherwise; IikCI_{ik}^C is an indicator variable equal to 11 if user ii trusts user kk and 00 otherwise; ∥⋅∥F\|\cdot\|_F denotes the Frobenius norm; and the trade-off hyperparameters are defined by variance ratios λC=σR2/σC2\lambda_C = \sigma_R^2 / \sigma_C^2, λU=σR2/σU2\lambda_U = \sigma_R^2 / \sigma_U^2, λV=σR2/σV2\lambda_V = \sigma_R^2 / \sigma_V^2, and λZ=σR2/σZ2\lambda_Z = \sigma_R^2 / \sigma_Z^2.

  2. Knowl 2 — Degree-Weighted Transformation for Social Trust Matrix

    equation

    In a directed online social trust network, explicitly declared trust values cik∈(0,1]c_{ik} \in (0, 1] between user ii and user kk contain noise and omit network structural authority. To account for local hub and authority effects—reducing confidence in trust asserted by users who trust large numbers of people (hubs) and increasing confidence in trust placed in users trusted by many peers (authorities)—the raw trust weight cikc_{ik} is transformed into an adjusted weight cik∗c_{ik}^*:

    cik∗=d−(vk)d+(vi)+d−(vk)×cikc_{ik}^* = \sqrt{\frac{d^-(v_k)}{d^+(v_i) + d^-(v_k)}} \times c_{ik}

    where d+(vi)d^+(v_i) is the outdegree of user node viv_i (number of users trusted by ii) and d−(vk)d^-(v_k) is the indegree of user node vkv_k (number of users who trust kk).

  3. Knowl 3 — Gradient Descent Optimization for SoRec

    algorithm

    The local minimum of the SoRec objective function L(R,C,U,V,Z)\mathcal{L}(R, C, U, V, Z) is computed by performing gradient descent over the latent vectors UiU_i, VjV_j, and ZkZ_k.

    The partial derivatives of L\mathcal{L} with respect to latent vectors are:

    ∂L∂Ui=∑j=1nIijRg′(UiTVj)(g(UiTVj)−rij)Vj+λC∑k=1mIikCg′(UiTZk)(g(UiTZk)−cik∗)Zk+λUUi\frac{\partial \mathcal{L}}{\partial U_i} = \sum_{j=1}^n I_{ij}^R g'(U_i^T V_j) \left( g(U_i^T V_j) - r_{ij} \right) V_j + \lambda_C \sum_{k=1}^m I_{ik}^C g'(U_i^T Z_k) \left( g(U_i^T Z_k) - c_{ik}^* \right) Z_k + \lambda_U U_i

    ∂L∂Vj=∑i=1mIijRg′(UiTVj)(g(UiTVj)−rij)Ui+λVVj\frac{\partial \mathcal{L}}{\partial V_j} = \sum_{i=1}^m I_{ij}^R g'(U_i^T V_j) \left( g(U_i^T V_j) - r_{ij} \right) U_i + \lambda_V V_j

    ∂L∂Zk=λC∑i=1mIikCg′(UiTZk)(g(UiTZk)−cik∗)Ui+λZZk\frac{\partial \mathcal{L}}{\partial Z_k} = \lambda_C \sum_{i=1}^m I_{ik}^C g'(U_i^T Z_k) \left( g(U_i^T Z_k) - c_{ik}^* \right) U_i + \lambda_Z Z_k

    where g′(x)=exp⁡(x)(1+exp⁡(x))2g'(x) = \frac{\exp(x)}{(1 + \exp(x))^2} is the derivative of the logistic function g(x)=11+exp⁡(−x)g(x) = \frac{1}{1 + \exp(-x)}.

    Input: Rating matrix RR, adjusted trust matrix C∗C^*, latent dimension ll, learning rate γ\gamma, regularization parameters λC,λU,λV,λZ\lambda_C, \lambda_U, \lambda_V, \lambda_Z
    Output: User latent matrix UU, item latent matrix VV, factor latent matrix ZZ
    Initialize U∈Rl×m,V∈Rl×n,Z∈Rl×mU \in \mathbb{R}^{l \times m}, V \in \mathbb{R}^{l \times n}, Z \in \mathbb{R}^{l \times m} with random values
    repeat
        for each user i∈{1,…,m}i \in \{1, \dots, m\} do
            Ui←Ui−γ∂L∂UiU_i \leftarrow U_i - \gamma \frac{\partial \mathcal{L}}{\partial U_i}
        end for
        for each item j∈{1,…,n}j \in \{1, \dots, n\} do
            Vj←Vj−γ∂L∂VjV_j \leftarrow V_j - \gamma \frac{\partial \mathcal{L}}{\partial V_j}
        end for
        for each factor node k∈{1,…,m}k \in \{1, \dots, m\} do
            Zk←Zk−γ∂L∂ZkZ_k \leftarrow Z_k - \gamma \frac{\partial \mathcal{L}}{\partial Z_k}
        end for
    until convergence criterion is met
    return U,V,ZU, V, Z

    Ratings for unobserved pairs are predicted as r^ij=g(UiTVj)⋅(Rmax⁡−1)+1\hat{r}_{ij} = g(U_i^T V_j) \cdot (R_{\max} - 1) + 1.

  4. Knowl 4 — Computational Complexity of SoRec Factorization

    theoretical result

    Let ρR\rho_R denote the number of observed non-zero entries in the rating matrix RR, ρC\rho_C denote the number of observed edges in the social network matrix CC, and ll denote the dimensionality of the latent feature space.

    Evaluating the objective function L(R,C,U,V,Z)\mathcal{L}(R, C, U, V, Z) requires O(ρRl+ρCl)\mathcal{O}(\rho_R l + \rho_C l) operations due to matrix sparsity. The computational complexities for evaluating the gradients are:

    • Gradient with respect to user matrices ∂L∂U\frac{\partial \mathcal{L}}{\partial U}: O(ρRl+ρCl)\mathcal{O}(\rho_R l + \rho_C l)
    • Gradient with respect to item matrices ∂L∂V\frac{\partial \mathcal{L}}{\partial V}: O(ρRl)\mathcal{O}(\rho_R l)
    • Gradient with respect to social factor matrices ∂L∂Z\frac{\partial \mathcal{L}}{\partial Z}: O(ρCl)\mathcal{O}(\rho_C l)

    The total computational complexity per gradient descent iteration is O(ρRl+ρCl)\mathcal{O}(\rho_R l + \rho_C l), which scales linearly with the total number of observed ratings and trust relationships in the sparse matrices.

  5. Knowl 5 — Mean Absolute Error Comparison Across Training Splits

    data/table

    The recommendation accuracy of SoRec was compared against Maximum Margin Matrix Factorization (MMMF), Probabilistic Matrix Factorization (PMF), and Constrained Probabilistic Matrix Factorization (CPMF) on the Epinions dataset across different training data percentages (99%, 80%, 50%, 20%) and latent dimensions (l=5l=5 and l=10l=10). Prediction accuracy is measured using Mean Absolute Error (MAE):

    MAE=∑i,j∣ri,j−r^i,j∣N\text{MAE} = \frac{\sum_{i,j} |r_{i,j} - \hat{r}_{i,j}|}{N}

    where ri,jr_{i,j} is the true rating, r^i,j\hat{r}_{i,j} is the predicted rating, and NN is the number of tested ratings. Parameter settings for SoRec are λC=10\lambda_C = 10 and λU=λV=λZ=0.001\lambda_U = \lambda_V = \lambda_Z = 0.001.

    Training Data Dimensionality = 5 Dimensionality = 10
    MMMF PMF CPMF SoRec MMMF PMF CPMF SoRec
    99% 1.0008 0.9971 0.9842 0.9018 0.9916 0.9885 0.9746 0.8932
    80% 1.0371 1.0277 0.9998 0.9321 1.0275 1.0182 0.9923 0.9240
    50% 1.1147 1.0972 1.0747 0.9838 1.1012 1.0857 1.0632 0.9751
    20% 1.2532 1.2397 1.1981 1.1069 1.2413 1.2276 1.1864 1.0944

    Across all training splits and latent dimensions, SoRec achieves lower MAE than all baseline methods. On average, SoRec improves accuracy relative to MMMF by 11.01%, PMF by 9.98%, and CPMF by 7.82%.

  6. Knowl 6 — Rating Prediction Superiority on Cold-Start and Sparse Users

    empirical result

    When users in the Epinions dataset are binned by the number of observed ratings in the training set into 10 categories ("=0", "1–5", "6–10", "11–20", "21–40", "41–80", "81–160", "160–320", "320–640", ">640"), SoRec demonstrates its largest performance margins on cold-start users and users with very sparse rating histories.

    For users with zero ratings in the training set ("=0"), traditional collaborative filtering methods (MMMF, PMF, CPMF) fail to compute personalized predictions effectively, whereas SoRec leverages social trust connections to generate accurate predictions. Specifically, for zero-rating users, SoRec improves MAE by more than 36.75% over MMMF, 40.82% over PMF, and 41.75% over CPMF. While baseline methods converge toward similar performance as user rating frequency increases, SoRec maintains superior or equal MAE across all user rating volume groups.

  7. Knowl 7 — Impact of Social Regularization Hyperparameter on Overfitting

    empirical result

    The parameter λC=σR2/σC2\lambda_C = \sigma_R^2 / \sigma_C^2 controls the relative weight between rating matrix factorization and social trust network factorization. When evaluated across λC∈{0.1,0.5,1,5,10,20,50,100}\lambda_C \in \{0.1, 0.5, 1, 5, 10, 20, 50, 100\} on the Epinions dataset for dimensions l=5l=5 and l=10l=10:

    1. Optimal performance is achieved in the range λC∈[10,20]\lambda_C \in [10, 20]. Small values (e.g., λC=0.1\lambda_C = 0.1) or overly large values (e.g., λC=100\lambda_C = 100) degrade prediction accuracy.
    2. Setting λC\lambda_C to small values (λC=0.1\lambda_C = 0.1 or λC=1.0\lambda_C = 1.0) causes gradient descent to begin overfitting after 200 to 300 iterations, resulting in rising test MAE. In contrast, larger values (λC=10\lambda_C = 10) prevent overfitting over more than 1,000 iterations.

  8. Knowl 8 — Epinions Social Recommendation Benchmark Dataset Characteristics

    experimental setup

    The empirical evaluation of SoRec uses a crawl of the Epinions knowledge sharing and review platform containing both user-item ratings and explicit user-user directed trust statements:

    • User-Item Matrix: m=40,163m = 40,163 users, n=139,529n = 139,529 distinct items, and 664,824664,824 ratings in the range {1,2,3,4,5}\{1, 2, 3, 4, 5\}. Matrix density is 0.01186%0.01186\%. User rating counts range from 11 to 1,0221,022 (mean 16.5516.55), with 46.87%46.87\% of users (18,82618,826 users) having submitted ≤5\le 5 reviews. Item rating counts range from 11 to 2,0182,018 (mean 4.764.76).
    • Social Trust Network: 487,183487,183 directed trust statements with power-law indegree and outdegree distributions.
    • Data Handling: Distrust ("block") statements are excluded due to privacy restrictions on the platform and distinct latent structural properties of distrust. Ratings are scaled to [0,1][0, 1] via f(x)=(x−1)/4f(x) = (x - 1)/4 during optimization and mapped back for MAE computation.

  9. Knowl 9 — Assumptions and Structural Limitations of SoRec

    limitation

    The SoRec model has three primary structural limitations:

    1. Linear Factor Combination: Preferences and trust relationships are modeled via inner products (UiTVjU_i^T V_j and UiTZkU_i^T Z_k) passed through a logistic function, assuming observed interactions are linear combinations of latent features rather than non-linear kernel mappings (e.g., Gaussian or polynomial kernels).
    2. Exclusion of Distrust Relationships: The framework incorporates only positive trust edges and cannot directly include distrust/blocking links because user distrust latent spaces may not align with user trust latent spaces.
    3. Static Social Links: The network factorization treats trust as static pairwise ties and ignores dynamic multi-step information diffusion or propagation processes across the social network graph.

Coverage note — None was omitted; all contributed models, loss formulations, gradient updates, theoretical complexity bounds, experimental setups, primary numerical results, parameter analyses, and model limitations are covered.

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Citation

MLA
Ma, H., et al. “SoRec”. Proceedings of the 17th ACM Conference on Information and Knowledge Management, 2008, pp. 931–40, https://doi.org/10.1145/1458082.1458205.
APA
Ma, H., Yang, H., Lyu, M. R., & King, I. (2008). SoRec. Proceedings of the 17th ACM Conference on Information and Knowledge Management, 931–940. https://doi.org/10.1145/1458082.1458205
Chicago
Ma, H., H. Yang, M. R. Lyu, and I. King. 2008. “SoRec”. Proceedings of the 17th ACM Conference on Information and Knowledge Management, 931–40. https://doi.org/10.1145/1458082.1458205.
Harvard
Ma, H. et al. (2008) “SoRec”, Proceedings of the 17th ACM conference on Information and knowledge management. ACM, pp. 931–940. Available at: https://doi.org/10.1145/1458082.1458205.
Vancouver
1. Ma H, Yang H, Lyu MR, King I (2008) SoRec. In: Proceedings of the 17th ACM conference on Information and knowledge management. ACM, pp 931–940

BibTeX

@inproceedings{Ma_2008, series={CIKM08}, title={SoRec: social recommendation using probabilistic matrix factorization}, url={http://dx.doi.org/10.1145/1458082.1458205}, DOI={10.1145/1458082.1458205}, booktitle={Proceedings of the 17th ACM conference on Information and knowledge management}, publisher={ACM}, author={Ma, Hao and Yang, Haixuan and Lyu, Michael R. and King, Irwin}, year={2008}, month=Oct, pages={931–940}, collection={CIKM08} }
Metadata:Crossref

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