PageRank-Weighted DPP & M-DPP Models

• PageRank-weighted DPPs are probabilistic models that combine diversity with network centrality by integrating PageRank scores into the DPP kernel.
• They extend traditional DPPs to Markov DPPs, ensuring temporal diversity by preserving novelty across sequential selections.
• The model is applied in recommendation systems and content curation, balancing high-quality, influential items with diverse subset selection through efficient inference and online learning.

A PageRank-weighted Determinantal Point Process (DPP) is a probabilistic model for subset selection that combines the inherent diversity-promoting properties of DPPs with the notion of item centrality, as measured by PageRank. By embedding PageRank scores within the quality terms of the DPP’s kernel, this model facilitates selection of subsets that are both diverse and preferentially include items of high (network) importance. The extension to time-dependent scenarios is formalized through Markov DPPs (M-DPPs), preserving diversity both within and across sequential selections.

1. Foundations of DPPs and Markov DPPs

A DPP on a finite base set $\mathcal{Y}$ defines a distribution over subsets $Y \subseteq \mathcal{Y}$ as

$P_L(Y) \propto \det(L_Y)$

where $L$ is a positive semidefinite matrix (“kernel”) and $L_Y$ is the principal submatrix indexed by $Y$. The determinant formulation ensures a balance between selecting high-quality items (when diagonal entries are large) and promoting diversity (when off-diagonal similarities are small) (Affandi et al., 2012, Fitzsimons et al., 22 May 2024).

Markov DPPs (M-DPPs) extend this framework to sequences $(Y_1,\dots,Y_T)$, introducing temporal structure. At each time $t$, the marginal of $Y_t$ remains DPP-distributed, but transitions are governed so that the union $Z_t = Y_t \cup Y_{t-1}$ is also DPP-distributed. Specifically, for L-ensemble DPPs, the Markov transition is written as

$$P(Y_t|Y_{t-1}) = \frac{\det(L_{Y_t \cup Y_{t-1}})}{\det(L_{[y] \setminus Y_{t-1}} + I)}$$

with $L$ as the DPP kernel and $[y]$ the base set. This construction enforces diversity not just within $Y_t$ but also between $Y_t$ and $Y_{t-1}$.

2. PageRank Integration into the DPP Kernel

The PageRank-weighted DPP adapts the quality component $q_i$ of each item $i$ to integrate PageRank, i.e., intrinsic network-derived centrality. The modified L-ensemble kernel becomes

$$L_{ij} = [\exp(\theta^\top f_i) \cdot PR(i)] \cdot (\phi_i^\top \phi_j) \cdot [\exp(\theta^\top f_j) \cdot PR(j)]$$

where:

• $\exp(\theta^\top f_i)$ models learned item quality from features $f_i$,
• $PR(i)$ is the PageRank score for item $i$,
• $\phi_i$ encodes item similarity. This structure preserves the DPP’s property of favoring both high-quality (now also high-PageRank) items and diverse selections, as the determinant is boosted by large $q_i$ values but penalizes highly similar items (Affandi et al., 2012, Fitzsimons et al., 22 May 2024).

In a MAP inference setting, the kernel can also be defined as

$L_{ij} = (p_i r_i)(p_j r_j) S_{ij}$

with $p_i$ the PageRank score, $r_i$ the relevance, and $S_{ij}$ an item similarity matrix (Chen et al., 2017).

3. Sampling and MAP Inference for PageRank-Weighted (M-)DPPs

Sampling from a PageRank-weighted DPP or M-DPP involves:

• Initial sampling: Draw $Y_1$ via standard DPP sampling algorithms based on the possibly weighted kernel.
• Sequential sampling: At each subsequent $t$, sample $Y_t$ from a conditional DPP with a kernel updated to reflect the exclusion of $Y_{t-1}$:

$$L^{(t)} = ((M + I_{[y] \setminus Y_{t-1}})_{Y^c})^{-1} - I$$

where $M = L(I-L)^{-1}$ and $L$ may be PageRank-weighted (Affandi et al., 2012).

In large-scale or real-time applications, MAP inference requires efficient algorithms. Incremental Cholesky-based updates can accelerate greedy MAP inference for DPPs. Each candidate addition relies on the update formulas:

• $c_i$: solution to $V c_i^\top = L_{Y_g, i}$
• $d_i^2 = L_{ii} - \|c_i\|_2^2$, marginal gain $\log d_i^2$ and after selection, Cholesky factors are efficiently updated in-place. Incorporation of PageRank modifies only the definition of $L$, not the inference algorithm (Chen et al., 2017).

4. Learning Quality Parameters and Integrating User Feedback

Quality parameters $\theta$ are updated online to adapt to user feedback. The core update rule is

$$\theta^{(t+1)} \leftarrow \theta^{(t)} + \eta \left(\frac{1}{|R_t|} \sum_{i \in R_t} f_i - \frac{1}{|S_t|} \sum_{i \in S_t} f_i \right)$$

where $R_t$ and $S_t$ are sets of preferred and non-preferred items, and $f_i$ is the feature vector of item $i$. When PageRank weighting is used, PageRank acts as a fixed multiplicative factor in $q_i$, biasing the system toward higher-centrality items while $\theta$ remains the parameter being incrementally learned (Affandi et al., 2012). This allows exploitation of both structural (PageRank) and contextual (user feedback) evidence.

5. Applications and Implications

PageRank-weighted (M-)DPPs are suitable for scenarios where subset relevance and coverage of important items are both paramount.

Application Domain	Role of PageRank-DPP	Temporal Extension (M-DPP)
Web/news recommendation	Ensures diverse, authoritative items via PageRank weighting	Prevents redundancy day-by-day
Scientific literature/citation networks	Favors influential works, diversified across topics	Sustains novelty in sequential curation
Social media, session-based search	Recommends influential yet distinct posts/pages	Adapts to user interests over sessions

A key implication is the simultaneous maximization of authority (by PageRank) and coverage (by DPP structure), reducing redundancy while exposing users to novel and high-impact items.

6. Extensions, Mathematical Formulations, and Limitations

The mathematical foundation encompasses both the marginal and L-ensemble perspectives:

• Marginal Markov process: $$P(Y_t = A | Y_{t-1} = B) = \det(K_A)\det(K_{A\cup B})$$, $A\cap B = \emptyset$
• L-ensemble DPP: $P(Y = A) = \det(L_A)/\det(L + I)$
• PageRank-weighted L: $$L_{ij} = [\exp(\theta^\top f_i) PR(i)] (\phi_i^\top \phi_j) [\exp(\theta^\top f_j) PR(j)]$$

Algorithmic complexity remains polynomial, typically $O(T N^3 + T N k_{\max}^3)$ for $T$ time steps (Affandi et al., 2012). An increase in PageRank dynamic range can affect the kernel’s conditioning; careful normalization is required.

In differentially-private settings, integrating PageRank can increase the kernel’s sensitivity, affecting the privacy-utility tradeoff. Regularization (“jitter”) is needed to ensure all eigenvalues are positive and the privacy loss remains bounded (Fitzsimons et al., 22 May 2024).

7. Practical Considerations and Implementation

• Kernel Definition: The practitioner must compute or obtain PageRank for all items; normalization or a nonlinear transformation (e.g., $\log$ scaling) may be necessary for numerical stability.
• Scalability: Incremental solvers and matrix factorizations are critical for handling large item sets.
• Learning Loop: Integration with online feedback is straightforward; the update rule for $\theta$ remains unchanged even with PageRank factors.
• Deployment: The model is applicable both in batch and sequentially-updating (M-DPP) settings. Sampling or MAP inference algorithms operate on the PageRank-weighted kernel.
• Parameter Selection: Weighting of PageRank vs. learned quality can be cross-validated to optimize the relevance-diversity tradeoff for a given application.

A plausible implication is that, by combining both structural network-derived insights (PageRank) and adaptive learning from user feedback, PageRank-weighted DPPs offer a flexible, robust mechanism for sequentially selecting diverse, high-importance subsets in recommendation, search, and content curation systems.