Personalized PageRank Algorithm

• Personalized PageRank is a modified PageRank algorithm that biases restart events toward selected nodes, ensuring personalized influence scores in a network.
• It employs multilayer and bipartite network structures to model complex relationships, enhancing tasks like risk propagation and community detection.
• The algorithm integrates a sparse influence matrix to restrict restarts, allowing more sensitive and scalable analysis compared to conventional default rate methods.

Personalized PageRank is a centrality and proximity algorithm that extends the classical PageRank to quantify node importance with respect to individual nodes or sets. In its traditional form, PageRank models a random walker who, at each step, either follows an outgoing edge or restarts from a specified distribution. Personalized PageRank (PPR) allows this restart to preferentially target certain nodes or classes of nodes, yielding node-specific influence scores that are fundamental for tasks such as recommendations, community detection, risk propagation, and signal diffusion in networks. The concept admits further generalization to multilayer and dynamic network settings, permitting fine control over restart events and propagation mechanisms.

1. Mathematical Formulation and Algorithmic Structure

The standard (single-layer) PageRank models a Markov chain with transition matrix $P$ and restart (teleportation) probability $1-\alpha$, typically written as: $\pi = \alpha P \pi + (1-\alpha) v$ where $v$ is the restart (personalization) vector and $\pi$ is the stationary distribution.

Personalized PageRank replaces the uniform restart with a bias towards specific nodes or sets. In multilayer settings, as introduced in (Bravo et al., 2020), the network is described as a multilayer adjacency tensor $M_{j\beta}^{i\alpha}$, with $i,j$ indexing nodes and $\alpha,\beta$ indexing layers. The walker's update equation generalizes to: $$p_{j\beta}(t+1) = R^{i\alpha}_{j\beta} p_{i\alpha}(t)$$ with the transition tensor: $$R^{i\alpha}_{j\beta} = r T^{i\alpha}_{j\beta} + \frac{1-r}{N \cdot L} u^{i\alpha}_{j\beta}$$ where $r$ is the intra-network move probability, $T^{i\alpha}_{j\beta}$ is the column-normalized supra-adjacency matrix (combining intra- and inter-layer transitions), and $u^{i\alpha}_{j\beta}$ is the influence matrix governing the restart dynamics.

Personalization is effected by making $u^{i\alpha}_{j\beta}$ sparse, with entries set to one only for desired influence nodes (e.g., known defaults). Random walk restarts sample exclusively from these nodes, restricting risk origin in applications like credit networks.

2. Adaptation for Multilayer and Bipartite Networks

Multilayer networks encode multiple, distinct relationship types in parallel layers (e.g., geographic vs. product layers in a lending network), with nodes potentially replicated in each layer. Each layer is often structured as a bipartite network (e.g., loans–districts or loans–products). The overall system's connectivity is represented as a supra-adjacency matrix with block structure: $$M_{j\beta}^{i\alpha} = \begin{bmatrix} A_j^i & I \ I & B_j^i \end{bmatrix}$$ where $A$ and $B$ are intra-layer bipartite adjacency matrices and $I$ encodes interlayer alignment (intrinsically, e.g., linking the same loan/node across different layers).

For algorithmic tractability, the rank-4 tensor is typically flattened into an $(N\cdot L) \times (N\cdot L)$ matrix, allowing the eigenproblem to be solved using standard matrix methods.

3. Temporal Application and Credit Risk Evolution

Personalized PageRank's multilayer extension can be used to model temporal evolution in dynamic systems such as financial credit networks. In the application of (Bravo et al., 2020), the dataset comprises approximately 70,000 agricultural loans recorded monthly over 15 years. The risk network at each timestep is modeled as a temporally indexed multilayer bipartite graph:

• Layer 1 (e.g., district): loans ↔ districts
• Layer 2 (e.g., product): loans ↔ products
• Interlayer edges: each loan is present in both layers, joined by an identity mapping.

At each rolling time window, the set of influence nodes is chosen as the set of loans that defaulted during that interval. The modified PPR computes steady-state scores per node/layer; aggregation over, for example, product-district pairs, yields risk influence time series.

Clustering (dynamic time warping with k-means) on these time series identifies synchronous evolution patterns among products/districts, unveiling risk propagation paths inaccessible to naive default rate analysis. Time series of network-driven risk can anticipate changes and provide greater sensitivity, underscoring the propagation and correlation aspects not captured by aggregated defaults.

4. Generalization: Influence Matrix and Personalization

In non-multilayer settings, restart personalization is usually carried out by setting the restart distribution $v$ directly. The influence matrix $u^{i\alpha}_{j\beta}$ in layered networks allows flexible specification of restart origins: for a given subnetwork or set of nodes (e.g., current defaulters), only block entries corresponding to those nodes are set to one, with zero elsewhere. This enables:

• Arbitrary restriction of restarts to any subset of the multilayered network (e.g., sectoral or geographic risk origins),
• Interlayer personalization—by activating influence only in a layer or block structure of interest.

A crucial implementation detail is adjusting the normalization denominator in the restart term to sum over only the nonzero entries in $u$, preserving the stochasticity of the transition process under sparse restarts.

5. Practical Implications and Comparisons

This personalized, multilayer PPR framework radically extends the classical approach:

• Credit risk and similar propagating phenomena can be modeled as dynamic processes across multiple, coupled dimensions (e.g., product, location).
• Risk influence is quantified not simply by static default rates but by propagation dynamics through the relationship structure—enabling earlier warnings and attribution of correlated risk.
• The algorithm is computationally scalable: the flattening of the supra-adjacency into a matrix ensures compatibility with standard sparse matrix solvers, and the personalization via a sparse influence matrix restricts random walk restarts to a small effective dimension.

By comparing aggregated network-based risk with conventional default rates, the paper demonstrates that the network-based metric can react more sharply and in advance, thus providing improved tools for proactive intervention in financial systems.

6. Methodological Advances and Generality

The introduction of a sparse, structured influence matrix for restarts—corresponding to selected nodes within specified layers—not only enables the construction of personalized PageRanks for generalized multilayer graphs but also allows seamless extension to:

• Arbitrary multilayer and multiplex systems with more than two relational dimensions,
• Bipartite or multipartite constructions within each layer,
• Situations requiring time-evolving influence sets to reflect dynamic origins (rolling default sets, epidemic sources, etc.).

Such frameworks are applicable not only in finance but in any setting where the propagation of influence, risk, or signal is inherently driven by complex, multi-relational structures.

7. Summary Table: Key Personalized PageRank Constructs in Multilayer Networks

Concept	Multilayer Formulation	Function
Supraadjacency tensor	$M_{j\beta}^{i\alpha}$	Encodes inter/intra-layer edges
Influence matrix	$u^{i\alpha}_{j\beta}$	Sparse; selects restart nodes/layers
Transition matrix	$R^{i\alpha}_{j\beta}$	Combines walk and personalized restart
Centrality per node	$\omega_i = \sum_\alpha \Omega_{i\alpha}$	Total steady-state probability over layers
Stationary propagation	Solve eigenproblem for principal eigenvector/eigentensor	Stationary risk influence or general diffusive variable

This methodology, validated on extensive agricultural lending data, substantiates the use of personalized, multilayer PageRank for high-fidelity risk propagation modeling, supporting segmentation, anticipation, and interpretation of network-driven systemic phenomena (Bravo et al., 2020).