Probabilistic Spreading Algorithm

• ProbS is a resource allocation and link prediction method that diffuses initial resources across bipartite networks to favor popular items.
• Hybridization with HeatS introduces a tunable balance between recommendation accuracy and diversity using the parameter λ.
• Heterogeneous initial resource configurations further refine performance, yielding improved ranking scores and targeted recommendation exposure.

The Probabilistic Spreading Algorithm (ProbS) is a resource allocation and link prediction method defined for bipartite user–object networks, including but not limited to recommender systems. In this framework, resource is initially assigned to objects collected by a user, and then reallocated throughout the network in analogy to diffusion dynamics. The process has formal connections to both probability spreading and heat spreading algorithms; recent research demonstrates that hybridization and the introduction of heterogeneity in the initial resource configuration enable simultaneous improvements in both recommendation accuracy and diversity.

1. Mathematical Formalism of Probabilistic Spreading

ProbS is formulated on bipartite graphs, where each user $i$ and object $\alpha$ are connected via binary indicator $a_{i\alpha}$ representing whether user $i$ collected object $\alpha$. The initial resource vector $f_0$ assigns each collected object a unit of resource (typically $f_\alpha^i = a_{i\alpha}$). The central operation is the diffusion step:

$f = W f_0$

where $W$ is a resource reallocation matrix. For standard ProbS,

$$W_{\alpha\beta} = \frac{1}{k_\beta} \sum_{i=1}^{m} \frac{a_{i\alpha} a_{i\beta}}{k_i}$$

Here, $k_\beta$ is the degree (popularity) of object $\beta$, $k_i$ is the degree of user $i$, and $m$ is the number of users.

The matrix $W$ thus encodes the redistribution of resource from objects to users and back, favoring popular objects by aggregating over their many connections. As a consequence, ProbS is oriented towards maximizing recommendation accuracy, tending to suggest items which are popular or more likely to be collected.

2. Hybridization with Heat Spreading

To address the trade-off between accuracy and diversity, an interpolation between ProbS and Heat Spreading (HeatS) algorithms was introduced. HeatS uses a weighting scheme that inverts the preference, favoring low-degree (unpopular) objects and thereby increasing diversity.

The hybrid resource reallocation matrix is defined by:

$$W_{\alpha\beta} = \frac{1}{k_\alpha^{1-\lambda} k_\beta^{\lambda}} \sum_{i=1}^m \frac{a_{i\alpha} a_{i\beta}}{k_i}$$

where $\lambda \in [0,1]$ controls the balance: $\lambda = 1$ selects pure ProbS (accuracy-focused), $\lambda = 0$ pure HeatS (diversity-focused). Real-world recommender systems implement this hybridization, tuning $\lambda$ for optimal accuracy/diversity outcomes.

3. Heterogeneous Initial Resource Configurations

Traditional implementations set the initial resource vector homogeneously, $f^i_\alpha = a_{i\alpha}$, assigning equal resource to all collected objects. The improved approach introduces heterogeneous allocation:

$f^i_\alpha = a_{i\alpha} k_\alpha^\eta$

with degree $k_\alpha$, and tunable parameter $\eta$. For $\eta < 0$, less resource is assigned to popular objects (emphasizing diversity); for $\eta > 0$, the reverse holds (emphasizing accuracy). This scheme enables granular control over the initial diffusion bias. Numerical results confirm that appropriately negative $\eta$ improves both diversity measures and overall accuracy.

4. Metrics of System Performance

Comprehensive evaluation relies on both accuracy and diversity metrics.

Accuracy:

• Ranking Score ($r$): Measures the average position of true probe links in the recommended list (lower is better).
• Precision ($P$): Fraction of items in top $L$ recommendations present in the probe set.
• Recall ($R$): Fraction of probe items appearing in top $L$ recommendations.

Diversity:

• Intra-user Diversity ($D_\text{intra}$): Average pairwise dissimilarity of items in a user's top recommended list, often using the Sørensen index.
• Inter-user Diversity ($D_\text{inter}$): Measures overlap between recommendation lists of different users; higher $D_\text{inter}$ implies increased personalization.

Explicit formulas and analysis are performed for these metrics, facilitating comparative studies and systematic tuning.

5. Numerical Results and Optimization

Experimental validation utilizes the MovieLens and Netflix datasets. The link sets are split (90% training, 10% probe) and recommendations are generated via the hybrid algorithm with varying $\lambda$ and $\eta$.

• Optimal values: On MovieLens, lowest ranking score at $\lambda_{\text{opt}} \approx 0.26$, $\eta_{\text{opt}} \approx -0.71$; on Netflix, $\lambda_{\text{opt}} \approx 0.21$, $\eta_{\text{opt}} \approx -0.51$.
• Performance gains: Using heterogeneous initial configuration yields ranking score improvements of $\approx 6.0\%$ (MovieLens) and $\approx 12.8\%$ (Netflix) over homogeneous approaches.
• Diversity robustness: Diversification metrics are robust or increased, especially for item recommendation scenarios sensitive to exposure of unpopular objects.
• Microscopic analysis: Popular items benefit from higher $\lambda$ and $\eta$; niche items from lower values, implying the possibility of class-specific parameterization.

6. Application and Practical Considerations

Implementations should adjust $\lambda$ and $\eta$ to simultaneously improve accuracy and diversity. The hybrid method is structurally robust and extensible to various recommendation systems and domains (including biomedical link prediction). By tuning $\eta$, systems may be tailored for strategic exposure objectives (boosting niche items or reinforcing known items).

Performance measures provide actionable targets for model calibration and monitoring post-deployment. The algorithm is scalable to large bipartite networks, with matrix operations being the computational bottleneck, but amenable to standard parallelization strategies.

7. Summary and Implications

ProbS defines resource reallocation on bipartite networks favoring accurate link prediction through popular object amplification. Hybridization with HeatS introduces a tunable trade-off between accuracy and diversity. The main innovation demonstrated is the introduction of a heterogeneous initial resource vector that further improves recommendation quality. The method attains robust improvements in ranking score, precision, recall, and diversity metrics and is validated on benchmark datasets. Practical recommender systems adopting this approach benefit from the algorithm’s flexibility, scalability, and ability to target explicit operational criteria for accuracy and diversity.

In conclusion, the Probabilistic Spreading Algorithm forms a principled and empirically verified backbone for high-performance, tunable network-based recommender systems, particularly when augmented by hybridization and degree-dependent initial resource allocation.