Technological Convergence Index (TCI)

• Technological Convergence Index (TCI) is a quantitative measure that captures the integration of previously distinct technological domains using methodologies such as heterogeneous graph learning, semantic embeddings, and bibliometric proximity.
• It combines multifaceted dimensions like depth—reflecting knowledge recombination—and breadth—gauging diverse technological coverage—through entropy weighting and similarity indices.
• TCI applications span R&D decision-making, innovation forecasting, and policy formulation to identify emerging convergence trends across patents, technology pairs, and topic landscapes.

Search arXiv for papers on "Technological Convergence Index heterogeneous graph semantic learning" and related TCI formulations. Technological Convergence Index (TCI) denotes several quantitative constructions for measuring technological convergence, understood as the phenomenon in which boundaries between technological areas and disciplines are increasingly blurred and previously distinct domains become integrated. In recent arXiv work, the term is used for at least three distinct but related formulations: a patent-level index that combines convergence depth and breadth through heterogeneous graph learning, semantic embeddings, Shannon diversity, and entropy weighting (Deng et al., 25 Sep 2025); a monthly pairwise proximity index for encryption technologies that sums keyword, citation, and collaboration sub-indices derived from OpenAlex attribution scores (Tavazzi et al., 2024); and, in a related semantic-graph setting, a formal aggregation of LLM-extracted topic co-occurrence into local and global Jaccard-based convergence indices over time (Sternfeld et al., 29 Oct 2025).

1. Conceptual scope and major formulations

Across these formulations, TCI is not a single standardized statistic. It may be defined at the level of individual patents, technology pairs observed month by month, or entire topic landscapes aggregated over temporal bins. The common objective is to quantify whether previously separate technological areas are becoming semantically linked, structurally connected, or jointly recombined in innovation outputs (Deng et al., 25 Sep 2025, Tavazzi et al., 2024, Sternfeld et al., 29 Oct 2025).

The principal differences concern the unit of analysis, the evidence source, and the aggregation rule.

Source	Unit of analysis	Core construction
(Deng et al., 25 Sep 2025)	Patent	Entropy-weighted combination of Depth and Breadth
(Tavazzi et al., 2024)	Technology pair by month	Un-normalized sum of five proximity sub-indices
(Sternfeld et al., 29 Oct 2025)	Topic graph by time period	Global average of pairwise Jaccard similarities

This heterogeneity has methodological consequences. A patent-level TCI emphasizes cross-field knowledge integration within a single inventive artifact; a pairwise monthly TCI emphasizes temporal proximity between two technologies; a graph-based TCI emphasizes system-wide acceleration of convergence. A plausible implication is that direct numerical comparison across these variants is generally inappropriate unless the underlying construction, scale, and normalization regime are aligned.

2. Patent-level TCI as a multidimensional index of depth and breadth

The patent-oriented formulation designs TCI around two fundamental dimensions: depth and breadth. Its depth dimension is built from a heterogeneous graph $G=(V,E,R)$ whose nodes include patent nodes, IPC nodes, topic nodes, and applicant nodes, and whose relations include Patent–IPC, IPC–IPC, Patent–Topic, and Applicant–Patent. Text-based nodes are initialized with Sentence-BERT embeddings,

$x_i^{(0)}=\mathrm{SBERT}(\mathrm{Text}_i),$

and updated through a Heterogeneous Graph Transformer layer,

$$h_v^{(\ell+1)}=\sigma\Biggl(\sum_{r\in R}\sum_{u\in\mathcal{N}_r(v)}\alpha_{uv}^r\,W_r\,h_u^{(\ell)}\Biggr).$$

After $L$ layers, IPC representations fuse structure and semantics by concatenating the structural embedding with the initial semantic embedding. The analytical inputs explicitly include IPC textual descriptions, patent titles and abstracts, applicants, and automatically extracted topics (Deng et al., 25 Sep 2025).

Depth is then decomposed into two components. The first, $D_1$, captures core-to-periphery extension from the main IPC to the most distant secondary IPC: $$S_{\mathrm{main,sec}}=\min_{i=1\ldots n}\frac{h_{\mathrm{main}}\cdot h_{\mathrm{sec}_i}}{\|h_{\mathrm{main}}\|\;\|h_{\mathrm{sec}_i}\|},\qquad D_1=1-S_{\mathrm{main,sec}}.$$ The second, $D_2$, captures heterogeneity among the auxiliary IPCs: $$S_{\mathrm{sec,sec}}=\alpha S_{\mathrm{avg}}+(1-\alpha)S_{\mathrm{max}},$$

$$S_{\mathrm{avg}}=\frac{1}{M}\sum_{i<j}\frac{h_{\mathrm{sec}_i}\cdot h_{\mathrm{sec}_j}}{\|h_{\mathrm{sec}_i}\|\;\|h_{\mathrm{sec}_j}\|},\qquad S_{\mathrm{max}}=\max_{i<j}\frac{h_{\mathrm{sec}_i}\cdot h_{\mathrm{sec}_j}}{\|h_{\mathrm{sec}_i}\|\;\|h_{\mathrm{sec}_j}\|},$$

$$D_2=1-S_{\mathrm{sec,sec}},\qquad \alpha=\frac{n}{n+k},\qquad M=\binom{n}{2}.$$

The exposition states

$x_i^{(0)}=\mathrm{SBERT}(\mathrm{Text}_i),$0

Its interpretation is explicit: depth measures how far a patent’s auxiliary knowledge extends beyond its core IPC and how heterogeneous that auxiliary set is.

Breadth is defined as portfolio diversity over IPC classes using the Shannon Diversity Index. With IPC-class shares

$x_i^{(0)}=\mathrm{SBERT}(\mathrm{Text}_i),$1

the unnormalized diversity is

$x_i^{(0)}=\mathrm{SBERT}(\mathrm{Text}_i),$2

with $x_i^{(0)}=\mathrm{SBERT}(\mathrm{Text}_i),$3 when all $x_i^{(0)}=\mathrm{SBERT}(\mathrm{Text}_i),$4. Breadth is normalized across the patent population as

$x_i^{(0)}=\mathrm{SBERT}(\mathrm{Text}_i),$5

The final fusion uses the Entropy Weight Method. For each patent $x_i^{(0)}=\mathrm{SBERT}(\mathrm{Text}_i),$6 and dimension $x_i^{(0)}=\mathrm{SBERT}(\mathrm{Text}_i),$7,

$x_i^{(0)}=\mathrm{SBERT}(\mathrm{Text}_i),$8

$x_i^{(0)}=\mathrm{SBERT}(\mathrm{Text}_i),$9

$$h_v^{(\ell+1)}=\sigma\Biggl(\sum_{r\in R}\sum_{u\in\mathcal{N}_r(v)}\alpha_{uv}^r\,W_r\,h_u^{(\ell)}\Biggr).$$0

and the Technological Convergence Index is

$$h_v^{(\ell+1)}=\sigma\Biggl(\sum_{r\in R}\sum_{u\in\mathcal{N}_r(v)}\alpha_{uv}^r\,W_r\,h_u^{(\ell)}\Biggr).$$1

This construction is explicitly multidimensional: depth quantifies boundary spanning and knowledge recombination, whereas breadth quantifies balanced multi-field coverage within the IPC portfolio.

3. Pairwise monthly TCI from text mining and bibliometric proximity

In the OpenAlex-based encryption study, TCI is defined for a technology pair $$h_v^{(\ell+1)}=\sigma\Biggl(\sum_{r\in R}\sum_{u\in\mathcal{N}_r(v)}\alpha_{uv}^r\,W_r\,h_u^{(\ell)}\Biggr).$$2 and month $$h_v^{(\ell+1)}=\sigma\Biggl(\sum_{r\in R}\sum_{u\in\mathcal{N}_r(v)}\alpha_{uv}^r\,W_r\,h_u^{(\ell)}\Biggr).$$3. The starting point is the attribution score $$h_v^{(\ell+1)}=\sigma\Biggl(\sum_{r\in R}\sum_{u\in\mathcal{N}_r(v)}\alpha_{uv}^r\,W_r\,h_u^{(\ell)}\Biggr).$$4, written as $$h_v^{(\ell+1)}=\sigma\Biggl(\sum_{r\in R}\sum_{u\in\mathcal{N}_r(v)}\alpha_{uv}^r\,W_r\,h_u^{(\ell)}\Biggr).$$5, which measures a paper’s association with a technology. These attribution weights propagate into all downstream proximity calculations (Tavazzi et al., 2024).

The keyword-based proximity index is

$$h_v^{(\ell+1)}=\sigma\Biggl(\sum_{r\in R}\sum_{u\in\mathcal{N}_r(v)}\alpha_{uv}^r\,W_r\,h_u^{(\ell)}\Biggr).$$6

where $$h_v^{(\ell+1)}=\sigma\Biggl(\sum_{r\in R}\sum_{u\in\mathcal{N}_r(v)}\alpha_{uv}^r\,W_r\,h_u^{(\ell)}\Biggr).$$7 is the average number of occurrences of keyword $$h_v^{(\ell+1)}=\sigma\Biggl(\sum_{r\in R}\sum_{u\in\mathcal{N}_r(v)}\alpha_{uv}^r\,W_r\,h_u^{(\ell)}\Biggr).$$8 in papers related to $$h_v^{(\ell+1)}=\sigma\Biggl(\sum_{r\in R}\sum_{u\in\mathcal{N}_r(v)}\alpha_{uv}^r\,W_r\,h_u^{(\ell)}\Biggr).$$9 or $L$0, $L$1 is the average cosine similarity of $L$2 to its host paper’s text via KeyBERT, and $L$3 is the average attribution score $L$4 over papers containing $L$5.

The citation-based proximity is directional: $L$6 with a symmetric $L$7. Collaboration-based proximity is computed in two variants,

$L$8

$L$9

where $D_1$0 is the average number of joint-technology papers authored in month $D_1$1, $D_1$2 is the non-incremental monthly $D_1$3-index, $D_1$4 is the incremental monthly $D_1$5-index, and $D_1$6 is the average attribution score over the author’s papers in the two technologies.

The resulting non-normalized TCI is the sum of five sub-indices: $D_1$7 The study states that additional normalization is deliberately omitted because dividing by volatile denominators would flatten the curves and obscure the rising trends that signal convergence.

The associated pipeline is explicit. OpenAlex is queried for 25 encryption-related concepts over 2002–2022, papers with missing references are removed, papers unlinked to any of the 25 technologies are dropped, suspicious monthly spikes are redistributed evenly, and deduplication keeps the most complete record per OpenAlex ID. Feature extraction then runs KeyBERT on title and abstract, computes monthly non-incremental and incremental $D_1$8-indices, maps citations, and identifies multi-technology authors. Index computation proceeds month by month for each technology pair, followed by cubic-polynomial interpolation of missing months, linear filling of endpoints, clipping of negatives to zero, and polynomial fitting of degree 0–10 by minimizing SMAPE.

4. Semantic-entity triple graphs and Jaccard-based TCI

A third line of work monitors convergence through a dynamic graph of technology-related entities and relations extracted from full text. Sternfeld et al. develop a pipeline that extracts subject–predicate–object triples from patents and preprints using LLMs, filters and groups technology-designating nouns, builds a semantic-entity graph, and measures temporal convergence through graph-derived indices. The exposition explicitly notes that the authors do not christen a single TCI; instead, it presents a formal TCI based on the temporal evolution of topic-pair co-occurrence as measured by the Jaccard similarity (Sternfeld et al., 29 Oct 2025).

At time $D_1$9, if $$S_{\mathrm{main,sec}}=\min_{i=1\ldots n}\frac{h_{\mathrm{main}}\cdot h_{\mathrm{sec}_i}}{\|h_{\mathrm{main}}\|\;\|h_{\mathrm{sec}_i}\|},\qquad D_1=1-S_{\mathrm{main,sec}}.$$0 denotes the set of extracted triples in which topic $$S_{\mathrm{main,sec}}=\min_{i=1\ldots n}\frac{h_{\mathrm{main}}\cdot h_{\mathrm{sec}_i}}{\|h_{\mathrm{main}}\|\;\|h_{\mathrm{sec}_i}\|},\qquad D_1=1-S_{\mathrm{main,sec}}.$$1 appears, then pairwise convergence is measured as

$$S_{\mathrm{main,sec}}=\min_{i=1\ldots n}\frac{h_{\mathrm{main}}\cdot h_{\mathrm{sec}_i}}{\|h_{\mathrm{main}}\|\;\|h_{\mathrm{sec}_i}\|},\qquad D_1=1-S_{\mathrm{main,sec}}.$$2

From this, the exposition defines a local index,

$$S_{\mathrm{main,sec}}=\min_{i=1\ldots n}\frac{h_{\mathrm{main}}\cdot h_{\mathrm{sec}_i}}{\|h_{\mathrm{main}}\|\;\|h_{\mathrm{sec}_i}\|},\qquad D_1=1-S_{\mathrm{main,sec}}.$$3

and a global index,

$$S_{\mathrm{main,sec}}=\min_{i=1\ldots n}\frac{h_{\mathrm{main}}\cdot h_{\mathrm{sec}_i}}{\|h_{\mathrm{main}}\|\;\|h_{\mathrm{sec}_i}\|},\qquad D_1=1-S_{\mathrm{main,sec}}.$$4

A rising $$S_{\mathrm{main,sec}}=\min_{i=1\ldots n}\frac{h_{\mathrm{main}}\cdot h_{\mathrm{sec}_i}}{\|h_{\mathrm{main}}\|\;\|h_{\mathrm{sec}_i}\|},\qquad D_1=1-S_{\mathrm{main,sec}}.$$5 is interpreted as acceleration of convergence across the technology landscape.

The full pipeline has six stages: data acquisition; preprocessing; semantic triple extraction; post-processing and filtering; noun stapling; and key-term extraction and clustering. The datasets comprise 278,625 arXiv preprints from 2017–2024 and 9,793 USPTO patent applications from 2018–2024. Preprocessing includes PDF-to-text conversion with PyMuPDF, citation removal, line-break and hyphen-join heuristics, and abbreviation expansion with the Schwartz–Hearst algorithm. Triple extraction compares spaCy/Textacy OpenIE with instruction-prompted LLMs, with Meta-Llama-3-8B-Instruct under few-shot prompting reported as best, achieving 100% correctly formatted triples, approximately 15.3 triples per paragraph, and less than 1% hallucination.

Post-processing normalizes terms, removes stopwords, discards tokens under three characters, and drops triples whose subject or object exceeds six words. Filtering combines a control-corpus frequency threshold $$S_{\mathrm{main,sec}}=\min_{i=1\ldots n}\frac{h_{\mathrm{main}}\cdot h_{\mathrm{sec}_i}}{\|h_{\mathrm{main}}\|\;\|h_{\mathrm{sec}_i}\|},\qquad D_1=1-S_{\mathrm{main,sec}}.$$6, a Gutenberg book-corpus term score $$S_{\mathrm{main,sec}}=\min_{i=1\ldots n}\frac{h_{\mathrm{main}}\cdot h_{\mathrm{sec}_i}}{\|h_{\mathrm{main}}\|\;\|h_{\mathrm{sec}_i}\|},\qquad D_1=1-S_{\mathrm{main,sec}}.$$7 retaining triples when at least one term lies in the top 10 percentile, and a cross-topic entropy filter $$S_{\mathrm{main,sec}}=\min_{i=1\ldots n}\frac{h_{\mathrm{main}}\cdot h_{\mathrm{sec}_i}}{\|h_{\mathrm{main}}\|\;\|h_{\mathrm{sec}_i}\|},\qquad D_1=1-S_{\mathrm{main,sec}}.$$8 retaining nouns with high entropy to ensure topic specificity. Semantically similar terms are grouped via SoftDICE with threshold $$S_{\mathrm{main,sec}}=\min_{i=1\ldots n}\frac{h_{\mathrm{main}}\cdot h_{\mathrm{sec}_i}}{\|h_{\mathrm{main}}\|\;\|h_{\mathrm{sec}_i}\|},\qquad D_1=1-S_{\mathrm{main,sec}}.$$9, using the soft-cardinality framework of Gali et al. (2019). The resulting time-binned graph $D_2$0 records edge weights $D_2$1, supports Louvain community detection at resolution $D_2$2, and enables convergence monitoring via the alternative Jaccard form

$D_2$3

5. Empirical validation and interpretation

The patent-level multidimensional TCI is validated through OLS regressions of patent quality proxies on TCI with controls for pages, claims, backward citations, and year fixed effects: $D_2$4 For First Claims, the reported coefficient is $D_2$5, $D_2$6, with $D_2$7. For Forward Citations, the coefficient is $D_2$8, $D_2$9, with $$S_{\mathrm{sec,sec}}=\alpha S_{\mathrm{avg}}+(1-\alpha)S_{\mathrm{max}},$$0. The study also compares seven baseline convergence measures, including IPC co-occurrence, pure SBERT, pure HGT, hybrid variants, and Rao–Stirling. The proposed TCI, labelled V8 in the exposition, shows Spearman rank correlations above 0.95 with the best composite baselines, highest or tied $$S_{\mathrm{sec,sec}}=\alpha S_{\mathrm{avg}}+(1-\alpha)S_{\mathrm{max}},$$1 values, and stronger and more consistent significance levels, which the study interprets as convergent validity and incremental explanatory power (Deng et al., 25 Sep 2025).

In the encryption case study, the pair Blockchain–Public-key cryptography exhibits an interpolation rate of 46% because few papers appear before 2012. From 2002–2016, all sub-indices are near zero. After 2017, sharp rises in keyword proximity and collaboration proximity with incremental $$S_{\mathrm{sec,sec}}=\alpha S_{\mathrm{avg}}+(1-\alpha)S_{\mathrm{max}},$$2-index indicate increasing overlap in vocabulary and authorship, while the citation components rise more modestly. The aggregate TCI shows an upward inflection beginning in 2017 and peaking in 2021. For 2017–2021, the reported median proximity indices are $$S_{\mathrm{sec,sec}}=\alpha S_{\mathrm{avg}}+(1-\alpha)S_{\mathrm{max}},$$3 for keywords, $$S_{\mathrm{sec,sec}}=\alpha S_{\mathrm{avg}}+(1-\alpha)S_{\mathrm{max}},$$4 for citations, and $$S_{\mathrm{sec,sec}}=\alpha S_{\mathrm{avg}}+(1-\alpha)S_{\mathrm{max}},$$5 for collaboration with incremental $$S_{\mathrm{sec,sec}}=\alpha S_{\mathrm{avg}}+(1-\alpha)S_{\mathrm{max}},$$6-index. The interpretation given is that shared keywords such as “digital signature,” “zero-knowledge proof,” and “elliptic curve” and joint authorships drive much of the observed proximity (Tavazzi et al., 2024).

In the semantic-triple framework, validation operates at the extraction and trend-detection levels. On 547 manually annotated triples, Llama-3 few-shot prompting attains 100% correct formatting, approximately 15.3 triples per paragraph, less than 1% hallucination, and 0.104 s/line, while spaCy yields approximately 3.9 triples per 15 lines. At corpus scale, the arXiv dataset produces 20,822,710 triples, 53,337 key terms, and 20 clusters; the USPTO dataset produces 3,027,121 triples and 10 clusters. The reported $$S_{\mathrm{sec,sec}}=\alpha S_{\mathrm{avg}}+(1-\alpha)S_{\mathrm{max}},$$7 shows a marked uptick in Q1 2023 on arXiv and begins rising in mid-2024 in patents, lagging arXiv by approximately one year. Emerging links include “Retrieval Augmented ↔ Conversational AI” in arXiv and “Source-code ↔ LLMs” and “On-device Speech ↔ Self-Attention” in patents (Sternfeld et al., 29 Oct 2025).

6. Methodological distinctions, common misconceptions, and applications

A recurrent misconception is that TCI denotes a single agreed-upon formula. The recent literature instead uses the same label for non-equivalent indices. In one case, TCI is a multidimensional patent-level measure based on Depth and Breadth with entropy-derived weights; in another, it is an un-normalized monthly sum of five proximity components for a technology pair; in the semantic-graph exposition, it is a formalized global average of pairwise Jaccard similarities, accompanied by the explicit note that the original authors did not themselves christen a single TCI (Deng et al., 25 Sep 2025, Tavazzi et al., 2024, Sternfeld et al., 29 Oct 2025). This suggests that any use of the term requires immediate specification of the corpus, temporal granularity, normalization rule, and convergence signal.

The normalization question is especially consequential. The patent-level framework normalizes Breadth to $$S_{\mathrm{sec,sec}}=\alpha S_{\mathrm{avg}}+(1-\alpha)S_{\mathrm{max}},$$8 and derives final dimensional weights through entropy. The encryption framework deliberately omits additional normalization in order to preserve rising trends in volatile monthly series. The semantic-graph framework normalizes pairwise overlap through Jaccard similarity and then averages across topic pairs. These choices are not merely technical details; they determine whether TCI behaves as a bounded patent attribute, an unbounded proximity trajectory, or a system-level convergence barometer.

The reported applications are correspondingly diverse. For R&D managers and inventors, the patent formulation proposes a two-dimensional map of Depth and Breadth and notes that thresholds such as above-median Depth in a technology section can guide resource allocation. For policymakers and funders, it proposes incorporating TCI into grant or tax-incentive criteria and notes that, in the green–digital twin transition, TCI can highlight structural gaps in Physics or Electricity. In the encryption setting, TCI curves are presented as inputs for identifying hybrid Blockchain–Public-key cryptography opportunities, white-space topics such as “post-quantum public key in blockchain,” strategic hires, partnerships, industry consortia, open-call themes, patent strategies, alliance formation, internal hiring, and M&A activity. In the semantic-graph setting, the principal application is technology forecasting, where rising Jaccard series and $$S_{\mathrm{sec,sec}}=\alpha S_{\mathrm{avg}}+(1-\alpha)S_{\mathrm{max}},$$9 are used to detect emerging convergence before downstream commercialization becomes visible (Deng et al., 25 Sep 2025, Tavazzi et al., 2024, Sternfeld et al., 29 Oct 2025).

Taken together, these formulations establish TCI not as a single metric but as a methodological class of convergence measures. What unifies them is the operationalization of cross-domain linkage through semantic similarity, graph structure, bibliometric interaction, diversity, or temporal co-occurrence. What differentiates them is the object being measured: convergence within a patent, between technologies, or across a topic ecosystem.