A central question in the topics of “the science of science” is the explanation of driving factors in the diffusion of scientific research related to documents and recorded knowledge (Coccia, 2018, 2021, 2022; Coccia & Bozeman, 2016; Coccia & Roshani, 2024; Coccia & Wang, 2016; Fortunato et al., 2018; Radicchi et al., 2008). Social studies of science and scientometrics analyze manifold factors to explain the drivers of duffusion of scientific results in science and society, such as the role of authorship networks on publication performance (Li et al., 2013), the impact of multiple funding on citation performance of papers (MacLean et al., 1998), etc. Yan et al. (2018) examine the association between research funding and citation impact in different disciplines. Quinlan et al. (2008) analyze the scientific outputs of four US-funded scientific research programs, whereas Pao (1991) shows the relation between funding and publication. Instead, Morillo (2020) investigates how open access publication affects the evolution of research fields. Roshani et al. (2021) analyze the relation between research funding and citation-based performance in three research fields: computer science, medicine and economics. Results suggest that journals in computer science publish more funded papers than unfunded ones. Journals in medicine tend to publish a similar share of funded and unfunded papers, whereas journals in economics, unlike computer science, have a proportion of unfunded publications higher than funded ones. Another finding of this study is that, in these three research fields (computer science, medicine and economics), the Matthew effect (Merton, 1988)-accumulated advantage- is for funded articles greater than unfunded documents. In this context, Mosleh et al. (2022) analyze how research funding affects the citations of scientific output in life sciences (including “agricultural and biological sciences”, “biochemistry, genetics, and molecular biology”, “Immunology and microbiology”, “neuroscience” and “pharmacology, toxicology and pharmaceutics”). Results reveal that, in scientific fields of life sciences, funded publications have higher citations than unfunded papers, though the number of unfunded published papers in journals is higher than published papers supported by research funding. Wang and Shapira (2015) show the role of research sponsorship on performance in nanotechnology papers.

The vast literature in these topics has analyzed many aspects of the relation between funding of research and papers’ citations in specific research fields. However, a generalization of the relationship between research funding and citations of papers between manifold research fields in basic and applied sciences is unknown. The study here reveals, for the first time, general empirical relationships, based on power law model, between citations of articles and their funding status both in basic and applied sciences: the laws of funding for scientific citations. Findings here can explain and generalize, whenever possible, the driving role of funding for the diffusion of science in order to guide policymakers in effective research and science policies to allocate economic resources towards promising research fields to support scientific development and related pervasive diffusion in science and society.

Table 1 shows documents retrieved from the database of Web of Science (2023) per funding status. The arithmetic mean of values between research fields, categorized in basic and applied sciences, suggests that: basic science has an average of 80.77% of papers funded with about 87% of total citations; these percentages are higher than applied sciences (65.71% of funded papers with 72.96% of total citations, cf., Table 1).

Table 2 provides information about the scaling exponent from OLS (α) of power law relationship between citations and related number of articles published in journals between fields of basic and applied research. In general, scaling exponent (α) of funded articles is greater than unfunded articles in journals, and this effect suggests a high Matthew effect: a higher change of citations (credit of papers) than the change of funded papers published in journals. Figure 1 shows that the expected increase in citations of papers with an increase of 1% in funded articles published in journals. Citations are higher in the research fields of sociology, geology and economics, whereas citations are lower in some basic sciences, such as physics and mathematics (Figure 1).

This study can clarify the increase of citations (C) -indicator of diffusion in scientific research- considering the funding of studies that are published in articles of journals (A). In particular, findings reveal the following general relationships of scientific change in citations of funded research that is published in articles categorized per basic and applied sciences:

Funded articles in Basic Sciences C=1.20 A^1.24

Funded articles in Applied Sciences C=1.68 A^1.23

Results are systematized in Figure 2 both for funded and unfunded papers in basic and applied sciences.

Figures 3-6 show a visualization of plot and estimated relationships for funded and unfunded research in basic and applied sciences.

The empirical results suggest some predictions for scientific diffusion and development and related diffusion:

□ 1^st Funded articles in journals receive higher citations than unfunded papers.

□ 2^nd Funded articles in journals have a super-linear change of citations. This finding generates a high Matthew effect in funded papers published in journals (given by a higher level of citations and consequential diffusion of scientific research) than unfunded papers.

□ 3^rd Invariant property of citations in basic and applied sciences is that:

- Funding generates both in basic and applied sciences a similar expected change of citations of about 1.23% by increasing 1% funded papers published in journals.

- Un-funding scientific research generates in basic and applied sciences a similar expected change of citations (lower than funding) of about 1.04% when un-funded articles published in journals grow by 1%.

Moreover, the close alignment between α_OLS and α_SMA values, particularly for funded research in basic and applied sciences (having a difference ∆ = α_SMA -α_OLS ≈0.10, see Table 2), seems to validate the overarching hypothesis that research funding significantly amplifies the impact and diffusion of scientific publications, leading to more citations in science and society. For instance, in basic sciences, the sciences of chemistry, physics and astronomy exhibit a noticeable congruence between the α_OLS and α_SMA values for funded research with ∆_{Basic Sciences} ≤0.10, underscoring the pronounced role of funding in higher citation dynamics. Similarly, the same results are for funded articles in applied sciences where the aggregate value shows ∆ _{Applied Sciences} = α_SMA -α_OLS ≈0.11. This critical result seems to be a strong confirmation of our general hypothesis that research funding significantly amplifies the impact of publications in science and society. The difference ∆ is higher for unfunded papers in basic and applied science in the range of 0.26-0.28 considering all research fields. These findings suggest some main observations. Basic research, when backed by adequate financial support, has the potential to increase the scientific impact by more citations in science domain. In applied sciences funding also remains a critical determinant for scientific diffusion and development, though the change of citations between funded and unfunded research in some research fields is less stark. This aspect could be due to the inherent nature of applied research, which even without direct funding, scientific production can show a consequential diffusion (with citations) of scientific and technological results having a potential commercialization in society.

In general, this main finding here supports the prediction that investments in scientific research have tangible and quantifiable returns in terms of a higher diffusion and impact of recorded knowledge in science and society. In addition, findings here suggest that funding scientific research that is published in journals, it generates a similar (invariant) effect for increasing citations and diffusion of knowledge both in basic and applied sciences. To put it differently, the funding effect for diffusion of scientific research is an invariant factor in basic and applied sciences, suggesting that research fields in science have a common evolutionary nature in scientific and social communities (Smith, 2000; Stephan, 1996).

However, dissimilarities in certain fields, especially for unfunded research, also suggest the complex dynamics of the interaction between publications and citations in research fields. In fact, the lower alignment of α_OLS and α_SMA for unfunded research published in journals of some fields may be due to exogenous and/or endogenous factors, which suggest further investigation is needed.

Findings of this study, suggest general laws of funding for scientific citations that (impact) provide several important observations and science policy implications:

● Uniform Funding Strategies: Given the invariant relationship and similar estimated coefficients (see Figure 2), policymakers and funding agencies should consider uniform approaches when allocating resources across basic and applied sciences. The consistent citation growth, irrespective of the nature of science, implies that research production in basic and applied fields of research has an equal potential impact and diffusion with research funding.

● Rethinking Research Evaluation: The similarity in citation growth for funded and unfunded papers in basic and applied sciences suggests also that the inherent value in science might be more similar, rather than allocate resources considering the traditional distinction of basic and applied sciences. This aspect could lead to a re-evaluation of how governments, policymakers, R&D managers, science analysts, etc. perceive and prioritize funding for scientific research, challenging conventional division between “basic” and “applied” research for a more comprehensive perspective of funding research fields that are multidisciplinary driven by converging pathways in applied and basic sciences for progress of human society (Coccia & Wang, 2016).

Although this study provides interesting results, it has also some limitations. For Instance, sources of data may only capture certain aspects of the ongoing dynamics of citations, as a consequence other confounding factors should be further investigated, such as networks of international collaborations, role of leading universities, open science, etc. (Adams, 2013; Coccia & Bozeman, 2016; Coccia & Wang, 2016; Newman, 2004; OECD, 2023). A significant concern that needs to be addressed is the potential bias related to funding acknowledgments in the Web of Science database. The database, though it has a vast coverage in natural science engineering and social science, may not always comprehensively or accurately capture funding sources for all articles. These technical issues may be due to the lack of standardization in funding information across different scientific fields and countries that can also lead to inaccuracies in data about funding indicated in publications (Pranckutė, 2021). In addition, the limited reporting of funding data in certain domains, such as in some social sciences, because of their lower reliance on grant funding compared to natural sciences, may result in incomplete data (Grassano, et al., 2017). In fact, some research may receive partial funding from multiple sources, or there might be instances where the funding acknowledgments are inadvertently omitted or inaccurately represented. Furthermore, smaller funding bodies might not be acknowledged or tracked accurately within database of articles and other scientific outputs. All these issues can be improved in the next study by also using data from other platforms of several literature search databases.

To conclude, findings here clearly illustrate a general prediction in scientific diffusion and related development: the driving role of research funding for increasing, with the same effect, the citations of articles and recorded knowledge in basic and applied sciences. Hence, these results here provide main information to policymakers and R&D managers that have to design effective research and science policies directed to allocate funding in research fields having a high potential of growth for increasing the diffusion of research for competitive advantage of nations and overall progress of human society (Coccia, 2018, 2019, 2019a, 2020; Hicks & Isett, 2020; Pagliaro & Coccia, 2021).

Mario Coccia (mario.coccia@cnr.it): Conceptualization and design (Equal), Formal analysis (Equal), Investigation (Equal), Methodology (Equal), Project administration (Lead), Supervision (Lead), Validation (Equal), Writing - original draft (Equal), Writing - review & editing (Equal); Saeed Roshani (Roshani@atu.ac.ir): Conceptualization and design (Equal), Data curation (Lead), Formal analysis (Equal), Investigation (Equal), Methodology (Equal), Writing - original draft (Equal), Writing - review & editing (Equal).