Product Disruption Index (PDI)

• PDI is a metric that quantifies disruptive innovation by mapping descendant product citations relative to inherited technological features.
• It employs methods like cosine similarity and thresholded networks to construct product genealogies and simulate supply chain disruptions.
• Empirical studies reveal that minor yet targeted design modifications can yield high PDI values, guiding innovation strategy and risk assessment.

The Product Disruption Index (PDI) is a quantitative metric formulated to capture and evaluate the degree to which a product innovation disrupts existing technological lineages or supply chain operations. PDI originates from methodologies established for measuring disruptive innovation in scientometrics and network science—particularly the “CD index” used for publications and patents—but is adapted for the specific context of product evolution, technological change, and supply chain resilience.

1. Mathematical Foundations and Conceptual Definition

At its core, the PDI generalizes the structure of disruption indices from the citation network domain to products and supply chains. In the scientometric context, disruption is measured by the extent to which a work is cited without co-citation of its references, formulated as:

$\text{CD}_p = \frac{N_i - N_j}{N_i + N_j + N_k}$

where:

• $N_i$: Number of later entities (e.g., products, papers) that reference only the focal item.
• $N_j$: Number of entities that reference both the focal item and its predecessors.
• $N_k$: Number of entities that reference only predecessors without referencing the focal item.

In the product context, as formalized in the automotive phylogenetic paper (He et al., 14 Jul 2024), this structure is preserved, with explicit mapping of products via technological similarity networks rather than formal citations. The PDI is computed as:

$\text{PDI} = \frac{n_i - n_j}{n_i + n_j + n_k}$

where:

• $n_i$: Number (or summative weight) of descendant products citing only the focal product.
• $n_j$: Number of descendants citing both the focal product and at least one ancestor.
• $n_k$: Number of descendants citing only the ancestor(s) without the focal product.

For supply chain disruption management, as in the MARE framework (Ramzy et al., 2022), the PDI can also be framed as a composite metric aggregating restoration performance measures:

$$\text{PDI} = \frac{\text{Cost Increase} + \alpha \cdot \text{Delay} + \beta \cdot \text{Unrecovered Quantity}}{\text{Total Planned Quantity}}$$

where $\alpha$ and $\beta$ are application-dependent weight factors.

2. Methodological Approaches for Calculating the PDI

A. Product Genealogy and Technological Feature Networks

The PDI implementation for products, particularly in domains without explicit citation behavior, requires construction of a phylogenetic (similarity) network:

• Each product is encoded as a “chromosome” vector of technological features (“genes”).
• Cosine similarity is computed between product vectors to form an adjacency matrix.
• A thresholding mechanism is applied (often determined by local product family triangles) to translate similarities above a cutoff into effective “citations.”
• Products are organized temporally into ancestor-focal-descendant triangles; links are classified as per the definitions of $n_i$, $n_j$, $n_k$.

Indicator functions formalize binary “citation” assignments: $$a_{k,j} = \begin{cases} 1, & \text{if } \text{sim}_{k,j} > \text{threshold} \ 0, & \text{otherwise} \end{cases}$$ (similarly for $f_{i,j}$, $b_{k,i}$).

B. Supply Chain Disruption Simulation

In supply chain assessment, the PDI aggregates outputs of semantic query-based performance metrics, leveraging ontologies and knowledge graphs to capture, for each disrupted order or plan:

• Extra cost incurred (“Recovery Cost Increase”)
• Delay or speed of recovery (“Recovery Speed”)
• Quantity and customer service loss (“Unrecovered Quantity” and customer-priority impact)

Semantic integration enables dynamic, query-driven evaluation over heterogeneous logistics, production, and procurement data sources.

C. Dynamic and Weighted Variants

Recent work in scientometrics proposes citation- or adoption-weighted disruption indices (Bentley et al., 2023): $$m\text{CD}_5(t) = \frac{1}{C_t}\sum_{i=1}^{N_t} c_{it} \cdot \text{CD}_5(i)$$ with $c_{it}$ as impact weights (e.g., annual citation or sales rates), mitigating dilution from exponential growth in network size or observation scope.

3. Biases, Temporal Dynamics, and Normalization Challenges

Empirical and computational studies have demonstrated that disruption indices, including PDI, are subject to several systematic biases:

• Structural bias (citation inflation): As the average number of references (or benchmarks) per entity grows, the denominator ($N_k$ or $n_k$) increases, systematically driving the index toward zero over time, independent of true disruptiveness (Petersen et al., 2023, Petersen et al., 21 Jun 2024).
• Behavioral bias: Shifts in self-citation, referencing strategy, or industry benchmarking (e.g., through greater triadic closure or supplier priority realignment) can alter measured disruptiveness without real innovation change.
• Temporal instability: Short observation windows (e.g., 3–5 years) yield noisy and unstable disruption classifications; stabilization >80% agreement with long-run values typically requires 10 years (Chen et al., 10 Apr 2025).
• Team or actor composition: In analogous research, team size inversely correlates with disruption index; small “teams” (or lone innovators) create more disruptive innovations, but this only becomes apparent over long evaluation windows (Lin et al., 31 Jan 2025).
• Analytical flexibility: Index specification (window length, inclusion thresholds, weighting schemes) introduces a “multiverse” of outcomes; full transparency requires reporting all defensible parameterizations (Leibel et al., 19 Jun 2024).

Addressing these challenges often requires normalization (e.g., citation/adoption-weighting, normalization to reference/sample size, or field-specific adjustment), as well as careful consideration of observation windows and data thresholds.

4. Empirical Results and Validation

Product Evolution Studies

Case studies in the automotive industry show that the PDI robustly reflects disruptive potential. For example, the 2014 Tesla Model S (with “small but not least” technological modifications relative to ancestors) registers a high PDI (0.424), in contrast to the Chevrolet Spark EV (–0.866), which, despite large changes, is comparatively less disruptive (He et al., 14 Jul 2024). Regression analyses reveal that minor, targeted modifications to inherited technological features maximize disruptiveness, aligning with the theory that “disruption arises from minor but crucial changes” rather than from wholesale overhauls.

Supply Chain Disruption

Simulation on synthetic and real supply chain data demonstrates that granular (establishment- and product-level) network models reveal greater shock propagation and higher disruption risk than aggregated firm- or sector-level models. Incorporating detailed product information exposes systemic vulnerabilities and enables the construction of more precise supply chain PDIs reflecting actual risk and substitutability (Inoue et al., 8 Oct 2024).

Validation in Scientometrics

In scholarly domains, weighted disruption indices reveal that—contrary to appearances from unweighted averages—disruptive impact remains stable or even increases when accounting for growth in publications and differential influence (Bentley et al., 2023). Best practices suggest restricting analysis to entities with minimum numbers of references/adoptions and allowing for longer stabilization windows (Leibel et al., 2023, Chen et al., 10 Apr 2025).

5. Practical Construction and Application Guidelines

Step	Key Considerations	Sources
Network construction	Use explicit citations, technological similarity, or semantic queries as appropriate to the domain.	(He et al., 14 Jul 2024, Ramzy et al., 2022)
Classification	Apply thresholding and family- or triangle-based approaches for descendant assignment.	(He et al., 14 Jul 2024)
Metric computation	Calculate PDI using well-defined, reference-normalized formulas or composite indices for supply chain.	(Ramzy et al., 2022, Bentley et al., 2023)
Bias control	Control for growth (citation/adoption/reference inflation), reporting window length, and normalization.	(Petersen et al., 2023, Petersen et al., 21 Jun 2024)
Sensitivity/multiverse	Report on the robustness of PDI to parameter choices; consider specification-curve or multiverse analysis.	(Leibel et al., 19 Jun 2024)

A best-practice schema involves: applying inclusion thresholds to avoid statistical artifacts; employing minimum observation windows for stabilization; correcting or normalizing for citation/reference inflation; and supplementing disruption measures with auxiliary impact metrics (such as sales, market adoption, or customer impact in supply chains).

6. Interpretive and Policy Implications

PDI offers stakeholders quantifiable, multi-dimensional insight into innovation and risk:

• Innovation strategy: Identifies which product modifications drive disruption (SBNL principle), informing R&D prioritization.
• Supply chain resilience: Enables real-time, semantically informed evaluation of supply chain restoration and performance, supporting targeted investment in strategic inventory, supplier diversification, or logistics.
• Research and market evaluation: Facilitates robust tracking of disruptive potential, provided that normalization and window-length practices are followed.
• Risk and resilience modeling: Informs policymakers and industry on systemic vulnerabilities, especially under non-substitutable production and supply conditions.

However, PDI values must be interpreted with respect to compound structural and behavioral biases, and their meaning is conditional on the chosen formula, observation window, and underlying network (phylogenetic or supply chain) parameters.

7. Open Research Directions and Limitations

Further refinement of PDI-related methodologies is ongoing, with key areas including:

• Dynamic normalization for inflation control, via statistical “deflator” approaches or benchmarking to extreme value/log-normal distributions (Petersen et al., 2023).
• Domain adaptation to other contexts (e.g., software, patents) and to hybrid “entity-based” indices using both textual and network features (Leibel et al., 2023).
• Specification transparency, advocating multiverse-style reporting to reveal sensitivity to formulaic and sampling choices (Leibel et al., 19 Jun 2024).
• Empirical validation through application to large-scale production networks, quantifying substitutability, concentration, and propagation characteristics (Inoue et al., 8 Oct 2024).
• Stabilization timing, establishing the minimum period or data threshold for reliable, bias-minimized evaluation (Chen et al., 10 Apr 2025).

Rigorous deployment of PDI thus requires adherence to robust mathematical, empirical, and procedural standards, along with domain-specific tuning and ongoing sensitivity analysis. In sum, while PDI constitutes a powerful metric for disruptive innovation in products and supply chains, its validity and interpretive power are inextricably linked to the fidelity of the modeling context, network construction, and normalization protocol.