Fame and obsolescence: Disentangling growth and aging dynamics of patent citations

K. W. Higham, M. Governale, A. B. Jaffe, and U. Zülicke

Phys. Rev. E 95, 042309 – Published 13 April 2017

Abstract

We present an analysis of citations accrued over time by patents granted by the United States Patent and Trademark Office in 1998. In contrast to previous studies, a disaggregation by technology category is performed, and exogenously caused citation-number growth is controlled for. Our approach reveals an intrinsic citation rate that clearly separates into an—in the long run, exponentially time-dependent—aging function and a completely time-independent preferential-attachment-type growth kernel. For the general case of such a separable citation rate, we obtain the time-dependent citation distribution analytically in a form that is valid for any functional form of its aging and growth parts. Good agreement between theory and long-time characteristics of patent-citation data establishes our work as a useful framework for addressing still open questions about knowledge-propagation dynamics, such as the observed excess of citations at short times.

Received 15 November 2016
Revised 6 February 2017

DOI:https://doi.org/10.1103/PhysRevE.95.042309

Physics Subject Headings (PhySH)

Econophysics Network formation & growth Preferential attachment

Interdisciplinary PhysicsNetworks

Authors & Affiliations

K. W. Higham¹, M. Governale¹, A. B. Jaffe², and U. Zülicke¹

¹Te Pūnaha Matatini, School of Chemical and Physical Sciences, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
²Te Pūnaha Matatini, Motu Economic and Public Policy Research, P.O. Box 24390, Wellington 6142, New Zealand

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Issue

Vol. 95, Iss. 4 — April 2017

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Images

Figure 1
Disentangling preferential-attachment-type growth from aging in the citation rate of category-4 USPTO patents granted in 1998. (a) Obsolescence time $τ$ extracted for each bin of patents with fixed number of citations $k$ from fitting the citation rate to the form $\bar{λ} (k, t) \propto exp (- t / τ)$ for time $t \geq t_{0} = 2.5$ years. (b) Preferential-attachment exponent $α$ , extracted for fixed times $t$ from fits of the citation rate to the form $\bar{λ} (k, t) \propto (k^{α} + f_{0})$ . (c) Inhomogeneous contribution $f_{0}$ to the preferential-attachment part of the citation rate, extracted for fixed times $t$ from fits to $\bar{λ} (k, t) \propto (k^{α} + f_{0})$ . Circles are fit-parameter values, solid lines their weighted averages, and black dashed (red dotted) curves show 95% confidence intervals for fit-parameter values (weighted averages).
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Figure 2
Distribution function $n (k, t)$ for patent citations, plotted as a function of the number $k$ of citations for fixed times $t = 2$ , 5, and 9 years. Empirical data for the cohort of category-4 USPTO patents granted in 1998 are shown as symbols. Curves have been calculated from the general theoretical expression in Eq. (4a) with functional forms for the aging function and the growth kernel given in Eqs. (2a) and (2b), respectively. To capture deviations from exponential aging at short times, the factor $Γ_{0}$ has been introduced as explained in the text. Parameters used are those listed for category 4 in Table 1.
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Figure 3
Average citation rate $Λ (t)$ for category-4 patents granted by USPTO in 1998. Symbols represent the data, and the red solid (black dashed) curve is the theoretical prediction based on applying preferential-attachment-type growth and exponential aging to the measured initial citation distribution $n_{0} (k)$ , with (without) correcting for short-time deviations from exponential aging using $Γ_{0} = 0.87$ .
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Figure 4
Fitting the time and citation-number dependences of the empirical citation rate. (a) Mean number of additional citations, $Δ k (t + Δ t)$ , as a function of time $t$ for fixed citation bins with logarithmic-scale midpoints at $k = 3$ , 10, and 25. Symbols are derived from the citation data, while the curves represent fits to the form of Eq. (1), with $A (t)$ given by Eq. (2a). Deviations from exponential-aging behavior occur only at short times. (b) Mean number of additional citations, $Δ k (t + Δ t)$ , as a function of cumulative number of citations $k (t)$ for various fixed times $t = 3$ , 6, and 9 years. Symbols are derived from the citation data, while the curves represent fits to the form of Eq. (1) with $f (k)$ given by Eq. (2b).
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Figure 5
Power-law vs exponential aging: The three-parameter power law (B1) fits data superficially well but yields very uncertain parameter values. For the case $k = 5$ shown in panel (a), the power-law fit (red solid curve) yields $β = 1.57 \pm 0.25$ and $τ^{'} = (2.40 \pm 1.30)$ years; relative uncertainties of 16% and 54% respectively at a 95% confidence level. This can be contrasted with the exponential fit (black dashed line) that gives $τ = (5.14 \pm 0.61)$ years—a relative uncertainty of 12%—while providing a fit that is basically indistinguishable from the power-law fit for times $t ≳ τ / 2$ . Parameters $τ^{'}$ and $β$ obtained from fitting the power-law aging model at different fixed $k$ are shown in panels (b) and (c), respectively. Note the consistently large uncertainties and systematic residual $k$ dependence, which severely limit the utility of the power-law aging model (B1) for analyzing our patent-citation data set. Solid red lines represents weighted averages, and black dashed (red dotted) lines represent 95% confidence intervals for the fit-parameter values (weighted averages). Results shown pertain to category-4 patents granted in 1998.
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